5.2: Quantum Ideal Gases - Low Density Expansions

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\)
\( \newcommand\VTheta{\vec\Theta}\)
\( \newcommand\VLambda{\vec\Lambda}\)
\( \newcommand\VXi{\vec\Xi}\)
\( \newcommand\VPi{\vec\Pi}\)
\( \newcommand\VSigma{\vec\Sigma}\)
\( \newcommand\VUps{\vec\Upsilon}\)
\( \newcommand\VPhi{\vec\Phi}\)
\( \newcommand\VPsi{\vec\Psi}\)
\( \newcommand\VOmega{\vec\Omega}\)
\( \newcommand\BA{\mib A}\)
\( \newcommand\BB{\mib B}\)
\( \newcommand\BC{\mib C}\)
\( \newcommand\BD{\mib D}\)
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\( \newcommand\BH{\mib H}\)
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\( \newcommand\Bvarpi{\mib\varpi}\)
\( \newcommand\Brho{\mib\rho}\)
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\( \newcommand\BBA{\boldsymbol\RA}\)
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\( \def\sbra#1{\langle \,  #1 \, |}\)
\( \def\sbraket#1#2{\langle \, #1  \, |  \, #2 \,  \rangle}\)
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\(\def\ket#1{\big| \, #1\, \big\rangle}\)
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\(\newcommand \gtwid{\approx}\)

\( \newcommand\index{\textsf{ind}}\)
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\( \def\bmapright#1{ \smash{\mathop{\hbox to 35pt{\rightarrowfill}}\limits_{#1}}\ }\)
\( \newcommand\bsqcap{\mbox{\boldmath{$\sqcap$}}}\)

\( \def\pabc#1#2#3{\left({\pz #1\over\pz #2}\right)\ns_{\!\!#3}}\)
\( \def\spabc#1#2#3{\big({\pz #1\over\pz #2}\big)\ns_{\!#3}}\)
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\( \newcommand\longrightleftharpoons{ \mathop{\vcenter{\hbox{\ooalign{\raise1pt\hbox{$\longrightharpoonup\joinrel$}\crcr  \lower1pt\hbox{$\longleftharpoondown\joinrel$}}}}}}\)
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\( \newcommand\veF{\ve\ns_\RF}\)
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\( \newcommand\SZ{\textsf Z}}\) \( \newcommand\kFd{k\ns_{\RF\dar}\)
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\( \newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}\)
\( \newcommand\xhiOZ{\xhi^\ssr{OZ}}\)
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Expansion in powers of the fugacity

From Equation [numeqn], we have that the number density \(n=N/V\) is \[\begin{split} n(T,z)&=\int\limits_{-\infty}^\infty\,\,\,d\ve\ {g(\ve)\over z^{-1}\,e^{\ve/\kT}\mp 1}\\ &=\sum_{j=1}^\infty (\pm 1)^{j-1}\,C\ns_j(T)\,z^j, \label{qvirn} \end{split}\] where \(z=\exp(\mu/\kT)\) is the fugacity and \[C\ns_j(T)=\,\,\impi d\ve\>g(\ve)\,e^{-j\ve/\kT}.\] From \(\Omega=-pV\) and our expression above for \(\Omega(T,V,\mu)\), we have \[\begin{split} p(T,z)&=\mp\, \kT\,\,\int\limits_{-\infty}^\infty\,\,\,d\ve\,g(\ve)\,\ln\,\Big(1\mp z\,e^{-\ve/\kT}\Big)\\ &=\kT\sum_{j=1}^\infty (\pm 1)^{j-1}\,j^{-1}C\ns_j(T)\,z^j. \label{qvirp} \end{split}\]

Virial expansion of the equation of state

Eqns. \ref{qvirn} and \ref{qvirp} express \(n(T,z)\) and \(p(T,z)\) as power series in the fugacity \(z\), with \(T\)-dependent coefficients. In principal, we can eliminate \(z\) using Equation \ref{qvirn}, writing \(z=z(T,n)\) as a power series in the number density \(n\), and substitute this into Equation \ref{qvirp} to obtain an equation of state \(p=p(T,n)\) of the form

\[p(T,n)=n\,\kT\,\Big(1+B\ns_2(T)\,n + B\ns_3(T)\,n^2 + \ldots\Big).\]

