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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!