# The Partition Function

Consider two systems (1 and 2) in thermal contact such that

• $$N_2 \gg N_1$$
• $$E_2 \gg E_1$$
• $$N= N_1 + N_2$$
• $$E = E_1 + E_2$$
• $$\text {dim} (x_1) \gg \text {dim} (x_2)$$

and the total Hamiltonian is just

$H (x) = H_1 (x_1) + H_2 (x_2)$

Since system 2 is infinitely large compared to system 1, it acts as an infinite heat reservoir that keeps system 1 at a constant temperature $$T$$ without gaining or losing an appreciable amount of heat, itself. Thus, system 1 is maintained at canonical conditions, $$N, V, T$$.

The full partition function $$\Omega (N, V, E )$$ for the combined system is the microcanonical partition function

$\Omega(N,V,E) = \int dx \delta(H(x)-E) = \int dx_1 dx_2 \delta (H_1(x_1) + H_2(x_2)-E)$

Now, we define the distribution function, $$f (x_1)$$ of the phase space variables of system 1 as

$f(x_1) = \int dx_2 \delta (H_1(x_1)+ H_2(x_2)-E)$

Taking the natural log of both sides, we have

$\ln f(x_1) = \ln \int dx_2 \delta (H_1(x_1) + H_2(x_2) - E)$

Since $$E_2 \gg E_1$$, it follows that $$H_2 (x_2) \gg H_1 (x_1)$$, and we may expand the above expression about $$H_1 = 0$$. To linear order, the expression becomes

\begin{align*} \ln f (x_1) &= \ln \int dx_2 \delta (H_2(x_2)-E) + H_1(x_1) \frac {\partial }{ \partial H_1 (x_1)} \ln \int dx_2 \delta (H_1(x_1) + H_2(x_2) - E) \vert _{H_1(x_1)=0} \\[4pt] &= \ln \int dx_2 \delta (H_2(x_2)-E) -H_1(x_1) \frac {\partial}{\partial E} \ln \int dx_2 \delta (H_2(x_2)-E) \end{align*}

where, in the last line, the differentiation with respect to $$H_1$$ is replaced by differentiation with respect to $$E$$. Note that

$\ln \int dx_2 \delta (H_2( _2)-E) =\frac {S_2 (E)}{k}$

$\frac {\partial}{\partial E} \ln \int dx_2 \delta (H_2(x_2)-E = \frac {\partial}{\partial E} \frac {S_2(E)}{k} = \frac {1}{kT}$

where $$T$$ is the common temperature of the two systems. Using these two facts, we obtain

$\ln f (x_1) = \frac {S_2 (E)}{k} - \frac {H_1 (x_1)}{kT}$

$f (x_1) = e^{\frac {S_2(E)}{k}}e^{\frac {-H_1(x_1)}{kT}}$

Thus, the distribution function of the canonical ensemble is

$f(x) \propto e^{\frac {-H(x)}{kT}}$

The prefactor $$exp (\frac {S_2 (E) }{k} )$$ is an irrelevant constant that can be disregarded as it will not affect any physical properties.

The normalization of the distribution function is the integral:

$\int dxe^{\frac {-H(x)}{kT}} \equiv Q(N,V,T)$

where $$Q (N, V, T )$$ is the canonical partition function. It is convenient to define an inverse temperature $$\beta = \frac {1}{kT}$$. $$Q (N, V, T )$$ is the canonical partition function. As in the microcanonical case, we add in the ad hoc quantum corrections to the classical result to give

$Q(N,V,T) = \frac {1}{N!h^{3N}} \int dx e^{-\beta H(x)}$

The thermodynamic relations are thus,