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The Partition Function

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  • 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,

    Contributors and Attributions

    Mark Tuckerman (New York University)