Note that the low density limit \(n\to 0\) yields the ideal gas law independent of the density of states \(g(\ve)\). This follows from expanding \(n(T,z)\) and \(p(T,z)\) to lowest order in \(z\), yielding \(n=C\ns_1\,z+\CO(z^2)\) and \(p=\kT \,C\ns_1\,z+\CO(z^2)\). Dividing the second of these equations by the first yields \(p=n\,\kT + \CO(n^2)\), which is the ideal gas law. Note that \(z=n/C\ns_1+\CO(n^2)\) can formally be written as a power series in \(n\).

Unfortunately, there is no general analytic expression for the virial coefficients \(B\ns_j(T)\) in terms of the expansion coefficients \(n\ns_j(T)\). The only way is to grind things out order by order in our expansions. Let’s roll up our sleeves and see how this is done. We start by formally writing \(z(T,n)\) as a power series in the density \(n\) with \(T\)-dependent coefficients \(A\ns_j(T)\):

\[z=A\ns_1\, n + A\ns_2 \, n^2 + A\ns_3\, n^3 + \ldots.\]

We then insert this into the series for \(n(T,z)\):

\[\begin{split} n&=C\ns_1\,z \pm C\ns_2\,z^2 + C\ns_3 z^3 + \ldots \\ &=C\ns_1\,\big(A\ns_1\,n + A\ns_2\,n^2 + A\ns_3\,n^3 + \ldots\big) \pm C\ns_2\,\big(A\ns_1\,n + A\ns_2\,n^2 + A\ns_3\,n^3 + \ldots\big)^2\\ &\qquad +C\ns_3\,\big(A\ns_1\,n + A\ns_2\,n^2 + A\ns_3\,n^3 + \ldots\big)^3 + \ldots. \end{split}\]

Let’s expand the RHS to order \(n^3\). Collecting terms, we have

\[n=C\ns_1 \,A\ns_1 \,n + \big(C\ns_1 \,A\ns_2 \pm C\ns_2\, A_1^2\big)\,n^2 + \big(C\ns_1 \,A\ns_3 \pm 2 C\ns_2 \,A\ns_1 A\ns_2 + C\ns_3 \,A_1^3\big)\,n^3 + \ldots\quad.\]

In order for this equation to be true we require that the coefficient of \(n\) on the RHS be unity, and that the coefficients of \(n^j\) for all \(j>1\) must vanish. Thus,

\[\begin{split} C\ns_1\,A\ns_1&=1\\ C\ns_1 \,A\ns_2 \pm C\ns_2\, A_1^2&=0\\ C\ns_1 \,A\ns_3 \pm 2 C\ns_2 \,A\ns_1 A\ns_2 + C\ns_3 \,A_1^3&=0. \end{split}\]

The first of these yields \(A\ns_1\):

\[A\ns_1={1\over C\ns_1}.\]

We now insert this into the second equation to obtain \(A\ns_2\):

\[A\ns_2=\mp{C\ns_2\over C^3_1}.\]

Next, insert the expressions for \(A\ns_1\) and \(A\ns_2\) into the third equation to obtain \(A\ns_3\):

\[A\ns_3={2 C_2^2\over C_1^5} - {C\ns_3\over C_1^4}.\]

This procedure rapidly gets tedious!

