tag:blogger.com,1999:blog-15005844440834997212024-03-27T15:04:34.445+01:00CJ on Mathematics and Sciencetowards understanding by critical constructive inquiryClaes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.comBlogger2160125tag:blogger.com,1999:blog-1500584444083499721.post-37217714835812957082024-03-26T20:29:00.128+01:002024-03-27T15:04:03.029+01:00Man-Made Universality of Blackbody Radiation <div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-fj32xZIJYt-iVhwhQuKgFizs17ixzleU8Dp43MImTdJoZTr1f3_ydYD8lj6EKFndPditd3LIHQ0OoUM_dCJa57jZZUvtGNRX3cUkDegfJZEBukR-XDBOaxg38blF7HbfzHT_bIDsMfH3vLjA4Rp1Fk_kbcJH83q8TjzoMCmSm6FOckAdYYml_xrKvWYk/s1142/Screenshot%202024-03-26%20at%2020.29.17.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="426" data-original-width="1142" height="194" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-fj32xZIJYt-iVhwhQuKgFizs17ixzleU8Dp43MImTdJoZTr1f3_ydYD8lj6EKFndPditd3LIHQ0OoUM_dCJa57jZZUvtGNRX3cUkDegfJZEBukR-XDBOaxg38blF7HbfzHT_bIDsMfH3vLjA4Rp1Fk_kbcJH83q8TjzoMCmSm6FOckAdYYml_xrKvWYk/w483-h194/Screenshot%202024-03-26%20at%2020.29.17.png" width="483" /></a></div><p>Pierre-Marie Robitaille is one of few physicists still concerned with the physics of blackbody radiation supposed to be the first expression of modern physics as presented by Max Planck in 1900, as expressed in <a href="https://arxiv.org/abs/physics/0507007">this article</a> and <a href="https://www.ptep-online.com/2008/PP-14-07.PDF">this article</a> and <a href="https://www.youtube.com/watch?v=9TOKo7Ik9f8">this talk.</a></p><p>Robitaille points to the fact that a <i>blackbody</i> is a <i>cavity/box</i> $B$ with interior walls covered with <i>carbon sooth or graphite.</i> Experiments show that the <i>spectrum </i>of the radiation $B_r$ from a little hole of such a cavity only depends on frequency $\nu$ and temperature $T$ according to <i>Planck's Law:</i></p><p></p><ul style="text-align: left;"><li>$B_r=\gamma T\nu^2$ if $\nu <\frac{T}{h}$ and $B_r=0$ else, (P) </li></ul><div>where $\gamma$ and $h$ are <i>universal constants, </i>and we refer to $\nu <\frac{T}{h}$ as <i>high-frequency cut-off. </i></div><div><br /></div><div>Experiments show that putting any material body $\bar B$ inside the cavity will not change (P), which is seen as evidence that the spectrum of $\bar B$ is the same as that of $B$ i<i>ndependent of the nature of </i>$\bar B$ as an expression of <i>universality. </i></div><div><br /></div><div>This is questioned by Robitaille, but not by main-stream physicists. Robitaille insists that the spectrum depends on the nature of the body. </div><div><br /></div><div>Let us see what we can say from our analysis in <a href="https://computationalblackbody.wordpress.com">Computational Blackbody Radiation.</a> We there identify a perfect blackbody to have a spectrum given by (P) with $\gamma$ maximal and $h$ minimal, thus by <i>maximal radiation and maximal cut-off.</i> By experiment we determine that graphite is a good example of a perfect blackbody. By maximality a blackbody spectrum dominates all greybody spectra.</div><div><br /></div><div>Let then a <i>greybody </i>$\bar B$ be characterised by different constants $\bar\gamma (\nu)=\epsilon (\nu)\gamma$ with $0<\epsilon (\nu) <1$ a coefficient of <i>emissivity = absorptivity possibly depending on </i>$\nu$, and $\bar h >h$. The radiation spectrum of $\bar B$ is given by </div><div></div><p></p><ul><li>$\bar B_r=\epsilon (\nu)\gamma T\nu^2$ if $\nu <\frac{T}{\bar h}$ and $\bar B_r=0$ else.</li></ul><p>This is not universality since $\epsilon (\nu)$ and $\bar h$ depend on the nature of $\bar B$. </p><p>But let us now put $\bar B$ at temperature $\bar T$ inside the cavity $B$ with graphite walls acting as a blackbody and let $B$ take on the the same temperature (assuming $\bar B$ has much bigger heat capacity than $B$) with thus</p><p></p><ul style="text-align: left;"><li>$\bar B_r=\epsilon (\nu)B_r$ for $\nu<\frac{\bar T}{\bar h}$ and $\bar B_r=0$ else.</li></ul><div>We then measure the spectrum of the radiation from the hole, which is the blackbody spectrum of $B_r$:</div><ul><li>$B_r=\gamma\nu^2$ for $\nu<\frac{\bar T}{h}$ and $B_r=0$ else.</li></ul><div>If we then insist that this is the spectrum of $\bar B$, which it is not, we get a false impression of universality of radiation. By maximality with $h<\bar h$ the cavity spectrum $B_r$ dominates $\bar B_r$.</div><div> </div><div>We conclude that the universality of blackbody radiation is a fiction reflecting a dream of physicists to capture existential essence in universal terms. It comes from using the cavity as a transformer of radiation from a greybody to a blackbody pretending that the strange procedure of putting objects into cavity with graphite walls to measure their spectrum, is not strange at all. </div><div><br /></div><div>We may compare with US claiming that the dollar $D$ represents a universal currency backing that by imposing an exchange rates $\epsilon <1$ for all other currencies $\bar D$, thus imposing the dollar as the universal currency for the the whole World forgetting that all currencies have different characteristics. This gives the FED a man-made maximal universally dominating role, which is now challenged... </div><div><br /></div><div><b>PS1</b> To meet criticism that painting the walls of the cavity with graphite may be seen as a rigging of the measurement of radiation through the hole, physicists recall that removing the graphite and letting the walls be covered with perfect reflectors, will give the same result, if only a piece of graphite is left inside the cavity. This shows to be true, but the piece of graphite is necessary and its effect can be understood from the maximality of blackbody radiation independent of object size. </div><div><br /></div><div><b>PS2</b> Recall radiation spectra of solid state is continuous while gasses have discrete spectra. Also recall that measuring spectra typically is done with instruments like bolometer or pyrgeometer, which effectively measure temperature from which radiation is computed according to some Planck law which may but usually does not represent reality. Atmospheric radiation spectra play an important role in climate modelling, and it is important to take them with a grain of salt, since what is de facto measured is temperature with radiation being computed according to some convenient formula serving man-made climate alarmism. </div><div><br /></div><div><b>PS3</b> The Sun has a continuous spectrum and so probably consists of liquid metallic hydrogen. Main-stream physics tells that it has a gaseous plasma state.</div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-37033977420524386302024-03-26T09:14:00.006+01:002024-03-26T10:18:01.886+01:00Thermodynamics of Friction <table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBbTUXzjq1MY5D3YuUFtSM7R_pfi7BrK8GXxwRYgzuFMg5-elu1KY_ExUStS-YpTeo3BMbnoegw7gN59i2CE_ei-wwJrDqtrsYdrSstq02c0uepGAEOMWamASVMP1KiW0OF_5H3Pq7bDiu_vpKqkSZYGuZTSXFAyPNsaf5ccpyAsKfra9gun5BcpmyboKm/s824/Screenshot%202024-03-26%20at%2010.15.58.png" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="714" data-original-width="824" height="277" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBbTUXzjq1MY5D3YuUFtSM7R_pfi7BrK8GXxwRYgzuFMg5-elu1KY_ExUStS-YpTeo3BMbnoegw7gN59i2CE_ei-wwJrDqtrsYdrSstq02c0uepGAEOMWamASVMP1KiW0OF_5H3Pq7bDiu_vpKqkSZYGuZTSXFAyPNsaf5ccpyAsKfra9gun5BcpmyboKm/w435-h277/Screenshot%202024-03-26%20at%2010.15.58.png" width="435" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Everything goes around in construction-deconstruction-construction...</td></tr></tbody></table><br /><p>In the previous post we considered <i>viscosity</i> in <i>laminar </i>and <i>turbulent flow</i> and <i>friction</i> between solid bodies as mechanisms for <i>irreversible transformation of large scale kinetic motion/energy into small scale kinetic motion/energy in the form of heat energy</i>, noting that the transformation cannot be reversed since the required very high precision cannot be realised, everything captured in a <i>2nd Law of Thermodynamics. </i> </p><p>Let us consider the generation of heat energy in friction when rubbing your hands or sliding an object over a floor or pulling the handbrakes of your bicycle. We understand that the heat energy is created from the <i>work</i> done by <i>force times displacement</i> (in the direction of the force), like pressing/pushing a sandpaper over the surface of a piece of wood to smoothen the surface by destroying its granular micro-structure. Work is thus done to <i>destroy more or less ordered micro-structure</i> and the work shows up as internal heat energy as unordered micro-scale kinetic energy. </p><p>The key is here destruction of micro-structure into heat energy in a process which cannot be reversed since the required precision cannot me met.</p><p>Skin friction between a fluid and solid acts like friction between solids. </p><p>Turbulent flow transforms large scale ordered kinetic energy into small-scale unordered kinetic energy as heat energy under the action of viscous forces. Laminar flow also generates heat energy from friction between layers of fluid of different velocity.</p><p>In all these cases heat energy is generated from destruction/mixing of order/structure in exothermic irreversible processes. This destruction is balanced by constructive processes like synchronisation of atomic oscillations into radiation and emergence of ordered structures like vortices in fluid flow and endothermic processes of unmixing/separation. </p><p>We thus see exothermic processes of destruction followed by endothermic construction, which is not reversed deconstruction, with different time scales where deconstruction is fast and brutal without precision and construction is slow with precision. This is elaborated in <a href="https://www.csc.kth.se/~cgjoh/theoryoftime.pdf">The Clock and the Arrow</a> in popular form. Take a look.</p><p> </p>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-81503368818838113172024-03-25T21:21:00.005+01:002024-03-25T21:31:45.869+01:00Norman Wildberger: Insights into Mathematics <div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOo3Sp7FCNq0In-6p2leKfT-zk0VdEQ01WOkuUCn3LMJtGd8Pvg15mrrROsRXe6U5mQA0Zq0KRZcLb7YIMMYDoeOYdCSaGI3zhq7jf2vp2-hSwNVmXj0Jnh2wZ-Nb4dNd00TEq2EFeAbkV4wkytOQ5_gOmpx77G5geka2iIR3ugUh9pFsm_2HTWHzlVtS0/s852/Screenshot%202024-03-25%20at%2021.28.59.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="472" data-original-width="852" height="177" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOo3Sp7FCNq0In-6p2leKfT-zk0VdEQ01WOkuUCn3LMJtGd8Pvg15mrrROsRXe6U5mQA0Zq0KRZcLb7YIMMYDoeOYdCSaGI3zhq7jf2vp2-hSwNVmXj0Jnh2wZ-Nb4dNd00TEq2EFeAbkV4wkytOQ5_gOmpx77G5geka2iIR3ugUh9pFsm_2HTWHzlVtS0/w381-h177/Screenshot%202024-03-25%20at%2021.28.59.png" width="381" /></a></div><br /><p>Mathematician Norman Wildberger presents an educational program for a wide audience as <a href="https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ">Insights into Mathematics</a> connecting to the principles I have followed in <a href="https://www.bodysoulmath.org">Body and Soul</a> and <a href="https://mathsimulationtechnology.wordpress.com">Leibniz World of Math.</a></p><p>A basic concern of Wildberger is how to cope with <i>real numbers </i>underlying a<i>nalysis</i> or c<i>alculus, geometry, algebra and topology, </i>since they appear to require working with aspects of <i>infinities </i>coming with difficulties, which have never been properly resolved, like computing with decimal expansions with infinitely many decimals and no last decimal to start a multiplication. Or the idea of an infinity of real numbers beyond countability.</p><p>I share the critique of Wildberger but I take a step further towards a resolution in terms of <i>finite precision computation, </i>which can be seen to<i> </i>be the view of an applied mathematician or engineer. In practice decimal expansions with a finite number of decimals are enough to represent the world and every representation can be supplied with a measure of quality as a certain number of decimals as a certain finite precision.<i> </i>This offers a foundation of mathematics without infinities in the spirit of Aristotle with infinities as never attained potentials representing "modes of speaking" rather than effective realities. </p><p>In particular the central concept of "continuum" takes the form of a computational mesh of certain mesh size or finite precision. With this view a "continuum" has no smallest scale yet is finite and there is a hierarchy of continua with variable mesh size. </p><p>The difficulty of infinities comes from an idea of <i>exact physical laws</i> and <i>exact solutions</i> to mathematical equations like $x^2=2$ expressed in terms of symbols like $\sqrt{2}$ and $\pi$. But this can be asking for too much, even if it is tempting, and so lead to complications which have to be hidden under the rug creating confusion for students.