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My statistical physics books reasons that the increase of entropy for a closed system arises naturally from statistics. Outline:

  1. Fundamental Assumption of Statistical Mechanics: For a system at equilibrium, each microstate is equally likely.

  2. The macrostate with the MOST microstates is the one where energy is homogeneously distributed across the system.

  3. As all microstates are equally likely, the system will approach the state where energy is homogeneously distributed. This is the equilibrium state, and the state of highest entropy (as it has more microstates).

You can probably see the circular reasoning here: we start with a postulate conditioned on the equilibrium state. We use this postulate to conclude that the equilibrium state will be reached.

This doesn't make sense! Is there another way to show that increase of entropy is basically just a statistical fact, or is the fundamental assumption of statistical mechanics valid for states away from their equilibrium as well?

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1) ... For a system at equilibrium, each microstate is equally likely.

This statement is missing some essential qualifiers. Here's a version with appropriate qualifiers:

1) For a system in equilibrium, each microstate that is compatible with all enforced conditions is equally likely.

The qualifiers are essential because there are many different possible equilibria, corresponding to different conditions that could be enforced, such as enforcing a given total volume and/or a given total energy.

2) The macrostate with MOST microstates is the one where energy is homogeneously distributed across the system.

This statement involves some implicit assumptions. Here's a version that makes the assumptions more explicit:

2) Among all macroscopically-distinct ways of distributing a given total energy among a large number of identical molecules, the one with the most microstates is the one in which the energy is homogeneously distributed as far as macroscopic observations can tell.

This is analogous to saying that among all million-digit binary numbers, most of them have an approximately equal number of $0$s and $1$s in any given thousand-digit interval.

3) ...the system will approach the state where energy is homogeneously distributed.

This statement is missing an important preamble. Here's a verion that includes the preamble:

3) If we remove whatever condition was previously keeping the distribution of energy inhomogeneous, then the system will approach the state where energy is homogeneously distributed.

Finallly, "will approach" assumes some kind of dynamics that causes the system to evolve through the set of microstates (the ones that are compatible with the new conditions) in a way that does not have any implicit preference for any particular subset of those microstates. This is a big assumption, but it's the least-presumptuous assumption we can make in the absence of additional information (or when faced with a hopelessly-complicated system of equations governing the microscopic dynamics), and apparently it works very well for many purposes. This is one way of expressing the Fundamental Assumption of Statistical Mechanics.

The statement that a system is "in equilibrium" means that the system has had enough time to explore microstates that differ arbitrarily much (within the new class of microstates) from those to which it was previously restricted.

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Is there another way to show that increase of entropy is basically just a statistical fact, or is the Fundamental Assumption of Statistical Mechanics valid for states away from their equilibrium as well?

Increase of thermodynamic entropy of a closed system when some constraint is removed is first and foremost a formalization and generalization of experience, such as experiments with heat engines. In its generality it wasn't ever proved or derived - it is the assumption in thermodynamics (or, from a different point of view, an implication of 2nd law of thermodynamics which is the one that is experience-based).

There is a common misunderstanding that this increase of entropy is a continuous process in time, where entropy continuously rises to its new equilibrium value.

While that point of view can be justified in some special cases, in general, it is too naive idealization, as thermodynamic entropy isn't defined for all possible intermediate states. The general "law of increase of entropy" is actually the statement that if an adiabatically isolated system that is perturbed from its initial equilibrium acquires new equilibrium, entropy of that state cannot be lower than entropy of the initial equilibrium state.

I suppose you are asking whether this behaviour of thermodynamic entropy for adiabatically isolated systems can be explained purely on the basis of molecular theory of matter and probability mathematics.

In a limited way, yes. One way is to reason based on information entropy (Gibbs entropy functional $\int \rho\ln \rho\, dqdp$): the basic assumption(there is always some assumption) is that thermodynamic entropy is the maximum possible value of that functional, given macroscopic state-implied constraints on the probability distribution $\rho$.

For full derivation, see Jaynes' Brandeis lectures, p.213 and subsequent ones:

https://bayes.wustl.edu/etj/articles/brandeis.pdf

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