Supersymmetry and fluctuation relations for currents in closed networks

We demonstrate supersymmetry in the counting statistics of stochastic particle currents and use it to derive exact nonperturbative relations for the statistics of currents induced by arbitrarily fast time-dependent protocols.

Figure 1

A closed graph with five nodes and six links, representing a Markov chain with five states. Links represent allowed transitions with rates $k_{\mathit{ij}}^{\alpha }={\mathit{ke}}^{\beta (E_i-W_{\alpha }+f_{\mathit{ij}}^{\alpha })}$. Nodes are characterized by energies of wells $E_i$, $i=1,\dots ,5$. Links are characterized by sizes of barriers $W_{\alpha }$, by parameters $F_{\alpha }\equiv F_{\mathit{ij}}^{\alpha }=\text{ln}(k_{\mathit{ij}}^{\alpha }/k_{\mathit{ji}}^{\alpha })=f_{\mathit{ij}}^{\alpha }-f_{\mathit{ji}}^{\alpha }+E_i-E_j$ and by counting parameters ${\chi }_{\alpha }\equiv {\chi }_{\mathit{ij}}^{\alpha }$ that are used to count the number of transitions through the link $\alpha$. Here $\alpha =1,\dots ,6$.

Figure 2

Equivalence relations among counting parameters.