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8.3: Weakly Inhomogeneous Gas

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\newcommand\Vbeta{\vec\beta}
\newcommand\Vgamma{\vec\gamma}
\newcommand\Vdelta{\vec\delta}
\newcommand\Vepsilon{\vec\epsilon}
\newcommand\Vvarepsilon{\vec\varepsilon}
\newcommand\Vzeta{\vec\zeta}
\newcommand\Veta{\vec\eta}
\newcommand\Vtheta{\vec\theta}
\newcommand\Vvartheta{\vec\vartheta}
\newcommand\Viota{\vec\iota}
\newcommand\Vkappa{\vec\kappa}
\newcommand\Vlambda{\vec\lambda}
\( \newcommand\Vmu
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\)
\( \newcommand\Vnu
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\)
\( \newcommand\Vxi
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\)
\( \newcommand\Vom
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\)
\( \newcommand\Vpi
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\)
\( \newcommand\Vvarpi
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\)
\( \newcommand\Vrho
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\)
\( \newcommand\Vvarrho
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\)
\( \newcommand\Vsigma
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\)
\( \newcommand\Vvarsigma
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\)
\( \newcommand\Vtau
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\)
\( \newcommand\Vupsilon
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\)
\( \newcommand\Vphi
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\)
\( \newcommand\Vvarphi
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\)
\( \newcommand\Vchi
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\)
\( \newcommand\Vpsi
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\)
\( \newcommand\Vomega
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\)
\( \newcommand\VGamma
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\)
\( \newcommand\VDelta
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\)
\newcommand\VTheta{\vec\Theta}
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\( \newcommand\xhihOZ
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\)
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\( \newcommand\labar
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    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.03:_Weakly_Inhomogeneous_Gas), /content/body/p[1]/span, line 1, column 23
\)
\newcommand\msa{m\ns_\ssr{A}}
\newcommand\msb{m\ns_\ssr{B}}
\newcommand\mss{m\ns_\Rs}
\newcommand\HBx{\hat\Bx}
\newcommand\HBy{\hat\By}
\newcommand\HBz{\hat\Bz}
\newcommand\thm{\theta\ns_m}
\newcommand\thp{\theta\ns_\phi}
\newcommand\mtil{\widetilde m}
\newcommand\phitil{\widetilde\phi}
\newcommand\delf{\delta\! f}
\newcommand\coll{\bigg({\pz f\over\pz t}\bigg)\nd_{\! coll}}
\newcommand\stre{\bigg({\pz f\over\pz t}\bigg)\nd_{\! str}}
\newcommand\idrp{\int\!\!{d^3\!r\,d^3\!p\over h^3}\>}
\newcommand\vbar{\bar v}
\newcommand\BCE{\mbox{\boldmath{$\CE$}}\!}
\newcommand\BCR{\mbox{\boldmath{$\CR$}}\!}
\newcommand\gla{g\nd_{\RLambda\nd}}
\newcommand\TA{T\ns_\ssr{A}}
\newcommand\TB{T\ns_\ssr{B}}
\newcommand\ncdot{\!\cdot\!}
\newcommand\NS{N\ns_{\textsf S}}

Consider a gas which is only weakly out of equilibrium. We follow the treatment in Lifshitz and Pitaevskii, §6. As the gas is only slightly out of equilibrium, we seek a solution to the Boltzmann equation of the form f=f^0+\delf, where f^0 is describes a local equilibrium. Recall that such a distribution function is annihilated by the collision term in the Boltzmann equation but not by the streaming term, hence a correction \delf must be added in order to obtain a solution.

The most general form of local equilibrium is described by the distribution f^0(\Br,\Gamma)=C\exp\bigg({\mu-\ve(\Gamma)+\BV\!\cdot\Bp\over \kT}\bigg)\ , where \mu=\mu(\Br,t), T=T(\Br,t), and \BV=\BV(\Br,t) vary in both space and time. Note that \begin{split} df^0&=\Bigg(d\mu+ \Bp\cdot d\BV+ (\ve-\mu-\BV\!\cdot\Bp)\>{dT\over T} -d\ve\Bigg)\,\bigg(\!-{\pz f^0\over \pz \ve}\bigg)\\ &=\Bigg( {1\over n} \> dp+ \Bp\cdot d\BV+ (\ve-h)\>{dT\over T}-d\ve\Bigg)\,\bigg(\!-{\pz f^0\over \pz \ve}\bigg) \end{split} where we have assumed \BV=0 on average, and used \begin{split} d\mu&=\pabc{\mu}{T}{p}dT + \pabc{\mu}{p}{T}dp\\ &= -s\,dT + {1\over n}\,dp\ , \end{split} where s is the entropy per particle and n is the number density. We have further written h=\mu+Ts, which is the enthalpy per particle. Here, c\ns_p is the heat capacity per particle at constant pressure5. Finally, note that when f^0 is the Maxwell-Boltzmann distribution, we have -{\pz f^0\over\pz\ve}={f^0\over k\ns_\RB T}\ .