</p><p>A more down-to-earth approach is then to give up exactness and be happy with finite precision not asking for infinities. </p>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-42914627609493004212024-03-25T11:31:00.008+01:002024-03-25T14:43:59.390+01:00How to Generate Heat Energy<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCFCfYQkXL5B6Iv7zZvJoVYbBMXcXOf3XEfuXMRkPEHjFZ9lk9VDDbWeemK_PJUka4N5ZPhVON22yuF_oBVxScI69Y_vDaUt-s0tWbbh5p2DVhxJ4i6vS7-u5VbOCU3OgQzDnMlzwHtKIVredHMJZ9gYYHBd5mMWIwQ-27jLVwSAqyxZ608HJ0PLxaGdVK/s706/Screenshot%202024-03-25%20at%2011.37.47.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="408" data-original-width="706" height="185" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCFCfYQkXL5B6Iv7zZvJoVYbBMXcXOf3XEfuXMRkPEHjFZ9lk9VDDbWeemK_PJUka4N5ZPhVON22yuF_oBVxScI69Y_vDaUt-s0tWbbh5p2DVhxJ4i6vS7-u5VbOCU3OgQzDnMlzwHtKIVredHMJZ9gYYHBd5mMWIwQ-27jLVwSAqyxZ608HJ0PLxaGdVK/w437-h185/Screenshot%202024-03-25%20at%2011.37.47.png" width="437" /></a></div><br /><p>We recall from a <a href="https://claesjohnson.blogspot.com/2022/05/second-coming-of-2nd-law.html">previous post</a>:</p><p></p><ul style="text-align: left;"><li><i>Heat energy</i> can be generated <i>from large scale kinetic energy</i> by <i>compression. </i></li><li><i>Kinetic energy</i> can be generated <i>from heat energy</i> by <i>expansion.</i></li></ul><p></p><p>More precisely, we saw in the previous saw that <i>heat energy</i> at <i>high temperature</i> can generate useful <i>mechanical work</i>. Heat energy at high temperature can be created by<i> nuclear/chemical reactions. </i></p><p>Heat energy<i> </i>typically at<i> </i>lower temperatures also appears as<i> losses</i> from electrical currents subject to <i>resistance, </i>fluid motion subject to turbulent/laminar <i>viscosity </i>and <i>friction </i>between solid bodies. These losses appear as <i>substantial</i>, <i>unavoidable</i> and <i>irreversible</i> as expressions of a 2nd Law.</p><p>We have seen that heat energy $\sim T\nu^2$ of <i>frequency </i>$\nu$ carried by an <i>atomic lattice of temperature </i>$T$ subject to <i>high-frequency cut-off</i> $\nu <\frac{T}{h}$ expressing ordered synchronised<i> atomic oscillation </i>or <i>kinetic motion, </i>can be <i>radiated. </i>Here $h$ is a constant. </p><p>We can view <i>turbulent dissipation</i> in fluid flow as a form of <i>high-frequency forcing</i> above present cut-off which cannot be reradiated and so is absorbed as <i>internal energy </i>in the form of unordered small scale kinetic energy. We can similarly view viscosity and friction as forms of high-frequency forcing supplying internal energy. </p><p>The contribution to internal energy increases the temperature and so allows unordered small scale motion to be synchronised to higher frequency and then radiated. </p><p>The key is thus that turbulent, viscous and frictional dissipation all represent high-frequency forcing above present cut-off, which cannot be represented and reradiated and so shows up as internal energy as small scale kinetic energy. </p><p>Rubbing hands is one way to transform large scale kinetic motion into small scale kinetic motion as heat energy. The brakes on your car work the same way. </p>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-72030290396297442172024-03-24T12:07:00.027+01:002024-03-25T09:29:41.321+01:00Exergy as Energy Quality <div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhn4IoMFb4vSlnpJXGFk66NJAuFgn-ox3pW7B0v-Khxe2xE2EQV9KtLcHlgNYJIL8O2iNQpp6lMwUCS4XW-WAqhwsf9yy_0Ipn2O5GH8qAm0C7CvHSzmjMIeSfY_7kMIdN9E7johXv9og6p4QhI1xk2I2K0ra1JfHy6KEOqpxw2grLAj-fvjAdeMLrAEeLj/s948/Screenshot%202024-03-24%20at%2012.10.37.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="456" data-original-width="948" height="154" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhn4IoMFb4vSlnpJXGFk66NJAuFgn-ox3pW7B0v-Khxe2xE2EQV9KtLcHlgNYJIL8O2iNQpp6lMwUCS4XW-WAqhwsf9yy_0Ipn2O5GH8qAm0C7CvHSzmjMIeSfY_7kMIdN9E7johXv9og6p4QhI1xk2I2K0ra1JfHy6KEOqpxw2grLAj-fvjAdeMLrAEeLj/w424-h154/Screenshot%202024-03-24%20at%2012.10.37.png" width="424" /></a></div><br /><p><i>Kinetic energy,</i> <i>electrical energy</i>,<i> chemical </i>and <i>nuclear energy</i> can all be converted fully into <i>heat energy</i>, while heat energy can only be partially converted back again. This is captured in the 2nd Law of Thermodynamics. We can thus say that heat energy is of <i>lower</i> <i>quality </i>compared with the other forms. More generally, the term <i>exergy </i>is used as a measure of <i>quality of energy </i>of fundamental importance for all forms of life and society as ability to do <i>work</i>.</p><p>We can make this more precise by recalling that the <i>quality of heat energy</i> comes to expression in <i>radiative </i>and <i>conductive heat transfer</i> from a body B1 of temperature $T_1$ to a neighbouring body B2 of lower temperature $T_2<T_1$ in basic cases according to <i>Stefan-Boltzmann's Law </i>or <i>Fourier's Law</i><i>:</i></p><p></p><ul style="text-align: left;"><li>$Q = (T_1^4-T_2^4)$ (SB)</li><li>$Q = (T_1-T_2)$ (F)</li></ul><div>with $Q$ heat energy per time unit. Heat energy of higher temperature thus can be considered to have higher quality than heat energy of lower temperature, which of course also plays role in conversion of heat energy to other forms of energy. The <i>maximal efficiency</i> of a <i>heat engine</i> operating between $T_1$ and $T_2$ and transforming heat energy to mechanical work, is equal to $\frac{T_1-T_2}{T_1}$ displaying the higher quality of $T_1$.</div><div><br /></div><div>Heat energy at high temperature is the major source for useful mechanical work supporting human civilisation, while heat energy at lower temperatures appears as a useless loss e g in the cooling of a gasoline engine.</div><div><br /></div><div>But what is the real physics behind (SB) and (F)? This question was addressed in a <a href="https://claesjohnson.blogspot.com/2024/03/heat-conduction-in-solids-as-radiative.html">previous post</a> viewing (F) to be a special case of (SB) with the physics behind (SB) displayed in the analysis of <a href="https://computationalblackbody.wordpress.com">Computational Blackbody Radiation</a>. </div><div><br /></div><div>The essence of this analysis is a <i>high-frequency cut-off</i> $\frac{T}{h}$ allowing a body of temperature $T$ to only emit frequencies $\nu <\frac{T}{h}$, where $h$ is a constant. This allows a body B1 of temperature $T_1$ to transfer heat energy to a body B2 of lower temperature $T_2$ via frequencies $\frac{T_2}{h}<\nu <\frac{T_1}{h}$, which cannot be balanced by emission from B2. </div><div><br /></div><div>High frequency cut-off increasing linearly with temperature represents <i>Wien's displacement law (W), </i>giving improved exergy with increasing temperature.</div><div><br /></div><div>The high-frequency cut-off can be seen as an expression of <i>finite precision </i>limiting the frequency being carried and emitted by an oscillating atomic lattice in<i> coordinated motion</i>, with frequencies above cut-off being carried internally as heat energy as <i>uncoordinated motion</i>. </div><div><br /></div><div>Higher temperature thus connects to higher quality heat energy or better exergy. The standard explanation of this basic fact is based on <i>statistical mechanics</i>, which is not physical mechanics. </div><div><br /></div><div><b>PS</b> Radiative heat transfer without high-frequency cut-off would boil down to (F), while (SB) is what is observed, which gives support to (W).</div><div><br /></div><div><br /></div><p></p>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-27840763234568235742024-03-22T13:15:00.011+01:002024-03-23T08:44:27.407+01:00Thermodynamics of War and Peace<p><br /></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAWlBLqV_PNDuCHNS5cniBRnhiMIIzbMGjJtFH2wQEM245MhcuM7BrH69CF5oLqnbJWJNykUmZ_AnUp7y4BkAhCFSIJ0PaNtNZiABd3Z5lvKghYL5zRDJF7_iPjwJ45dD6ROKkwziCBYs259x4t0W_ArsMs-G5hIbnQK8FDb4aMpY2BoEkN-9hJ52yZPd7/s1308/Screenshot%202024-03-22%20at%2013.03.29.png" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="880" data-original-width="1308" height="215" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAWlBLqV_PNDuCHNS5cniBRnhiMIIzbMGjJtFH2wQEM245MhcuM7BrH69CF5oLqnbJWJNykUmZ_AnUp7y4BkAhCFSIJ0PaNtNZiABd3Z5lvKghYL5zRDJF7_iPjwJ45dD6ROKkwziCBYs259x4t0W_ArsMs-G5hIbnQK8FDb4aMpY2BoEkN-9hJ52yZPd7/s320/Screenshot%202024-03-22%20at%2013.03.29.png" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Opposing ordered armies at the moment before turbulent destruction. </td></tr></tbody></table><p>The recent posts on 2nd law of thermodynamics describe a process where <i>increasing spatial gradients </i>eventually reach a level (from convection and opposing flow) where further increase is no longer possible because it would bring the process to brutal stop, and so some form of<i> equilibration</i> of spatial differences must set in where </p><p></p><ul style="text-align: left;"><li><i>each particle tends to take on the mean-value of neighbouring particles. </i>(M)</li></ul><p></p><p>This is the process in <i>turbulent fluid flow </i>transforming ordered <i>large scale kinetic energy</i> into <i>small scale disordered kinetic energy</i> taking the form <i>internal heat energy </i>in a <i>turbulent cascade </i>of <i>turbulent dissipation. </i>Here (M) is necessary to avoid break-down into a stop. The flow or show must go on.</p><p>(M) is also the essence of the diffusion process of <i>heat conduction </i>seeking to <i>decrease gradients, </i>even if not absolutely necessary as in turbulent fluid flow.</p><p>It is natural to connect turbulence to the violent break-down of large scale ordred structures into <i>rubble </i>in a <i>war </i>necessarily resulting from<i> escalation of opposing military forces in direct confrontation </i>which at some level cannot be further escalated and so have to be dissipated in a war. </p><p>It is then natural to connect the equilibration (M) in heat conduction to a geopolitical/parliamentary process in peace time, where each country/party takes on the mean value of neighbouring countries/parties keeping gradients small. </p><p>While (M) is necessary in turbulence to let the flow go on, one may ask what the physics of (M) in the case of heat conduction, and find <a href="https://claesjohnson.blogspot.com/2024/03/heat-conduction-in-solids-as-radiative.html">answer in this post.</a> </p><p>The mathematics is elaborated in: </p><p></p><ul style="text-align: left;"><li><a href="https://www.dropbox.com/s/yz7cng6n8xyhwn4/%28Am%20Bs%204%29%20Johan%20Hoffman%2C%20Claes%20Johnson%20-%20Computational%20Turbulent%20Oncompressible%20Flow-Springer%20%282007%29.pdf?dl=0">Computational Turbulent Incompressible Flow</a></li><li><a href="https://www.dropbox.com/s/pt4up03axk52ahs/ambsthermo.pdf?dl=0">Computational Thermodynamics</a></li><li><a href="https://computationalblackbody.wordpress.com">Computational Blackbody Radiation</a></li></ul><div>The geopolitical/parliamentary situation today evolves towards sharpened gradients, while politicians refuse to follow (M) and so there is a steady march towards break-down... </div><p></p><p><br /></p>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com5tag:blogger.com,1999:blog-1500584444083499721.post-32146518164461974422024-03-20T11:19:00.007+01:002024-03-21T21:04:24.619+01:00Secret of Conductive and Radiative Heat Transfer<p>This is a continuation of the previous post on <a href="https://claesjohnson.blogspot.com/2024/03/heat-conduction-in-solids-as-radiative.html">Heat Conduction in Solids as Radiative Heat Transfer</a> with clarifying analysis from <a href="https://www.csc.kth.se/~cgjoh/ambsblack.pdf">Mathematical Physics of Blackbody Radiation</a> and <a href="https://computationalblackbody.wordpress.com">Computational Blackbody Radiation.</a></p><p>The key aspect of both conductive and radiative heat transfer is <i>interaction in a coupled system of weakly damped oscillators</i> of different <i>frequencies</i> tending to an <i>equilibrium</i> with <i>all oscillators having the same temperature </i>as the <i>system temperature.</i> The damping can be <i>frictional</i> (1st order time derivative) or <i>radiative</i> (3rd order time derivative) </p><p>There are two main questions: (i) Why do different systems take on the same temperature? (ii) Why do oscillators with different frequencies in a system take on the same temperature? </p><p>The answer is hidden in the interaction between<i> incoming</i> radiation, <i>oscillator </i>and <i>outgoing</i> radiation in a <i>weakly radiatively damped oscillator</i> analysed in detail in the above texts. The essence is that under <i>near resonance</i> between incoming frequency and oscillator frequency, </p><p></p><ul style="text-align: left;"><li><i>incoming radiation</i> is balanced by <i>outgoing radiation </i>plus <i>internal heating.