H.2 Euler-Maclaur in Sum Formula
19.7 Partition Function and Density of States
Under suitable circumstances, the energy levels of the quantum states of a system can be treated as quasi-continuous. Specifically, the spacing between levels must be small compared tokBT, which is often possible for large systems if the temperature is not too low. Under those circumstances, the sum overstatesthat is used to calculate the partition function, namely18
Z(β)=
j
exp(−βEj), (19.142)
18Zwill generally depend on other parameters such as the volumeV but we suppress these variables for simplicity.
can be approximated by an integral of the form Z(β)= ∞
0
e−βED(E)dE, (19.143)
whereD(E)is known as thedensity of statesand accounts for the spacing and degeneracy of the quantum states. Specifically,D(E)is a distribution function such thatD(E)dEis the number of quantum states in the energy interval betweenEandE+dE. Equation (19.143) has the same form as a Laplace transform with transform variableβ. Therefore, one can use the Laplace inversion formula
D(E)= 1 2πi
Br
eβEZ(β)dβ (19.144)
to computeD(E)from a knowledge ofZ(β). In Eq. (19.144),βis regarded as a complex vari- able and the integration is over a contourBrin the complex plane known as the Bromwich contour. This contour starts out atβ = −i∞, goes to the right of all singularities19ofZ(β) and ends up atβ=i∞. One can use Cauchy’s theorem to deform the contour and thus calculateD(E)by standard methods of contour integration.
Example Problem 19.3. Calculate the Laplace transformZ(β)of the partition function forN atoms of a monotonic ideal gas to determine its density of statesD(E)and relateD(E)to the corresponding function(E)of the microcanonical ensemble.
Solution 19.3. By combining Eq. (19.48) with Eq. (19.56), we see that the partition function forNatoms of a monatomic ideal gas is given by
Z(β)=(VnQ)N N! =VN
N! m
2πh¯2β 3N/2
. (19.145)
Thus,
D(E)=VN N!
m 2πh¯2
3N/2 1 2πi
Br
eβE
β3N/2 dβ. (19.146) The integrand certainly has a singularity atβ=0 but ifN is an odd integer, one also needs a branch cut, usually taken fromβ=0 toβ= − ∞along the real axis to make it analytic. ButN is large so we do not really care if it is odd or even. Therefore, we temporarily pretend that it is even, in which case the integrand has a pole of order 3N/2 at the origin. We can therefore close the contour in the left half plane and apply Cauchy’s theorem to shrink the contour to a small circle aroundβ=0. The result of integration is then well known to be
Br
eβE
β3N/2 dβ=2πiResidue eβE
β3N/2
=2πi E3N/2−1
(3N/2−1)!, (19.147) where Residue means to extract the coefficient of 1/β. Thus,
D(E)= VN N!(3N/2−1)!
mE 2πh¯2
3N/2 1
E. (19.148)
19Such singularities are poles whereZ(β)becomes infinite or branch cuts needed to make it single-valued.
332 THERMAL PHYSICS
We note thatD(E)has dimensions of 1/Eso thatD(E)dEis dimensionless, as a probability should be. In the present case, we can easily check our result because Eq. (16.44) gives an expression for, which is the total number of microstates having energies less thanE.
Differentiation with respect toEshows that(∂/∂E)N,V=D(E)as it should (see Eq. (19.154) for more detail).
Note that Eq. (19.148) can be written in terms of the gamma function in the form D(E)= VN
N!(3N/2) mE
2π¯h2
3N/2 1
E. (19.149)
Of course(3N/2)makes sense even when 3N/2 is a half integer, so we suspect that Eq. (19.149) might hold in general. Substitution into Eq. (19.143) shows that this conjecture is true.
We remark that this same Laplace transform relationship holds between the density of statesD1(ε)of a single particle and its partition functionz(β). Thus for an ideal gas,
z(β)=VnQ=V m
2πh¯2β 3/2
, (19.150)
so
D1(ε)= V (3/2)
mε 2π¯h2
3/21
ε = V
(1/2)π1/2 mε
2π¯h2 3/21
ε, no spin degeneracy, (19.151) where we have used(x+1)=x(x)and(1/2) = π1/2. This result is the same as the density of statesG(ε)/2 given by Eq. (25.13), where the division by 2 is necessary because G(ε)contains a factor of 2 due to spin degeneracy. SinceD1(ε)is proportional toV, one often deals with the intensive quantity
D1(ε)
V = 1
(1/2)π1/2 mε
2πh¯2 3/21
ε, (19.152)
which is also called the density of states and has units of (volume energy)−1. One must therefore be careful to ascertain from the context just what density of states is being used!
