Postulates of classical statistical mechanics | Statistical mechanics

Postulates of classical statistical mechanics

It is based on the following postulates

Postulates of equal a priori probability

According to this postulate the probability of finding the phase point for given system in any region of the Γ-space is identical with that for any other region of equal extension or volume.
Or For a system in equilibrium, all accessible microstates corresponding to a given macro-state are equally probable.
Thus in thermodynamic equilibrium the system under consideration is a member of an ensemble (micro-canonical ensemble) with a density function ρ (p, q) and the value of density function is given according to the following rule.
If E < H (p, q) < E + ΔE, ρ (p, q) = constant, otherwise ρ (p, q) = 0
Thus all members of the ensemble have the same number of particles N and same volume V.
Let f (p, q) represents any measurable property of the system.
If the postulate of equal a priori probability is useful, then the average value of f (p, q) from different methods have the same results.

(i) Most probable value of f (p, q)

It is that value of f (p, q) which is possessed by the largest number of systems in the ensemble.

(ii) Ensemble average of f (p, q)

The ensemble average of f (p, q) i.e, < f > is given by

$\left<f \right>=\frac{\int d^{3N}pd^{3N}qf(p,q)\rho (p,q)}{\int d^{3N}pd^{3N}q\rho (p,q)}$

Both the values of f (p, q) i.e., the most probable value and the ensemble average are nearly equal, if the mean square fluctuation (MSF) is small i.e.,

$MSF=\frac{<f^{2}>-<f>^{2}}{<f>^{2}}<<1$

In all physical cases, MSF is of the order of 1/N
Since N → ∞ and therefore MSF << 1
Thus the most probable value of f (p, q) and ensemble average of f (p, q) both have identical values.

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