^{1}

^{1}

^{*}

Motivated by developing a simple model to calculate the diffusion coefficient in moderate friction region, a simplified model is proposed to deal with the diffusion of Brownian particles in a periodic potential. Where the internal noise is a Gaussian white noise, and the basic cell of the periodic potential is composed of a parabolic potential linked with a harmonic potential. When the particles cross the joint point of the potential, a time coarse-graining scheme is used to obtain a simple analytical expression of the probability distribution. The particles drift and diffuse from the first barrier to the second barrier, the passing probability over the second barrier corresponding to the escape rate becomes decrease serves as the long-jump probability. The theoretical result is confirmed by numerical simulation results. The approach can be extended to color noise case.

The diffusion of Brownian particles in a spatially periodic potential is a topic of great interest in many scientific areas of physics, chemistry, and biology [

The matrix-continued-fraction method was employed to investigate the Brownian motion in one and two-dimensional periodic potentials. The diffusion coefficient was obtained through numerical calculation of the dynamic structure factor [

Many diffusion features of Brownian particles in a periodic potential have been revealed by numerical simulations. For nonseparable and anisotropic potentials, molecular dynamics simulation found that the diffusion coefficient presents different dependence on friction in low friction regime as compared with separable potentials, which is directly related to the occurrence of long jumps [

Some analytical approaches have been developed to study the diffusion of Brownian particles in a periodic potential. By expanding the distribution function into suitable eigenfunctions, a general method was given in Ref. [

Although some analytical methods have been developed, simple and exact method is still deserve explored. Combine the physical picture of diffusion and the random walk model, a model to calculate the diffusion coefficient in the turnover region of damping is proposed in the present work. The proposed method provides a simple and exact approach to calculate the diffusion coefficient. Based on this approach, the diffusion of Brownian particles in the usual cosine periodic potential can be deal with by resort to the perturbation theory. The theoretical results for the applied periodic potential are confirmed by Langevin simulation results.

We consider a Brownian particle moving in a periodic potential with a basic cell composed of a parabolic potential barrier linked smoothly with a harmonic potential well, which is subjected to a Gaussian white noise. The equation of motion of the particle reads

x ˙ = v , v ˙ = − γ v − 1 m ∂ V ∂ x + 1 m ξ ( t ) , (1)

where m is the mass of the Brownian particle, γ is the damping coefficient, and V ( x ) is the periodic potential, its basic cell is given by

V ( x ) = { V b − 1 2 m ω b 2 x 2 , regionI ; 1 2 m ω 0 2 ( x − 2 ) 2 , regionII ; V b − 1 2 m ω b 2 ( x − 4 ) 2 , regionIII . (2)

The three parts in a basic cell of the piecewise potential are connected smoothly at x = x c 1 , x c 2 (

The probability density function in every potential barrier or potential well can be obtained exactly. However, the exact form of probability density after several step jumps is a high dimension integration due to the particle passes through the joint points with stochastic times. The problem is in essence a complex nonlinear one. To simplify the calculation, we construct a time coarse-grain model: the particle passes the joint point x c 1 with mean passage time of the corresponding potential barrier region. The initial velocity of the particle starting diffusion from the barrier top is set to the average velocity calculated by the Kramers formula. We label the first potential barrier, the first potential well, and the second potential barrier as region I, II, and III, respectively, as shown in

W 1 ( x , v , t ; x 0 , v 0 ,0 ) = N 1 exp [ − α 1 x 2 − β 1 v 2 − γ 1 x v ] , (3)

where

α 1 = 1 / 2 σ 11 − 1 ( t ) , β 1 = 1 / 2 σ 22 − 1 ( t ) , γ 1 = σ 12 − 1 ( t ) , N 1 = 1 2 π [ 4 α 1 β 1 − γ 1 2 ] 1 / 2 . (4)

The expressions of the second order moments σ i j can be found in Ref. [

W 01 ( x , v , t 1 ) = N 20 exp [ − b 1 ( v − v ¯ 1 ) 2 ] δ ( x − x c 1 ) , (5)

with b 1 = β 1 , v ¯ 1 = − γ 1 x c 1 / ( 2 β 1 ) , and N 20 = 1 / 2 β 1 / π . The mean first passage time t 1 is given by

t 1 = ∫ 0 ∞ d t ∫ 0 1 d x ∫ ∞ d v W 1 ( x , v , t ; x 0 , v 0 , 0 ) = ∫ 0 ∞ d t N 1 π 4 α 1 β 1 − γ 1 2 erf ( α 1 − γ 1 2 4 β 1 x c 1 ) (6)

The total probability for the particle appearing in region I for x > 0 has been normalized as 1. The transition probability density [

P 2 ( x , v , t ; x c 1 , v 1 , t 1 ) = N 2 exp ( − α 2 ( x − x ¯ ′ 2 ) 2 − β 2 ( v − v ¯ ′ 2 ) 2 − γ 2 ( x − x ¯ ′ 2 ) ( v − v ¯ ′ 2 ) ) . (7)

where

x ¯ ′ 2 = G 11 x c 1 + G 12 v 1 , v ¯ ′ 2 = G 21 x c 1 + G 22 v 1 , α 2 = 1 / 2 σ 11 − 1 ( t ) , β 2 = 1 / 2 σ 22 − 1 ( t ) , γ 2 = σ 12 − 1 ( t ) , N 2 = 1 2 π [ 4 α 2 β 2 − γ 2 2 ] 1 / 2 , (8)

all of these quantities are defined in region II. The probability density in region II is then obtained