Strictly speaking, one should haveD(E)=(1/N!) ∂VR/∂E, whereVRgiven by Eq. (16.39) is the total number of microstates for all energies≤ E. But(1/N!)VR =(E)/F, whereF is given by Eq. (16.40). Therefore,
D(E)=∂(/F)
∂E = 1 F
∂
∂E + F2
3NE 2E2 exp
−3NE 2E
. (19.153)
The second term in Eq. (19.153) is negligible compared to the first (because of the exponential) andF ≈1, so
D(E)≈∂(E)
∂E (19.154)
to an excellent approximation.
Finally, we make one more connection between the microcanonical ensemble and the canonical ensemble as follows. For the microcanonical ensemble, we haveS=kBln(E); however, for the canonical ensemble
S=U−F T =kB
lnZ+ U kBT
=kBln
ZeU/kBT
. (19.155)
IfEandUare nearly the same, we should have
ln((E))∼ln((˜ U))≡ln(ZeU/kBT). (19.156) However, Eq. (19.156) must be interpreted very carefully because the systems being compared are not quite the same. (E) relates to the microcanonical ensemble for which the energy E of each microstate is specified, whereas (˜ U) relates to the canonical ensemble for which the temperature is specified, so only the average energy U(T) is specified. Therefore, if we exponentiate both sides of Eq. (19.156) we obtain
(E)∼ ˜(U)≡ZeU/kBT, (19.157) which only holds to the extent that ln(E) ∼ ln(˜ U)when sub-extensive terms are neglected.
For example, for an ideal gas, for whichU=(3/2)NkBT, we have (˜ U)= 1
N!VN mU
3πh¯2N 3N/2
e3N/2. (19.158)
According to Eq. (16.44), we have
(E)=VN(mE/2π¯h2)3N/2
N!(3N/2)! . (19.159)
We observe that the factors that multiplyU3N/2and E3N/2 are not quite the same, but sinceN is large we can use(3N/2)! ∼ N3N/2e−3N/2√
3πN to write Eq. (19.158) in the form
(˜ U)∼√
3πNVN(mU/2π¯h2)3N/2
N!(3N/2)! . (19.160)
Thus, in the thermodynamic limit of extremely largeN, we have
ln(˜ U)=ln(E)+(1/2)ln(3πN) (19.161) in which the last term is sub-extensive, and therefore negligible. It is also illuminating to use Eq. (19.148) withE→Uto express(˜ U)in terms of the density of states evaluated at energyU, which results in
(˜ U)∼√ 2π U
√3N/2D(U)=√ 2π
3N/2kBTD(U). (19.162)
334 THERMAL PHYSICS
In view of Eq. (19.89), we recognize the factor√
3N/2kBT=
(E)2to be a measure of the spread of energy at temperatureT. Thus Eq. (19.162) can be written
(˜ U)∼√ 2π
(E)2D(U), (19.163)
which demonstrates clearly that the density of states D(U) must be multiplied by the spread of energy to approximate the number of microstates(˜ U).
For a single ideal gas particle, the correspondence implied by Eq. (19.157) would give 1(ε)∼zexp(ε/kBT)=ze3/2, (19.164) which illustrates thatz is essentially a measure of the number of states available to an individual particle at temperatureT.
Another way of evaluating˜ in Eq. (19.157) is to evaluate approximately the partition function
Z= ∞
0 D(E)e−βEdE= ∞
0
e[−βE+lnD(E)]dE (19.165) by expanding about the most probable state. To do this, we recognize thatD(E)is a rapidly increasing function ofEand e−βEis a rapidly decreasing function ofE. Thus the integrand has a sharp maximum at the most probable valueE∗that satisfies
0= ∂
∂E[−βE+lnD(E)]E∗= −β+ [lnD(E∗)], (19.166) where the prime indicates a derivative. We can therefore expand the exponent in the right- hand integrand in Eq. (19.165) to second order to obtain
−βE+lnD(E)= −βE∗+lnD(E∗)−(1/2)α(E−E∗)2+ ã ã ã, (19.167) where
α:= − [lnD(E∗)]>0 (19.168) is positive becauseE∗corresponds to a sharp maximum. Thus withξ=E−E∗,
Z ≈D(E∗)e−βE∗ ∞
−E∗ e−(α/2)ξ2dξ≈√ 2π 1
√αD(E∗)e−βE∗, (19.169) where the lower limit in the second integral has been approximated by −∞because the peak is so sharp. See Widom [17, Eq. 1.25] for an equivalent result with hisδE=√
2/α. Thus ˜ ∼√
2π 1
√αD(E∗)e−β(E∗−U). (19.170) But the difference20betweenE∗andUis of orderkBT, so the exponential in Eq. (19.170) gives a numerical factor of order 1 and the result greatly resembles Eq. (19.162).
