# Postulate of a-priori probabilities

In Statistical Mechanics, we often postulate that for an isolated system, the phase-space density of all accessible microstates (i.e all microstates consistent with the energy) is the same. This is equivalent to assuming that the system is ergodic. This postulate leads us to the assertion that at any given time, the system is most likely to be found in that macrostate which has the maximum number of consistent microstates, and from then on, we calculate the entropy for this macrostate and get the fundamental relation for entropy, and hence, other thermodynamic quantities. My question now is: Is this assumption of equal a-priori probabilities too strong to obtain the second assertion? In other words, can we not say that the macrostate with the maximum number of microstates is the observed thermodynamic state, while being non-committal as to whether indeed they are all equally probable or not? Or am I losing some information by not considering their probability distribution (maybe, say fluctuations)?

First, although it is common in some textbooks, I don't think it is a good thing to necessarily relate the equiprobability postulate to ergodicity.

Second, what this postulate enables is to estimate the probability distribution for the macrovariable you want to look at. You can of course look at the most probable value for this macrostate and this will correspond to a "thermodynamic interpretation" of what you can expect to observe.

However, in statistical mechanics, what you can expect is not the most probable value but rather the average value and they need not be the same.

Moreover, you may want more than simply the average value; you can also want to predict what is the free energy difference between one value of the macrostate and another in some kind of transformation in your sytem and for this you are in trouble if you don't try to get the probabilities right.

Correct entropy in the analysis of complex systems: What is the consequence of rejecting the postulate of equal A priori probabilities? Delas, N. doi: 10.15587/1729-4061.2015.47332