W 2 ( x , v , t ) = ∫ − ∞ ∞ d v 1 ∫ − ∞ ∞ d x 1 W 01 ( x 1 , v 1 , t 1 ) P ( x , v , t ; x 1 , v 1 , t 1 ) = N 0 N 2 π D 2 exp ( − a 2 ( x − x ¯ 2 ) 2 − b 2 ( v − v ¯ 2 ) 2 − c 2 ( x − x ¯ 2 ) ( v − v ¯ 2 ) ) , (9)

where

x ¯ 2 = G 11 x ¯ 1 + G 12 v ¯ 1 , v ¯ 2 = G 21 x ¯ 1 + G 22 v ¯ 1 , D 2 = b 1 + α 2 G 12 2 + β 2 G 22 2 + γ 2 G 12 G 22 , a 2 = 4 b 1 α 2 + ( 4 α 2 β 2 − γ 2 2 ) G 22 2 ( t 2 − t 1 ) 4 D 2 , b 2 = 4 b 1 β 2 + ( 4 α 2 β 2 − γ 2 2 ) G 12 2 ( t 2 − t 1 ) 4 D 2 , c 2 = 4 b 1 γ 2 − 2 ( 4 α 2 β 2 − γ 2 2 ) G 12 ( t 2 − t 1 ) G 22 ( t 2 − t 1 ) 4 D 2 . (10)

The transition probability density in region III is given by

P 3 ( x , v , t ; x c 2 , v 2 , t 2 ) = N 3 exp ( − α 3 ( x − x ¯ ′ 3 ) 2 − β 3 ( v − v ¯ ′ 3 ) 2 − γ 3 ( x − x ¯ ′ 3 ) ( v − v ¯ ′ 3 ) ) . (11)

Where

x ¯ ′ 3 = G 11 x c 2 + G 12 v 2 , v ¯ ′ 3 = G 21 x c 2 + G 22 v 2 , α 2 = 1 / 2 σ 11 − 1 ( t ) , β 2 = 1 / 2 σ 22 − 1 ( t ) , γ 2 = σ 12 − 1 ( t ) , N 2 = 1 / ( 2 π det ( σ ) ) , (12)

All these quantities are defined in region III. The probability density in region III is expressed as

W 3 ( x , v , t ) = ∫ t 1 t d t 2 ∫ − ∞ ∞ d v 2 v 2 W 2 ( x c 2 , v 2 , t 2 ) P 3 ( x , v , t ; x c 2 , v 2 , t 2 ) . (13)

The escape probability at time t crossing over the second barrier top is

P e ( t ) = ∫ 0 ∞ d x ∫ − ∞ ∞ d v W 3 ( x , v , t ) , (14)

Performing the Gaussian integrations over x and v, the escape probability can be expressed as

P e ( t ) = ∫ t 1 t d t 2 ∫ − ∞ ∞ d v 2 v 2 1 2 π D 2 N 0 N 2 erfc [ − α 3 − γ 3 2 4 β 3 x ¯ ′ 3 ] × exp [ − a 2 ( x c 2 − x ¯ 2 ) 2 − b 2 ( v 2 − v ¯ 2 ) 2 − c 2 ( x c 2 − x ¯ 2 ) ( v − v ¯ 2 ) ] . (15)

In random walk model, the diffusion coefficient is expressed as [

D = k 〈 l 2 〉 = k ∑ n ( n d ) 2 P n , (16)

where d is the spatial periodic of the potential, P n is the probability of n-step jumps, and k is the Kramers rate calculated by the Kramers formula in spatial diffusion regime, which is still valid for damping out of spatial diffusion regime due to the parabolic potential barrier. When the center of the probability packet moves toward to the second barrier, the passing probability over the barrier top

increases rapidly at almost a constant rate k e = d P e d t , as shown by the numerical

results. When the center of the probability packet moves back to the potential well, the passing probability over the second potential barrier only increases due to diffusion and then becomes slow. We take the critical probability P c that the increase of P e ( t ) as a function of time from rapid to slow as the long jump probability (more than one step). Thereafter the process in the first basic cell is repeated periodically, which is a part of our simplified model. The probabilities for n-step jumps is then given by

P n = P c n − 1 ( 1 − P c ) ( n = 1 , 2 , ⋯ ) (17)

such a geometric progression jump probability distribution is a good approximation for several step jumps, as shown by simulation results (see Fig. 6 of Ref. [

To check the accuracy of the diffusion coefficient obtained by the simplified model,

we simulate the Langevin Equations (1) by second order Runge-Kutta algorithm. The number of test particles and the time step are taken as N = 3 × 10 5 and Δ t = 2 × 10 − 3 , respectively. As shown in

A simple model is proposed to calculate the diffusion coefficient for Brownian particles moving in a periodic potential. The basic cell of the periodic potential is composed of a parabolic potential barrier linked with a harmonic potential well smoothly, which can serve as a zero-order approximation of a cosine periodic potential. Further theoretical results for the common cosine periodic potential can be obtained by perturbation theory. The theoretical result for the applied potential is confirmed by the simulation result in moderate friction region, which is an often encountered region in surface diffusion problem. The proposed approach can be generalized conveniently to color noise case.

The authors declare no conflicts of interest regarding the publication of this paper.

Zhang, C.H. and Bai, Z.W. (2021) A Simple Model for the Calculation of Diffusion Coefficient in a Periodic Potential. Journal of Modern Physics, 12, 91-98. https://doi.org/10.4236/jmp.2021.122009