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5.2: Quantum Ideal Gases - Low Density Expansions

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Expansion in powers of the fugacity

From Equation [numeqn], we have that the number density n=N/V is n(T,z)=dε g(ε)z1eε/kBT1=j=1(±1)j1Cj(T)zj,

where z=exp(μ/kBT) is the fugacity and Cj(T)=dεg(ε)ejε/kBT.
From Ω=pV and our expression above for Ω(T,V,μ), we have p(T,z)=kBTdεg(ε)ln(1zeε/kBT)=kBTj=1(±1)j1j1Cj(T)zj.

Virial expansion of the equation of state

Eqns. ??? and ??? 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 ???, writing z=z(T,n) as a power series in the number density n, and substitute this into Equation ??? to obtain an equation of state p=p(T,n) of the form

p(T,n)=nkBT(1+B2(T)n+B3(T)n2+).

Note that the low density limit n0 yields the ideal gas law independent of the density of states g(ε). This follows from expanding n(T,z) and p(T,z) to lowest order in z, yielding n=C1z+O(z2) and p=kBTC1z+O(z2). Dividing the second of these equations by the first yields p=nkBT+O(n2), which is the ideal gas law. Note that z=n/C1+O(n2) can formally be written as a power series in n.

Unfortunately, there is no general analytic expression for the virial coefficients Bj(T) in terms of the expansion coefficients nj(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 Aj(T):

z=A1n+A2n2+A3n3+.

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

n=C1z±C2z2+C3z3+=C1(A1n+A2n2+A3n3+)±C2(A1n+A2n2+A3n3+)2+C3(A1n+A2n2+A3n3+)3+.

Let’s expand the RHS to order n3. Collecting terms, we have

n=C1A1n+(C1A2±C2A21)n2+(C1A3±2C2A1A2+C3A31)n3+.

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 nj for all j>1 must vanish. Thus,

C1A1=1C1A2±C2A21=0C1A3±2C2A1A2+C3A31=0.

The first of these yields A1:

A1=1C1.

We now insert this into the second equation to obtain A2:

A2=C2C31.

Next, insert the expressions for A1 and A2 into the third equation to obtain A3:

A3=2C22C51C3C41.

This procedure rapidly gets tedious!

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

pkBT=C1(A1n+A2n2+A3n3+)±12C2(A1n+A2n2+A3n3+)2+13C3(A1n+A2n2+A3n3+)3+=C1A1n+(C1A2±12C2A21)n2+(C1A3±C2A1A2+13C3A31)n3+NN=n+B2n2+B3n3+

We can now write

B2=C1A2±12C2A21=C22C21B3=C1A3±C2A1A2+13C3A31=C22C412C33C31.

It is easy to derive the general result that \boldsymbol{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,

(p+aN2V2)(VNb)=NkBT.

This may be recast as

p=nkBT1bnan2i=nkBT+(bkBTa)n2+kBTb2n3+kBTb3n4+,

where n=N/V. Thus, for the van der Waals system, we have B2=(bkBTa) and Bk=kBTbk1 for all k3.

Ballistic Dispersion

For the ballistic dispersion ε(p)=p2/2m we computed the density of states in Equation ???. One finds Cj(T)=gSλdTΓ(d/2)0dttd21ejt=gSλdTjd/2.

We then have B2(T)=2(d2+1)g1SλdTB3(T)=(2(d+1)3(d2+1))2g2Sλ2dT.
Note that B2(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)=±gSλdTLid2(±z)

and p(T,z)=±gSkBTλdTLid2+1(±z),
where Liq(z)n=1znnq
is the polylogarithm function2. Note that Liq(z) obeys a recursion relation in its index, viz. zzLiq(z)=Liq1(z),
and that Liq(1)=n=11nq=ζ(q).


This page titled 5.2: Quantum Ideal Gases - Low Density Expansions is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

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