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Pressure and work virial theorems

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    5227
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    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

    \( \langle P_{\rm int} \rangle\) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

    \(-{1 \over \Delta}C_N\int_0^{\infty}dVe^{-\beta PV} \int d{\rm x}{\partial H \over \partial V}e^{-\beta H({\rm x})}\)

    $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

    \({1 \over \Delta} C_N\int_0^{\infty}dVe^{-\beta PV} \int d{\rm x}kT {\partial \over \partial V}e^{-\beta H({\rm x})}\)

    $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

    \({1 \over \Delta} \int_0^{\infty}dVe^{-\beta PV} kT {\partial \over \partial V} Q(N,V,T)\)

    Doing the volume integration by parts gives

    \(\langle P_{\rm int} \rangle\) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

    \({1 \over \Delta} \left[e^{-\beta PV} kT Q(N,V,T) \right]\vert _0^{\infty} - {1 \over \Delta } \int _0^{\infty}dVkT \left({\partial \over \partial V} e^{-\beta PV} \right) Q(N,V,T)\)

    $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

    \(P{1 \over \Delta} \int_0^{\infty}dVe^{-\beta PV} Q(N,V,T) = P\)

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

    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 \(-\partial H/\partial V\)

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

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

    \(\langle P_{\rm int}V\rangle\) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

    \({1 \over \Delta} \left[e^{-\beta PV} kTV Q(N,V,T) \right]\vert _0^{\infty} - {1 \over \Delta} \int _0^{\infty}dVkT \left({\partial \over \partial V}Ve^{-\beta PV} \right)Q(N,V,T)\)

    $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

    \({1 \over \Delta} \left[-kT \int_0^{\infty}dVe^{-\beta PV} Q(V) + P \int_0^{\infty}dVe^{-\beta PV} VQ(V)\right]\)

    $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

    \(-kT + P\langle V \rangle\)

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

    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.


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