Monte Carlo time definition

Here Fq is the cooling rate, and t the Monte Carlo time (measured in Monte Carlo steps (MCS) see the definition below). In the simulation the cooling rate was varied over two orders of magnitude from Fq = 4 x lO" to Fq = 4 x 10 (measured in MCS ). 

The Monte Carlo method permits simulation, in a mathematical model, of stochastic variation in a real system. Many industrial problems involve variables which are not fixed in value, but which tend to fluctuate according to a definite pattern. For example, the demand for a given product may be fairly stable over a long time period, but vary considerably about its mean value on a day-to-day basis. Sometimes this variation is an essential element of the problem and cannot be ignored. 

Despite its bad reputation as an analytical tool, XRF is potentially a traceable method according to the CCQM definition and could be a primary method although it was not selected as such, and won t be for a long time. In fact, it is the only microanalytical method which can at present be considered as a candidate for accurate microscopic elemental analysis. Proof of this statement follows from Monte Carlo calculations in which experimental XRF spectra can be accurately modelled starting from first principles [23], This is not an easy approach but with computing power now available it is feasible, though not worth the effort for bulk chemical analysis where other alternatives are available. 

The usual approach to dynamic Monte Carlo simulations is not based on the master equation, but starts with the definition of some algorithm. This generally starts, not with the computation of a time, but with a selection of a site and a reaction that is to occur at that site. We will show here that this can be extended to a method that also leads to a solution of the master equation, which we call the random-selection method (RSM). [31]... 

A longstanding ambition has been to obtain dynamical information from Monte Carlo simulations. Of coinse, formally MC has no inherent timescale and is a, albeit sophisticated, phase space sampling procedure. Huitma and van Eerden have looked at this issue again. They ascribed the following definition of the physical time per MC step,... 

Here we move definitively out of the realm of systematic classification and into the realm of ingenuity. The possibilities for non-local moves are almost endless, but it is very difficult to find one which is useful in a Monte Carlo algorithin. There are two reasons for this Firstly, since a non-local move is very radical, the proposed new walk usually violates the self-avoidance constraint. (If you move a large number of beads around, it becomes very likely that somewhere along the walk two beads will collide.) It is therefore a nontrivial problem to invent a non-local move whose acceptance probability does not go to zero too rapidly as Af —> oo. Secondly, a non-local move usually costs a CPU time of order N (or in any case W with p > 0), in contrast to order 1 for a local or bilocal move. It is nontrivial to find moves whose effects justify this expenditure (by reducing more than they increase Tcpu)-... 

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