Week 4: The Microcanonical Ensemble

The microcanonical ensemble is built upon the so called postulate of equal a priori probabilities:

Postulate of equal a priori probabilities: For an isolated macroscopic system in equilibrium, all microscopic states corresponding to the same set of macroscopic observables are equally probable.

To intuitively absorb the idea of ensemble let’s consider a system and its phase space vector \(\mathbf{x}(t)\), evolving in time at constant total energy, volume and number of particles. The configurations sampled, i.e. the realizations of \(\mathbf{x}\) that constitute \(\mathbf{x}(t)\) are sampling the the constant energy hypersurface and are therefore equally probable.

The number of equally probable states \(\Omega(N,V,E)\) defines the partition function for the Microcanonical ensemble.

The partition function of the microcanonical ensemble is proportional to the integral in phase space over all the states at constant Energy:

\[ \Omega(N,V,E)\propto\int{d\mathbf{x}}\delta(H(\mathbf{x})-E) \]

the proportionality constant is equal to: \(C_N=E_0/N!h^{3N}\)

Here \(h\) is a constant with units Energy\(\times\)Time, and \(E_0\) is a constant having units of energy. The extra factor of \(E_0\) is needed because the \(\delta\) function has units of inverse energy. Such a constant however has no effect at all on any properties). Thus, \(\Omega(N,V,E)\) is non-dimensional.

The microcanonical partition function quantify the number of microstates accessible to a system at constant energy. Boltzmann identified \(\Omega(N,V,E)\) as the microscopic quantity defining \(S\) - the Entropy - which we was introduced as a postulate while stating the second law of Thermodynamics. As \(S\), \(\Omega(N,V,E)\) is a natural function of \(N\), \(V\) and \(E\).

The famous Boltzmann’s relation between \(S\) and \(\Omega(N,V,E)\) is:

\[ S(N,V,E)=k\ln{\Omega(N,V,E)} \]

where k is Boltzmann’s constant.

This is a key relation, as it establishes a connection between the macroscopically measurable thermodynamic properties of a system and its microscopic details.

In particular, by recalling the relationships introduced in Week 1, defining T and the forces associated with the extensive mechanical variables we get:

\[ k\left(\frac{\partial{\ln{\Omega}}}{\partial{V}}\right)_U=\left(\frac{\partial{S}}{\partial{V}}\right)_U=-\left(\frac{\partial{S}}{\partial{U}}\right)_V=\frac{P}{T} \]

\[ k\left(\frac{\partial{\ln{\Omega}}}{\partial{n_i}}\right)_U=\left(\frac{\partial{S}}{\partial{n_i}}\right)_U=-\left(\frac{\partial{S}}{\partial{U}}\right)_{n_i}=-\frac{\mu_i}{T} \]

and

\[ k\left(\frac{\partial{\ln{\Omega}}}{\partial{U}}\right)_{n_i,V}=\left(\frac{\partial{S}}{\partial{U}}\right)_{n_i,V}=\frac{1}{T} \]