Pumping Restriction Theorem for Stochastic Networks

We formulate an exact result, which we refer to as the pumping restriction theorem (PRT). It imposes strong restrictions on the currents generated by periodic driving in a generic dissipative system with detailed balance, and provides a universal nonperturbative approach to explore the stochastic pump effect in nonadiabatically driven systems.

Figure 1

Five-state graph with one loop [2342].

Figure 2

The total flux passed through the link 2–3 by time $t$. Variable parameters are $E_1(t)=E_1+E\sin (\omega t)$, and $W_{45}(t)=W_{45}+W\cos (\omega t)$, for $t\in (0,2\pi /\omega )$, and $E_1(t)=E_1$, $W_{45}(t)=W_{45}+W$ for $t>2\pi /\omega$, and we distinguish between $E_1(t)$ and $E_1$ etc. Choice of constant parameters is $\omega =1$, $k=1$, $E=1$, $W=2$, $E_1=-1$, $E_2=0.2$, $E_3=-0.05$, $E_4=0.1$, $E_5=-0.25$, $W_{45}=0.3$, $W_{12}=0$, $W_{23}=0.2$, $W_{34}=-0.2$, $W_{24}=0.25$, $W_{ij}=W_{ji}$ for any $i\mathrm{\text{−}}j$.

Figure 3

The total flux passed through the link 2–3 by time $t$. $W_{ij}$ is varied along one of the loop links. Variable parameters, indicated on a graph legend, change with time according to $E_i(t)=E_i+E\sin (\omega t)$, and $W_{ij}(t)=W_{ij}+W\cos (\omega t)$, for $t\in (0,2\pi /\omega )$, and $E_i(t)=E_i$, $W_{ij}(t)=W_{ij}+W$ for $t>2\pi /\omega$. Constant parameters are as in Fig. 2.

Figure 4

The total flux passed through the link 2–3 by time $t$, when only $E_i$ were allowed to vary on one of the loop sites and $W_{ij}$ is varied along external links. Variable parameters, indicated on a graph legend, change with time according to $E_i(t)=E_i+E\sin (\omega t)$, and $W_{ij}(t)=W_{ij}+W\cos (\omega t)$, for $t\in (0,2\pi /\omega )$, and $E_i(t)=E_i$, $W_{ij}(t)=W_{ij}+W$ for $t>2\pi /\omega$. Constant parameters are as in Fig. 2.

Figure 5

A six state Markov chain.