Classical Statistical Mechanics

Consider an isolated system with unchanging $N, E$.

Figure: Phase Space

Phase Space - The $6N$-dimensional space defined by the $3N$ canonical co-ordinates and $3N$ canonical momenta. In this space $H(p,q) = E$ defines an energy surface. An energy conserving (isolated) system always stays on the same surface. A system defined by $E_0 \le E \le E_0 + \Delta E_0$ occupies a region like the shaded one in the phase-space shown in figure[1]. The dynamics of the system is completely determined by the set of $6N$ equations of motion, given by,

$$\frac{\partial H(p,q)}{\partial p_i} = \dot{q_i}$$ (1)

$$\frac{\partial H(p,q)}{\partial q_i} = - \dot{p_i},$$ (2)

where $H(p,q)$ is the Hamiltonian of the system. The phase-space density function, $\rho(p,q)$, is defined such that $\rho(p,q)dp dq$ is the number of representative points contained in the volume $dp dq$ located at $(p,q)$ in the phase-space (denoted by $\Gamma$).

Lioville's Theorem - The local density of representative points in the phase-space remain the same for an observer moving with a point. In other words, the distribution of the phase-space points move like an incompressible fluid. This implies,

$$\frac{\partial \rho}{\partial t} + \sum\limits ^{3 N}_{i=1}... ...}{\partial q_i} \frac{\partial \rho}{\partial p_i} \right) = 0$$ (3)

If $\rho (p, q) = \rho'(H(p,q))$ then $\frac{\partial \rho}{\partial t} = 0$, i.e, the density of states is time-independent. A particular class of choice, in which $\rho(p,q)$ is constant (equal a priori probability), gives the micro-canonical ensemble. This is evident for systems with a constant energy.

Postulate of Equal a Priori Probability - All the microscopic states corresponding to a particular macroscopic state of a system in thermal equilibrium are equally likely.

Consequence - Any macroscopic state in thermodynamic equilibrium is a member of the micro-canonical ensemble with the density function,

$$\rho (p,q) = 1 \mbox{for } E < H (p,q) < E+ \delta$$ (4)

$$= 0 \quad \mbox{otherwise.}$$ (5)

Ensemble Average of any macroscopic quantity $f$ is defined as,

$$\langle f \rangle = \frac{\int dp dq {\it {f}} (p,q) \rho (p,q) }{\int dp dq \rho (p,q)}.$$ (6)

The ensemble average is very close to the most probable value if the fluctuation is small. The mean square deviation of $f$ is given by,

$$\langle f \rangle ^2 = \frac{1}{N} \sum\limits_ i ( {\it f}_i - \bar{\it f})^2 = \langle f^2 \rangle - \langle f \rangle^2$$ (7)

and, therefore, the fluctuation is given by,

$$\delta = \frac{\langle f^2 \rangle - \langle f\rangle^2}{ \langle f \rangle ^2} \ll 1 \sim \frac{1}{N}.$$ (8)

Hence, for large $N$ the average value is equal to the most probable value. The Postulate of Classical Statistical Mechanics is established experimentally. The logical justification actually comes from Quantum Mechanics.