G25.2651: Statistical Mechanics

Problem set #2

Due: Feb. 17, 2003

1.

Gibbs proposed the following definition of entropy:

where is the phase space distribution function that satisfies the Liouville equation:

a.

Show that for the canonical ensemble, the Gibbs definition leads to the correct expression for the entropy:

where Q(N,V,T) is the canonical partition function.

b.

Show that for an arbitrary choice of , dS/dt=0, hence S(t)-S(0)=0 and S(t) cannot increase. Is this in violation of the second law of thermodynamics?

c.

The distribution, , is known as a fine-grained distribution function. This means that it contains all of the detailed microstructure of phase space, which cannot be resolved in reality. Consider, therefore, the following coarse-graining operation: Divide phase space up into small cells. Each cell will be characterized by a coarse-grained phase space probability density . Further divide each cell into small subcells, such that each subcell has a probability density . f and are chosen such that at t=0, . For t;SPMgt;0, a transformation from f to is made by transferring probability from subcells where to subcells where until the whole cell is at the same probability .

Define a coarse-grained entropy:

Show that .

: Show that the entropy change upon transferring probability from one small subcell to another is positive. Then, sum the contributions from all subcells to get the full contribution to .

d.

Even if were possible to measure the entropy with the greatest possible precision, the quantum mechanical uncertainty principle states that the positions and momenta can never be resolved with uncertainties less than or equal to , i.e. . Does this suggest that the second law of thermodynamics might have a purely quantum mechanical origin?

2.

Consider an N-particle system that is separable in each of the particles, i.e., the Hamiltonian can be written as

where h is the single-particle Hamiltonian and and are the ith Cartesian momentum and position variables.

a.

Show that the canonical partition function Q(N,V,T) can be written as

where q(V,T) is the one-particle partition function

b.

Show that the chemical potential is given by

where Q(N-1,V,T) is the partition function for the corresponding N-1 particle system.

Hint: Stirling's approximation for is and is valid when N is a very large number.

c.

Show that the result of part (b) is valid even if the Hamiltonian is not separable.

3.

Consider a classical system of N noninteracting diatomic molecules enclosed in a box of volume V that is held at a fixed temperature T. The Hamiltonian for a single molecule is taken to be

where .

a.

Calculate the partition function.

b.

Calculate the Helmholtz free energy.

c.

Calculate the heat capacity at constant volume.

d.

Calculate the mean-square molecular diameter

4.

Use the Legendre transform to determine the energy that results by transforming from volume, V to pressure, P starting from the microcanonical ensemble. What thermodynamic function does this energy represent? What energy results if, instead, you transform from particle number, N to chemical potential, , starting from the microcanonical ensemble?

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