Renormalization group analysis for an asymmetric simple exclusion process

A perturbative renormalization group method is used to obtain steady-state density profiles of a totally asymmetric simple exclusion process with particle adsorption and evaporation. This method allows us to obtain a globally valid solution for the density profile without the asymptotic matching of bulk and boundary layer solutions. In addition, we show a nontrivial scaling of the boundary layer width with the system size close to specific phase boundaries.

Figure 1

The phase diagram of the particle nonconserving totally asymmetric simple exclusion process (TASEP-LK) with ${\omega }_a={\omega }_d$. HM and LM denote high-density-maximal-current and low-density-maximal-current phases where the density profile has a linear part [similar to high- or low-density phases; see Eq. (3)] along with a constant profile (similar to the maximal current phase). In a similar way, LMH denotes a phase where the density profile has parts similar to that of low-density (L), high-density (H), and maximal (M) current phases.

Figure 2

A representative diagram of the density profile in the low-density phase with the boundary layer appearing near $x=1$ boundary. The boundary layer is described by the $tanh$ solution that saturates to the linearly varying bulk part as $\tilde{x}\to -\infty$. The saturation of the $tanh$ solution on the other side (i.e., as $\tilde{x}\to \infty$) happens beyond the physically acceptable region.

Figure 3

A representative diagram of the density profile in the high-density phase with the boundary layer appearing near $x=0$ boundary. The boundary layer is described by a $tanh$-type solution that saturates to the linearly varying bulk part as $\tilde{x}\to \infty$.

Figure 4

A representative diagram of the density profile in the high-density phase with the boundary layer appearing near $x=0$ boundary. The boundary layer is described by a $coth$-type solution that saturates to the linearly varying bulk part as $\tilde{x}\to \infty$.