Physics 418: Statistical Mechanics I
Prof. S. Teitel ----- Spring 2004

Problem Set 2

Due Tuesday February 24 in lecture

  • Problem 1 [30 points]

    This problem is meant to get you to think about the relationship between different ensembles and their corresponding free energies, and how one can often change ensembles to remove a constraint and make a calculation simpler.

    Consider the following model for an elastic string in a plane. The height of the string at position x is the function y(x). To make the calculation doable, imagine discretizing the horizontal axis into L discrete units, xi=i, i=0, 1, 2, ..., L. The height at position xi is then yi.


    The energy to stretch the string is then determined by the height differences according to the following Hamiltonian:

    H[yi] = halfalpha L
    (yi - yi-1)2

    a) [15 pts] The string is constrained so that its endpoints are fixed at:

    y0=0,       yL=Y

    so that the string has an average slope Y/L.

    Compute the partition function

    Z(Y) = ( L-1
    dyi) e-H[yi]/kBT

    Compute the corresponding free energy,

    F(Y) = - kBT ln Z(Y)

    There are two tricks you need to sum the partition function. One is to figure out how to deal with the fact that yi interacts with both yi+1 and yi-1. The second is to figure out how to deal with the constraint of the fixed enpoints. If you get desparate, here is a hint.

    b) [15 pts] Now consider the partition function in a new ensemble

    Z'(eta) = +infty
    dY Z(Y) e-eta Y/kBT

    Z'(eta) is the Laplace transform of Z(Y).

    In this ensemble, the average slope of the string, (yL-y0)/L is no longer fixed, but fluctuates. Its average value, however, is determined by the parameter eta.

    Compute the corresponding free energy

    F'(eta) = -kBT ln Z'(eta)

    Show that the average height difference of the endpoints, Y=yL-y0, is given by

    <Y> = partialF'(eta)/partialeta

    Now take the Legendre transform of F'(eta) to get,

    F(Y) = F'(eta) - etaY

    Show that F(Y) above agrees with what you found in part (a), provided you take the thermodynamic limit of L -> infty. That is, show that the free energies per unit length, F(Y)/L, are equal as L -> infty.

    You should find that the calculation in (b) is much easier and more straight forward to do than the calculation in (a). This shows that by going to the new ensemble, in which the original constraint of fixed Y was lifted by introducing its conjugate variable eta, the calculation of F(Y) becomes much easier.

  • Problem 2 [25 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). [5 points]

    (b) Find the temperature T as a function of energy E and number N. Show that T will be negative if E>0. [5 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? [5 points]

    (d) Working in the canonical ensemble, find the Helmholtz free energy A(T, N) as a function of temperature T and number N. [5 points]

    (e) By making the appropriate transformations on A(T, N), find the canonical entropy as a function of the average energy E and number N. Show that your result agrees with your answer for part (a), in the large N limit. [5 points]

  • Problem 3 [ 10 points]

    In lecture we derived the canonical partition function QN(V,N) for the idea gas using a factorization method. One can also directly compute it by taking the Laplace transform of the microcanonical partition function BigOmega(E,V,N). Using our result from lecture,

    BigOmega(E,V,N) = [ V
    (2pimE)3/2 ]N             1            
    [(3N/2) - 1]!  N!

    compute directly the Laplace transform,

    QN(T,V) = infty

    and show that you get the same result as found in lecture.

Last update: Wednesday, August 22, 2007 at 8:48:40 AM.