Harvey Gould and Jan Tobochnik: Statistical and Thermal Physics: With Computer Applications, Second Edition

Problem 1.8. The demon and the ideal gas (a) The applet/application at simulates a demon that exchanges energy with an ideal gas of N particles moving in d spatial dimensions. Because the particles do not interact, the only coordinate of interest is the velocity of the particles. In this case the demon chooses a particle at random and changes its velocity in one of its d directions by an amount chosen at random between -A and +A. For simplicity, the initial velocity of each particle is set equal to tune, where up = (2Ep/m)1/2/N, En is the desired total energy of the system, and m is the mass of the particles. For simplicity, we will choose units such that m = 1. Choose d = 1, N = 40, and En = 10 and determine the mean energy of the demon E, and the mean energy of the system E. Why is E # Es? (b) What is Z, the mean energy per particle of the system? How do z and E, compare for various values of Eo? What is the relation, if any, between the mean energy of the demon and the mean energy of the system? (c) Choose N = 80 and En = 20 and compare & and Ea. What conclusion, if any, can you make? (d) Run the simulation for several other values of the initial total energy En and determine how Z depends on E. for fixed N. (e) From your results in part (d), what can you conclude about the role of the demon as a thermometer? What properties, if any, does it have in common with real thermometers? (f) Repeat the simulation for d = 2. What relation do you find between Z and Es for fixed N? (g) Suppose that the energy momentum relation of the particles is not e = p'/2m, but is e = cp. where c is a constant (which we take to be unity). Determine how Z depends on Ed for fixed N and d = 1. Is the dependence the same as in part (d)? (h) Suppose that the energy momentum relation of the particles is = = Ap]/2, where A is a constant (which we take to be unity). Determine how Z depends on E, for fixed N and d = 1. Is this dependence the same as in part (d) or part (g)? (i) The simulation also computes the probability P(E.)SE that the demon has energy between Ey and Eg+6E. What is the nature of the dependence of P( E.) on Ea? Does this dependence depend on the nature of the system with which the demon interacts? There are finite size effects that are order 1/N