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Lattice of 1D anharmonic oscillators (Cannonical Ensemble)

  1. Oct 19, 2013 #1
    1. The problem statement, all variables and given/known data

    I have a system of N non-interacting anharmonic oscillators whose potential energy is given by,

    [tex] V(q) = cq^2 -gq^3 -fq^4 [/tex]

    where [itex] c,f,g > 0 [/itex] and [itex] f,g [/itex] are small.

    2. Relevant equations

    The Hamiltonian is given by,

    [tex] H = \sum_{i=1}^N \big ( \frac{p^2_i}{2m} + V(q_i) \big ) [/tex]

    And the corresponding partition function is,

    [tex] Z = \int \prod_{i=1}^N\frac{dq_idp_i}{h}e^{-\beta H} [/tex]

    3. The attempt at a solution
    I'm trying to calculate the partition function, from which everything else will essentially follow. Substituting H into my integral, and re-arranging things a bit, I find,

    [tex] Z = \frac{1}{h^N}\prod_{i=1}^N \int dq_ie^{-\beta V(q_i)} \int dp_i e^{-\beta\frac{p^2_i}{2m}} [/tex]

    The integral over the [itex] p_i[/itex]'s is easy, it's the one over the [itex]q_i[/itex] which has me stuck, because it seems like it has to diverge... written out in full,

    [tex] \int dq_ie^{-\beta V(q_i)} = \int dq_ie^{-\beta(cq^2 -gq^3-fq^4)} [/tex]

    The problem wants things in leading orders of [itex] f,g [/itex], but I still don't see how the integral is not going to explode since I'm integrating over all possible phase-space... :(

    Can someone show me what I'm missing here?
  2. jcsd
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