H.2 Euler-Maclaur in Sum Formula
16.1 Fundamental Hypothesis of Statistical Mechanics
For an isolated thermodynamical system, we consider allmicrostatescompatible with a macrostate. For the sake of illustration, we will specify this macrostate by its total energy1 E, its volumeV, and its number of particlesN; more complex systems can be treated by adding additional extensive state variables, specifying subsystems, etc. We consider our system to be governed by quantum mechanics and therefore associate each microstate with a stationary quantum state for the given quantities E,V,N. Since our system is macroscopic, the difference betweenEand the energy levels of one of its particles is very large compared to the differences among the energy levels of a particle. Thus, there is massive degeneracy and hence a huge number of microstates for each macrostate. This ensemble is usually referred to as themicrocanonical ensemble.
Thefundamental hypothesisis that every microstate of an isolated thermodynamic system that is compatible with a given macrostate of such a system isequally probable. If is the total number of compatible microstates for a given macrostate, then the probability of each microstate is 1/.
It follows from this hypothesis that the expected value yof any property y of the system is given by
y =(1/) ν=1
yν, (16.1)
whereyν is the value ofyin the microstate with labelν. Furthermore, the entropy of the system is defined to be
S=kBln, (16.2)
where kB is Boltzmann’s constant. Equation (16.2) is exactly the function given by Eq. (15.14) that was based on the disorder function D{pi} for the case pi=1/. The classical counterpart to Eq. (16.2) was proposed by Boltzmann and will be discussed in the next chapter.
SinceSis a monotonically increasing function of, maximizingis consistent with the second law of thermodynamics, according to whichSfor an isolated system will be a
1This is the total internal energy of the system which excludes macroscopic kinetic energy associated with motion of the center of mass. Here, we use the symbolEinstead ofUto make a subtle distinction because we specifythe energy rather than calculating its average value from a knowledge of other state variables, for example, the temperature.
maximum at equilibrium, with respect to variations of its internal extensive parameters and subject to any internal constraints. To better illustrate what this means, we consider a composite system made up of two subsystems with energiesE1andE2, volumesV1andV2, and particle numbersN1andN2. Furthermore, we assume that the subsystems are very weakly interacting and statistically independent so that=1(E1,V1,N1)2(E2,V2,N2), where the total quantitiesE=E1+E2,V=V1+V2, andN=N1+N2. So even withE,V, and Nheld constant,(E,V,N,E1,V1,N1)still depends on the partitioning of energy, volume, and particle numbers between the two subsystems. Since=12, the total entropy is simply additive, resulting in
S=kBln=kBln(12)=kBln(1)+kBln(2)=S1+S2. (16.3) WithE,V,N,V1, andN1held constant, maximizing lnwith respect toE1gives
∂ln
∂E1 = 1 1
∂1
∂E1 + 1 2
∂2
∂E2
∂E2
∂E1 = 1 1
∂1
∂E1 − 1 2
∂2
∂E2 =0. (16.4)
But (1/1)∂1/∂E1=(1/kB)∂S1/∂E1=T1/kB and similarly (1/2)∂2/∂E2=T2/kB, so Eq. (16.4) is equivalent to equality of absolute temperature,T1=T2. Similarly, maximizing with respect toV1 gives equality of pressure, and maximizing with respect toN1 gives equality of chemical potential. If a system cannot be decomposed into statistically independent subsystems, will not factor and it will be difficult to enumerate the microstates, but the maximization of and hence S will still be a valid criterion for equilibrium.
The considerations of the preceding paragraph are still valid if the two subsystems are actually portions of the same system that are initially isolated from one another by constraints that forbid exchange of energy, volume, and mole numbers. Then =(E,V,N)(E,V,N). As these constraints are gradually relaxed, each subsys- tem can proceed through a series of equilibrium states until=is maximized. We can therefore say thatis proportional to the probability of a macrostate. The quantity 1/is the probability of each microstate of the ensemble that represents that macrostate.
For further support thatis proportional to the probability of a macrostate, the reader is referred to Section 19.1.3 in which the number of waysWensof constructing an ensemble from members, each of which is in a single eigenstatei, is related to the probabilityPi
of occurrence of that eigenstate in the ensemble. If every member of the ensemble has thesameenergy, as it would for the microcanonical ensemble,Wens will be a maximum when all probabilities are equal. See also Chapter 22 where the entropy of any ensemble is discussed in terms of maximizing a probability. Moreover, sinceSgiven by Eq. (16.2) is proportional to the maximum value of the disorder functionD{qi}given by Eq. (15.11), a macrostate of maximum entropy, and hence maximum , is a state of maximum disorder, equivalent to a state with minimum information content, compatible with constraints.
In some books on statistical mechanics, there is an attempt to justify the fundamental assumption of equal probability of each compatible microstate from other considerations.
260 THERMAL PHYSICS
The basic idea for classical systems, see Landau and Lifshitz [7], is that a macroscopic system in equilibrium is assumed to progress in time through phase space so that it visits every allowed volume of phase space with equal probability. This is sometimes called the ergodic hypothesis. The quantum analog would be to assume that every microstate is visited with equal probability over a timeτ that is long compared to some characteristic relaxation time. The measurement of some physical property of a thermodynamic system is given by a time average of the form
¯ y=1
τ τ
0
y(t)dt. (16.5)
Thus, the time average would be equal to the ensemble average, that is,
y = ¯y (16.6)
for a system in equilibrium. In this book, we shall assume that Eq. (16.6) is true. In the last analysis, Eqs. (16.1), (16.2), and (16.6) are hypotheses that have borne up under the test of experiment. In Chapter 26, we introduce the statistical density operator and give a more detailed discussion of ensemble averages and time averages in the context of quantum mechanics.
Under conditions for which a macroscopic system can be described approximately by a set of N particles that obey classical mechanics, we do not have access to the concept of stationary quantum states. As discussed in the next chapter, we taketo be proportional to the volume of phase space (volume of momentum space times volume of actual space) in a thin shell near a hypersurface that corresponds to the total energy, E. It turns out that the volume of the shell itself is not important. Indeed, isolation of a system is only an idealization and therefore an approximation, so there will always be a small uncertainty in its energy. Landau and Lifshitz [7] refer to such systems as being quasi-isolated. Nevertheless, to agree ultimately with quantum mechanics, it is necessary to assume that the number of microstates in a volume element (dpdq)3N is given by (dpdq/h)3N whereh is Planck’s constant.2This is an artificial prescription that has no real basis in classical mechanics, for which Planck’s constant is effectively 0.
The microcanonical ensemble is easy to define but very hard to use because of the difficulty in cataloging and enumerating the microstates that are compatible with specification of the macrostate. For simple systems this is possible, as we shall illustrate for two-state subsystems, simple harmonic oscillators and ideal gases. The main value of the microcanonical ensemble is its theoretical importance, and we shall use it later to derive the canonical ensemble and the grand canonical ensemble which are much more tractable.
2This result holds foridentical but distinguishable particles. For an ideal gas, the number of microstates is approximately(dpdq/h)3N(1/N!)at high temperature and low density. The extra Gibbs factor of 1/N!is needed to make the entropy an extensive function and follows from quantum mechanics for identical indistinguishable particles. SeeSection 16.4for further details.