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It is understood that all members of the ensemble have the same number of particles N and the same volume V. Suppose f(p, q) is a measurable property of the system, such as energy or momentum. When the system is in equilibrium, the observed value of f(p, q) must be the result obtained by averaging f(p, q) over the microcanonical ensemble in some manner. If the postulate of equal a priori probability is to be
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useful, all manners of averaging must yield essentially the same answer. Two kinds of average values are commonly introduced: the most probable value and the ensemble average. The most probable value of f(p, q) is the value of f(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average of f~p, q) is defined by
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The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e., if
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STATISTICAL MECHANICS
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observed value of f may be calculated. When it is not, we should question the validity of statistical mechanics. In all physical cases we shall find that mean square fluctuations are of the order of liN. Thus in the limit as N ~ 00 the ensemble average and the most probable value became identical. Strictly speaking, systems in nature do not obey classical mechanics. They obey quantum mechanics, which contains classical mechanics as a special limiting case. Logically we should start with quantum statistical mechanics and then arrive at classical statistical mechanics as a special case. We do this later. It is only for pedagogical reasons that we begin with classical statistical mechanics. From a purely logical point of view there is no room for an independent postulate of classical statistical mechanics. It would not be logically satisfactory even if we could show that the postulate introduced here follows from the equations of motion (6.2), for, since the world is quantum mechanical, the foundation of statistical mechanics lies not in classical mechanics but in quantum mechanics. At present we take this postulate to be a working hypothesis whose justification lies in the agreement between results derived from it and experimental facts.
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In the microcanonical ensemble every system has N molecules, a volume V, and an energy between E and E + ~. It is clear that the average total momentum of the system is zero. We show that it is possible to define quantities that correspond to thermodynamic quantities. The fundamental quantity that furnishes the connection between the microcanonical ensemble and thermodynamics is the entropy. It is the main task of this section to define the entropy and to show that it possesses all the properties attributed to it in thermodynamics. Let f ( E) denote the volume in f space occupied by the microcanonical ensemble:
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The dependence of f(E) on N, V, and ~ is understood. Let L(E) denote the volume in f space enclosed by the energy surface of energy E:
(6.11)
Then
f(E) = L(E
+ ~) - L:(E)
(6.12) (6.13)
is so chosen that
E, then
f(E) = w(E)~
CLASSICAL STATISTICAL MECHANICS
where w ( E) is called the density of states of the system at the energy E and is defined by
w(E)
The entropy is defined by
a[(E) aE
(6.14)
8(E, V) == k log f(E)
(6.15)
where k is a universal constant eventually shown to be Boltzmann's constant. To justify this definition we show that (6.15) possesses all the properties of the entropy function in thermodynamics, namely,
( a) 8 is an extensive quantity: If a system is composed of two subsystems
whose entropies are, respectively, 8 1 and 8 2 , the entropy of the total system is 8 1 + 8 2 , when the subsystems are sufficiently large. ( b) 8 satisfies the properties of the entropy as required by the second law of the thermodynamics. To show the extensive property, let the system be divided into two subsystems which have N 1 and N2 particles and the volumes V1 and V2 , respectively. * The energy of molecular interaction between the two subsystems is negligible compared to the total energy of each subsystem, if the intermolecular potential has a finite range, and if the surface-to-volume ratio of each subsystem is negligibly small. The total Hamiltonian of the composite system accordingly may be taken to be the sum of the Hamiltonians of the two subsystems:
.Yt(p, q) = .Yt1(P1' q1)
+ .Yt2(P2' q2)
(6.16)
where (P1' q1) and (P2' q2) denote, respectively, the coordinates and momenta of the particles contained in the two subsystems. Let us first imagine that the two subsystems are isolated from each other and consider the microcanonical ensemble for each taken alone. Let the energy of the first subsystem lie between E 1 and E 1 + ~ and the energy of the second subsystem lie between E 2 and E 2 + ~. The entropies of the subsystems are, respectively, 81(E 1, V1) = k log f 1(E 1)
82(E2, V2) = k log f 2(E2) where f 1(E 1 ) and f 2 (E2 ) are the volumes occupied by the two ensembles in their respective f spaces. They are schematically represented in Fig. 6.1 by the volumes of the shaded regions, which lie between successive energy surfaces that differ in energy by ~. Now consider the microcanonical ensemble of the composite system made up of the two subsystems, and let the total energy lie b'etween E and E + 2~.
"For simplicity we assume that the same N1 , N z particles are always confined respectively to the volumes Vi' Vz . The proof is therefore invalid for a gas, for which S has to be modified (See Section 6.6).