Fundamental Function for the Lattice-based Random Walk Simulation in One Dimension

Fundamental Function for the Random Walk Simulation under the External Field in One Dimension

“…[1][2][3][4][5][6][7] Recently, we reported the fundamental distribution functions for the lattice-based random walk model in one dimension for the nonreactive and the Smoluchowski boundary conditions. 8 We also obtained the discrete version of the survival probability for the Smoluchowski boundary condition, 9 which can reduce to the well-known continuum version result. 8 The previous results only consider the case where the probability of the movement of the walker is completely random or not affected by other external fields, which include gravitational, magnetic, and electric effects.…”

“…8 We also obtained the discrete version of the survival probability for the Smoluchowski boundary condition, 9 which can reduce to the well-known continuum version result. 8 The previous results only consider the case where the probability of the movement of the walker is completely random or not affected by other external fields, which include gravitational, magnetic, and electric effects. Since these external field effects on diffusion-reaction systems are ubiquitous in a broad range of chemical and biological systems, [9][10][11][12][13][14][15][16][17][18] it is of use to generalize the previous results to include the external field effects.…”

“…Note that a can be obtained by a = tanh −1 (2p − 1) from Eq. (8). Therefore, we can obtain the exact results more efficiently from Eq.…”

Monte Carlo Simulation Methods for External Field Effects on Diffusion-Influenced Reactions

“…[12][13][14][15][16][17] To study the field effects on diffusionreaction systems, one should make the simulation methods produce consistent results with the known analytical ones of diffusion-reaction equations. The relation between the magnitude of the external field and the hopping probability can be found to utilize the energy difference between lattice points as in the conventional Metropolis method.…”

Fundamental Function for the Random Walk Simulation with a Perfect Mirror in One Dimension

“…Recently, we have reported the fundamental probability distribution functions for the lattice-based random walk simulations with the nonreactive and the Smoluchowski (or absorbing) boundary conditions in one dimension. 1,2 The analytical fundamental functions are superior to the computer simulation methods that cannot circumvent the statistical noise.…”