onsdag 7 juli 2010

Thermodynamics of a Pulsating Universe

We consider a cosmological model in the form of a reactive gas subject to gravitation
and exotermic fusion and and endotermic fission. The 2nd Law of Thermodynamics can be expressed in the form
  • dE = - W + D + Q
  • dK + dP = W - D
  • E = heat energy = small scale kinetic energy
  • K = large scale kinetic energy
  • W = work; W positive in expansion, W negative in contraction
  • D = turbulent dissipation (positive); transforms large scale kinetic energy to small scale ditto
  • Q = latent heat energy; Q positive for fusion, Q negative for fission
  • P = potential gravitational energy
and d denotes differentiation with respect to time.

We observe that the total energy E + K + P stays constant if the reaction Q cancels over time
(energy released by fusion = energy required for fission).

We now consider two cases:

I. Without Gravitation/Reaction

In the case P=0 and Q=0, the 2nd Law tells us that energy can shift from E to K
only by expansion with W positive, since D is positive.

On the other hand, energy steadily shifts from K to E by positive turbulent diffusion D.

This means that a system initialized in a hot high pressure concentrated rest state with K=0, will get into motion by expansion driven by high pressure, but will cool and eventually find a new expanded rest state with K = 0 from which it cannot move. This system cannot experience periodic motion with energy oscillating between E and K.

II. With Gravitation/Reaction

We now add gravitation/reaction and start the system in the same an initial hot high pressure concentrated rest state now ready for exotermic fusion with Q positive, and all energy as small scale heat E. When released the system will expand driven by high pressure with W positive and now Q can keep E from decaying so that the expansion pressure is maintained.

During the expansion the gravitational energy P grows like the elastic energy a spring under extension.

If now the fusion Q is shut off, then the expansion will slow down as the temperature drops, and a maximal inflated state with K = 0 may be reached, assuming the expansion does not continue for ever. Then gravitational pull will take over and start contracting the system drivern by negative dP, increasing first K and then E through D while balancing a negative Q in fission.

Will this system return to the initial concentrated hot high pressure state?

Doesn't the positivity of D mean that there is an irreversible flow from K +P to E so that periodic motion is impossible?

In a way, yes: Certainly there must be more E at the end of the contraction and less K+P, than before the expansion, since there is a positive flow D from K+P to E.

But we now come to a key point: What is the difference between K and E? We have said that K is large scale kinetic energy and E small scale kinetic energy. But in a hot concentrated state
after fission with everything smashed into pieces (in a big crunch), the difference between large scale and small scale disappears, since there are only small scales.

This means that we can use the surplus of E to restore K+P.

The net result is a system which may keep oscillating back and forth between a hot concentrated state (big bang) and an expanded cooler state, with the expansion driven by fusion, the gravitational pull taking care of the contraction and finally the fission transforming E into K +P and restoring initial data (in a big crunch resetting clocks).

We see that all the ingredients of reactive thermodynamics with gravitation come into play
to allow a Universe pulsating for ever; connecting to the old idea of Alexander Friedman.

In previous posts we have seen that also global climate can be described as reactive thermodynamics with gravitation. We have seen that in this case the periodic motion consists of ascending and descending air transporting heat from the Earth surface to the top of the atmosphere. In this case the motion is driven by and exterior source in the form of radiation from the Sun.

This post is just a little exercise testing the 2nd Law in the form proposed in Computational Thermodynamics (without any relativity). Nothing serious...

Inga kommentarer:

Skicka en kommentar