# Justification of acceptance probability in simulated annealing

In simulated annealing the acceptance probability for a new state in step $$k$$ is traditionally defined as $$P(\text{accept new})= \begin{cases} \exp(-\frac{\Delta}{T_k}), & \text{ if } \Delta \geq 0 \\ 1, & \text{ if } \Delta < 0 \end{cases},$$ where $$\Delta = f(\text{new}) - f(\text{old})$$ is the change in objective function $$f$$ which is to be minimised and $$T_k$$ is a strictly decreasing positive sequence with $$\lim_{k\rightarrow \infty} T_k=0.$$

## Question

1. Is there a fundamental or conceptional justification for using specifically $$\exp(-\frac{\Delta}{T_k})$$ in the acceptance probability instead of any other term which fulfills the "obvious" requirements, i.e. being decreasing in $$\Delta$$ and converging to zero for $$k\rightarrow\infty$$ and $$\Delta\rightarrow\infty$$?
2. In which way is the analogy with statistical physics, where $$f$$ is energy and $$T_k$$ temperature, anything more than a superficial or "inspiring" analogy?

I would equally appreciate direct answers or references.

• 1. Check metropolis-hastings 2. the analogy is purely nominal. – Xi'an Dec 7 '18 at 13:24