[NOTES/SM-03001] Fundamental Postulate of Statistical Mechanics

There are two approaches to the statistical mechanics. In the first one the Boltzmann relation is taken as the fundamental postulate. In the second approach the postulate of equal a priori probabilities is the starting point. We briefly describe the two approaches.

Let \(\Omega\) denote the number of microstates of system with energy. In equilibrium a system is completely occupied by \(E,V,N\). Additional variable, collectively represented by \(\alpha\), must be specified in case of non equilibrium states.

Postulate of equal a priory probability

This postulate states that all states with a given set of constraints of the system,  \(E,V,N\), \textcolor{blue}{\em all microstates}  have equal a priory probability. Thus the probability that a system will be found in a \textcolor{blue}{\em macrostate} specified by \(E,V,N, \alpha\) is proportional to \textcolor{blue}{\em the number of microstates} \(\Omega(E,V,N,\alpha)\). In an approach where this postulate is taken as the starting point of statistical mechanics, the equilibrium state of the system corresponds a value of \(\alpha\) which makes the number of microstates for a given \(E,V,N\) maximum.

Boltzmann Entropy

The Boltzmann relation, relation between entropy and the number of microstates  \begin{equation}\Label{BRel} S = k \ln \Omega(E,V,N,\alpha), \end{equation} is fundamental relation. It defines the entropy of a macrostate specified by \(E,V,N,\alpha\), where \(k\) is the Boltzmann constant. he Boltzmann relation forms one of the possible  starting point for statistical mechanics. The Boltzmann relation \eqRef{BRel} gives the entropy a meaning in terms of mircoscopic properties of the system. The statements about \(\Omega\) for equilibrium state then translate into a statement about the entropy. 

Principle of Increase in Entropy

 In a real process the entropy of an isolated system always increases. In the equilibrium state the entropy attains a maximum value.

Equilibrium conditions
Once the entropy is written in terms of the number of microstates, the equilibrium condition is obtained by maximizing  the entropy as given by the Boltzmann formula.