</i> </li></ul><div>This is a non-trivial basic fact reflecting that the forcing and oscillator are <i>out-of-phase</i> with a<i> shift of half a period</i> as a consequence of small radiative damping and near resonance. </div><div><br /></div><div>Two coupled oscillators thus interact with outgoing from one oscillator acting as incoming for the other and vice versa and so are led to take on the same temperature, which is then spread over the oscillators of a system and also over systems. </div><div><i><br /></i></div><div>The essential components in this equilibration process are thus</div><div><ul style="text-align: left;"><li>weakly damped oscillators generating outgoing radiation and internal heating </li><li>out-of-phase balance between forcing and damping from near resonance</li><li>high-frequency cut-off increasing with temperature from <i>finite precision computation</i>. </li></ul><div>This analysis connects to Planck's derivation of his law of radiation with statistics replaced by finite precision thus replacing non-physics by physics. </div></div><p></p>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com5tag:blogger.com,1999:blog-1500584444083499721.post-46158521347580844922024-03-19T21:11:00.015+01:002024-03-21T13:32:59.856+01:00Sverige Måste Vinna Kriget mot Ryssland?<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEietJWibEUkz_LvdSVUS7ioWJpTnwAhjnM9cSyH96o7MCMoOs7O8C-dULD3xnqgQFhiwobPt4-49PUeIKdPiNdMJsSWYIyPEoB-oXNbw-Cy_O-wSVjHhQfXnoK3NhQzsKR3CuD67YGoDwy7GbNwbxgqcvVaCK_d8FcDIfkjqtBTlpLPJ22o96cxxTyPzkSv/s958/Screenshot%202024-03-19%20at%2021.13.21.png" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="436" data-original-width="958" height="176" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEietJWibEUkz_LvdSVUS7ioWJpTnwAhjnM9cSyH96o7MCMoOs7O8C-dULD3xnqgQFhiwobPt4-49PUeIKdPiNdMJsSWYIyPEoB-oXNbw-Cy_O-wSVjHhQfXnoK3NhQzsKR3CuD67YGoDwy7GbNwbxgqcvVaCK_d8FcDIfkjqtBTlpLPJ22o96cxxTyPzkSv/w452-h176/Screenshot%202024-03-19%20at%2021.13.21.png" width="452" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Sverige besegrar Ryssland i slaget vid Narva 1701 under ledning av Karl XII. </td></tr></tbody></table><p>Idag skriver ett antal svenska ambassadörer och militärer <a href="https://www.svd.se/a/76XxQ3/debattorer-vinner-putin-vantar-en-storre-konflikt">på SvD Debatt:</a></p><p></p><ul style="text-align: left;"><li><i>Regeringen bör skyndsamt hitta formerna för att <b>mångfalt öka det militära biståndet till Ukraina.</b> </i></li><li>Inför den <b>ryska aggressionen</b> befinner sig de västliga demokratierna som de befann sig inför <b>Hitler.</b></li><li><i><b>Kriget kommer inte att sluta så länge Putin sitter vid makten.</b> </i></li><li><i><b>Enda sättet att avsluta kriget är alltså att Putin förlorar makten.</b></i></li><li><i><b>Det kan ske om han lider ett så svidande nederlag...</b></i></li><li><i><b>Putins avlägsnande är bara ett nödvändigt...</b><b>kännbart ryskt nederlag i Ukraina.</b></i></li><li><i>Ett sådant nederlag har den <b>västliga demokratiska alliansen</b> i sina händer</i>. <i>Hos den finns de <b>vapen och de resurser</b> med vilka Ukraina kan vända kriget och <b>tillfoga Ryssland ett nederlag.</b></i></li><li><i><b>Ukraina måste segra och Ryssland måste förlora.</b></i></li><li><b><i>Vi vädjar till regeringen att tänka utanför den ärvda boxen – det gäller att skyndsamt hitta formerna för att mångfalt öka det militära biståndet till Ukraina.</i></b></li></ul><p></p><div dir="ltr">Vi ser här att segern i Narva 1701 är i kärt minne i den "ärvda boxen", medan förlusten i Poltava 1711 och förlusten av Finland 1809 är bortglömda. Tanken är alltså att vi kan besegra Ryssland, som är den starkaste kärnvapenmakten i världen med nyvunnen konventionell militär styrka om mer än en million tränade soldater vida överlägsen den som västliga demokratier kan skrapa ihop, där Sverige kan bidra med kanske 3000 otränade pojkar och flickor, och därmed genomföra den <b>erövring av Ryssland </b>som Karl XII, Napoleon och Hitler misslyckades med. </div><div dir="ltr"><br /></div><div dir="ltr">Speglar detta stämningarna hos det ledande skiktet av svenska politiker, ämbetsmän, militärer och företagare och svenska folket? Alla partier verkar vara inställda på krig även V och Mp. Fredsrörelsen har somnat. </div><div dir="ltr"><br /></div><div dir="ltr">USA är på väg att dra sig ur, vilket ger Sverige en ledande roll som krigförande, naturligtvis tillsammans med Finland som är bra på krigföring i snö.</div><div dir="ltr"><br /></div><div dir="ltr">Påven ber Ukraina hissa vit flagg och gå till förhandlingsbordet där Putin väntar. Fortsatt krig gör bara saken värre för Ukraina och Väst. Ett mångfalt ökat militärt bistånd till Ukraina leder till WW3. </div><div dir="ltr"><br /></div><div dir="ltr">Hur kan någon tro att Sverige kan vinna krig mot Ryssland idag? Eller tänker man att eftersom WW2 var bra för Sverige, så kan WW3 också vara det? I så fall bör man tänka en gång till.</div><div dir="ltr"><br /></div><div dir="ltr">En stark kärnvapenmakt kan inte besegras på hemmaplan. Det bästa resultat som kan uppnås är oavgjort som MAD<a href="https://en.wikipedia.org/wiki/Mutual_assured_destruction"> Mutual Assured Destruction.</a></div><div dir="ltr"><br /></div><div dir="ltr"><a href="https://tntradio.live/shows/pelle-neroth-taylor/">Lyssna på Pelle Neroth show med Mats Nilsson som förklarar saker o ting.</a></div><div dir="ltr"><br /></div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-26071273556777240512024-03-19T14:51:00.023+01:002024-03-24T05:30:17.258+01:00Heat Conduction in Solids as Radiative Heat Transfer<p><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisCNG5R136zD52atJjmma6KPPi7Fh7_F_mloegEvvzDxInfJyY_vk0IRLt-RVGUgAv7qZByNXKexU-_8Fv6Lfxqf4ZMOuOHuR7-m5mYen1HVdc-MftWwNLGHjFvLVInT-JZi6-wEurB7hZvVhefN-EcBozYZpnJm34cKYNO91lTR7BJ3LL421zsNRQ3mPt/s950/Screenshot%202024-03-19%20at%2015.03.27.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="492" data-original-width="950" height="166" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisCNG5R136zD52atJjmma6KPPi7Fh7_F_mloegEvvzDxInfJyY_vk0IRLt-RVGUgAv7qZByNXKexU-_8Fv6Lfxqf4ZMOuOHuR7-m5mYen1HVdc-MftWwNLGHjFvLVInT-JZi6-wEurB7hZvVhefN-EcBozYZpnJm34cKYNO91lTR7BJ3LL421zsNRQ3mPt/w439-h166/Screenshot%202024-03-19%20at%2015.03.27.png" width="439" /></a></div><p>What is the physics of heat conduction in a solid like a metal? The trivial story is that "heat flows from warm to cold" or "there is a flux of heat from warm to cold" which scales with the temperature difference or gradient. </p><p>But <i>heat is not a substance</i> like water in a river flowing from high-altitude/warm to low-altitude/cold, which connects to the <i><a href="https://en.wikipedia.org/wiki/Caloric_theory">caloric theory</a></i> and also to <a href="https://en.wikipedia.org/wiki/Phlogiston_theory">phlogiston theory</a> presenting<i> fire as form of substance,</i> both debunked at the end of the 18th century. </p><p>In any case there is a law of physics named <i>Fourier's Law:</i></p><p></p><ul style="text-align: left;"><li>$q =- \nabla u$ (F)</li></ul><div>which combined with a <i>law of conservation </i></div><div><ul style="text-align: left;"><li>$\nabla\cdot q = f$</li></ul><div>leads to the following<i> heat equation </i>(here in stationary state for simplicity) in the form of <i>Poisson's equation</i></div></div><div><ul style="text-align: left;"><li>$-\Delta u = f$. (H)</li></ul><div>where $u(x)$ is <i>temperature</i> and $f(x)$ <i>heat source</i> depending on a space variable $x$, and $q(x)$ is named "heat flux" although it has no physical meaning; heat is <i>not any substance </i>which <i>flows</i> or is in a state of <i>flux.</i> </div></div><div><br /></div><div>Let us now seek the physics of (F) in the case of metallic body as a <i>lattice of atoms, </i>and so seek an explanation of the observation that the temperature distribution $u(x)$ of the body tends to an equilibrium state with $u(x)=U$ with $U$ a constant (assuming no interaction with the surrounding and no internal heating for simplicity). </div><div><br /></div><div>We thus ask: </div><div><ul style="text-align: left;"><li>What is the physics of the process towards equilibrium with constant temperature?</li><li>How is heat transferred from warm to cold?</li><li>Why is (F) valid?</li></ul><div>We then recall our analysis of <i>radiative transfer</i> of energy at distance in a system of bodies/parts separated in space which (without external forcing) leads to an equilibrium state with all bodies having the same temperature, based on the following physical model:</div><div><ul style="text-align: left;"><li>Each body is a <i>vibrating lattice</i> of atoms described by a <i>wave equation</i> with <i>small radiative damping. </i></li><li>The bodies<i> interact</i> by <i>electromagnetic waves</i> through <i>resonance. </i></li><li>There is a <i>high-frequency cut-off increasing linearly with temperature </i>with the effect that heat transfer mediated by electromagnetic waves between two bodies, is <i>one-way from high temperature to low temperature. </i></li></ul>The key is here the high-frequency cut-off increasing with temperature, which makes heat transfer one-way. The cut-off can be seen as a form of f<i>inite precision threshold</i> allowing coordinated lattice vibration only below cut-off, thus allowing a high temperature lattice to carry higher frequencies. It is like a warmed-up opera soprano being able to reach higher frequencies. </div><div><br />We can view metallic body as a system composed of parts/atoms interacting by </div><div>electromagnetic waves at small distance. </div><div><br />Heat conduction will then come out as a special case of electromagnetic heat transfer </div><div>between atoms of different temperature with high-frequency cut-off guaranteeing one-way transport as expressed by (F) and exposed above. <br /><br />Note that the reference text<i> Conduction of Heat in Solids by Carslaw and Jaeger </i>presents (F) as an ad hoc physical law without physics.</div></div><div><br /></div><div>Recall that the standard explanation of radiative heat transfer from warm to cold is based on statistics without physics, which if used to explain heat conduction would again invoke statistics without physics thus not very convincing. </div><div><br /></div><div>Also note that the standard explanation of heat transfer in a gas involves collisions of molecules of different kinetic energy, which is not applicable to a metal with atoms in a lattice.</div><div><br /></div><div><b>PS1</b> Fluid flow in a river from higher to lower altitude is driven by pressure. "Heat flow" from warm to cold is not driven by pressure and so the physics is different. </div><div><br /></div><div><b>PS2 </b>Also compare with one-way osmotic transport of material driven by pressure. </div><div><br /></div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-48018849092884810852024-03-17T15:42:00.019+01:002024-03-18T13:27:40.836+01:00Universality of Radiation with Blackbody as Reference <div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEHUXbeLoliIE1PC5qnx11ePBTY5Lqz41Afd3iINEeycv-s-ts_vQy5YbKFJsvRC-zj7JS2LyAK6oxgiBUqERHQFi6tW1s5-cu8qWkCSkEBSXIrwe3flA6HKD0TalXwxK_mbjAzG0ftG2rr6FhG-IyEHzjJIdLrvXlGAG-yk-1ngYwH4bjk4EAu8CXM6hL/s956/Screenshot%202024-03-17%20at%2016.14.19.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="354" data-original-width="956" height="118" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEHUXbeLoliIE1PC5qnx11ePBTY5Lqz41Afd3iINEeycv-s-ts_vQy5YbKFJsvRC-zj7JS2LyAK6oxgiBUqERHQFi6tW1s5-cu8qWkCSkEBSXIrwe3flA6HKD0TalXwxK_mbjAzG0ftG2rr6FhG-IyEHzjJIdLrvXlGAG-yk-1ngYwH4bjk4EAu8CXM6hL/s320/Screenshot%202024-03-17%20at%2016.14.19.png" width="320" /></a></div><br /><p>One of the unresolved mysteries of classical physics is why the <i>radiation spectrum</i> of a material body <i>only depends</i> on <i>temperature</i> and <i>frequency</i> and not on the physical nature of the body, as an intriguing example of <i>universality. </i>Why is that? The common answer given by Planck is <i>statistics of energy quanta</i>, an answer however without clear physics based on <i>ad hoc assumptions</i> which cannot be verified experimentally as shown by <a href="https://edisciplinas.usp.br/pluginfile.php/48089/course/section/16461/qsp_chapter10-plank.pdf">this common argumentation. </a></p><p>I have pursued a path without statistics based on clear physics as <a href="http://computationalblackbody.wordpress.com/">Computational Blackbody Radiation</a> in the form of<i> near resonance</i> in a <i>wave equation</i> with <i>small radiative damping</i> as <i>outgoing radiation,</i> subject to <i>external forcing</i> $f_\nu$ depending on frequency $\nu$ , which shows the following<i> radiance spectrum</i> $R(\nu ,T)$ (with more details <a href="http://claesjohnson.blogspot.com/2014/05/is-blackbody-radiation-universal.html">here</a>) characterised by a <i>common temperature</i> $T$, <i>radiative damping</i> parameter $\gamma$, and $h$ defines a <i>high-frequency cut-off.