The Boltzmann equation is written \bigg({\pz \over\pz t} + {\Bp\over m}\cdot{\pz\over\pz\Br}+\BF\cdot{\pz \over\pz\Bp} \bigg) \big(f^0+\delf\big)=\coll\ . The RHS of this equation must be of order \delf because the local equilibrium distribution f^0 is annihilated by the collision integral. We therefore wish to evaluate one of the contributions to the LHS of this equation, \begin{split} {\pz f^0\over\pz t} + {\Bp\over m}\cdot{\pz f^0\over\pz\Br} + \BF\cdot{\pz f^0\over\pz\Bp}&=\bigg(\!-{\pz f^0\over\pz \ve}\bigg) \Bigg\{{1\over n}\,{\pz p\over\pz t} + {\ve - h\over T}\,{\pz T\over\pz t} + m\Bv\ncdot\Big[(\Bv\ncdot\bnabla)\,\BV\Big] \label{LHSA}\\ &\qquad+\Bv\cdot\bigg(m\,{\pz \BV\over\pz t} + {1\over n}\,\bnabla p\bigg) + {\ve - h\over T}\>\Bv\cdot\bnabla T - \BF\cdot\Bv\Bigg\}\ . \end{split} To simplify this, first note that Newton’s laws applied to an ideal fluid give \rho {\dot\BV}=-\bnabla p, where \rho=mn is the mass density. Corrections to this result, e.g. viscosity and nonlinearity in \BV, are of higher order.

Next, continuity for particle number means {\dot n} + \bnabla\ncdot (n\BV)=0. We assume \BV is zero on average and that all derivatives are small, hence \bnabla\ncdot (n\BV)=\BV\ncdot \bnabla n + n\,\bnabla\ncdot \BV\approx n\,\bnabla\ncdot \BV. Thus, {\pz\ln n\over\pz t}={\pz\ln p\over\pz t} - {\pz\ln T\over\pz t} = -\bnabla\ncdot \BV\ , \label{ptea} where we have invoked the ideal gas law n=p/\kT above.

Next, we invoke conservation of entropy. If s is the entropy per particle, then ns is the entropy per unit volume, in which case we have the continuity equation {\pz (ns)\over\pz t} + \bnabla\cdot(ns\BV)=n\bigg({\pz s\over\pz t} + \BV\ncdot\bnabla s\bigg) + s\bigg( {\pz n\over\pz t} + \bnabla\cdot (n\BV)\bigg)=0\ . The second bracketed term on the RHS vanishes because of particle continuity, leaving us with {\dot s} + \BV\ncdot\bnabla s\approx {\dot s}=0 (since \BV=0 on average, and any gradient is first order in smallness). Now thermodynamics says \begin{split} ds&=\pabc{s}{T}{p}dT + \pabc{s}{p}{T}dp \\ &={c\ns_p\over T} \> dT - {\kB\over p}\>dp\ , \end{split} since T\big({\pz s\over\pz T}\big)\nd_p=c\ns_p and \big({\pz s\over\pz p}\big)\nd_T=\big({\pz v\over\pz T}\big)\nd_p, where v=V/N. Thus, {c\ns_p\over\kB}\,{\pz \ln T\over\pz t} - {\pz\ln p\over\pz t}=0\ . \label{pteb} We now have in eqns. [ptea] and [pteb] two equations in the two unknowns {\pz\ln T\over\pz t} and {\pz\ln p\over\pz t}, yielding \begin{aligned} {\pz\ln T\over\pz t}&=-{\kB\over c\ns_V}\>\bnabla\ncdot \BV\\ {\pz\ln p\over\pz t}&=-{c\ns_p\over c\ns_V}\>\bnabla\ncdot \BV\ .\end{aligned} Thus Equation [LHSA] becomes \begin{split} {\pz f^0\over\pz t} + {\Bp\over m}\cdot{\pz f^0\over\pz\Br} + \BF\cdot{\pz f^0\over\pz\Bp}&=\bigg(\!-{\pz f^0\over\pz \ve}\bigg) \Bigg\{ {\ve(\Gamma)-h\over T}\>\Bv\cdot\bnabla T + m \, v\ns_\alpha v\ns_\beta \, \CQ\ns_{\alpha\beta} \\ &\qquad + {h-T c\ns_p-\ve(\Gamma)\over c\ns_V/k\ns_\RB}\,\bnabla\ncdot \BV -\BF\cdot\Bv\Bigg\}\ , \end{split} where \CQ\ns_{\alpha\beta}={1\over 2}\,\bigg({\pz V\ns_\alpha\over\pz x\ns_\beta} + {\pz V\ns_\beta\over\pz x\ns_\alpha} \bigg)\ .

Therefore, the Boltzmann equation takes the form \Bigg\{ {\ve(\Gamma)-h\over T}\>\Bv\cdot\bnabla T+ m \, v\ns_\alpha v\ns_\beta \, \CQ\ns_{\alpha\beta} - {\ve(\Gamma)-h+T c\ns_p\over c\ns_V/k\ns_\RB}\,\bnabla\ncdot\BV -\BF\cdot\Bv\Bigg\}\,{f^0\over\kT} + {\pz\,\delf\over\pz t}=\coll\ . \label{bwig} Notice we have dropped the terms \Bv\cdot{\pz\,\delf\over\pz\Br} and \BF\cdot{\pz\,\delf\over\pz\Bp}, since \delf must already be first order in smallness, and both the {\pz\over\pz\Br} operator as well as \BF add a second order of smallness, which is negligible. Typically {\pz\,\delf\over\pz t} is nonzero if the applied force \BF(t) is time-dependent. We use the convention of summing over repeated indices. Note that \delta\ns_{\alpha\beta}\,\CQ\ns_{\alpha\beta}=\CQ\ns_{\alpha\alpha}=\bnabla\ncdot\BV. For ideal gases in which only translational and rotational degrees of freedom are excited, h=c\ns_\Rp T.


This page titled 8.3: Weakly Inhomogeneous Gas is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

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