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, $h=c\ns_\Rp T$.