2nd Coming of the 2nd Law

The 2nd Law of Thermodynamics has remained as a main mystery of physics ever since it was first formulated by Clausius in 1865 as non-decrease of entropy, despite major efforts by mathematical physicists to give it a rational understandable meaning.

The view today is, based on the work by Ludwig Boltzmann, that the 2nd Law is a statistical law expressing a lack of precise human knowledge of microscopic physics, rather than a physical law independent of human observation and measurement. This view prepared the statistical interpretation of quantum mechanics as the basis of modern physics.

Modern physics is thus focussed on human observation of realities, while classical physics concerns realities independent of human observation. To involve the observer into the observed makes physics subjective which means a depart from the essence of physics of objectivity. A 2nd Law based on statistics thus comes along with many difficulties, which ended Boltzmann's life, and it is natural to seek a formulation in terms of classical physics without statistics.

Such a formulation is given in Computational Thermodynamics based on the Euler equations for an ideal compressible gas solved by finite precision computation. In this formulation the 2nd Law is a consequence of the following equations expressing conservation of kinetic energy K and internal (heat) energy E:

• dK/dt = W - D
• dE/dt = - W + D
• D > = 0,  

where W is work and D is nonnegative turbulent dissipation (rates). The crucial element is the turbulent dissipation rate D which is non-negative, and thus signifies one-way transfer of energy from kinetic energy K into heat energy E.

The work W, positive in expansion and negative in compression, allows a two-way transfer between K and E, while turbulent diffusion D >= 0 can only transfer kinetic energy K into heat energy E, and not the other way.

We compare dE/dt = - W + D or rewritten as dE/dt + W = D as an alternative formulation of the 2nd Law, with the classical formulation found in books on thermodynamics:

• dE + pdV = TdS = dQ
• dS > = 0, 

where p is pressure, V is volume (with pdV corresponding to W), T is temperature, S is entropy and dQ added heat energy.

We see that D >= 0 expresses the same relation as dS >= 0 since T > 0, and thus the alternative formulation expresses the same effective physics as the classical formulation.

The advantage of the alternative formulation is that turbulent dissipation rate D with D >= 0 has a direct physical meaning, while the physical meaning of S and dS >= 0 has remained a mystery.

The alternative formulation thus gives a formulation in terms of physical quantities without any need to introduce a mysterious concept of entropy, which cannot decrease for some mysterious reason. A main mystery of science can thus be put into the wardrobe of mysteries without solution and meaning, together with phlogistons.

Notice the connection to Computational Blackbody Radiation with an alternative proof of Planck's radiation law with again statistics replaced by finite precision computation.

For a recent expression of the confusion and mystery of the 2nd Law, see Ludwig Boltzmann: a birthday by Lubos.

PS1 The reason to define S by the relation dE + pdV = TdS is that for an ideal gas with pV = RT this makes dS = dE/T + pdV/T an exact differential, thus defining S in terms of T and p. The trouble with S thus defined, is that it lacks direct physical meaning.

PS2 Lubos refers to Bohr's view of physics:

• There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature...

This idea has ruined modern physics by encouraging a postmodern form medieval mysticism away from rational objectivity as the essence of science, where the physical world is reduced to a phantasm in the mind of the observer busy counting statistics of non-physical micro states.

PS3 Recall that statistics was introduced by Boltzmann to give a mathematical proof of the 2nd law, which appeared to be impossible using reversible Newtonian micromechanics, followed by Planck to prove his law of radiation, followed by Born to give the multidimensional Schrödinger equation an interpretation. But this was overkill. It is possible to prove a 2nd law and law of radiation using instead of full statistics a concept of finite precision computation as shown in Computational Thermodynamics and Computational Blackbody Radiation, which maintains the rationalism and objectivity of classical mechanics, while avoiding the devastating trap of reversible micromechanics.