Brownian Asymmetric Simple Exclusion Process

We study the driven Brownian motion of hard rods in a one-dimensional cosine potential with a large amplitude compared to the thermal energy. In a closed system, we find surprising features of the steady-state current in dependence of the particle density. The form of the current-density relation changes greatly with the particle size and can exhibit both a local maximum and minimum. The changes are caused by an interplay of a barrier reduction, blocking, and exchange symmetry effect. The latter leads to a current equal to that of noninteracting particles for a particle size commensurate with the period length of the cosine potential. For an open system coupled to particle reservoirs, we predict five different phases of nonequilibrium steady states to occur. Our results show that the particle size can be of crucial importance for nonequilibrium phase transitions in driven systems. Possible experiments for demonstrating our findings are pointed out.

Figure 1

Driven Brownian motion of interacting particles of size $\sigma$ in a cosine potential with barrier height $U_0$ and period length $\lambda$ under drag force $f$.

Figure 2

(a) Current-density relations for various fixed particle sizes $\sigma$. The solid and dashed black lines mark the currents $j_0(\rho )$ and $j_{\mathrm{ASEP}}(\rho )$ for noninteracting particles and the corresponding ASEP, respectively. (b) Particle size dependence of the current change $\mathrm{\Delta }j(\rho ,\sigma )=[j(\rho ,\sigma )-j_0(\rho )]/j_0(\rho )$ due to hard-core interactions for different fixed densities $\rho$. (Inset) The curve ${\sigma }_{\times }(\rho )$, which separates the region of current enhancement (blue area) and reduction (red area) and the dependence of ${\rho }_{\sim }$, ${\rho }_{\max }$, and ${\rho }_{\min }$ on $\sigma$.

Figure 3

Particle trajectories in the BASEP for $\rho =0.75$, and (a) $\sigma =0.25$ and (b) $\sigma =0.75$. The horizontal dotted lines indicate the positions of the potential minima. (a) A cascadelike propagation of double occupancies marked by the circles is demonstrated by the arrows. (b) Potential wells are vacated and filled in a sequential process.

Figure 4

Phase diagrams of the open BASEP for particle sizes (a) $\sigma =0.58$, (b) $\sigma =0.62$, (c) $\sigma =0.75$, and (d) $\sigma =0.98$. Dashed and solid lines indicate phase transitions of first and second order, respectively. The regions labeled I–V in (a) mark two left-boundary induced phases (I and V), a right-boundary induced phase (III), a maximal current phase (II), and a minimal current phase (IV). These phases are equally colored in all graphs.