3.2: The Partition Function
- Page ID
- 5169
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
\[H (x) = H_1 (x_1) + H_2 (x_2) \nonumber \]
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
\[\Omega(N,V,E) = \int dx \delta(H(x)-E) = \int dx_1 dx_2 \delta (H_1(x_1) + H_2(x_2)-E) \nonumber \]
Now, we define the distribution function, \(f (x_1)\) of the phase space variables of system 1 as
\[ f(x_1) = \int dx_2 \delta (H_1(x_1)+ H_2(x_2)-E) \nonumber \]
Taking the natural log of both sides, we have
\[ \ln f(x_1) = \ln \int dx_2 \delta (H_1(x_1) + H_2(x_2) - E) \nonumber \]
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
\[ \ln \int dx_2 \delta (H_2( _2)-E) =\frac {S_2 (E)}{k} \nonumber \]
\[ \frac {\partial}{\partial E} \ln \int dx_2 \delta (H_2(x_2)-E = \frac {\partial}{\partial E} \frac {S_2(E)}{k} = \frac {1}{kT} \nonumber \]
where \(T\) is the common temperature of the two systems. Using these two facts, we obtain
\[\ln f (x_1) = \frac {S_2 (E)}{k} - \frac {H_1 (x_1)}{kT} \nonumber \]
\[f (x_1) = e^{\frac {S_2(E)}{k}}e^{\frac {-H_1(x_1)}{kT}} \nonumber \]
Thus, the distribution function of the canonical ensemble is
\[f(x) \propto e^{\frac {-H(x)}{kT}} \nonumber \]
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:
\[\int dxe^{\frac {-H(x)}{kT}} \equiv Q(N,V,T) \nonumber \]
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
\[ Q(N,V,T) = \frac {1}{N!h^{3N}} \int dx e^{-\beta H(x)} \nonumber \]
The thermodynamic relations are thus,