H.2 Euler-Maclaur in Sum Formula
19.1.1 Derivation from Microcanonical Ensemble I
We derive the canonical ensemble from the microcanonical ensemble by applying the fundamental hypothesis to an isolated total system with fixed energyET, consisting of a reservoirRand a systemIof interest. The systemImay, itself, be very large and consist of a number of subsystems, or particles, that may interact with one another. We assume that the systemIhas quantum statesEiand that its extensive macrovariables, other than energy, are fixed. The indexiindicates a specific quantum state, so it actually represents a complete set of quantum numbers.
Suppose that the systemI is in adefinite quantum state i having energy Ei. Then the reservoir has energyET−Ei. For the total system, the number of microstates can be expressed as a product of the number of microstates of the reservoir,R, and the number of microstates of the system of interest,, in the form
Thermal Physics.http://dx.doi.org/10.1016/B978-0-12-803304-3.00019-3 305
Copyright © 2015 Elsevier Inc. All rights reserved.
306 THERMAL PHYSICS
iT=R(ET−Ei)(Ei)=R(ET−Ei)×1=R(ET−Ei). (19.1) In other words, the systemIis in a single definite microstate, so for the system of interest, (Ei)=1; therefore, only the number of microstates of the reservoir must be counted. An equation similar to Eq. (19.1) holds if the systemIis in the quantum statej. As explained in Section 16.1, the probability of a system being in a given macrostate with energyE, volume V, and number of particlesN is proportional to(E,V,N), which is the sum of its number of equally probable microstates. Therefore, the ratio of the probabilityPiof systemIbeing in the eigenstateito the probabilityPjof systemIbeing in the eigenstatejis
Pi
Pj =R(ET−Ei)
R(ET−Ej) =exp[SR(ET−Ei)/kB]
exp[SR(ET−Ej)/kB], (19.2) whereSR(ER)is the entropy of the reservoir in a state having energyER. We now assume that the reservoirRis very large so that|Ej−Ei| |ET−Ej|for any states ofI. Then by expanding in a Taylor series we obtain1
SR(ET−Ei)=SR[(ET−Ej)+(Ej−Ei)] =SR(ET−Ej)+(Ej−Ei)∂SR
∂ER+ ã ã ã
=SR(ET−Ej)+Ej−Ei
TR + ã ã ã, (19.3)
whereTRis the temperature of the reservoir. Substitution into Eq. (19.2) and cancellation of the factor exp[SR(ET−Ej)/kB]gives
Pi
Pj =exp(−Ei/kBTR)
exp(−Ej/kBTR). (19.4)
Equation (19.4) states that the probability Pi of system I being in eigenstate i is proportional to its Boltzmann factor exp(−Ei/kBT)where we have dropped the subscript RonT for simplicity.2 We can obtain a normalized probability by dividing by the total partition function
Z =
j
exp(−βEj) (19.5)
to obtain
Pi=exp(−βEi)
Z , (19.6)
whereβ=1/(kBT). In Eq. (19.5), the sum is over all of the quantum states of the system of interest. Equation (19.5) resembles our former equation for the occupation probabilities pi=exp(−βεi)/zof weakly interacting distinguishable subsystems except that we are now
1The ratio of the second-order term to the first-order term is−(Ej−Ei)/(2CRTR), whereCRis the heat capacity of the reservoir. We assume thatCRis so large that this term and higher order terms are negligible. This is essentially the definition of a heat reservoir.
2We must still bear in mind, however, that the canonical ensemble applies to a system in contact with a heat reservoir of constant temperatureT. Given that other extensive variables of the system are held constant in this derivation, the canonical ensemble will relate thermodynamically to the Helmholtz free energy.
dealing with the states and energy levels of awhole system. The internal energy of our system is
U=
i
PiEi= −∂lnZ
∂β , (19.7)
which resemblesU = −N∂lnz/∂βexcept that the factor ofN is now missing because we are dealing withZfor the whole system.
Finally, we obtain the Helmholtz free energyF of system I. Since F=U −TS and S= −∂F/∂T, we see thatFsatisfies the differential equation
F−T∂F
∂T =U, (19.8)
which, in terms ofβ, can be rewritten in the form F+β∂F
∂β = −∂lnZ
∂β . (19.9)
The left-hand side of Eq. (19.9) is recognized immediately to be∂(βF)/∂β, so it may be integrated to obtain
F= −1
βlnZ+a
β, (19.10)
whereais a function of integration (independent ofβ). The entropy is therefore S= U−F
T = U
T +kBlnZ−kBa. (19.11)
But whenT→0, only the ground state with degeneracyg0and energyE0is occupied, so Z→g0exp(−βE0)and lnZ→ lng0−βE0. Similarly, asT→0 we haveU→E0, so Eq. (19.11) becomes
S(T→0)=kBlng0−kBa. (19.12) Consistent with Eq. (16.2), however, we require3
S(T→0)=kBlng0, (19.13)
which means that the function of integrationa=0. Thus Eq. (19.10) becomes F= −1
βlnZ (19.14)
3Note that the value ofS(T→0)according to Eq. (19.13) is not zero due to the possibility of degeneracy of the ground state. It would be strictly zero for a nondegenerate ground state for whichg0=1. If this degeneracy were massive, say of orderg0=qN, whereqis some integer andNis the number of subsystems or “particles” in the system, thenS(T→0)would beNkBlnqwhich would be extensive and significant. Otherwise,S(T→0)is practically zero.
308 THERMAL PHYSICS
which resembles our former result for weakly interacting identical but distinguishable subsystems withN missing andzreplaced byZ. Equation (19.14) can also be written in the form
j
exp(−βEj)=exp(−βF), (19.15)
which shows the relationship between the microscopic picture (on the left) and the macroscopic picture (on the right). Moreover,
S=U T −F
T =kBβU+kBlnZ= −kB κ
i=1
PilnPi= −kBβ2 ∂
∂β lnZ
β
. (19.16)
Note that the quantity −κ
i=1PilnPi=D{Pi} is the disorder function of information theory discussed in Section 15.1.