Is the common formulation for (Ising Model) Monte Carlo simulations a bit off? A common [Ising Model] Monte Carlo simulation repeats the following algorithm:

*

*randomly pick a [flip] event

*compute change in energy $\Delta E$

*if new energy is lower than old energy, accept the change. Otherwise, accept the change with probability $\exp(-\Delta E/kT)$
In this formulation, all “downhill” energy moves are treated the same (occur with equal probability). In a physical system, wouldn’t larger “downhill” events occur more frequently? An easy reformulation that better matches with my expectations about a real system would be:


*if new energy is lower than old energy, accept the change with probability $1 - \exp(-|\Delta E|/kT)$. Otherwise, accept the change with probability $\exp(-|\Delta E|/kT)$.

The thinking is that if an event has a probability $p$ of going to a higher energy state due to thermal energy, then it should have a probability of $1-p$ of staying in a higher energy state due to thermal energy.
 A: The goal of any sampling algorithm has to be that it is unbiased -- i.e., the random walk it describes is in a configuration $\vec s$ with relative probability $e^{-E(\vec s)/kT}$.
This is achieved by the standard Metropolis sampling you describe.
It is also achieved by any other random process where the ratio of probabilities to go from $\vec s$ to $\vec r$ vs. back satisfies
$$
\frac{p_{\vec s\to \vec r}}{p_{\vec r\to\vec s}} = \frac{e^{-E(\vec r)/kT}}{e^{-E(\vec s)/kT}}\ ,
\tag{*}
$$
the so-called "detailed balance" condition.
In particular, you could choose the transition probabilities equal to $e^{-E(\vec r)/kT}$ and $e^{-E(\vec s)/kT}$, respectively. This is not the formula you give (which will not work), but maybe more along the lines of your intuition.
Now why don't we choose the probabilities as above? Well, we want to maximize the probability for a move to be accepted (otherwise, we have to take far more samples), keeping the detailed balance condition (*). Thus, it is best to rescale the transition probabilities as much as possible -- that is, until the larger of them becomes equal to 1. This is precisely what the standard Metropolis update, as you describe it at the beginning of your question, does.
