Derivation of the Boltzmann distribution (Dr. David Tong)

[soviet1100]

Hello!

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

$\Omega (E_{total}) = \sum\nolimits_{n} \Omega_{R}(E_{total}-E_n)$

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

$\Omega (E_{total}) = \Omega_{R}(E_{total}-E_{n})\,. \Omega_{A}(E_{n})$

Grateful for any help, thanks!

 

[DrClaude]

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.

 

[soviet1100]

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.

thank you, DrClaude. That cleared my question.