And we’re only half way done. We still must express \(p\) in terms of \(n\):

\[\begin{split} {p\over\kT}&=C\ns_1 \,\big(A\ns_1 \, n + A\ns_2\,n^2 + A\ns_3\,n^3 + \ldots\big)\pm\half C\ns_2 \, \big(A\ns_1 \, n + A\ns_2\,n^2 + A\ns_3\,n^3 + \ldots\big)^2\\ &\qquad\qquad + \third C\ns_3\,\big(A\ns_1 \, n + A\ns_2\,n^2 + A\ns_3\,n^3 + \ldots\big)^3+\ldots\\ &=C\ns_1\,A\ns_1\,n + \big(C\ns_1\,A\ns_2 \pm\half C\ns_2\, A_1^2\big) n^2 +\big(C\ns_1\,A\ns_3\pm C\ns_2 \,A\ns_1\,A\ns_2 +\third\,C\ns_3\,A_1^3\big)\,n^3 + \ldots\bvph\\ &=n+B\ns_2 \,n^2 + B\ns_3\,n^3 + \ldots \end{split}\]

We can now write

\[\begin{split} B\ns_2&=C\ns_1\,A\ns_2\pm \half C\ns_2 A_1^2=\mp {C\ns_2\over 2 C_1^2}\\ B\ns_3&=C\ns_1\,A\ns_3\pm C\ns_2 \,A\ns_1\,A\ns_2+\third\,C\ns_3\,A_1^3={C_2^2\over C_1^4}-{2 \,C\ns_3\over 3 \,C_1^3}. \end{split}\]

It is easy to derive the general result that \(B^\ssr{F}_j=(-1)^{j-1} B^\ssr{B}_j\), where the superscripts denote Fermi (F) or Bose (B) statistics.

We remark that the equation of state for classical (and quantum) interacting systems also can be expanded in terms of virial coefficients. Consider, for example, the van der Waals equation of state,

\[\bigg(p+{aN^2\over V^2}\bigg)\big(V-Nb)=N\kT.\]

This may be recast as

\[\begin{split} p&={n\kT\over 1-bn} -an^2\vph\\ &=n\kT + \big(b\,\kT-a\big) \,n^2 + \kT\,b^2 n^3 + \kT\, b^3n^4+ \ldots, \end{split}\]

where \(n=N/V\). Thus, for the van der Waals system, we have \(B\ns_2=(b\,\kT-a)\) and \(B\ns_k=\kT\,b^{k-1}\) for all \(k\ge 3\).

Ballistic Dispersion

For the ballistic dispersion \(\ve(\Bp)=\Bp^2/2m\) we computed the density of states in Equation \ref{BDOS}. One finds \[C\ns_j(T)={\,\Sg\nd_S\,\lambda_T^{-d}\over\RGamma(d/2)}\int\limits_0^\infty\,\,dt\>t^{{d\over 2}-1}\,e^{-jt}=\Sg\nd_S\,\lambda_T^{-d}\,j^{-d/2}.\] We then have \[\begin{split} B\ns_2(T)&=\mp\, 2^{-\left({d\over 2}+1\right)}\cdot\Sg^{-1}_S\,\lambda_T^d\\ B\ns_3(T)&=\Big(2^{-(d+1)} - 3^{-\left({d\over 2}+1\right)}\Big)\cdot 2\,\Sg^{-2}_S\,\lambda_T^{2d}. \end{split}\] Note that \(B\ns_2(T)\) is negative for bosons and positive for fermions. This is because bosons have a tendency to bunch and under certain circumstances may exhibit a phenomenon known as Bose-Einstein condensation (BEC). Fermions, on the other hand, obey the Pauli principle, which results in an extra positive correction to the pressure in the low density limit.

We may also write \[n(T,z)=\pm\Sg\nd_S\,\lambda_T^{-d}\>\,{Li}\ns_{d\over 2}(\pm z)\] and \[p(T,z)=\pm\Sg\nd_S\,\kT\,\lambda_T^{-d}\>\,{Li}\ns_{{d\over 2}+1}(\pm z),\] where \[{Li}\ns_q(z)\equiv\sum_{n=1}^\infty {z^n\over n\nsub^q}\] is the polylogarithm function². Note that \({Li}\ns_q(z)\) obeys a recursion relation in its index, viz. \[z\,{\pz\over\pz z}\,{Li}\ns_q(z)={Li}\ns_{q-1}(z), \label{zetarec}\] and that \[{Li}\ns_q(1)=\sum_{n=1}^\infty {1\over n\nsub^q}=\zeta(q).\]