I'm trying to understand the physical meaning of the proposal step generating function in Metropolis algorithm.
In the original paper, and most derivations I found, it seems that it's not much important as long as it's symmetric (for detailed-balance) and allows a good exploration of the configuration space (e.g. $\sim 0.5$ acceptance ratio). Usually it's either a uniform or a normal distribution.
Now when studying Simulated Annealing, most papers assume a common use of a normal distribution with $\sigma = T$.
Given simulated annealing is deeply rooted in physical analogy, if we put the temperature there, there must be some physical sense we're trying to model. I mean, we could use a fixed uniform distribution with the proper width and still get good enough parameter space exploration and boltzmann distribution in the result, so why we use a gaussian and why the temperature dependence?
Also, when Cauchy distribution is introduced (e.g. by Szu and Hartley, 1987) they say the temperature used as a control parameter is kind of artificial, implying that the one in the Gaussian was not.
So what's the physical phenomenon with Gaussian, temperature dependent, steps we're modeling?
I know it all can be better explained with markov chains, priors, posteriors and likelihood, but that way you lose the physical analogy...