Thermodynamics of gravitational systems

In this last (first?) post of the year I would like to express some confusion I have with respect to applying thermodynamic reasoning to cosmology or in general situations governed by gravity. The main puzzle I would like to understand is the question regarding the entropy balance of the universe: According to the second law of thermodynamics, entropy is never decreasing (I hope this is the correct sign, I can never remember it. Let's see, S is minus trace rho log rho. If rho is proportional to the projector on an N dimensional subspace we have S = - N 1/N log(1/N) = log(N). Thus it is increasing if the probability spreads over a larger subspace. Good). So if it is increasing, it should have been minimal at the big bang which seems to be at conflict with the universe being a hot soup of all kinds of fluctuations right after the big bang.

With the popular science interpretation of entropy as a measure of disorder or negative information the early universe must have been highly ordered and should have contained maximal information, a notion which is highly counter intuitive. So this needs some clearing up.

The simplest resolution would be that it is compatible with observation to assume that the universe has infinite volume and if it has a finite entropy density the entropy is infinite and any discussion of increasing or decreasing entropy is meaningless as it will be infinite at any time and it does not make sense to talk about more or less infinite entropy.

We could however try to still make sense in a local, desitised version: We could make the usual cosmology assumption of the universe being pretty much homogeneous and talk solely of entropy densities (after all, we only observe a Hubble sized ball of it and should thus only make appropriate local statements). But since the universe is expanding should we use co-moving or constant volumes when computing the densities when applying a desitised second law? But I don't think this is the real problem.

I am much more worried about another point: I am not convinced it makes sense to apply thermodynamic reasoning to situations that involve gravity! Obviously, the universe as we see it is not in thermal equilibrium, all the interesting stuff we see are local fluctuations. So standard textbook equilibrium thermodynamics does not apply: Remember for example, temperature is a property of an equilibrium, the fact it is well defined is sometimes called the zeroth law and out of equilibrium situations do not have a temperature! Only if locally things are not too different from an equilibrium state one can assign something like a local temperature. But things are even worse: The usual systems that we are used to describe thermodynamically (steam engines, containers of gas etc) have the property that the equilibrium is an attractor of the dynamics: All kinds of small, local perturbations diffuse away exponentially fast. This is in line with our intuitive understanding of the second law: The homogeneous state is the one with the highest entropy and thus the diffusion is governed by the second law.

This is not the case anymore as soon as gravity is the dominating force: What is different here is that gravity is always attractive. Thus if you have a nearly homogeneous matter distribution with small local fluctuations, over-dense regions will gravitate even stronger and thus will be even denser while under-dense regions will gravitate less and will become even emptier. Thus the contrast is increasing over time (a feature which is of course essential to structure formation of galaxies, stars etc). But this means the equilibrium is unstable. This is at least in conflict with the naive understanding of the second law above.

Some deeper inspection reveals that when you axiomatise thermodynamics you usually make some assumption on convexity (or concavity, depending on whether you use intensive or extensive variables of state) of your favorite thermodynamic potential (free energy etc). IIRC this is something you impose. Your system has to fulfill this property in order to be described by thermodynamics. And it seems that gravity does not have this property (the stability) and it quite possible (if I am not mistaken, sitting here in a train without any books or internet access) thermodynamic arguments do not apply to gravity.

Note well that I am talking classically (actually even only about the weak field situation in which the fluctuations are well described by Newtonian gravity), I have not even mentioned black holes and their negative heat capacity due to Hawking radiation which should make you even more uneasy about thermodynamic stability.

There is however a related problem my classically relativistic friends told me about: When discussing cosmology, it is usually a good first approximation that the universe is homogeneous which supposedly it is at large scales. At small scales however, this is obviously not the case with voids, galaxies, stars, stones etc. But for the evolution at large scales you average all those local fluctuations and replace everything by the cosmological fluid.

The problem with the non-linear theory of gravity is however that it is by far not obvious that this averaging commutes with time evolution: That, starting from good initial conditions it does not matter if you first average and then compute the time evolution of the averaged matter density or if you first compute the time evolution and then to the spatial averaging. The first thing is of course what we always compute while the second thing is what really happens. An incarnation of this problem was an argument that was discussed a few years ago that what looks like the cosmological constant in our local patch of the universe is just a density fluctuation with a super-horizon wave length. At first you would reject such a suggestion since something that happens over regions that are causally disconnected from us should not influence our local observations. However, due to the non-linear nature of gravity this argument is too fast and needed a more thorough inspection. My impression is that eventually it was decided that this idea does not work. I would be happy to be informed by somebody follows these things more closely.

To wrap up, I feel that I would need to have to understand much more basic things about thermodynamics applied to gravity before I could make sensible statements about the entropy of the universe or Boltzmann brains and the similar.