coulomb excitation data analysis codes; gosia 2007 - Physics and ...

configurations with a probability e −E/kT (Where E is the energy of the configuration, T is the temperatureand k is the Boltzmann constant). These configurations are used in computing statistical properties ofthe system by averaging the value of such properties in the space of these configurations, and weightingthe configurations evenly. In order to accomplish this, new configurations are generated from an initialconfiguration by considering motions of each particle, and accepting steps for which ∆E 0 with probability e −E/kT . This method accurately reproduced theequation of state for a two-dimensional system (the ratio PA/nkT as a function of n, the average density ofparticles at the surface of one particle).The extension of this method to minimization procedures is obvious: since the Boltzmann-like energyweighting reproduces physical thermodynamic properties, it is theoretically possible, by slowly lowering thetemperature of the simulated numeric system, to find the global minimum of the system, analogously tothe process of annealing of crystals. In the following discussion, the method of Simulated Annealing willbe described for problems of one dependent variable only. The adaptation of this method to multi-variableminimization problems will be discussed later.The Simulated Annealing method in its most general terms is described as follows:Set starting value of temperature TLOOP For some period of time:generate a step ∆xcalculate change in cost function ∆C(x + ∆x)Calculate probability of accepting a step with ∆C(x + ∆x): G(∆C, T)accept step (x = x + ∆x) orreject(x unchanged) based on the value of G(∆C, T)After sufficient time/loop executions:Reduce temperature TReturn to LOOP section aboveFollowing the thermodynamic analogy strictly, the probability G(∆C, T) is the Boltzmann factor G =e −∆C/T whereTisnowanon-physicalparameterinthesameunitsasthecostC. Note that for ∆C