</i> Radiative equilibrium with incoming = outgoing radiation shows to satisfy:</p><div><ul><li>$R(\nu ,T)\equiv\gamma T\nu^2 =\epsilon f_\nu^2$ for $\nu\leq\frac{T}{h}$,</li><li>$R(\nu ,T) =0$ for $\nu >\frac{T}{h}$,</li></ul><div>where $0<\epsilon\le 1$ is a coefficient of <i>absorptivity </i>= <i>emissivity, </i>while frequencies above <i>cut-off</i> $\frac{T}{h}$ cause heating. The radiation can thus be described by the coefficients $\gamma$, $\epsilon$ and $h$ and the temperature scale $T$. </div><div><br /></div><div>Here $\epsilon$ and $h$ can be expected to depend on the physical nature of the body, with a <i>blackbody </i>defined by $\epsilon =1$ and $h$ minimal thus with <i>maximal cut-off. </i></div><div><i><br /></i></div><div>Let us now consider possible universality of the radiation parameter $\gamma$ and temperature $T$.</div><div><br /></div><div>Consider then two radiating bodies 1 and 2 with different characteristics $(\gamma_1,\epsilon_1, h_1, T_1)$ and $(\gamma_2,\epsilon_2, h_2, T_2)$, which when brought into <i>radiative equilibrium </i>will satisfy (assuming here for simplicity that $\epsilon_1=\epsilon_2$):</div><div><ul><li>$\gamma_1T_1\nu^2 = \gamma_2T_2\nu^2$ for $\nu\leq\frac{T_2}{h_2}$ </li><li>assuming $\frac{T_2}{h_2}\leq \frac{T_1}{h_1}$ </li><li>and for simplicity that 2 reflects frequencies $\nu > \frac{T_2}{h_2}$. </li></ul><div>If we choose body 1 as<i> </i>reference<i>,</i> to serve as an <i>ideal reference blackbody,</i> defining a <i>reference temperature </i>scale $T_1$, we can then <i>calibrate</i> the temperature scale $T_2$ for body 2 so that </div><div><ul><li>$\gamma_1T_1= \gamma_2T_2$,</li></ul><div>thus effectively <i>assign </i>temperature $T_1$ and $\gamma_1$ to body 2 by radiative equilibrium with body 1 acting as a<i> reference thermometer.</i> Body 2 will then mimic the radiation of body 1 in radiative equilibrium and a form of<i> universality with body 1 as reference </i>will be achieved, with independence of $\epsilon_1$ and $\epsilon_2$.</div><div><br /></div><div>The analysis indicates that the critical quality of the reference blackbody is maximal cut-off (and equal temperature of all frequencies), and not necessarily maximal absorptivity = emissivity = 1. </div><div><br /></div><div>Universality of radiation is thus a consequence of radiative equilibrium with a specific reference body in the form of a blackbody acting as reference thermometer. </div><div><br /></div><div>Note that the form of the radiation law $R(\nu ,T)= \gamma T\nu^2$ reflects that the radiative damping term in the wave equation is given by $-\gamma\frac{d^3}{dt^3}$ with a third order time derivative as universal expression from oscillating electric charges according to<a href="https://en.wikipedia.org/wiki/Larmor_formula"> Larmor.</a></div><div><br /></div><div>In practice body 1 is represented by a <i>small piece of graphite</i> inside a <i>cavity with reflecting walls </i>represented by body 2 with the effect that the cavity will radiate like graphite independent of its form or wall material. Universality will thus be reached by mimicing of a reference, viewed as an ideal blackbody, which is perfectly understandable, and not by some mysterious deep inherent quality of blackbody radiation. Without the piece of graphite the cavity will possibly radiate with different characteristics and universality may be lost.</div><div><br /></div><div>We can compare many local currencies calibrated to the dollar as common universal reference. </div><div><ul><li><i>All dancers which mimic Fred Astaire, dance like Fred Astaire, but all dancers do not dance like Fred Astaire. </i><i> </i> </li></ul></div></div><div><b>PS1</b> The common explanation for the high frequency cut-off is that they have <i>low probability</i>, which is not physics, while I suggest that high frequencies cannot be represented because of <i>finite precision,</i> which can be physics. </div></div></div><div><br /></div><div><b>PS2</b> Note that high-frequency cut-off increasing with temperature gives a 2nd Law expressing that energy is radiated from warm to cold and to no degree from cold to warm, thus acting like semi-conductor allowing an electrical current only if a voltage difference is above a certain value.</div><div><br /></div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-72361609401342602092024-03-14T14:02:00.016+01:002024-03-15T07:16:47.636+01:00Geopolitics of Mathematical Physics <p><br /></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQmewnvtOzCg1-vM4qgzTHDxoE1wuYK_Wj5V9ub6y3_5HYLNnVwG9qYdMaK_Iq24yIqXorSE1RrB9sc9adWeXTxTDILQKJXjQYaosiRM30wAsDORbyWeIAUrAT_Xlt40hlEnKkzQfQwShJzO1nAWrzI5RMWgVQU5QyJGMmFg9EF98AGTK8BoZQQsL7EXTj/s946/Screenshot%202024-03-14%20at%2014.51.29.png" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="464" data-original-width="946" height="157" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQmewnvtOzCg1-vM4qgzTHDxoE1wuYK_Wj5V9ub6y3_5HYLNnVwG9qYdMaK_Iq24yIqXorSE1RrB9sc9adWeXTxTDILQKJXjQYaosiRM30wAsDORbyWeIAUrAT_Xlt40hlEnKkzQfQwShJzO1nAWrzI5RMWgVQU5QyJGMmFg9EF98AGTK8BoZQQsL7EXTj/w455-h157/Screenshot%202024-03-14%20at%2014.51.29.png" width="455" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Swedish Prime Minister offering to build 1000 Swedish Tank90.</td></tr></tbody></table><p>Let us identify some leading scientist/period/country in the history of <i>mathematical physics </i>as the foundation of science:</p><p></p><ul style="text-align: left;"><li>Archimedes: Greece </li><li>Song Dynasty: China </li><li>Galileo: Renaissance Italy</li><li>Leibniz and Euler: Calculus Scientific Revolution Holy Roman Empire</li><li>Newton: Calculus Scientific Revolution England </li><li>Cauchy and Fourier: Scientific/French Revolution France</li><li>Maxwell: Industrial Revolution England </li><li>Boltzmann: statistical mechanics German Empire </li><li>Hilbert: mathematics German Empire </li><li>Planck: radiation German Empire</li><li>Einstein: relativity German Empire, Weimar Republic</li><li>Schrödinger: quantum mechanics atomic physics Weimar Republic </li><li>Oppenheimer: atomic bomb USA</li><li>Feynman: subatomic physics USA</li><li>Gell Mann, Weinberg: subatomic physics USA</li><li>Witten: string theory subsubatomic physics USA</li><li>China? India? Third World?</li></ul><div>We see that geopolitical dominance comes along with leading science. No wonder, since science gives industrial and military power. We can follow a shift of power from the Holy Roman Empire (Germany) to France/England back to German Empire over Weimar Republic followed by USA until China and India and the Third World now are ready to step in. </div><div><br /></div><div>We can also follow a development from macroscopis of engines to microscopics of atoms to the string theory of today, along with an expansion from our Solar system to cosmology of the Universe. </div><div><br /></div><div><i>Modern physics </i>is viewed to have started with Planck's derivation of his radiation law based on <i>statistical mechanics</i> as a break-away from the rationality of the classical deterministic mechanics serving the scientific/industrial revolution, which after 20 years of incubation developed into the new quantum mechanics of atom physics, with the atomic bomb as triumph. </div><div><br /></div><div>Modern physics thus in 1900 took a step away from deterministic to statistical mechanics, from <i>rationality</i> to <i>irrationality</i> apparently breaking with the tradition from Archimedes to Hilbert carried by rational mathematical thinking, which led the way into the 20th century. </div><div><br /></div><div>Questions:</div><div><ol style="text-align: left;"><li>Can it be that the roots of the observed irrationality of the 20th century with two World Wars, and today a 21st century on its way to a third, can be traced to the irrationality of modern physics? </li><li>Will the dominance of USA persist with the present focus on string theory?</li><li>Will the rationalism of ancient China lead to a take over?</li></ol></div><div>A more detailed perspective on modern physics politics is given in <a href="https://www.csc.kth.se/~cgjoh/mysticism.pdf">Dr Faustus of Modern Physics.</a></div><p></p>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-82002309124111950162024-03-14T10:47:00.004+01:002024-03-14T10:48:23.148+01:002nd Law or 2-way Radiative Heat Transfer?<p>In the present discussion of the 2nd Law of Thermodynamics let us go back to <a href="http://claesjohnson.blogspot.com/2011/10/two-way-transfer-of-heat-as-dlr.html">this post from 2011</a> exposing the 19th century battle between<b> 1-way</b> transfer (<a href="https://en.wikipedia.org/wiki/Raoul_Pictet">Pictet</a>) vs 2<b>-way</b> <b>transfer</b> (<a href="https://en.wikipedia.org/wiki/Pierre_Prevost_(physicist)">Prevost</a>) of <b>heat energy by radiation</b> playing a central role in climate science today. </p><p>In 1-way transfer a warm body heats a colder body in accordance with the 2nd Law. </p><p>In 2-way heat transfer both warm and cold bodies are viewed to heat each other, but the warm heats more and so there is a net transfer of heat energy from warm to cold. But a cold body heating a warm body violates the 2nd Law of thermodynamics, and so there is something fishy here. </p><p>Yet the basic mathematical model of radiative heat transfer in the form of <a href="http://claesjohnson.blogspot.com/search/label/Schwarzschild">Schwarzschild's equation</a> involve 2-way transfer of heat energy, in apparent violation of the 2nd Law. </p><p>I have discussed this situation at length on this blog with tags such as <a href="http://claesjohnson.blogspot.com/search/label/2nd%20law%20of%20thermodynamics">2nd law of thermodynamics</a> and <a href="http://claesjohnson.blogspot.com/search/label/radiative%20heat%20transfer">radiative heat transfer</a> with more on <a href="https://computationalblackbody.wordpress.com">Computational Blackbody Radiation</a>.</p><p>If you worry about the 2nd Law, you can ask yourself how 2-way radiative heat transfer is physically possible, when it appears to violate the 2nd Law? What is false here: 2nd Law or 2-way heat transfer?</p><p>What is your verdict? </p><p><br /></p>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-54470491342995679802024-03-12T15:23:00.008+01:002024-03-13T21:12:22.976+01:00Philosophy of Statistical Mechanics?<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9-ZFpl_c72PECnuEU7I5Kb_UM69i7VDbDcq0gf1aSa-NJMwZV3QdGuqXk6T_Km527HObpjF3IQWwJnquswWjODmZxX4l4pinDBY8BubpPia9i1hu0w9sXdTPRVQMRYjLZn0XCuYlf8cOZli9onrmmRGVn_V1Bhyphenhyphen7mq8mVL1ib0DTPxs3ou7FyOOLEVcBT/s1288/Screenshot%202024-03-12%20at%2015.39.14.png" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="848" data-original-width="1288" height="211" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9-ZFpl_c72PECnuEU7I5Kb_UM69i7VDbDcq0gf1aSa-NJMwZV3QdGuqXk6T_Km527HObpjF3IQWwJnquswWjODmZxX4l4pinDBY8BubpPia9i1hu0w9sXdTPRVQMRYjLZn0XCuYlf8cOZli9onrmmRGVn_V1Bhyphenhyphen7mq8mVL1ib0DTPxs3ou7FyOOLEVcBT/w423-h211/Screenshot%202024-03-12%20at%2015.39.14.png" width="423" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Collapsed pillars of modern building.</td></tr></tbody></table><p>Let us continue with the post <a href="https://claesjohnson.blogspot.com/2024/03/three-elephants-in-modern-physics.html">Three Elephants of Modern Physics</a> taking a closer look at one of them. </p><p>We learn from Stanford Encyclopedia of Philosophy the following about <a href="https://plato.stanford.edu/entries/statphys-statmech/">Statistical Mechanics</a> SM: </p><ul><li><i>Statistical Mechanics is the <b>third pillar of modern physics</b>, next to <b>quantum theory </b>and <b>relativity theory</b>. </i></li><li><i>Its aim is to account for the </i><b>macroscopic behaviour </b><i>of physical systems </i>in terms of<i> dynamical laws governing the <b>microscopic constituents</b> of these systems and <b>probabilistic assumptions.</b></i></li><li><i>Philosophical discussions in statistical mechanics face an immediate difficulty because unlike other theories, <b>statistical mechanics has not yet found a generally accepted theoretical framework </b>or a canonical formalism. </i></li><li><i>For this reason, a review of the philosophy of SM <b>cannot </b>simply <b>start with a statement of the theory’s basic principles</b> and then move on to different interpretations of the theory.</i></li></ul><div>This is not a very good start, but we continue learning: </div><ul><li><i>Three broad theoretical umbrellas: “Boltzmannian SM” (<b>BSM</b>), “Boltzmann Equation” (<b>BE</b>), and “Gibbsian SM” (<b>GSM</b>).</i></li><li>BSM enjoys <b>great popularity in foundational debates</b> due to its clear and intuitive theoretical structure. Nevertheless, <b>BSM faces a number of problems and limitations</b></li><li>There is no way around recognising that BSM is mostly used in foundational debates, but it is <b>GSM that is the practitioner’s workhorse.</b></li><li><i>So what <b>we’re facing is a schism</b> whereby the day-to-day work of physicists is in one framework and foundational accounts and explanations are given in another framework.</i></li><li><i>This would not be worrisome <b>if the frameworks were equivalent</b>, or at least inter-translatable in relatively clear way...