5.2: Pressure and Work Virial Theorems

Last updated
Save as PDF

Page ID: 5227

Mark Tuckerman
New York University

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

As noted earlier, the quantity \( -\partial H/\partial V \) is a measure of the instantaneous value of the internal pressure \( P_{\rm int} \). Let us look at the ensemble average of this quantity

\[ \begin{align*} \langle P_{\rm int} \rangle &= -{1 \over \Delta}C_N\int_0^{\infty}dVe^{-\beta PV} \int d{\rm x}{\partial H \over \partial V}e^{-\beta H({\rm x})} \\[4pt] &= {1 \over \Delta} C_N\int_0^{\infty}dVe^{-\beta PV} \int d{\rm x}kT {\partial \over \partial V}e^{-\beta H({\rm x})} \\[4pt] &= {1 \over \Delta} \int_0^{\infty}dVe^{-\beta PV} kT {\partial \over \partial V} Q(N,V,T) \end{align*}\]

Doing the volume integration by parts gives

\[ \begin{align*} \langle P_{\rm int} \rangle &= {1 \over \Delta} \left[e^{-\beta PV} kT Q(N,V,T) \right]\vert _0^{\infty} - {1 \over \Delta } \int _0^{\infty}dVkT \left({\partial \over \partial V} e^{-\beta PV} \right) Q(N,V,T) \\[4pt] &=P{1 \over \Delta} \int_0^{\infty}dVe^{-\beta PV} Q(N,V,T) \\[4pt] &= P \end{align*}\]

Thus,

\[ \langle P_{\rm int}\rangle = P \nonumber \]

This result is known as the pressure virial theorem. It illustrates that the average of the quantity \( -\partial H/\partial V\) gives the fixed pressure \(P\) that defines the ensemble. Another important result comes from considering the ensemble average \(-\partial H/\partial V\)

\[ \langle P_{\rm int} V\rangle = {1 \over \Delta} \int_0^{\infty}dVe^{-\beta PV} kTV {\partial \over \partial V}Q(N,V,T) \nonumber \]

Once again, integrating by parts with respect to the volume yields

\[ \begin{align*} \langle P_{\rm int}V\rangle &= {1 \over \Delta} \left[e^{-\beta PV} kTV Q(N,V,T) \right]\vert _0^{\infty} - {1 \over \Delta} \int _0^{\infty}dVkT \left({\partial \over \partial V}Ve^{-\beta PV} \right)Q(N,V,T) \\[4pt] &={1 \over \Delta} \left[-kT \int_0^{\infty}dVe^{-\beta PV} Q(V) + P \int_0^{\infty}dVe^{-\beta PV} VQ(V)\right] \\[4pt] &=-kT + P\langle V \rangle \end{align*}\]

or \[ \langle P_{\rm int} V\rangle + kT = P\langle V\rangle \nonumber \]

This result is known as the work virial theorem. It expresses the fact that equipartitioning of energy also applies to the volume degrees of freedom, since the volume is now a fluctuating quantity.