Behavior of maximum entropy states in a closed system?

As I understand it, entropy is a measure of the number of permutations of microstates of a system possible, without changing the observed 'macrostate variables' / measurements or properties of a system (as opposed to the microstates specifically) such as temperature or volume. I could say entropy is a measure of 'number of permutations assuming some combination', permitting some simplifications.

In this view it is easy to see (self-evident almost) why entropy should increase - statistically, a system is more likely to transition into a configuration belonging to a 'large group' of configurations than a 'small group' of configurations.

It is possible that some groups of configurations cannot transition between eachother or that the transition is one-way - it may even depend on how 'configuration' and 'microstate' are defined, to my knowledge this can be arbitrary though this is fine so long as the definitions are consistent.

situations where the available microstates are finite in nature

One could imagine an ideal gas in an isolated container, and fairly easily imagine the lowest-energetic state of such a gas - the particles are very likely to have low velocities, resulting in lower rates of particle-particle and particle-container collisions / interactions (low pressure, temperature).

The space of possible microstates is small, because the possible configurations at lower and lower energies just become permutations of eachother with fewer and fewer distinct combinations. This is the low-entropy side of things...

Because this view of entropy depends on permutations vs combinations, when the possible combinations are finite, the distribution of entropy across all possible configurations the system can be in, becomes very similar to a binomial distribution. The feature of a binomial distribution is it is symmetrical with peak entropy in the mid-range of possible configurations.

So then I ask (all the same question):

What is the behavior of an ideal gas at maximum entropy in an isolated, fixed container - does it represent the 'most common' states possible? What would it mean to add energy (eg increase particle average speeds) into a maximally entropic state such as this; how would the properties of the gas change? Does the entropy of a maximal-entropy system go down if more energy is introduced into it?

• Entropy is the amount of heat of a system that is not available to do useful work at a specific temperature divided by that temperature. There, isn't that much easier and infinitely more useful? The "self-evidence" of the statistical mechanics interpretation is marred by the fact that one has to assume ergodicity, which replaces another item of faith, that heat never flows from cold to warm without something else happening in return. – CuriousOne Jan 7 '15 at 5:52
• I've seen that definition around but not familiar with it yet (another reason to ask). Anyway, maybe I'm missing something but isn't heat only relevant in the context where there is heat flow? Is it right to talk about heat of a system that isn't doing any work? – Xeren Narcy Jan 7 '15 at 22:11