<b>this is not the case</b>.</i></li><li>The <b>crucial conceptual question</b>s (concerning BE) at this point are: <b>what exactly did Boltzmann prove with the H-theorem</b>?</li></ul><div><span style="color: #1a1a1a; font-family: serif; font-size: small;">This is the status today of the third pillar of modern physics formed by <a href="https://en.wikipedia.org/wiki/Elementary_Principles_in_Statistical_Mechanics" style="caret-color: rgb(26, 26, 26);">Boltzmann 1866-1906</a> and <a href="https://en.wikipedia.org/wiki/Elementary_Principles_in_Statistical_Mechanics" style="caret-color: rgb(26, 26, 26);">Gibbs 1902</a> as still being without a generally accepted theoretical framework, despite 120 years of deep thinking by the sharpest brains of modern physics. </span></div><div><span style="color: #1a1a1a; font-family: serif; font-size: small;"><br /></span></div>Is this something to worry about? If one the pillars apparently is shaky, what about the remaining two pillars? Who cares?<div><span style="color: #1a1a1a; font-family: serif; font-size: small;"><br /></span></div><div><span style="color: #1a1a1a; font-family: serif;">Recall that SM was introduced to rationalise the <i>2nd Law of Thermodynamics</i> stating <b>irreversibility of macroscopic systems</b> based on <b>deterministic reversible exact microscopics</b>. This challenge was taken up by Boltzmann facing the question: If all components of a system are reversible, how can it be that the system is irreversible? From where does the irreversibility come? The only way forward Boltzmann could find was to replace exact determinism of microscopics by <i>randomness/statistics</i> as a form of <i>inexactness. </i></span></div><div><span style="color: #1a1a1a; font-family: serif;"> </span></div><div><span style="color: #1a1a1a; font-family: serif;">In the modern digital world the inexactness can take the form of <i>finite precision computation</i> performed with a certain number of digits (e g single or double precision). Here the microscopics is deterministic up the point of keeping only a finite number of digits, which can have more or less severe consequences on macroscopic reversibility. This idea is explored in <i><a href="https://www.dropbox.com/s/pt4up03axk52ahs/ambsthermo.pdf?e=1&dl=0">Computational Thermodynamics</a> </i>offering a 2nd Law expressed in the physical quantities of<i> kinetic energy, internal energy, work </i>and <i>turbulent dissipation </i>without need to introduce any concept of<i> entropy.</i></span></div><div><span style="color: #1a1a1a; font-family: serif;"><i><br /></i></span></div><div><span style="color: #1a1a1a; font-family: serif;">Replacing SM by precisely defined finite precision computation gives a more solid third pillar. But this is new and not easily embraced by analytical/theoretical mathematicians/physicists not used to think in terms of computation, with Stephen Wolfram as notable exception. </span></div><div><span style="color: #1a1a1a; font-family: serif;"><i><br /></i></span></div><div><span style="color: #1a1a1a; font-family: serif;"><b>PS1</b> To meet criticism that the <a href="https://plato.stanford.edu/entries/statphys-Boltzmann/#2">Stosszahlansatz</a></span><span style="color: #1a1a1a; font-family: serif;"> underlying the H-theorem stating that particles before collision are uncorrelated, simply assumes what has to proved (<span style="caret-color: rgb(26, 26, 26);">irreversibility), Boltzmann argued:</span></span></div><ul style="text-align: left;"><li><i>But since this consideration has, apart from its tediousness, not the slightest difficulty, nor any special interest, and because the result is so simple that one might almost say it is self-evident I will only state this result.</i></li></ul><div>Convincing?</div><div><br /></div><div><b>PS2</b> Connecting to the previous post, recall that the era of quantum mechanics was initiated in 1900 by Planck introducing statistics of "energy quanta" inspired by Boltzmann's statistical mechanics, to explain observed atomic radiation spectra, opening the door to Born's statistical interpretation in 1927 of the Schrödinger wave function as the "probability of finding an electron" at some specific location in space and time, which is the text book wisdom still today. Thus the pillar of quantum mechanics is also weakened by statistics. The third pillar of relativity is free of statistics, but also of physics, and so altogether the three pillars offer a shaky foundation of modern physics. Convinced? </div><div><br /></div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-62339342919486436302024-03-11T18:52:00.017+01:002024-03-12T08:21:52.328+01:00The 2nd Law as Radiative Heat Transfer<p><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgm4RP-JVzRzrC1aVetVFrqGX7ehWjWV8PaBHUjTA1n4_YcAfvB5nCuQFRwpE6rsuVLZPzm5YH47MrDVGrmTBiyBofwXkjRHKUmbzsrLjA079QExJ4GStF8mFgo73DcJmQbDWSxH7GzPRSv6_xi0QWxzBLqxKX5FiAFMbDOo2DzCQn3n66GsetrmzEwi5bc/s824/Screenshot%202024-03-11%20at%2019.04.05.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="392" data-original-width="824" height="152" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgm4RP-JVzRzrC1aVetVFrqGX7ehWjWV8PaBHUjTA1n4_YcAfvB5nCuQFRwpE6rsuVLZPzm5YH47MrDVGrmTBiyBofwXkjRHKUmbzsrLjA079QExJ4GStF8mFgo73DcJmQbDWSxH7GzPRSv6_xi0QWxzBLqxKX5FiAFMbDOo2DzCQn3n66GsetrmzEwi5bc/w453-h152/Screenshot%202024-03-11%20at%2019.04.05.png" width="453" /></a></div><p>The 2nd Law of Thermodynamics states that heat energy $Q$ without forcing, is transferred from a body of temperature $T_1$ to a body of temperature $T_2$ with $T_1>T_2$ by <i>conduction </i>according to <i>Fourier's Law </i>if the bodies are in contact:<i> </i></p><p></p><ul style="text-align: left;"><li>$Q =\gamma (T_1-T_2)$ </li></ul><p></p><p>and/or by <i>radiation</i> according <i>Stephan-Boltzmann-Planck's Law </i>if the bodies are not in contact as <i>radiative heat transfer</i>: </p><p></p><ul style="text-align: left;"><li>$Q=\gamma (T_1^4-T_2^4)$ (SBP)</li></ul><div>where $\gamma > 0$.</div><div><br /></div><div>The energy transfer is <i>irreversible </i>since<i> </i>it has <i>a direction from warm to cold </i>with $T_1>T_2$. It is here possible to view conduction as radiation at close distance and thus reduce the discussion to radiation. </div><div><br /></div><div>We can thus view the 2nd Law to be a consequence of (SBP), at least in the case of two bodies of different temperature: There is an irreversible transfer of heat energy from warm to cold. </div><div><br /></div><div>To prove 2nd Law for radiation thus can be seen to boil down to prove (SBF). This was the task taken on by the young Max Planck, who after a long tough struggle presented a proof in 1900, which he however was very unhappy with, since it like Boltzmann's H-theorem from 1872 was based on <i>statistical mechanics </i>and not classical deterministic <i>physical mechanics.</i></div><div><i><br /></i></div><div>But it is possible to prove (SBF) by replacing statistics with an assumption of <i>finite precision computation</i> in the form of <a href="https://computationalblackbody.wordpress.com">Computational Blackbody Radiation</a>. Radiative heat transfer is here seen to be geared as a deterministic <i>threshold phenomenon</i> like a <i>semi-conductor</i> allowing heat transfer only one-way from warm to cold. </div><div><br /></div><div>Another aspect of radiation is that it is impossible to completely turn off or block by shielding of some sort. It connects to the <i>universality of blackbody radiation </i>taking the same form independent of material matter as shown <a href="https://computationalblackbody.wordpress.com/universality/">here</a>. </div><div><br /></div><div>We are thus led to the following form of the 2nd Law without any statistics:</div><div><ul style="text-align: left;"><li><b>Radiative heat transfer from warm to cold is unstoppable and irreversible. </b></li></ul><div>The finite precision aspect here takes the form of a threshold, thus different from that operational in the case of turbulent dissipation into heat energy connecting to complexity with sharp gradients as discussed in recent posts.</div></div><div><br /></div><div><b>PS</b> To learn how statistical mechanics is taught at Stanford University by a world-leading physicist, listen to <a href="https://www.bing.com/videos/riverview/relatedvideo?q=is+statistical+mechanics+real+physics%3f&mid=E04722F0DEDA93625D44E04722F0DEDA93625D44&FORM=VIRE">Lecture 1</a> and ask yourself if you get illuminated:</div><div><ul style="text-align: left;"><li><i>Statistical mechanics is useful for predictions in cases when you do not know the initial conditions nor the laws of physics.</i></li></ul></div><div></div><div><br /></div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-75999052491463360472024-03-11T14:23:00.015+01:002024-03-11T16:00:41.995+01:002nd Law for Cosmology<p>A mathematical model of the Universe can take the form of<i> Euler's equations for a gas</i> supplemented with <i>Newton's law of gravitation</i> as stated in <a href="https://www.dropbox.com/s/pt4up03axk52ahs/ambsthermo.pdf?e=1&dl=0">Chap 32 Cosmology of Computational Thermodynamics</a>. </p><p>Computational solutions of these equations satisfy the following <i>evolution equations</i> as laws of thermodynamics depending on time $t$ </p><p></p><ul style="text-align: left;"><li>$\dot K(t)=W(t)-D(t)-\dot\Phi (t)$ (1)</li><li>$\dot E(t)=-W(t)+D(t)$, (2)</li></ul><div>where $K(t)$ is total <i>kinetic energy,</i> $E(t)$ total<i> internal energy</i> (<i>heat energy</i>), $W(t)$ is total <i>work</i>, $D(t)\ge 0$ is total <i>turbulent dissipation, </i>$\Phi (t)$ is total <i>gravitational energy </i>and the dot signifies differentiation with respect to time. Adding (1) and (2) gives the following <i>total energy </i>balance:</div><div><ul style="text-align: left;"><li>$K(t)+E(t)-\Phi(t)= constant.$ (3)</li></ul><div>Further (1) and (2) express an irreversible transfer of energy from kinetic to internal energy with $D(t)>0$, and so serve as a<i> 2nd Law for Cosmology</i> giving<i> time a direction. </i>Recall that the theoretical challenge is to tell/show why turbulent dissipation is unavoidable. </div><div><i><br /></i></div><div>Computations may start from a hot dense state at $t=0$ which is seen to expand/cool (<a href="https://editor.p5js.org/claesjohnson/full/5ViZK_NVp">run code</a>) (Big Bang) to maximal size and then contract/warm back to a hot dense state (Big Crunch) (<a href="https://editor.p5js.org/claesjohnson/sketches/NF8o1fkx">run code</a>) in an irreversible sequence of expansions/contractions until some final stationary equilibrium state with $E(\infty )=P(\infty )$. Compare with<a href="https://claesjohnson.blogspot.com/2011/10/cosmology-as-thermodynamics-with.html"> post from 2011</a>.</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-OWtutR2fONYVmqcicZcTsS9KTW8pttQS1gzMG-kZuVeJ3jvVMAGsfPFu-mG_8j4NrJojays8EuSVLfT6_aDNsuEN8Zev5x2RH1PiMWtlKN2c8fFLHEA9GCxbgBmK6pjzsROx9pBOJv5wr0hwzjAPT22_d5Zc9bFLZEw-aqjlSBvm5-OBlnep5fJdV4dx/s774/Screenshot%202024-03-11%20at%2014.22.38.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="478" data-original-width="774" height="238" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-OWtutR2fONYVmqcicZcTsS9KTW8pttQS1gzMG-kZuVeJ3jvVMAGsfPFu-mG_8j4NrJojays8EuSVLfT6_aDNsuEN8Zev5x2RH1PiMWtlKN2c8fFLHEA9GCxbgBmK6pjzsROx9pBOJv5wr0hwzjAPT22_d5Zc9bFLZEw-aqjlSBvm5-OBlnep5fJdV4dx/w478-h238/Screenshot%202024-03-11%20at%2014.22.38.png" width="478" /></a></div></div><div><br /></div></div><p></p>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-7371035195209993082024-03-11T10:01:00.004+01:002024-03-11T10:08:23.917+01:00Dark Matter as Axions as 85% of All Matter?<p style="text-align: left;">Sabine Hossenfelder in <a href="http://backreaction.blogspot.com/2024/03/exploding-stars-made-of-dark-matter.html">Exploding stars made of dark matter could heat up universe</a> informs us about some new speculations about the physics of <i>dark matter, </i>believed to make up 85% of all matter in the Universe, in the form of </p><p style="text-align: left;"></p><ul style="text-align: left;"><li><i>axions </i>or<i> axion particles </i></li></ul><div>able to form </div><div><ul style="text-align: left;"><li><i>axion stars</i></li></ul><div>able to <i>explode </i>and so able to </div></div><div><ul style="text-align: left;"><li><i>heat surrounding gas </i></li></ul><div>which could be a detectable phenomenon. Sabine ends asking how it is possible that physicists can be paid for this kind of speculation. </div></div><div><br /></div><div>Compare with the idea I have suggested that matter with density $\rho (x,t)=\Delta \phi (x,t)$ is formed from a gravitational potential $\phi (x,t)$ locally in space-time with coordinates $(x,t)$ from <i>differentiation</i> expressed by the <i>Laplacian differential operator</i> $\Delta$, and that <i>dark matter</i> corresponds to large regions where the potential is <i>smooth </i>in the sense that<i> </i>$\Delta \phi (x,t)$ is not large enough to create matter which is visible. </div><div><br /></div><div>It is conceivable that such large regions could concentrate gravitationally and even form stars which could explode as in the above scenario. Is anyone willing to pay for this idea? Does it make sense? </div><p></p>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-50437434332342673072024-03-10T13:26:00.020+01:002024-03-10T21:18:38.230+01:00Three Elephants in Modern Physics <table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2BPsTXWgvNxkr2zdd-UBvGtYlwsTO78XprHJ9ePYqSzQPxiQKPcu0uHb5O3Is0kfXRRTY1yH3cmzppCXRaVAWbzM43R6j3wrnYq5hs1YjXomlUuEtyF5E5iYk5Wo2Q41vpYJMkGK5yA8pIEcGrOBVZQ_1tJDvKGnaDTH4y1sy6lSXlsHrxj87YpwFLRlE/s954/Screenshot%202024-03-10%20at%2013.22.00.png" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="492" data-original-width="954" height="165" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2BPsTXWgvNxkr2zdd-UBvGtYlwsTO78XprHJ9ePYqSzQPxiQKPcu0uHb5O3Is0kfXRRTY1yH3cmzppCXRaVAWbzM43R6j3wrnYq5hs1YjXomlUuEtyF5E5iYk5Wo2Q41vpYJMkGK5yA8pIEcGrOBVZQ_1tJDvKGnaDTH4y1sy6lSXlsHrxj87YpwFLRlE/w410-h165/Screenshot%202024-03-10%20at%2013.22.00.png" width="410" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Modern theoretical physicists busy at work handling the crisis.