Various Statistical Ensembles
4.2 Microcanonical Ensemble
Taking the average, we find (∆rmsG)2=
M i=1
∑
M
∑
j=1(gi−g)(gj−g). (4.10) We have already assumed that the interaction among the adjacent parts is sufficiently weak, implying that we may regard each part as statistically independent of the adjacent ones. Thus, ifi= j,
(gi−g)(gj−g)=gi−ggj−g=0. (4.11) However, ifi=j,
(gi−g)(gj−g)=(gi−g)2=:(∆rmsg)2 (4.12) and (4.10) becomes
(∆rmsG)2=M(∆rmsg)2. (4.13) Combining (4.7) and (4.13), we arrive at
∆rmsG G ∼
!1
M . (4.14)
If we fix the number of particles included in each of the parts, into which we divided our original system, thenM∼N, and hence
∆rmsG G ∼
!1
N (4.15)
as advertised.
4.2 Microcanonical Ensemble
Equation (4.5) indicates that the distribution of the system energy is sharply peaked aroundHfor macroscopic systems withN∼1024. This being the case, the canon- ical ensemble can be approximated by a new ensemble, called microcanonical ensemble, in which all the copies have practically the same energy.
4.2.1 Expression for ρ
To define a statistical ensemble precisely, we must give an explicit expression forρ. For this purpose, let us start from a canonical ensemble and then imagine selecting from the ensemble only those copies whose energy lies within an extremely narrow
semi-open interval(E−∆E,E]. (That is,Hof a copy must satisfyE−∆E<H≤E.) Those copies so selected form the microcanonical ensemble. Now, we recall that copies in the original canonical ensemble are distributed in the phase space accord- ing to the densityρwhich depends only on the energy of the system. Since∆E/Eis extremely small, we can regardρwithin the interval(E−∆E,E]as constant with- out any noticeable loss of accuracy. The microcanonical ensemble constructed from the canonical ensemble is therefore characterized by
ρ(qf,pf)dqfdpf = ( 1
CM
dqfdpf
hfP ifE−∆E<H(qf,pf)≤E
0 otherwise. (4.16)
Aside from the normalization factor, this expression forρ whenE−∆E<H≤E follows from (3.165) by replacing the Boltzmann factore−βHby a constant.
There is nothing special about our choice of the interval forH. We could have very well chosen[E−∆E,E)instead. The particular choice we made here, how- ever, will affect the definition ofW(E)and our choice for the step function to be introduced in what follows. The reason for using a half-open interval becomes clear in (4.23).
If we take a phase point A, the number of copies in the ensemble whose rep- resentative phase points fall inside the volume element dqfdpf taken around A is given byNρ(qf,pf)dqfdpf, whereN is the total number of copies in the ensem- ble. According to (4.16), if the dqfdpf lies entirely inside the regionE−∆E <
H(qf,pf)≤E, this number isN dqfdpf/hfPCMregardless of exactly where A is located. On the other hand, if dqfdpf is taken entirely outside the indicated region, the number is strictly zero. Thus, the copies in the ensemble are distributed with uniform density in the region of the phase space corresponding to(E−∆E,E]. In other words, all states satisfying E−∆E<H(qf,pf)≤E are equally probable.
This result is known as theprinciple of equal weight.
We arrived at this principle by applying a canonical ensemble to a macroscopic system. In a more common approach to statistical mechanics, one starts with a microcanonical ensemble. In this case, the principle plays a more central role of being the logical foundation of statistical mechanics.
We recall thatρ must be normalized:
ρ(qf,pf)dqfdpf =1. (4.17) Substituting (4.16), we find
CM= 1 hfP
E−∆E<H≤E
dqfdpf. (4.18)
We refer toCMas themicrocanonical partition function. Note that the integral in (4.18) is the volume of the region in the phase space (or the phase volume for short) that is compatible withE−∆E<H≤E, whilehfPis the phase volume occupied by a single (quantum mechanically distinct) state. Thus,CMis the number of states consistent with the condition thatE−∆E<H≤E.
