Statistical Mechanics - Fundamental Postulate

Fundamental Postulate

The fundamental postulate in statistical mechanics (also known as the equal a priori probability postulate) is the following:

Given an isolated system in equilibrium, it is found with equal probability in each of its accessible microstates.

This postulate is a fundamental assumption in statistical mechanics - it states that a system in equilibrium does not have any preference for any of its available microstates. Given Ω microstates at a particular energy, the probability of finding the system in a particular microstate is p = 1/Ω.

This postulate is necessary because it allows one to conclude that for a system at equilibrium, the thermodynamic state (macrostate) which could result from the largest number of microstates is also the most probable macrostate of the system.

The postulate is justified in part, for classical systems, by Liouville's theorem (Hamiltonian), which shows that if the distribution of system points through accessible phase space is uniform at some time, it remains so at later times.

Similar justification for a discrete system is provided by the mechanism of detailed balance.

This allows for the definition of the information function (in the context of information theory):

When all the probabilities ρ_i (rho) are equal, I is maximal, and we have minimal information about the system. When our information is maximal (i.e., one rho is equal to one and the rest to zero, such that we know what state the system is in), the function is minimal.

This information function is the same as the reduced entropic function in thermodynamics.

Mark Srednicki has argued that the fundamental postulate can be derived assuming only that Berry's conjecture (named after Michael Berry) applies to the system in question. Berry's conjecture is believed to hold only for chaotic systems, and roughly says that the energy eigenstates are distributed as Gaussian random variables. Since all realistic systems with more than a handful of degrees of freedom are expected to be chaotic, this puts the fundamental postulate on firm footing. Berry's conjecture has also been shown to be equivalent to an information theoretic principle of least bias.