Explore BrainMass

Explore BrainMass

    Multiplicity function of the two state paramagnet

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    For a single large two-state paramagnet, the multiplicity function is very sharply peaked about N/2.

    (a) Use Stirling's approximation to estimate the height of the peak in the
    multiplicity function.

    (b) Derive a formula for the multiplicity function in the vicinity of the peak.

    (c) How wide is the peak in the multiplicity function?

    (d) Suppose you flip 1,000,000 coins. Would you be surprised to obtain 501,000
    heads and 499,000 tails? Would you be surprised to obtain 510,000 heads and
    490,000 tails? Explain.

    © BrainMass Inc. brainmass.com October 9, 2019, 6:49 pm ad1c9bdddf

    Solution Preview

    Some comments (the solution is in the attached file):

    In this problem we keep the sqrt[2 pi N] factor in Stirling's approximation. I'm also doing (a) and (b) at once. I don't ...

    Solution Summary

    The multiplicity of a two state paramagnet is Omega = (n1 + n2)!/(n1! n2!) where n1 and n2 are the number of up and down spins, respectively. In terms of the total number of spins N = n1 + n2 and the excess number of up spins eta = n1 - N/2, this can be written as Omega = Binomial[N, N/2 + eta]. Using Striling's formula, we derive an expression for Omega valid for large N and small eta/sqrt(N).