4.2 Microcanonical Ensemble 173 We introduced a microcanonical ensemble as an approximation to a canonical ensemble. Thus, it is sensible to expect that Gibbs’s entropy formula continues to apply to the microcanonical ensemble. Using (4.16) in (3.168) and noting that xlnx→0 asx→0,20we have
−S/kB=
ρln(hfPρ)dqfdpf =
ρln 1
CMdqfdpf=ln 1
CM . (4.19) The last step follows from (4.17) and the fact that CM is a constant. This is the famousBoltzmann’s entropy formula:
S=kBlnCM. (4.20)
4.2.2 Choice of ∆ E
It is clear from (4.18) that, once the functional form ofHis specified and the inte- gration is carried out, the resultingCMand henceSdepend on f,E,∆E, and the lim- its of integrations. Because the integration in (4.18) includes only those microstates compatible withE−∆E<H≤Eand∆E/Eis extremely small, the internal energy may be identified withE:
U:=H ≈E. (4.21)
For a system ofN particles contained in a three-dimensional box, f =3Nand the limits of integrations for their coordinates usually show up only in the form of the system volumeV in the final expression forCM. This being the case,
S=S(U,V,N,∆E). (4.22)
Of all these quantities, upon whichScan depend,∆Ealone does not have any cor- responding quantity in thermodynamics. We now show that the value of∆Ecan be taken quite arbitrarily within a reasonable bound with no quantitative consequence to the value ofS. Thus,Sis practically a function only ofU,V, andN.
To see thatS is in fact insensitive to∆E, letW(E,V,N)denote the number of states satisfyingH≤Efor given values ofVandN. Clearly,
CM(E,V,N,∆E) =W(E,V,N)−W(E−∆E,V,N). (4.23) Because∆E is extremely small compared toE, we expand the secondW into a Taylor series to obtain
CM(E,V,N,∆E)≈W(E,V,N)−
W(E,V,N)−∂W
∂E ∆E
= ∂W
∂E ∆E=:Ω(E,V,N)∆E, (4.24)
where
Ω(E,V,N):=∂W
∂E (4.25)
is thedensity of states. This terminology is quite appropriate since an integration ofΩover a certain interval of energy yields the total number of states whose energy falls within that interval.
Using (4.20), we obtain
S=kBlnΩ(E,V,N)∆E, (4.26) which is still a function ofE,V,N, and∆E.
Based on the consideration that led us to microcanonical ensemble, it seems quite reasonable to set∆E=E/√
N, for which the entropy is evaluated as
S1=kBlnΩ(E,V,N) E
√N . (4.27)
An extreme choice for ∆E would be E itself. For this choice, the Taylor series expansion used in arriving at (4.24) cannot be justified. Nevertheless, if we proceed blindly, we find
S2=kBlnΩ(E,V,N)E. (4.28) Thus,
S2−S1 kBN = 1
2NlnN. (4.29)
For a macroscopic body with N∼1024, this quantity is approximately 3×10−23 even though the two choices for∆Ediffer by a factor of√
N∼1012. For macro- scopic systems, therefore, the precise value of∆E is unimportant. What happens if Nis not large enough? In this case, the use of microcanonical ensemble is of little interest. The ensemble was introduced only as an approximation applicable for a largeN.
4.2.3 Isolated System
In a microcanonical ensemble, all copies of the ensemble have nearly the same amount of energy. If we recall howρwas constructed from the long-time behavior of a single system, this implies that its energy fluctuates very little. In fact, the width of this fluctuation relative to the energy itself is of the order of 1/√
N andE of the system is a constant within the accuracy of any practical means of measuring E. We recall that the energy of an isolated system is a constant of motion. Thus, a microcanonical ensemble is a natural choice for describing a system that can be regarded as isolated within the accuracy of experiments.
This correspondence between a microcanonical ensemble and an isolated sys- tem is quite satisfactory from the point of view of thermodynamics. We recall that entropy played a prominent role in the condition of equilibrium of an isolated sys-