20In this Gaussian approximation, there is negligible difference betweenE∗andUif the lower limit of the integral in Eq. (19.169) can be approximated by−∞.
Specifically for a monatomic ideal gas, Eq. (19.148) shows that D(E)=AE3N/2−1 so E∗=(3N/2−1)kBTandα=(3N/2−1)/(E∗)2. SinceU=(3N/2)kBT, Eq. (19.170) becomes
˜ ∼√
2π E∗
√3N/2−1D(E∗)e=√
2π U
√3N/2−1D(U)e(E∗/U)3N/2. (19.171) But
(E∗/U)3N/2=
3N/2−1 3N/2
3N/2
=
1− 2 3N
3N/2
∼
e−2/3N3N/2
=e−1. (19.172) Therefore, Eq. (19.170) reduces to
˜ ∼√
2π U
√3N/2−1D(U)=√ 2π
(E)2D(U), (19.173) in excellent agreement with Eq. (19.163) because the 1 in the square root is negligible.
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20
Classical Canonical Ensemble
For the canonical ensemble,1the temperature rather than the energy is fixed. Therefore, the members of the ensemble have various energies. Members of the ensemble having a given energy must still obey the Liouville theorem and hence Eq. (17.11). This possibility can be accommodated by choosing the densityρ in phase space to be some function of the classical HamiltonianH, in which case Eq. (17.11) becomes
{ρ(H),H} = dρ
dH{H,H} =0. (20.1)
Proceeding with the same arguments as in the quantum mechanical case, it can be inferred for a system in contact with a heat reservoir at temperatureTthat the probability of a system having energyE is proportional to the Boltzmann factor exp(−βE), where β = 1/(kBT) as usual. Since H = E for such a system, the appropriate probability distribution function is
P(p,q):=exp[−βH(p,q)]
ZC , (20.2)
where wherepandqare 3N-dimensional vectors representing the canonical momenta and coordinates, respectively. The function
ZC:=
exp[−βH(p,q)]dω, (20.3)
where dω ≡ d3Npd3Nq and the integration is over all phase space. P(p,q)dω is the probability that the system will be in the volume element dωof phase space centered about the pointp,q. The factorZC in the denominator of Eq. (20.2) is needed to insure normalization, that is,
P(p,q)dω=1. (20.4)
IfY(p,q)is some function ofpandq, then the average value ofY is Y:=
Y(p,q)P(p,q)dω. (20.5)
1Those interested in the historical development of classical statistical mechanics are encouraged to read the original work of J.W. Gibbs [4]. Based on Hamilton’s classical dynamical equations that we discussed in Chapter 17, Gibbs developed the classical canonical ensemble in Chapter IV, the microcanonical ensemble in Chapter X, and the grand canonical ensemble in Chapter XV. The integral form of Liouville’s theorem that we presented in Section 17.1 is what Gibbs called the “conservation of probability of phase.” Ifρ=eηis the probability density function in phase space, Gibbs calledηthe “index of probability.” Then he referred to a “canonical distribution”
as one in which the index of probability is a linear function of the energy.
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338 THERMAL PHYSICS
Comparison of Eqs. (20.2) and (20.3) with Eqs. (19.5) and (19.6) shows that the function ZCplays the role of a classical partition function. In fact, the formula Eq. (19.7) for the average internal energy has exactly the same form in the classical case. Thus,
U:= H =
H(p,q)P(p,q)dω= − 1 ZC
∂ZC
∂β = −∂lnZC
∂β . (20.6)
But in some other respects, the correspondence ofZCwith the quantum mechanical parti- tion function is incorrect. Unlike the quantum partition function,ZCis not dimensionless and does not account for the number of quantum states that need to be associated with a volume of phase space. ForN identical particles that occupy the same volume, one can define a dimensionless classical partition function
ZC∗:=ZC
ω0 = 1 ω0
exp[−βH(p,q)]dω. (20.7)
The factorω0is the same factor as in Eq. (17.14) that allows us to convert from volume of phase space to microscopic states. For identical distinguishable particles we haveω0= h3N and for identical indistinguishable classical particles we have approximatelyω0 = h3NN!. In other words, dωhas been replaced by the dimensionless quantity dω/ω0, which is the differential of the number of microscopic states. Doing this gives rise to the correct entropy constant at high temperatures, where classical statistics are valid approximately.
In this respect, we could view Eq. (20.2) in the form P(p,q)dω=exp[−βH(p,q)]
ZC∗
dω ω0
. (20.8)
In this manner, we can also relate properly to the Helmholtz free energy, namely
F= −kBTlogZC∗, (20.9)
and the entropy will be correctly given by S= −∂F
∂T. (20.10)