</td></tr></tbody></table><p>There are three elephants in the crisis room of modern physics:</p><p></p><ol style="text-align: left;"><li>special relativity</li><li>2nd law of thermodynamics</li><li>foundations of quantum mechanics </li></ol><p></p><p></p><div>which since many years are no longer discussed in the physics community, not because they have since long been resolved, but because they remain open problems with no progress for 100 years. </div><div><br /></div><div>Only crackpot amateur physicists still debate these problems on fringe sites like <a href="https://www.youtube.com/watch?v=Pv7naqxiak0&t=292s">John Chappell Natural Philosophy Society</a>. Accordingly these topics are longer part of core theoretical physics education, since questions from students cannot be answered. This may seem strange, but it has come to be an agreement within the physics community to live with. Discussion closed. Research submitted to leading journals on these topics will be rejected, without refereeing.</div><div><br /></div><div>Is it possible to understand why no progress has been made? <a href="https://www.quantamagazine.org/crisis-in-particle-physics-forces-a-rethink-of-what-is-natural-20220301/">Why is there a crisis?</a></div><div><br /></div><div>As concerns special relativity the reason is that it is not a theory about real physics, but a theory about perceived observer perceptions of "events" identical to space-time coordinates without physical meaning. This makes special relativity to a game following some ad hoc rules without clear physics pretending to describe a world, which shows to be very very strange. Unless of course you realise that it is just a game and not science, but then nothing to teach at a university. </div><div><br /></div><div>As concerns topics 2 and 3 the reason is the introduction of <i>statistical mechanics </i>by Boltzmann followed by Planck as last rescue when deterministic physics seemed to fail. Again the trouble is that statistical mechanics is physics in the eyes of observers as probabilities of collections of physical events without cause-effect, rather than specific real events with cause-effect. Like special relativity this makes statistical mechanics into a game according to some ad hoc rules without physical meaning. </div><div><br /></div><div>In all three cases the observer is given a new key role as if the world depends on observation and cannot as in classical deterministic physics, itself go ahead without. This makes discussion complicated since there is no longer any common ground to all observers, and so eventually discussion dies. Nothing more to say. Everybody agrees that that there is nothing to disagree about. Everything in order. </div><div><br /></div><div>Schrödinger as the inventor of the Schrödinger equation for the Hydrogen atom with one electron as a deterministic classical continuum mechanical model, was appalled by Born's statistical interpretation of the multi-dimensional generalisation of his equation to atoms with several electrons defying physical meaning, and so gave up and turned to more fruitful pastures like the physics of living organisms.</div><div><br /></div><div>But the elephants are there even if you pretend that they are not, and that is not a very healthy climate for scientific progress. You find my attempts to help out on this blog. </div><div> </div><p></p>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-53853349607364116922024-03-09T12:00:00.012+01:002024-03-09T12:34:30.613+01:00Challenges to the 2nd Law <p>The book <a href="https://link.springer.com/book/10.1007/1-4020-3016-9">Challenges to the Second Law of Thermodynamics</a> by Capek and Sheenan starts out describing the status of this most fundamental law of physics as of 2005:</p><ul style="text-align: left;"><li><i>For more than a century this field has lain fallow and beyond the pale of <b>legitimate scientific inquiry </b>due both to a dearth of scientific results and to a <b>surfeit of peer pressure against such inquiry. </b></i></li><li><i>It is remarkable that 20th century physics, which embraced several radical paradigm shifts, was <b>unwilling to wrestle with this remnant of 19th century physics</b>, whose <b>foundations were admittedly suspect</b> and largely unmodified by the discoveries of the succeeding century. </i></li><li><i>This failure is due in part to the many strong imprimaturs placed on it by prominent scientists like Planck, Eddington, and Einstein. There grew around the second law a <b>nearly inpenetrable mystique </b>which only now is being pierced.</i></li></ul><div>The book then continues to present 21 formulations of the 2nd Law followed by 20 versions of entropy and then proceeds to a large collection of challenges, which are all refuted, starting with this background:</div><div><ul style="text-align: left;"><li><i><b>The 2nd Law has no general theoretical proof.</b></i></li><li><i>Except perhaps for a dilute gas (Boltzmann's statistical mechanics), its absolute status rests squarely on<b> empirical evidence. </b></i></li></ul><div>We learn that modern physics when confronted with the main unresolved problem of classical physics reacted by denial and oppression as cover up of a failure of monumental dimension. The roots of the present crisis of modern physics may hide here. </div><div><br /></div><div><a href="https://www.dropbox.com/s/pt4up03axk52ahs/ambsthermo.pdf?e=1&dl=0">Computational Thermodynamics</a> seeks to demystify the 2nd Law as a result of<i> finite precise computation </i>meeting systems developing<i> increasing complexity</i> like <i>turbulence</i> in slightly viscous flow. </div><div><br /></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEc9Yk14z6YdT8chnz66AmLLIMyQaK-vqtjOaA7G1cbXK8cV877FNWkQp1wP_hZvj2nvumdL7EqZJIN-APuuODnvdE0QZPcio2GhceTFw0-GVpLd-2j_c_v7pzhZDGUL1QYzGr-mSRdyW5RbPuidNa3kE4Ec77CQmK5nJ6RnLG-IRjr4wfUhyphenhyphenPQRNq5y8H/s788/Screenshot%202024-03-09%20at%2012.32.12.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="788" data-original-width="612" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEc9Yk14z6YdT8chnz66AmLLIMyQaK-vqtjOaA7G1cbXK8cV877FNWkQp1wP_hZvj2nvumdL7EqZJIN-APuuODnvdE0QZPcio2GhceTFw0-GVpLd-2j_c_v7pzhZDGUL1QYzGr-mSRdyW5RbPuidNa3kE4Ec77CQmK5nJ6RnLG-IRjr4wfUhyphenhyphenPQRNq5y8H/w339-h320/Screenshot%202024-03-09%20at%2012.32.12.png" width="339" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Physicists confronted with proving the 2nd Law. </td></tr></tbody></table><br /><div><br /></div><div><br /></div></div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-20124208229893499042024-03-08T10:23:00.015+01:002024-03-09T09:55:03.385+01:002nd Law vs Friction <p><i>Perpetual motion</i> is viewed to be impossible because according to the <i>2nd Law of Thermodynamics</i> (see<a href="https://claesjohnson.blogspot.com/search/label/2nd%20law%20of%20thermodynamics"> recent posts</a>)</p><p></p><ul style="text-align: left;"><li><b>There is always some friction. </b>(F)</li></ul><div>Is this true? It does not appear to be true in the microscopics of atoms in stable ground state, where apparently electrons move around (if they do) without losing energy seemingly without friction, see <a href="https://physicalquantummechanics.wordpress.com">Real Quantum Mechanics</a>. </div><div><br /></div><div>But is it true in macroscopics of many atoms or molecules such as that of fluids? A viscous fluid in motion meets a flat plate boundary with a<i> skin friction</i> depending on the <i>Reynolds number</i> $Re\sim \frac{1}{\nu}$ with $\nu$ <i>viscosity</i> as follows based on observation:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2NMJ64-AGnuBkh0zJesGPwFAbDImiHKJc1ubaKmdUxTNunXUHjz-Dz0l39s8Gl5dPKHOQ1OgWT8K7DiLtJ_Kch6f44jGwTQVKTBVl-Ec6hh1cuIggvNHIu7-sBFt0a9-AnsnE3C27wrCxYIlprvaeW-MrtyIfYfrd1kE7A1og80LYX4ehE1lHH1m8thMN/s1274/Screenshot%202024-03-08%20at%2009.46.11.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="710" data-original-width="1274" height="178" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2NMJ64-AGnuBkh0zJesGPwFAbDImiHKJc1ubaKmdUxTNunXUHjz-Dz0l39s8Gl5dPKHOQ1OgWT8K7DiLtJ_Kch6f44jGwTQVKTBVl-Ec6hh1cuIggvNHIu7-sBFt0a9-AnsnE3C27wrCxYIlprvaeW-MrtyIfYfrd1kE7A1og80LYX4ehE1lHH1m8thMN/w501-h178/Screenshot%202024-03-08%20at%2009.46.11.png" width="501" /></a></div><br /><div>We see that skin friction coefficient $c_f$ in a laminar boundary layer tends to zero as $Re$ tends to infinity (or viscosity tends to zero), which is supported by basic mathematical analysis. </div><div><br /></div><div>We also that there is a transition from laminar to turbulent boundary layer for $Re > 5\times 10^5$ with larger turbulent skin friction coefficient $c_f\approx 0.002$ with a very slow decay with increasing $Re$ with no observations of $c_f<0.001$. </div><div><br /></div><div>The transition to a turbulent boundary layer is the result of <i>inherent instability </i>of the motion of a fluid with small viscosity, which is supported by mathematical analysis, an instability which cannot be controlled in real life, see<a href="https://www.dropbox.com/s/yz7cng6n8xyhwn4/(Am%20Bs%204)%20Johan%20Hoffman%2C%20Claes%20Johnson%20-%20Computational%20Turbulent%20Oncompressible%20Flow-Springer%20(2007).pdf?e=1&dl=0"> Computational Turbulent Incompressible Flow.</a></div><div><br /></div><div>We thus get the message from both observation and mathematical analysis that (F) as concerns skin friction is true: Skin friction in a real fluid does not tend to zero with increasing $Re$, because there is always transition to a turbulent boundary layer with $c_f>0.001$. </div><div><br /></div><div>Laminar skin friction is vanishing in the limit of infinite $Re$, but not turbulent skin friction.</div><div><br /></div><div>We can thus can rationalise the 2nd Law for fluids as presence of unavoidable skin friction from turbulent motion resulting from uncontrollable instability, which does not tend to zero with increasing $Re$ , reflecting presence of a non-zero limit of turbulent dissipation in the spirit of Kolmogorov. </div><div><br /></div><div>Connecting to finite precision computation discussed in recent posts, we understand that computational resolution to physical scale is not necessary, which makes turbulent flow computable without turbulence modelling. </div><div><br /></div><div>In bluff body computations (beyond drag crisis) it is possible to effectively set skin friction to zero as a slip boundary condition thus avoiding having to resolve turbulent boundary layers. In pipe flow skin friction cannot be set to zero.</div><div><br /></div><div><b>PS1</b> A hope of vanishingly small laminar skin friction has been nurtured in the fluid mechanics community, but the required instability control has shown to be impossible, as an expression of the 2nd Law. </div><div><br /></div><div><b>PS2</b> One may ask if motion without friction is possible at all? <a href="https://claesjohnson.blogspot.com/2024/02/motion-vs-appearance-or-emergence.html">Here we face</a> the question what motion is, connecting to wave motion as an illusion. In Newtonian gravitation of a Platonic ideal world there is no dissipation/friction, but in the real world there is: The Moon is slowly receding from the Earth because of tidal motion energy loss. What then about the supposed motion of photons at the speed of light seemingly without energy loss from friction? Is this motion illusion, with necessary energy loss to receive a light signal? <a href="https://claesjohnson.blogspot.com/2024/02/motion-vs-appearance-or-emergence.html">Computational Blackbody Radiation</a>ֶ says yes! The 2nd Law can thus be formulated: </div><div><ul style="text-align: left;"><li><b>There is no motion without friction. Photons in frictionless motion is illusion.