# necessary and sufficient conditions for an isolated dynamical system which can approach thermal equilibrium automatically

Given an isolated $N$-particle system with only two body interaction, that is $$H=\sum_{i=1}^N\frac{\mathbf{p}_i^2}{2m}+\sum_{i<j}V(\mathbf{r}_i-\mathbf{r}_j)$$

In the thermodynamic limit, that is $N\gg 1$ and $N/V=$constant, it seems that not all two body interaction can make system approach thermal equilibrium automatically. For example, if the interaction is inverse square attractive force, we know the system cannot approach thermal equilibrium.

Although there is Boltzmann's H-theorem to derive the second law of thermodynamics, it relies on the Boltzmann equation which is derived from Liouville's equation in approximation of low density and short range interaction.

My question:

1. Does it mean that any isolated system with low density and short range interaction can approach thermal equilibrium automatically? If not, what's the counterexample?

2. For long range interaction or high density isolated system, what's the necessary and sufficient conditions for such system can approach thermal equilibrium automatically? What's about coulomb interaction of plasma(i.e. same number of positive and negative charge)?

3. How to prove rigorously that a pure self-gravitational system cannot approach equilibrium? I only heard the hand-waving argument that gravity has the effect of clot, but I never see the rigorous proof.

I know there is maximal entropy postulate in microscopic ensemble.I just want to find the range of application of this postulate of equilibrium statistical mechanics. I'm always curious about the above questions but I never saw the discussion in any textbook of statistical mechanics. You can also cite the literature in which I can find the answer.

1. Does it mean that any isolated system with low density and short range interaction can approach thermal equilibrium automatically? If not, what's the counterexample?

No, you cannot guarantee equilibrium is always reached. Example: the ideal gas of point particles in a perfectly rigid container. As discussed in the question How does a gas of particles with uniform speed reach the Maxwell-Boltzmann distribution?, you need, e.g., non-ideal walls or finite particles to reach equilibrium in this system.

In general, a system can reach thermodynamical equilibrium when it's ergodic or mixing, so that the "fast degrees of freedom" can be averaged out, and the systems can be described by the thermodynamical quantities alone. To know when a system exhibits these properties, check Are there necessary and sufficient conditions for ergodicity?.

1. (a) For long range interaction or (b) high density isolated system, what's the necessary and sufficient conditions for such system can approach thermal equilibrium automatically? (c) What's about coulomb interaction of plasma(i.e. same number of positive and negative charge)?

a) Rigorously, systems with long-range interactions don't reach equilibrium. More specifically: Relaxation towards equilibrium is extremely slow and the relaxation time diverges with the number of particles.

These systems can actually have two phases of relaxation. A fast phase (sometimes called "violent" in astrophysics), followed by a second, (divergingly) slow one, which can also present quasi-stationary states. Other unusual features include non-additivity and regions of negative specific heat. For details, you might check, e.g.,

b) In high-density systems, short-range interactions tend to dominate and the answer to the previous question (namely, mixing leads to equilibrium) applies.

c) In a plasma the charge is shielded and the interactions are not truly (Coulombian) long-range.

1. How to prove rigorously that a pure self-gravitational system cannot approach equilibrium? I only heard the hand-waving argument that gravity has the effect of clot, but I never see the rigorous proof.

Since gravity is a long-range interaction, this last question is a particular case of the previous one. At any rate, a recent reference on the subject is Melkikh's Can we use thermodynamics in the systems with gravity? (e-print).

Regardless of the equilibrium question, the subject is very relevant, especially in the context of quantum gravity. See, e.g., Brown et al.'s Thermodynamic ensembles and gravitation (e-print) and Martinez' The postulates of gravitational thermodynamics (arXiv).

As for the "effect of clot", that might refer to dissipative (such as protoplanetary) systems.