In the canonical ensemble, we have the state of system $x_s$ and the state of the environment $x_e$. The probability of the total system is $$P(x_s,x_e)= const.$$ and that is independent of the states $x_s$ and $x_e$.

Now the probability to find the system in a certain state $x_s$ is $$P(x_s)= \sum_e P(x_s,x_e)$$ which is equal to some constant times the number of accessible states of the environment.

I don't understand how the probability the of system is equal to the sum of states of the environment on the total probability.


The probability $\rho$ to find your system $S$ with energy $E_S$ in a certain state in phase-space is proportional to the phase space volume of the reservoir ("environment") at energy $E_R$

$$\rho \propto \Gamma(E_R=E-E_S)$$

with $E=E_R+E_S$

Try to look at it this way: For every single state of your system $S$ with energy $E_S$, you have all states of the reservoir at energy $E_R=E-E_S$ available. The sum of all states in phase space is nothing but the phase space volume $\Gamma$.


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