</b></li></ul><div>Do you buy this argument? Can you explain why? Instability? What is a photon in motion without friction? Illusion?</div></div><div><br /></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzD9UvORiqZFvaSAY99OULCj-MkJ5TuPRa33jCHCmM5wJYX_8AbJFx0mGpLQauMzEv0yfGyRWdtTwUsNr2DcsxnUEf7yQ5ryOTw6PtNEhyphenhyphenrrIHgzDxIuy92t7RPF9mqRvNPMUjHLOS060TmLQRlpeZgTVP85vxcZ15BTqW7XERit48CPB-yxNjNrKrTsRJ/s1266/Screenshot%202024-03-08%20at%2015.23.20.png" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="580" data-original-width="1266" height="147" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzD9UvORiqZFvaSAY99OULCj-MkJ5TuPRa33jCHCmM5wJYX_8AbJFx0mGpLQauMzEv0yfGyRWdtTwUsNr2DcsxnUEf7yQ5ryOTw6PtNEhyphenhyphenrrIHgzDxIuy92t7RPF9mqRvNPMUjHLOS060TmLQRlpeZgTVP85vxcZ15BTqW7XERit48CPB-yxNjNrKrTsRJ/w432-h147/Screenshot%202024-03-08%20at%2015.23.20.png" width="432" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Why is there always some friction?</td></tr></tbody></table><br /><div><br /></div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-59282680290080631422024-03-06T20:57:00.015+01:002024-03-08T07:55:26.098+01:002nd Law for Radiative Heat Transfer as Finite Precision Physics<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdSy1Uo_TzqdcYrWBSoSdmsujvbzsc4RLpbpolBy1J2D2sTQuH2wtGFpcoZqlzFWbwqdXD-cThfKypZdD4K31EDdkIztw1_ETEaJgwQ8dd9vGuIj9jIx5zbUeXwclO_mH6cE5RAcjlBT7xIuqm1MgyOxKQfPg956yk_1K8X08HYYMNr8utj9tfvXhVsq0C/s1226/Screenshot%202024-03-07%20at%2019.33.41.png" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="606" data-original-width="1226" height="158" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdSy1Uo_TzqdcYrWBSoSdmsujvbzsc4RLpbpolBy1J2D2sTQuH2wtGFpcoZqlzFWbwqdXD-cThfKypZdD4K31EDdkIztw1_ETEaJgwQ8dd9vGuIj9jIx5zbUeXwclO_mH6cE5RAcjlBT7xIuqm1MgyOxKQfPg956yk_1K8X08HYYMNr8utj9tfvXhVsq0C/w462-h158/Screenshot%202024-03-07%20at%2019.33.41.png" width="462" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Transfer of heat energy from warm to cold by electromagnetic waves.</td></tr></tbody></table><p>This is a continuation of <a href="https://claesjohnson.blogspot.com/search/label/2nd%20law%20of%20thermodynamics">recent posts on the 2nd Law of Thermodynamics.</a></p><p>There is a 2nd Law for <i>radiative heat transfer </i>expressing: </p><p></p><ul style="text-align: left;"><li>Heat energy is transferred by electromagnetic waves from a body with higher temperature to a body with lower temperature, not the other way. (*) </li></ul><div>Why is that? Standard physics states that it is a consequence of <i>Plank's law</i> of radiation based on <i>statistics of energy quanta,</i> as an analog of Boltzmann's proof of a 2nd Law based on <i>statistical mechanics</i>. The objections raised to Boltzmann's proof carry over to that of Planck, who was very unhappy with his proof but not as unhappy as Boltzmann with his. </div><div><br /></div><div>An approach without statistics is presented on<a href="https://computationalblackbody.wordpress.com"> Computational Blackbody Radiation</a> where (*) appears as a <i>high frequency cut-off </i>increasing with temperature. The effect is that only frequencies above cut-off for the body with lower temperature have a heating effect resulting in one-way transfer of heat from warm to cold. For more details check-out<a href="https://www.youtube.com/watch?v=chs4M5c6OzQ"> this presentation. </a></div><div><br /></div><div>The high-frequency cut-off can be seen as an expression of <i>finite precision</i> increasing with temperature of atomic oscillation as heat energy. One-way heat transfer is thus a <i>threshold phenomenon </i>connected to finite precision.</div><div><br /></div><div>Similarly, the photoelectric effect can be explained as a threshold phenomenon connected to finite precision, where only light of sufficiently high frequency can produce electrons. </div><div><br /></div><div>A 2nd Law based on <i>finite precision physics</i> thus can serve a role both in both fluid mechanics, and electromagnetics, and also quantum mechanics as discussed in <a href="https://claesjohnson.blogspot.com/2024/03/2nd-law-vs-perpetual-motion.html">this post.</a> </div><div><br /></div><div>In other words, finite precision physics in analog or digital form appears as the crucial aspect giving meaning to a universal 2nd Law, which is missing in standard physics with infinite precision. </div><div><br /></div><div>The general idea is to replace statistical physics, which is not real physics, by finite precision computation, which can be both analog and digital physics. </div><div><br /></div><div>Of course, this idea will not be embraced by analytical mathematicians or theoretical physicists working with infinite precision...</div><p></p>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-7045544213533856632024-03-06T11:09:00.016+01:002024-03-06T12:54:48.083+01:00The 2nd Law in a World of Finite Precision<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpsvCrCUX8oZrJ4wW_SFY-BEyn5gQ2cnIj-CpgAkRjPqohF4q7KypwwmEikSyQu1nflQgsnhsHIE_-y97v_nXfKPWlgqCUJ7Y9YOOvq6TZGZ_NGqN1ss355M6apRD9cnGgtxQehgj9nCuz0gVp9PeD0fZaiKhcuXvIXzI_WXQVwKMbSqF6_w6DVeegK7j_/s1088/Screenshot%202024-03-06%20at%2011.35.55.png" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="724" data-original-width="1088" height="213" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpsvCrCUX8oZrJ4wW_SFY-BEyn5gQ2cnIj-CpgAkRjPqohF4q7KypwwmEikSyQu1nflQgsnhsHIE_-y97v_nXfKPWlgqCUJ7Y9YOOvq6TZGZ_NGqN1ss355M6apRD9cnGgtxQehgj9nCuz0gVp9PeD0fZaiKhcuXvIXzI_WXQVwKMbSqF6_w6DVeegK7j_/w426-h213/Screenshot%202024-03-06%20at%2011.35.55.png" width="426" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Let there be a World of Finite Precision.</td></tr></tbody></table><p>Here is a summary of aspects of the <i>2nd Law of Thermodynamics</i> discussed in recent posts: </p><p></p><ul style="text-align: left;"><li>2nd Law gives an<i> arrow of time </i>or <i>direction of time. </i></li><li>A <i>dissipative</i> system satisfies a 2nd Law.</li><li>A dissipative system contains a <i>diffusion</i> mechanism <i>decreasing sharp gradients by averaging.</i> </li><li><i>Averaging is irreversible</i> since an average does not display how it was formed. </li><li>Averaging/diffusion<i> destroys ordered structure/information irreversibly.</i> </li><li>Key example: <i>Destruction of large scale ordered kinetic energy</i> into <i>small scale unordered kinetic energy </i>as<i> heat energy </i>in<i> turbulent viscous dissipation. </i> </li></ul><div>To describe the World, it is not sufficient to describe dissipative destruction, since also processes of <i>construction</i> are present. These are processes of <i>emergence </i>where structures like <i>waves</i> and <i>vortices </i>with <i>velocity gradients </i>are formed in <i>fluids, </i><i>solid ordered structures</i> are formed by <i>crystallisation</i> and<i> living organisms develop. </i></div><div><br /></div><div>The World then appears as combat between <i>anabolism</i> as <i>building of ordered structure </i>and<i> metabolism as destruction of ordered structure into unordered heat energy. </i></div><div><i><br /></i></div><div>The 2nd Law states that destruction cannot be avoided. Perpetual motion is impossible. There will always be some friction/viscosity/averaging present which makes real physical processes irreversible with an arrow of time. </div><div><br /></div><div>The key question is now why some form of friction/viscosity/averaging cannot be avoided? There is no good answer in classical mathematical physics, because it assumes <i>infinite precision </i>and with infinite precision<i> </i>there is no need to form averages since all details can be kept. In other words, in a <i>World of Infinite Precision</i> there would be no 2nd Law stating unavoidable irreversibility, but its existence would not be guaranteed. </div><div><br /></div><div>But the World appears to exist and then satisfy a 2nd Law and so we are led to an idea of an <i>Analog World of</i> <i>Finite Precision, </i>which possible can be mimicked by a<i> Digital World of Finite Precision</i> (while a possibly non-existing World of infinite precision cannot).<i> </i></div><div><i><br /></i></div><div>The Navier-Stokes equation for a fluid/gas with positive viscosity as well as Boltzmann's equations for a dilute gas are dissipative systems satisfying a 2nd Law with positive dissipation. But why positive viscosity? Why positive dissipation?</div><div><br /></div><div>The Euler equations describe a fluid with zero viscosity, which formally in infinite precision is a system without dissipation violating the 2nd Law. </div><div><br /></div><div>We are led to consider the Euler equations in Finite Precision, which we approach by digital computation to find that computational solutions are turbulent with positive turbulent dissipation independent of mesh size/precision once sufficiently small. We understand that the presence of viscosity/dissipation is the result of a necessary averaging to avoid the flow to blow-up from increasing large velocity gradients emerging form convection mixing high and low speed flow. </div><div><br /></div><div>We thus explain the emergence of positive viscosity in a system with formally zero viscosity as a necessary mechanism to allow the system to continue to exist in time. </div><div><br /></div><div>The 2nd Law thus appears as being a mathematical necessity in an existing World of Finite Precision. </div><div><br /></div><div>The mathematical details of this scenario in the setting of Euler's equations id described in the books <a href="https://www.dropbox.com/s/yz7cng6n8xyhwn4/(Am%20Bs%204)%20Johan%20Hoffman%2C%20Claes%20Johnson%20-%20Computational%20Turbulent%20Oncompressible%20Flow-Springer%20(2007).pdf?e=1&dl=0">Computational Turbulent Incompressible Flow</a>, <a href="https://www.dropbox.com/s/pt4up03axk52ahs/ambsthermo.pdf?e=1&dl=0">Computational Thermodynamics</a> and <a href="https://www.dropbox.com/s/v31ndxamugc9xlm/EulerRightBook21.pdf?e=1&dl=0">Euler Right</a>.</div><div><br /></div><div><b>PS</b> A related question is if in a <a href="https://claesjohnson.blogspot.com/2024/02/a-logical-analysis-of-western-politics.html">World of Very Limited Human Intelligence a WW3 destruction is necessary</a>?</div><div><br /></div><p></p>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-51999310994643895542024-03-05T20:04:00.013+01:002024-03-06T09:46:14.852+01:00No Mathematical Proof of the 2nd Law of Thermodynamics<p>ChatGPT informs me that there is no mathematical proof of 2nd Law of thermodynamics and so it is simply an empirical law albeit:</p><p></p><ul style="text-align: left;"><li><i>supported by a wealth of empirical evidence</i></li><li><i>deeply ingrained in our understanding of how physical systems behave</i></li><li><i>while mathematical frameworks like statistical mechanics provide a basis for explaining the law, the principle itself is derived from the consistency of these explanations with real-world observations.</i></li></ul><div>ChatGPT is useful in the sense that it reports what it has learned from physics literature, while it is not intelligent enough to cover up like a real theoretical physicist, who would never admit anything like that.</div><div><br /></div><div>In any case we learn that that there is no mathematical proof/explanation of the 2nd Law. </div><div><br /></div><div>It means that this could be added to the list of Clay Millennium Problems, or even better replace the closely related Navier-Stokes problem, which is still open without any progress to a solution, see <a href="https://claesjohnson.blogspot.com/2024/03/2nd-law-vs-clay-millennium-problem-on.html">previous post.</a></div><div><br /></div><div>What do you think? Is there a mathematical proof? Or not? </div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLF2x3h2fnnRFPVQJ36X6Mx1hKclqrjdAAixD2XuTiIuuiVS56pPusa4sXiAlJC-4EpwmjTnrdvmyaEmUNuJ_9fj2nOvBDa1qUTz-MTllK7bmTgVgv9pRoK20ajNY7GqqsPkKvDlhJ5AM9OIhwVSH196HG8uvMtpxbiCZTTjZUsvJGQ4xWhDjoY8iqxEGP/s788/Screenshot%202024-03-05%20at%2020.12.26.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="788" data-original-width="768" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLF2x3h2fnnRFPVQJ36X6Mx1hKclqrjdAAixD2XuTiIuuiVS56pPusa4sXiAlJC-4EpwmjTnrdvmyaEmUNuJ_9fj2nOvBDa1qUTz-MTllK7bmTgVgv9pRoK20ajNY7GqqsPkKvDlhJ5AM9OIhwVSH196HG8uvMtpxbiCZTTjZUsvJGQ4xWhDjoY8iqxEGP/w474-h320/Screenshot%202024-03-05%20at%2020.12.26.png" width="474" /></a></div><b>PS</b> Recall that Boltzmann's <i>H-theorem</i> stating a steady progress to a <i>Maxwellian equilibrium</i> in a dilute gas attempted as a mathematical proof of the 2nd Law, is based on Boltzmann's <i>Stosszahlansatz </i>asking two particles about to collide to be uncorrelated as an assumption of statistical nature, which however cannot be verified nor assumed to be true in any generality. <div><div><br /></div><p></p></div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-87950225058137414912024-03-05T12:31:00.009+01:002024-03-05T20:31:05.295+01:002nd Law vs Clay Millennium Problem on Navier-Stokes Equations <p>The <a href="https://www.claymath.org/millennium/navier-stokes-equation/">Clay Institute Millennium Problem on Navier-Stokes equations </a>is introduced as follows:</p><ul><li><i>This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also <b>understanding.