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5.2: Pressure and Work Virial Theorems

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    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

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


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

    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.

    This page titled 5.2: Pressure and Work Virial Theorems is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Mark Tuckerman.