## Quick review on Monte Carlo Now we have two states, or configurations, $\nu$ and $\nu'$. Define $P_\nu$ as the probability that at a given time the trajectory is in the state $\nu$. $\omega_{\nu\nu'}$ as the probability ***per unit time*** of $\nu$ jump to $\nu'$, then we have$\dot{P_\nu} = \sum_{\nu'}-\omega_{\nu\nu'}P_\nu +\omega_{\nu'\nu} P_{\nu'} $ >[!Note] > this can be interpreted as the summation of all other states. With addition of those jump away from $\nu$ and substruction of those jump from $\nu'$ to $\nu$. And probability $P_\nu \propto e^{-\beta E_\nu}$, so we have $\frac{\omega_{\nu\nu'}}{\omega_{\nu'\nu}} = \frac{P_{\nu'}}{P_\nu} = e^{-\beta \Delta E_{\nu\nu'}}$ With $\beta = \frac{1}{kT}$, the [[Ensemble#^boltzmann-factor|Boltzmann factor]], $\Delta E_{\nu\nu'}$ is energy difference. In this case, the basis for metropolis Monte Carlo, $ \begin{equation} \omega_{\nu\nu'} =\begin{cases} 1, & \text{if $E_{\nu\nu'} \leq 0$}.\\ e^{-\beta \Delta E_{\nu\nu'}}, & \text{otherwise}. \end{cases} \end{equation} $ 8 years just it's the move against energy decrease, then we accept the move. However, this energy is not decreased or equal. We check the Boltzmann factor and compare it with the random number in $(0, 1]$, call the random number $\rho$, if $\rho \leq e^{-\beta \Delta E_{\nu\nu'}}$, we accept the move. >[!Notice] >Remember it is when the random number smaller than this $e^{-\beta \Delta E_{\nu\nu'}}$ we accept, cause if $\Delta E$ increase, the possibility is small. Notice that metropolitan does not have time inside. So if we want to obtain rage related information, we have to consider [[Kinetic Monte Carlo|kinetic Monte Carlo]].