# 8.3: Weakly Inhomogeneous Gas


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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.