Deriving the Boltzmann distribution

[raeed]

I was reading the derivation of Boltzmann distribution using the reservoir model.
lets call the reservoir by index R and the tiny system by index A.
In the derivation they proposed that the probability for being at energy e (for A) is proportional to the number of states in reservoir. I didn't understand this completely and i would be happy to get some help!
here is my take on it, and please correct me if I'm wrong.
- The temperature of the whole system is T and it's constant therefor the number of states for the whole systems g is also constant
- both A and R are independent of each other therefor g = g_A ⋅ g_R
- if g_R goes up then g_A has to go down meaning g_A ∝ g_R
- P(e) ∝ 1/g_A → P(e) ∝ g_R
I'm not really convinced by my explanation so if someone could explain it and perhaps give me an intuitive physical explanation, I'd be happy. Thank you

 

[DrClaude]

The starting point is that all microstates are equally probable. Then if the system A is made up a single particle, its state has no influence on the total probability of the state of A + R (the multiplicity of A is always 1), so you can focus on the states of the reservoir only.