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Physics 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu ----- Spring 2005

Problem Set 2

Due Friday, February 18 noon, in my mailbox outside the physics department main office

• Problem 1 [15 points]

  Consider a box of volume V. The box is split exactly in half by a thermally conducting imovable wall. Equal quantities of the same type of ideal gas fill each half of the box, and the system is in equilibrium. The total energy of the gas is fixed at E_T.

  a) Show that the pressure of the gas on each side is the same.

  b) Using the formular derived in lecture for the number of states of an ideal gas at energy E, (E), find the most likely value for the energy of the gas on one side of the box.

  c) Using (E), if P(E) is the probability distribution for the gas on one side of the box to have energy E, show that the relative width (i.e. the width divided by the average) of P(E) is proportional to 1/N^1/2, for sufficiently large N, where N is the number of particles in the gas. This shows that the fluctuation of E away from its average becomes negligibly small in the thermodynamic limit of N going to infinity.

• Problem 2 [15 points]

  Consider a system of N distinguishable non-interacting objects, each of which can be in one of two possible states, "up" and "down", with energies +e and -e. Assume that N is large.

  (a) Working in the microcanonical ensemble, find the entropy of the system S(E, N) as a function of fixed total energy E and number N (Hint: it is useful to consider the numbers N₊ and N_- of up and down objects).

  (b) Find the temperature T as a function of energy E and number N. Show that T will be negative if E>0.

  (c) What happens if such a system (1) with T₁<0 comes into thermal contact with another such system (2) with T₂>0? Does T₁ increase or decrease? Does T₂ increase or decrease? In which direction does the heat flow?