Monday, June 23, 2014

Monte Carlo algorithm to simulate the evolution of a solid in a heat bath until it reached its thermal equilibrium

4 4. .  S Si im mu ul la at te ed d  A An nn ne ea al li in ng g  Metropolis et al. [14] proposed an algorithm to simulate the evolution of a solid in a heat bath until it reached its thermal equilibrium. The Monte Carlo method was used to simulate the process, which started from a certain thermodynamic state of the system, defined by a certain energy ever and temperature. Then, the state was slightly perturbed. If the change in energy produced by this perturbation was negative, the new configuration was accepted.  If it was positive, it was accepted with a probability given by е-ΔT ⁄ kT , where k is the so-called Boltzmann constant, which is a constant of nature that relates temperature to energy [15]. This process is repeated until a frozen state is achieved [16,17]. Thirty years after the publication of Metropolis’ approach, Kirkpatrick et al. [18] and Cerny [19] independently pointed out the analogy between this “annealing” process and combinatorial optimization.  These researchers indicated several important analogies: a system state is analogous to a solution of the optimization problem; the free energy of the system (to be minimized) corresponds to the cost of the objective function to be optimized; the slight perturbation1 imposed on the system to change it to another state corresponds to a movement into a neighboring position (with respect to the local search state); the cooling schedule corresponds to the control mechanism adopted by the search algorithm; and the frozen state of the system corresponds to the final solution generated by the search algorithm (using a population size of one). These important analogies led
E E. .  T Te ei im mo ou ur ry y, ,  H H. .  M Mi ir rz za ah ho os ss se ei in ni ia an n  & &  A A. .  K Ka ab bo ol li i  / /  A A  M Ma at th he em ma at ti ic ca al l  M Me et th ho od d  f fo or r  M Ma an na ag gi in ng g  I In nv ve en nt to or ri ie es s  i in n  a a  D Du ua al l  C Ch ha an nn ne el l  …
…                      35 
to the development of an algorithm called “Simulated Annealing”.

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