V - = u

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(6.1)

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where the specific volume u is a given finite number.

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STATISTICAL MECHANICS

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The system will be regarded as isolated in the sense that the energy is a constant of the motion. This is clearly an idealization, for we never deal with truly isolated systems in the laboratory. The very fact that measurements can be performed on the system necessitates some interaction between the system and the external world. If the interactions with the external world, however, are sufficiently weak, so that the energy of the system remains approximately constant, we shall consider the system isolated. The walls of the container containing the system (if present) will be idealized as perfectly reflecting walls. A state of the system is completely and uniquely defined by 3N canonical coordinates ql' q2' ... ,q3N and 3N canonical momenta PI' P2' ... ,P3N. These 6N variables are denoted collectively by the abbreviation (p, q). The dynamics of the system is determined by the Hamiltonian .Yt(p, q), from which we may obtain the canonical equations of motion

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J.Yt(p,q) Jp; J.Yt(p,q) Jqj

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(6.2)

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It is convenient to introduce, as we did in 3, the 6N-dimensional r space, or phase space, of the system, in which each point represents a state of the system, and vice versa. The locus of all points in r space satisfying the condition .Yt(p, q) = E defines a surface called the energy surface of energy E. As the state of the system evolves in time according to (6.2) the representative point traces out a path in r space. This path always stays on the same energy surface because by definition energy is conserved. F or a macroscopic system, we have no means, nor desire, to ascertain the state at every instant. We are interested only in a few macroscopic properties of the system. Specifically, we only require that the system has N particles, a volume V, and an energy lying between the values E and E + ~. An infinite number of states satisfy these conditions. Therefore we think not of a single system, but of an infinite number of mental copies of the same system, existing in all possible states satisfying the given conditions. Anyone of these system can be the system we are dealing with. The mental picture of such a collection of systems is the Gibbsian ensemble we introduced in 3. It is represented by a distribution of points in r space characterized by a density function p(p, q, t), defined in such a way that p(p, q, t) d 3Npd 3Nq = no. of representative points contained in the volume element (6.3) d 3Npd 3Nq located at (p, q) in r space at the instant t

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We recall Liouville's theorem:

~: + j~

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; : :;;)

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(6.4)

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CLASSICAL STATISTICAL MECHANICS

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In geometrical language it states that the distribution of points in r space moves like an incompressible fluid. Since we are interested in the equilibrium situation, we restrict our considerations to ensembles whose density function does not depend explicitly on the time and depends on (p, q) only through the Hamiltonian. That is,

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p( p, q)

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p' ( .l't ( p, q))

(6.5)

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where p' (.l't) is a given function of .l't. It follows immediately that the second term on the left side of (6.4) is identically zero. Therefore

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aiP(p, q) = 0

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Hence the ensemble described by p(p, q) is the same for all times. Classical statistical mechanics is founded on the following postulate.

(6.6)

Postulata of Equal a Priori Probability When a macroscopic system is in thermodynamic equilibrium, its state is equally likely to be any state satisfying the macroscopic conditions of the system. This postulate implies that in thermodynamic equilibrium the system under consideration is a member of an ensemble, called the microcanonical ensemble, with the density function

_ {const. p ( p, q ) - 0