interval $\Delta t$ of the TASEP/LK can be written in terms of transition rules:

For $1<j<L$: $$\begin{array}{lll}
\mbox{Hopping}: & 10\to 01 & \quad\mbox{with probability } p_j\Delta t \\
\mbox{Attachment}: & 0\to 1 & \quad\mbox{with probability }\omega_a \Delta t\\
\mbox{Detachment}: & 1\to 0 & \quad\mbox{with probability } \omega_d \Delta t
\end{array}$$

for $j=1$: $$\begin{array}{lll}
\mbox{Hopping}: & 10\to 01 & \quad\mbox{with probability } p_1\Delta t \\
\mbox{Entry}: & 0\to 1 & \quad\mbox{with probability } \alpha \Delta t \\
\mbox{Detachment}: & 1\to 0 & \quad\mbox{with probability } \omega_d \Delta t
\end{array}$$

and for $j=L$: $$\begin{array}{lll}
\mbox{Attachment}: & 0\to 1 & \quad\mbox{with probability } \omega_a\Delta t \\
\mbox{Exit}: & 1\to 0 & \quad\mbox{with probability } \beta \Delta t
\end{array}$$ Other transitions are prohibited. Here “0” represents empty and “1” occupied sites. We can write the time evolution of the density $\rho_j=\langle \tau_j\rangle$ as $$\begin{aligned}
  \frac{d\rho_j}{dt}(t) &=& p_{j-1}\langle   \tau_{j-1}(t)(1-\tau_j(t))
     \rangle-p_j\langle \tau_j(t)(1-\tau_{j+1}(t))\rangle \nonumber \\ 
  &+& \omega_a   (1-\rho_j(t))-\omega_d \rho_j(t)\end{aligned}$$ in the bulk and $$\begin{aligned}
  \frac{d\rho_1}{dt}(t) &=& -p_1\langle \tau_1(t)(1-\tau_2(t))\rangle
  \nonumber \\ 
  &+& \alpha(1-\rho_1(t))-\omega_d \rho_1(t) \\ 
\frac{d\rho_L}{dt}(t) &=& p_{L-1}\langle
  \tau_{L-1}(t)(1-\tau_L(t))\rangle \nonumber \\ 
  &-& \beta \rho_L(t)+\omega_a (1-\rho_L(t))\end{aligned}$$ at the left and right boundary, respectively. The parameters $\alpha,\beta$ correspond to the generic boundary rates defined before. The source term is $s(\rho)=\omega_a(1-\rho)-\omega_d\rho$. 