# Understanding the approach to equilibrium for a statistical system

I have studied statistical mechanics using F. Reif's book, and learned a lot, but there are still a couple very fundamental questions which still elude me. If anyone would be willing to share some insight, I would really appreciate it!

So Reif early on discusses the fundamental postulate of statistical mechanics:
"A system in equilibrium has equal probability of being found in any of it's accessible states."

But here are my two questions (they sound kind of simple, but seem very tricky when I really try to think about them)
1. What exactly is equilibrium, in a rigorous sense? When the average energy $$\bar E$$ of the statistical system is time independent (constant)? When the "external parameters" such as volume are constant? I can't seem to come up with a totally inclusive definition of equilibrium.
2. What says that a system should relax to equilibrium to begin with? Is this a second postulate, or is there something simple I'm over looking? Reif never really touched on this point.

Thanks for any help!

"Equilibrium" usually means equilibrium of a macroscopic system. The system keeps changing on a microscopic scale, but the distribution of the micro states in the ensemble is independent of time.

As I understand it, equilibrium means that the intensive parameters (pressure, temperature, and chemical potential) are constant throughout the system. The entropy is maximal.

It is the Le Chatêlier principle that says that a system always goes back into a stable equilibrium. The stability conditions are

- thermal stability: the isochore heat capacity is greater than or equal to zero.
If heat is given into a system, then its temperature rises and it emits heat into its environment, so the temperature falls again.

- mechanical stability: the isothermal compressibility is greater than or equal to zero.
If the volume of a subarea is expanded, then the pressure in it decreases. Its environment has a higher pressure and compensates this.

heat naturally flows from a greater to a lesser temperature gradient until there is no more difference in temperature. for example, if i drop an ice cube in a cup of hot water (in a closed system), heat naturally flows from the hotter body of water to the cooler ice cube until both bodies of water reach the same temperature and form a homogeneous body of luke-warm water.

what says that heat should flow naturally from a greater to lesser temperature gradient?

paco1955 said:
what says that heat should flow naturally from a greater to lesser temperature gradient?

Fourier law:

$$\mathbf{q} \ = - \kappa \ \nabla T$$

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MalleusScientiarum
There is actually one derivation of the partition function that relies on a certain definition of entropy in terms of the density operator, and furthermore that in equilibrium the density operator commutes with the Hamiltonian. This would correspond to every particle falling into an energy eigenstate at the microscopic level.