Fundamentals Of Statistical Mechanics by B B Laud

By B B Laud

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Let the number of these states be denoted by (Er ). The prime on the symbol emphasizes the fact that its functional form will depend on the nature of the reservoir; of course, the details of this dependence are not going to be of any particular relevance to our final results. Now, the larger the number of states available to the reservoir, the larger the probability of the reservoir assuming that particular energy value Er (and, hence, of the system A assuming the corresponding energy value Er ).

15) 30 Chapter 2 . 3 The microcanonical ensemble In this ensemble the macrostate of a system is defined by the number of molecules N, the volume V , and the energy E. 4, we may prefer to specify a range of energy values, say from E − 12 to E + 12 , rather than a sharply defined value E. With the macrostate specified, a choice still remains for the systems of the ensemble to be in any one of a large number of possible microstates. In the phase space, correspondingly, the representative points of the ensemble have a choice to lie anywhere within a “hypershell” defined by the condition E− 1 2 ≤ H(q, p) ≤ E + 1 2 .

Interpret the result physically. ] Consider a particle of energy E moving in a one-dimensional potential well V (q), such that m dV dq {m(E − V )}3/2 . Show that the allowed values of the momentum p of the particle are such that p dq = n + 1 h, 2 where n is an integer. 6. The generalized coordinates of a simple pendulum are the angular displacement θ and the angular momentum ml2 θ˙ . Study, both mathematically and graphically, the nature of the corresponding trajectories in the phase space of the system, and show that the area A enclosed by a trajectory is equal to the product of the total energy E and the time period τ of the pendulum.

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