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Homework Help: Statistical Mechanics

  1. Feb 26, 2012 #1
    [itex]\Omega[/itex]^(0)(E) = [itex]\Omega[/itex](E)[itex]\Omega[/itex](E^(0) - E),

    a) Write this equation in terms of entropy

    b)Taylor series expand this resulting equation to 2nd order in the individual energies.Use the fact that the subsystems are in equilibrium with a total xed energy to simplify the resulting expression.

    c) How can you show the resulting expression for problem b that the variance of (E) is proportional to N, the number of particles in the system.

    The way I approached this question was...

    a) I know that Helmholtz free energy is A = U - TS, where U= (E), the total energy and S is the entropy, than A = E - TS, therefore S=-kTlnZ, however how can I apply when E^(0) is the total energy of the two subsystems. In other words how can I mathematically solve this problem?

    b) I assumed subsystem A has some mean energy E about which there are fluctuations. Therefore since the macroscopic state of maximal entropy for the system is the one where all micro-states are equally likely to occur, with probability 1 / [itex]\Omega[/itex](E), during the system's fluctuations, so S is equal kln ([itex]\Omega[/itex](E)). Am I right? how can I compute this?

    c) No clue. So far I could establish that the fluctuations
    from equilibrium, of the number of accessible states scales could be 1/[itex]\sqrt{}N[/itex].
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
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