  Probably one of the most striking examples of the close connections between
global optimization processes and statistical physics is the simulated
annealing method, inspired by the famous Monte Carlo algorithm devised by
Metropolis et al. in the middle of the last century. In this paper we show how
the tools of linear kinetic theory allow to describe this gradient-free
algorithm from the perspective of statistical physics and how convergence to
the global minimum can be related to classical entropy inequalities. This
analysis highlight the strong link between linear Boltzmann equations and
stochastic optimization methods governed by Markov processes. Thanks to this
formalism we can establish the connections between the simulated annealing
process and the corresponding mean-field Langevin dynamics characterized by a
stochastic gradient descent approach. Generalizations to other selection
strategies in simulated annealing that avoid the acceptance-rejection dynamic
are also provided.
