- $\qquad p_i = \frac{e^{\beta E_i}}{Z}$
- \item markov process: \\
- \begin{itemize}
- \item $P(A,t)$: probability of configuration $A$ at time $t$
- \item $W(A \rightarrow B)$: transition probability
- \[
- \begin{array}{l}
- P(A,t+1) = P(A,t) + \\
- \sum_B \Big( W(B \rightarrow A) P(B,t) - W(A \rightarrow B) P(A,t) \Big)
- \end{array}
- \]
- \end{itemize}
+ $\qquad p_i = \frac{e^{- \beta E_i}}{Z}$
+ \item detailed balance \\[6pt]
+ sufficient condition for equilibrium: \\
+ \[
+ W(A \rightarrow B) p(A) = W(B \rightarrow A) p(B)
+ \]
+ $\Rightarrow \frac{W(A \rightarrow B)}{W(B \rightarrow A)} = \frac{p(B)}{p(A)} = e^{\frac{- \Delta E}{k_B T}}$ \\[6pt]
+ with $\Delta E = E(B) - E(A)$