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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.

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

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