</b></i></li><li><i>Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an <b>understanding</b> of solutions to the Navier-Stokes equations. </i></li><li><i>Although these equations were written down in the 19th Century, our<b> understanding</b> of them remains <b>minimal.</b> The challenge is to make <b>substantial progress</b> toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.</i></li></ul>The problem is still open. No solution is even in sight after 24 years. No progress at all.<div><br /><div>Another <a href="https://claesjohnson.blogspot.com/2024/03/no-mathematical-proof-of-2nd-law-of.html">main open problem</a> of mathematical physics is the 2nd Law of Thermodynamics, which in particular applies to the flow of fluids such as water and air as governed by Navier-Stokes equations. </div><div><br /></div><div>It is thus possible to view the Clay Navier-Stokes Problem as an instance of the 2nd Law of Thermodynamics, and so be reformulated into:</div><div><ul style="text-align: left;"><li>Mathematical proof of the 2nd Law of Thermodynamics for fluids. (P)</li></ul><div>This version has a more obvious significance and it is possible that a solution can be found and so increase understanding in the spirit of Clay.</div><div><br /></div><div>A resolution to (P) is presented in <a href="https://claesjohnson.blogspot.com/search/label/2nd%20law%20of%20thermodynamics">recent posts on the 2nd Law</a>. </div></div><div><br /></div><div>I have sent the following letter to the President of the Clay Institute and will report reaction:</div><div><br /></div><div><span style="font-family: Helvetica; font-size: 12px;">Dear President </span><div style="font-family: Helvetica; font-size: 12px;"><br /></div><div style="font-family: Helvetica; font-size: 12px;">No progress towards a solution to the Clay Millennium Problem on Navier-Stokes equations has been </div><div style="font-family: Helvetica; font-size: 12px;">made over a period of 24 years. A reformulation into a problem which possibly can be solved may better </div><div style="font-family: Helvetica; font-size: 12px;">meet the stated Clay objective of increasing understanding. </div><div style="font-family: Helvetica; font-size: 12px;"><br /></div><div style="font-family: Helvetica; font-size: 12px;">Thus I suggest a reformulation into a mathematical proof of the 2nd Law of Thermodynamics for fluids as </div><div style="font-family: Helvetica; font-size: 12px;">expressed here:</div><div style="font-family: Helvetica; font-size: 12px;"><br /></div><div style="font-family: Helvetica; font-size: 12px;"><a href="https://claesjohnson.blogspot.com/2024/03/2nd-law-vs-clay-millennium-problem-on.html">https://claesjohnson.blogspot.com/2024/03/2nd-law-vs-clay-millennium-problem-on.html</a></div><div style="font-family: Helvetica; font-size: 12px;"><br /></div><div style="font-family: Helvetica; font-size: 12px;">Sincerely</div><div style="font-family: Helvetica; font-size: 12px;">Claes Johnson</div><div style="font-family: Helvetica; font-size: 12px;">prof em applied mathematics Royal Institute of Technology Stockholm</div></div><div><br /></div></div>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-25654339710156964452024-03-04T10:58:00.022+01:002024-03-05T11:29:14.747+01:002nd Law vs Finite Precision Computation <p>Recent posts present an approach to the 2nd Law of Thermodynamics based on a notion of <i>finite precision computation </i>in both <i>analog physical</i> and <i>digita</i>l form as it appears in the basic case of <i>slightly viscous fluid flow </i>carrying the phenomenon of <i>turbulence </i>as described in detail in <a href="https://www.dropbox.com/s/yz7cng6n8xyhwn4/(Am%20Bs%204)%20Johan%20Hoffman%2C%20Claes%20Johnson%20-%20Computational%20Turbulent%20Oncompressible%20Flow-Springer%20(2007).pdf?e=1&dl=0">Computational Turbulent Incompressible Flow</a> and <a href="https://www.dropbox.com/s/pt4up03axk52ahs/ambsthermo.pdf?e=1&dl=0">Computational Thermodynamics</a>. </p><p>In analog physical form finite precision connects to the <i>smallest physical scale</i> present, and in digital form to the <i>mesh size</i> of a <i>computational mesh. </i> </p><p>In fluid flow the <i>smallest physical scale</i> can be viewed to be determined by the viscosity $\nu$ with normalisation of velocity and spatial dimension, with corresponding Reynolds number $Re =\frac{1}{\nu}$. A basic case concerns the <i>drag</i> of body as the force of resistance to motion through the fluid, which is captured in a <i>drag coefficient</i> $C_D$ depending on the shape of the body, with $C_D\approx 0.4$ for a sphere. The flow around a bluff body like a sphere attaches as laminar and separates in a <i>wake </i>of turbulent flow. </p><p>A critical question concerns the dependence of $C_D$ on $Re$ as the dependence on the smallest physical scale $\frac{1}{Re}$, with the following typical dependence:</p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQnrp9VrfZVedmAtryRiZ3SQhpV-y_7H0fsubIbIeG68HFdje36FHrvt6KSHG5Ipo9VYEauQVTAwCfRXmdgkgtq8i-9zFvQvGmRJnbtD0sSra63A0KKuMrrUEhqepsJNnTkiw3e7oL4dUwgqq7boM3rVyGhw80K3tYMh4nVOA1_aAYTyIJWYgt1hH9vyFc/s1594/Screenshot%202024-03-04%20at%2010.01.28.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1594" height="181" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQnrp9VrfZVedmAtryRiZ3SQhpV-y_7H0fsubIbIeG68HFdje36FHrvt6KSHG5Ipo9VYEauQVTAwCfRXmdgkgtq8i-9zFvQvGmRJnbtD0sSra63A0KKuMrrUEhqepsJNnTkiw3e7oL4dUwgqq7boM3rVyGhw80K3tYMh4nVOA1_aAYTyIJWYgt1hH9vyFc/w432-h181/Screenshot%202024-03-04%20at%2010.01.28.png" width="432" /></a></div><p>We see that $C_D$ overall varies little with $Re$, but has a substantial dip in the wide interval $10^5 <Re <10^6$ referred to as <i>drag crisis, </i>which covers many cases of practical interest in aero/hydrodynamics. </p><p>We see that whether within the interval of drag crisis or outside, $C_D$ varies little with analog computational precision and so is a robust quantity. In particular, the reduced drag in the interval of of the drag crisis depends to a switch from no-slip to slip boundary condition which decreases the width of the turbulent wake but not its intensity. </p><p>Let us now turn to the the precision in digital simulation form as the<i> mesh size</i> $h$. To capture the physical scale would seem to require $h<\nu$ and so $h<10^6$ in typical aero/hydrodynamics, which in 3d is beyond the capacity of any foreseeable computer. This is the status of standard Computational Fluid Dynamics CFD today: Turbulent flow is uncomputable, because resolution to physical scale is impossible. Turbulence modelling is necessary, but seemingly impossible. </p><p>As concerns the 2nd Law, we could stop here: Drag is roughly independent of finest physical scale and in particular does not to go to zero under resolution going to zero. Turbulent dissipation cannot be avoided. The 2nd Law is valid.</p><p>But is it really true that turbulent flow is uncomputable? Is it necessary to resolve the flow to smallest physical scale to capture drag? Maybe not, since the plot above gives hope: Except for the drag crisis $C_D$ is roughly independent of physical scale and so the mesh size can maybe be larger then $10^{-6}$, maybe $h=10^{-3}$ could suffice? </p><p>And yes, this turns out to be true as shown in detail in the book <a href="https://www.dropbox.com/s/yz7cng6n8xyhwn4/(Am%20Bs%204)%20Johan%20Hoffman%2C%20Claes%20Johnson%20-%20Computational%20Turbulent%20Oncompressible%20Flow-Springer%20(2007).pdf?e=1&dl=0">Computational Turbulent Incompressible Flow</a> where in particular drag crisis can be captured using a slip boundary condition along with $h=10^{-3}$ showing that <i>turbulent flow is computable today on a laptop. </i></p><p>We sum up:</p><p></p><ul style="text-align: left;"><li>The 2nd Law is true in the sense that turbulent dissipation is substantial independent of smallest physical scale (<i>Kolmogorov's conjecture</i> confirmed), and cannot be avoided because of inherent instability. </li><li>Turbulent flow is computable because resolution to smallest physical scale is not necessary. </li></ul><div>By artificially adding viscosity a mathematical proof of a 2nd Law stating energy dissipation is direct. The real challenge is to prove <i>why viscosity must be present with a substantial effect,</i> which is done in the book. </div><div><br /></div><div><b>PS </b>Turbulent dissipation at smallest scale is given by $\nu (\frac{du}{dx})^2\sim 1$ with Reynolds number $\frac{du\times dx}{\nu}\sim 1$ with $du$ velocity variation of length scale $dx$, which gives $du\sim \nu^{\frac{1}{4}}$ and $dx\sim \nu^{\frac{3}{4}}$ reflecting <i>Lipschitz continuity</i> of velocity with exponent $\frac{1}{3}$ in accordance with <i>Onsager's conjecture</i>. Turbulent dissipation takes place mainly at smallest scale because energy is transferred in a cascade from large to smaller scales. </div><p></p>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0tag:blogger.com,1999:blog-1500584444083499721.post-17715813140098454972024-03-03T15:20:00.011+01:002024-03-05T10:16:17.015+01:002nd Law vs Perpetual Motion<p><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqLV9UtlO0xErhxTxz6uZmyP7-Jd1VPMh9C_Bbizf-RXiwO6d-kgQlsBGlOkE0unllYqENn7aUzznZE7JkPW2SNgSiCs4_yXAijkIQ2FTRtby4TobC8Dcbvp-ve_gcQaC2FXcJjTtLb34Al2acImfM9ytkPz2rWBtQ3zQK0z5DRxc0okR2bmyk2O0qmYd3/s1132/Screenshot%202024-03-03%20at%2015.28.26.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="622" data-original-width="1132" height="176" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqLV9UtlO0xErhxTxz6uZmyP7-Jd1VPMh9C_Bbizf-RXiwO6d-kgQlsBGlOkE0unllYqENn7aUzznZE7JkPW2SNgSiCs4_yXAijkIQ2FTRtby4TobC8Dcbvp-ve_gcQaC2FXcJjTtLb34Al2acImfM9ytkPz2rWBtQ3zQK0z5DRxc0okR2bmyk2O0qmYd3/w443-h176/Screenshot%202024-03-03%20at%2015.28.26.png" width="443" /></a></div><br /><p>A physicist would say that <i>perpetual motion</i> is impossible, because by the <i>2nd Law of Thermodynamics </i>there is always some positive <i>loss</i> <i>of energy</i> from some form of <i>friction,</i> which even a best possible engineer cannot turn off. In a fluid the friction energy takes the form of <i>turbulent dissipation</i> into <i>heat energy.</i> But on the question <i>why </i>there must be some loss, the answer would be vague probably with some reference to <i>statistics,</i> <i>Boltzmann's H-theorem</i> and <i>entropy</i> as <i>disorder.</i></p><p>Let us see if the <i>Physical 2nd Law</i> I have described in recent posts can give an answer with more physical substance. The basic idea is that real <i>physics is a form of analog computation, </i>which can be <i>mimicked in digital computation</i> and that in both cases the computation has <i><b>finite precision.</b> </i>In real physics that may be set by the <i>smallest physical scale</i> and in computation by the <i>computational mesh. </i>A most remarkable conjectured by <i>Kolmogorov,</i> is that turbulent dissipation rate is independent of the absolute size of the smallest scale, because turbulent energy is transferred in a cascade to dissipate at smallest scale. </p><p>This is confirmed in the book <a href="https://www.dropbox.com/s/yz7cng6n8xyhwn4/(Am%20Bs%204)%20Johan%20Hoffman%2C%20Claes%20Johnson%20-%20Computational%20Turbulent%20Oncompressible%20Flow-Springer%20(2007).pdf?e=1&dl=0">Computational Turbulent Incompressible Flow</a>, which means that <i>turbulent flow is computable</i> without mesh resolution to physical scale and so opens a new window in fluid mechanics. </p><p>It also shows that it is impossible to decrease the loss/turbulent dissipation by refining the precision into a finer model and finer mesh. In other words, it is impossible to bring loss to zero and realise perpetual motion. Because of finite precision.</p><p>The notion of finite precision present in both analog and digital physics thus opens to a new understanding of the 2nd Law and why it makes perpetual motion impossible to realise. More substance is given in <a href="https://www.dropbox.com/s/pt4up03axk52ahs/ambsthermo.pdf?e=1&dl=0">Computational Thermodynamics.</a></p><p>The 2nd Law does not apply to the microscopics of a <i>hydrogen atom in ground state</i> with an electronic change in time like a harmonic oscillator without change of electron density and then without friction as described on <a href="https://physicalquantummechanics.wordpress.com">Real Quantum Mechanics.</a> But a radiating atom with electron density changing over time is subject to radiative loss, which must be balanced by an exterior force in sustained motion. The 2nd Law thus is relevant on all scales, not just for macroscopic ensembles of many. </p><p><br /></p>Claes Johnsonhttp://www.blogger.com/profile/07411413338950388898noreply@blogger.com0