Canonical Distribution

DUE: Monday, January 17, 2005

1.

Consider the binomial coefficient







For large values of N and m, show that it has its largest value when

m=N/2 (HINT: use Stirling’s formula). What is the maximum value? For

values of m near N/2, show that the binomial coefficient is

approximately Gaussian in form (HINT: Expand the log of the binomial

coefficient in a Taylor series in m around the maximum value). What

is the magnitude of the width of the Gaussian? Given these results,

would using the maximum term method for series involving the binomial

coefficients be valid? Why?



2.



a)

Hill, Introduction to Statistical Thermodynamics, problem 1-5 (p.32).



b)

Rather than using the equal-apriori probability postulate, Gibbs assumed
that the entropy is always given by







where the sum is over all the things which can fluctuate in the

ensemble. In a system with constant T, N, and V, the thermodynamic
equilibrium state is that which minimizes the Helmholtz free-energy,

; minimize the free-energy with respect to

compare your result with the canonical distribution. (Remember that

the probabilities must sum to unity.)



3.

Use the expression for the energy levels of a particle in a box to show
that







the pressure associated with the j’th state.

Average both sides of the equation and use the result (to be shown

derive the ideal gas equation

of state.



4.

A model for a spin in a magnetic field can be obtained from statistical
mechanics. The energy of a spin-I nucleus in a magnetic field is:







a constant called the

"gyromagnetic ratio", H is the external field,isandthe

projection of the total spin along the direction of the external

field (i.e., it is the magnetic quantum number and can take on values

).



a)

Work out an expression for the canonical partition function. HINT:
Recall that for geometric series:







and that





b)

By using a canonical ensemble, derive an expressions for the average
magnetization energy and entropy per spin in the presence of the
external field. Ignore all interactions between different spins.



c)

What is the expression for the constant external field heat capacity?
How does it behave at high and low temperatures?



a function of temperature.



5.

An approximate canonical partition function for a dense gas is:







where m is the mass of the particles and a and b are molecular

parameters (which are independent of temperature). Calculate the

energy, entropy, pressure, and chemical potential for this system.

To what well known thermodynamic approximate equation of state does

your answer correspond?