Simulating system dynamics with arbitrary time step

Abstract

We suggest a dynamic simulation method that allows efficient and realistic modeling of kinetic processes, such as atomic diffusion, in which time has its actual meaning. Our method is similar in spirit to widely used kinetic Monte Carlo (KMC) techniques; however, in our approach, the time step can be chosen arbitrarily. This has an advantage in some cases, e.g., when the transition rates change with time sufficiently fast over the period of the KMC time step (e.g., due to time dependence of some external factors influencing kinetics such as moving scanning probe microscopy tip or external time-dependent field) or when the clock time is set by some external conditions, and it is convenient to use equal time steps instead of the random choice of the KMC algorithm in order to build up probability distribution functions. We show that an arbitrary choice of the time step can be afforded by building up the complete list of events including the ``residence site'' and multihop transitions. The idea of the method is illustrated in a simple ``toy'' model of a finite one-dimensional lattice of potential wells with unequal jump rates to either side, which can be studied analytically. We show that for any choice of the time step, our general kinetics method reproduces exactly the solution of the corresponding master equations for any choice of the time steps. The final kinetics also matches the standard KMC, and this allows better understanding of this algorithm, in which the time step is chosen in a certain way and the system always advances by a single hop.