Mean-field approach for diffusion of interacting particles

A nonlinear Fokker-Planck equation is obtained in the continuous limit of a one-dimensional lattice with an energy landscape of wells and barriers. Interaction is possible among particles in the same energy well. A parameter $\gamma$, related to the barrier's heights, is introduced. Its value is determinant for the functional dependence of the mobility and diffusion coefficient on particle concentration, but has no influence on the equilibrium solution. A relation between the mean-field potential and the microscopic interaction energy is derived. The results are illustrated with classical particles with interactions that reproduce fermion and boson statistics.

Figure 1

Energy landscape of wells $E_i$ and barriers $B_{i+1/2}$. From top to bottom $\gamma =0$, 1/2, and 1 $\forall i$. In this illustration, $C$ is constant and $U_i=0\forall i$.

Figure 2

Equilibrium concentration $n$ for classical fermions with $U\left(x\right)=-Fx, \beta Fa=0.05$, and total concentration $0.5$. The curve corresponds to the Fermi-Dirac distribution and the dots to Monte Carlo simulations with different values of $\gamma$: $\gamma =0$ (circles), $\gamma =1/2$ (squares), and $\gamma =1$ (diamonds). The same distribution is attained for all values of $\gamma$. Simulations were made with multiple occupancy of sites, $n_i=m_i/N$. The following are the other parameters: number of samples, 1000; MC steps for each sample, ${10}^7$ ($\gamma =0,1$ and $N=100$) or ${10}^8$ ($\gamma =1/2$ and $N=500$); fixed boundary conditions; and system size, $L=100a$.

Figure 3

Fermion's mobility $\mu$ against concentration $n$ for different values of $\gamma$, obtained from a system with homogeneous concentration, constant force $\beta Fa=0.05$, and periodic boundary conditions (see the text for more details). The curves correspond to Eq. (15). Dots correspond to Monte Carlo simulations with $\gamma =0$ (circles), $\gamma =1/2$ (squares), and $\gamma =1$ (diamonds). The following are the other parameters: number of samples, 1000; MC steps, between ${10}^5$ and ${10}^6$; $N=100$; and system size, $L=100a$.

Figure 4

Fermion's diffusion coefficient $D$ against concentration $n$ for different values of $\gamma$, obtained from a system with unequal fixed conditions at the ends and zero force, in a nonequilibrium stationary state (constant current $J$; see the text for more details). The curves correspond to Eq. (16). Dots correspond to Monte Carlo simulations with $\gamma =0$ (circles), $\gamma =1/2$ (squares), and $\gamma =1$ (diamonds). The following are the other parameters: number of samples, 1000; MC steps, between ${10}^8$ and ${10}^9$; $N=500$; system size, $L=100a$; $n\left(0\right)=0.8$; and $n\left(L\right)=0$.

Figure 5

Equilibrium concentration $n$ for bosons with energy $U\left(x\right)=-Fx, \beta Fa=0.03$, and total concentration $0.5$. The curve corresponds to the Bose-Einstein distribution and the dots to Monte Carlo simulations with different values of $\gamma$: $\gamma =0$ (circles), $\gamma =1/2$ (squares), and $\gamma =1$ (diamonds). In order to reduce simulation fluctuations, we define the concentration $n_i$ at site $i$ as $n_i=m_i/N$, where $m_i$ is the number of particles in this site. The following are the other parameters: number of samples, 1000; MC steps for each sample, ${10}^7$; $N=10$; fixed boundary conditions; and system size, $L=100a$.

Figure 6

Boson's mobility $\mu$ against concentration $n$ for different values of $\gamma$, obtained from a system with homogeneous concentration, constant force $\beta Fa=0.05$, and periodic boundary conditions. The curves correspond to Eq. (15). Dots correspond to Monte Carlo simulations with $\gamma =0$ (circles), $\gamma =1/2$ (squares), and $\gamma =1$ (diamonds). The following are the other parameters: number of samples, 5000; MC steps, between ${10}^5$ and $6\times {10}^6$; $N=20$; system size, $L=100a$.

Figure 7

Boson's diffusion coefficient $D$ against concentration $n$ for different values of $\gamma$, obtained from a system with unequal fixed conditions at the ends and zero force, in a nonequilibrium stationary state. The curves correspond to Eq. (16). Dots correspond to Monte Carlo simulations with $\gamma =0$ (circles), $\gamma =1/2$ (squares), and $\gamma =1$ (diamonds). The following are the other parameters: number of samples, 5000; MC steps, between ${10}^7$ and ${10}^8$; $N=100$; system size, $L=100a$; $n\left(0\right)=3.5$, and $n\left(L\right)=0$.