Problem Set 3

Problem Set 3

Due Thursday, February 27, uploaded to Blackboard by 11:59pm

Problem 1 [10 points total]
Consider a box containing N extremely relativistic particles, moving so fast that their speed can be taken as essentially the speed of light c, but the relativisitc factor γ = 1/(1-v²/c²)^1/2 is finite. For such particles the energy-momentum relation is well approximated by ε = pc. Equivalently, you can think of these as massless particles (think photon!) which travel with speed c, carry momentum p, and have energy ε = pc.
Using a kinetic theory for such particles, as in Notes 2-1, find the relation between the pressure P, volume V, number of particles N, and the average energy per particle <ε>. You should assume that a particle is equally likely to be traveling in any spatial direction. If we assume that the ideal gas law continues to hold for such a gas (it does), what is the relation between the average energy per particle and the temperature?-->
Problem 2 [15 points total]
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 (remember, the wall is imovable, so you can't just appeal to mechanical equilibrium).
b) Using the formular derived in lecture for the number of states Ω(E) of an ideal gas at total energy E, find the most likely value for the energy of the gas on one side of the box.
c) Using Ω(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. (For "width" you may use half width at half height or any other reasonable definition.)
Problem 3 [25 points total]
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 +ε and −ε. Assume that N is large.
(a) Working in the microcanonical ensemble, find the number of states Ω(E,N), and then the entropy of the system S(E, N), as a function of fixed total energy E and number N (Hint: it may be useful to consider the numbers N₊ and N₋ of up and down objects). Sketch S(E,N) as a function of E for fixed N, and show that it is not a monotonic increasing function of E. What is the key feature of this system that causes the dependence of S on E to be so qualitatively different from that of an ideal gas? [10 points]
(b) Using your result from (a) find the temperature T as a function of energy E and number N. Show that T will be negative if E>0. Sketch T vs E for fixed N. [7 points]
(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? [8 points]