8/13/2019 Sm Homework 4

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Statistical Mechanics HW 4 : Due next Monday, Oct. 7, 2013

1. a) The entropy according to information theory is written as = {}

{}

Show that the probability distribution of a microstate of a paricle system, {} thatmaximizes the entropy subject to constraints {} = 1 and {}{} =,where {}is the energy of the microstate, is the canonical, i.e.,

{} [{}].

With another constraint added, {}{} = , for the average number ofparticles, show that the distribution that maximizes the entropy is grand canonical,

{} [({} {})].

b) Show that the variances of the energy and the particle number in grand canonical

ensemble are given by 2= and 2=

respectively

2. A Beryllium solid at a temperature T=200K, which is much lower than its Debye

temperature = 1440,has the heat capacity,

What is the entropy of the solid? It is in equilibrium with its own vapor, which can be

regarded as classical at the temperature T. Evaluate the vapor pressure in the unit of atm.

3. Consider the adsorption of atoms on M lattice sites each with area aas shown right. Asdiscussed in the class, the canonical partition function of N(