Design Of The Lattice Monte Carlo Simulation Algorithm And Its Accuracy Analysis For Biased Diffusion Model

Biased diffusion are frequently encountered in numerous physical,chemical and biological systems.Lattice Monte Carlo(LMC)simulation is an important and effective method to solve such kinds of problems when it is difficult to get analytical solutions or necessary to track the trajectory of particles.In principle,a LMC simulation will arbitrarily approach the solution of continuous systems if the algorithm is correctly designed and the lattice step is small enough.The Fokker-Planck equation is one way to depict diffusion problems.As far as the equation is concerned,the LMC algorithm with three transition probabilities has already been devised although it cannot be extended to the more general three dimensional case.Moreover,it is already proved that the traditional fixed time step LMC algorithm can reproduce both the mean and the mean square displacement correctly in the long time limit.However,it is found that the traditional algorithm always produces a non-zero skewness when there is a drift.The error introduced in the skewness converges to zero very slowly in the long time limit.The limited accuracy of the traditional fixed time step LMC algorithm is potentially related with the three transition probabilities because a particle can only jump to its adjacent lattices or stay put each time.Obviously,this does not agree with the nature of biased diffusion.To further improve the accuracy of the traditional algorithm,it is suggested to devise a new fixed time step LMC algorithm with five transition probabilities to exactly reproduce the first three moments of the distribution.As for the new fixed time step LMC algorithm,its probabilities are given analytically according to the theory of random walks on lattices,statistics and first passage time.The valid scope of the lattice step and the time step can be determined numerically once the drift and the diffusivity are given.Finally,the accuracy of the new LMC algorithm is verified by numerical calculations of some typical biased diffusion models.It is proved that the numerical results can indeed reproduce the first three moments exactly.It is partially proved that the accuracy of the LMC algorithm can be improved if more transition probabilities are used with the cost of more numerical calculations.Further,it is found that the accuracy characteristics of the new algorithm are quite different from the traditional one.A particular lattice step can be found to make the first step error and the maximum error equal.The maximum error is V-shaped with the increase of the valid time step and an optimal time step can be found so that the maximum error reaches its minimum.