Homework 1

PHGN530 Statistical Mechanics

Note: Please collaborate with your classmates. Your Midterm and Final Exams will be based primarily on your homework.

Due date: 09/08/2016 These problems are from the textbook (Statistical Mechanics by Pathria)

1. The statistical basis of thermodynamics (300 points total, 150 each):

a) Show that, for two large systems in thermal contact, the number 0 0 1( , )E E of

Section 1.2 in the textbook (Statistical Mechanics by Pathria) can be expressed a Gaussian in the variable 1E . Determine the root-mean-square deviation of 1E from

the mean E in terms of other quantitites pertaining to the problem. b) Make an explicit evaluation of the root-mean-square deviation of 1E in the special

case when the systems 1A and 2A are ideal classical gas.

2. Connection to thermodynamics (150 points):

Assuming that the entropy S and the statistical number of a physical system are related through an arbitrary functional form ( )S f , show that the additive character of

S and the multiplicative character of necessarily require that the function ( )f be

of the form lnS k .

3. Effect of brining things together (150 points): Two systems of A and B , of identical composition, are brought together and allowed to exchange both energy and particles, keeping volumes AV and BV constant. Show that the

minimum value of the quantity /A AdE dN is given by A B B A

B A

T T

T T

; where ' s and 'T s

are the respective chemical potentials and temperatures.

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