Variable time step Monte Carlo

For a theory of variable time step Monte Carlo algorithm for time dependent rate eonstants the reader is referred to Ref. 51... 

A variable time step dynamic Monte Carlo simulation algorithm was constructed in order to model temporal surface reactivity and kinetics The algorithm explicitly accounts for all... 

The whole solution time for the non-linear solution (i.e. 1.04 s) was divided into 370 steps of variable length. The Full Newton-Raphson method was used for solving this non-linear problem. Solution of 500 Monte Carlo simulations (calculated simultaneously on four different parallel computers with 26 CPU)... 

Key equations needed in Monte Carlo simulations include the interevent time Atg representing the time spent from q—l to q Monte Carlo steps, coalescence kernel Kij, normalized probability ptj for a successful collision between particles i and j, and a number of intermediate variables. The coalescence kernel can be divided into particle property independent part Kc and dependent part ky (X X ) as follows ... 

The software allows the user to customize the Monte Carlo-DSTE method implemented in MUSTADEPT for the representation and propagation of the uncertainty associated to the imprecise information provided by the experts. In particular, the user has to set two parameters the length of the time steps partitioning the time horizon of interest and the number of Monte Carlo simulations. These variables determine the level of precision of the final results and the computational efforts required to carry out the analysis the... 

There are five important sources of error in these first diffusion Monte Carlo calculations (1) statistical or sampling error associated with the limited number of independent sample energies used in determining the energy from an average of variable potential energies, (2) the use of a finite time step At rather than an infinitesimal time step as required for the exact simulation of a differential equation, (3) numerical error associated with trimcation and/or round-off... 

In MD, time is a clearly singled out variable in a deterministic simulation based on a postulated force field and on the classical equations of motion. For the simulation of an evolving crystal aggregate, MD has the obvious advantage that the kinetics of the process is transparent, as accretion rates can be immediately described as a function of computational time, although the rate of any molecular process is obviously dependent on the postulated force model. In contrast, there is no apparent time variable in an MC simulation, because evolution steps are random and may randomly affect molecular evolutions which in reality happen on different timescales. If, as is often the case, time in MC is taken as proportional to the number of moves, one is implicitly assuming that all molecular moves occur on the same timescale, perhaps not a very severe approximation in studies of molecular aggregates bound by nearly isotropic van der Waals forces. In a variant of the MC formulation, called kinetic Monte Carlo (KMC)... 

Note For a realistic rigorous solution, the BTE must be solved for electrons, phonons, and photons. Each of these functions depends on six variables, that is, three space and three momentum (energy). The time scales of electron-phonon and phonon-phonon interactions vary by two orders of magnitude. Monte Carlo solution is one approach to solve these problems, which is time consuming. Another approach (two-step model) has been to propose hydrodynamic-type equation for modeling electron and phonon transport, which is discussed in the following section. 

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