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Derivation of the Boltzmann distribution (Dr. David Tong)

  1. Mar 8, 2015 #1

    Dr. David Tong, in his statistical physics notes, derives the boltzmann distribution in the following manner.

    He considers a system (say A) in contact with a heat reservoir (say R) that is at a temperature T. He then writes that the number of microstates of the combined system (A and R) is

    [itex] \Omega (E_{total}) = \sum\nolimits_{n} \Omega_{R}(E_{total}-E_n) [/itex]

    where the summation is over all states of the system A (states of A are labelled as |n>, each of which has energy E_n )

    Can anyone help me understand how he arrives at the equation above? What about the microstates of the system A itself? I was of the understanding that the number of microstates of the composite system would be

    [itex] \Omega (E_{total}) = \Omega_{R}(E_{total}-E_{n})\,. \Omega_{A}(E_{n}) [/itex]

    Grateful for any help, thanks!
  2. jcsd
  3. Mar 8, 2015 #2


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    Staff: Mentor

    I don't know the notes you are referring to, but in derivations I've seen, you take the system A to be a simple particle with a non-degenerate, discrete energy spectrum. The multiplicity of the system is then 1.
  4. Mar 10, 2015 #3
    thank you, DrClaude. That cleared my question.
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