### thermodynamics

It seems that there is some confusion about several issues in thermodynamics, so the following might be helpful.

1) If a system is not in thermodynamic equilibrium, certain macroscopic quantities may not be well defined, e.g. temperature as mean kinetic energy. However, entropy as a measure of our ignorance about the micro state is in general defined even far away from equilibrium. Otherwise we would not have the 2nd law of thermodynamics, because dS/dt ~ 0 if a system is in equilibrium.

2) The heat capacity of a gravitating system (Newtonian gravity) is in general negative. As an example consider a star radiating energy away, which will cause it to heat up due to gravitational contraction. This can be confusing, but there is nothing wrong with thermodynamics if one includes Newtonian gravity.

In general, the 0th law does not always hold and things can get funny, but this does not affect the 1st and 2nd law.

3) If we consider Newtonian mechanics carefully, we find that no classical system is stable and thus no purely classical system can be in thermodynamic equilibrium. This was historically the reason for Bohr to propose the first version of quantum mechanics.

4) In general, we do not know how to calculate the entropy of a particular spacetime. There is the proposal of Penrose to equate it with the Weyl curvature; However, there are problems with this proposal.

Things can get quite funny if one considers a spacetime which contains a naked singularity or closed timelike loops. Unfortunately, current state-of-the-art is still that one has to remove such geometries by hand on the grounds that things get quite funny otherwise.

5) In quantum theory, if a system is in a pure state the corresponding entropy is zero. If one assumes that the 'wave function of the universe' was initially in a pure state, it would remain in a pure state, assuming unitary evolution for quantum gravity (as suggested by the AdS-CFT correspondence). There is thus a problem for (some) many worlds interpretations in my opinion.

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## 3 comments:

The ignorance of some string theorists about physics outside their narrow expertise is amazing.

Lubos wrote some harsh words about his collegue, but actually, I find it quite refreshing if people acknolwedge that they don't know something. And usually physicists are pretty good at being honest (nobody is perfect).

But compare it to politicians who always have an answer for everything, even the most difficult issues.

Wolfgang, I'd like to make some remarks on your point 1) above.

a. I find it better not to conflate 'temperature', 'entropy', and other quantities in thermodynamics and 'temperature', 'entropy', and other quantities in statistical mechanics. They are two different theories. And even if some physicists make that conflation, others don't; so better stay on the safe side. The identification of the two groups of quantities in the two theories is merely based on analogy (Maxwell explains this with wonderful sincerity in Trans. Cambridge Philos. Soc. 12 (1879), pp. 547-570; see 'Recapitulation' on pp. 726-728). I don't mean to say that I don't believe that thermodynamic temperature corresponds somehow to an averaged kinetic energy in the standard case (though I believe there are cases in which the identification is different); but that correspondence still lacks rigorous proof. The usual trick is to introduce, in the microscopic description, another system with a Hamiltonian weakly coupled to the main system's one. The second system is called 'thermometer'. Now they must still convince me that such a description is really appropriate for the mercury thermometer that I plunge into a glass of water.

b. If a thermodynamic system is not in equilibrium, temperature may still be 'well defined', and also thermodynamic entropy in general. In the theory that describes non-equilibrium thermodynamical or thermomechanical systems the temperature usually depends on time and position, and the second law is usually called the 'entropy inequality' or the 'Clausius-Duhem inequality'. This theory, thermomechanics, is taught in every engineering department and in most departments of applied mathematics around the world and, as far as I know, is used e.g. in the design and realisations of satellites and interplanetary probes. Just take a textbook for a course in thermoelasticity for example. Thermostatics, as usually taught to most physicists, is a special case of it. Unfortunately all this is not mentioned in most thermodynamics textbooks for physicists. This reflects the sad sectoring of science today. Most physicists just teach and publish what they learnt from their teachers, and so on for a couple of generations; as a result, they are sixty years behind engineers and applied mathematicians.

Some systems, however, like e.g. liquid crystals, have a thermodynamic entropy that is defined not modulo a constant, as it happens in thermostatics, but modulo a function.

c. There is also a statistical mechanics of non-equilibrium systems, formulated in full around the eighties, which is beginning to be applied only recently though. The basic and very simple idea is to use a Gibbs distribution not in phase space, but in the space of 'paths' or 'histories' of the system. It is thus not important whether the system's dynamics is an 'equilibrium' one or not. In this theory the statistical temperature is naturally promoted to a space-time-dependent field - exactly as it happens for the non-equilibrium thermodynamic temperature in thermomechanics. See Jaynes, 'Macroscopic prediction', http://bayes.wustl.edu/etj/node1.html, and references therein, in particular Grandy's review in Phys. Rep. 62 (1980), pp. 175-266.

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