Duality and fluctuation relations for statistics of currents on cyclic   graphs
  We consider stochastic motion of a particle on a cyclic graph with arbitrarily periodic time dependent kinetic rates. We demonstrate duality relations for statistics of currents in this model and in its continuous version of a diffusion in one dimension. Our duality relations are valid beyond detailed balance constraints and lead to exact expressions that relate statistics of currents induced by dual driving protocols. We also show that previously known no-pumping theorems and some of the fluctuation relations, when they are applied to cyclic graphs or to one dimensional diffusion, are special consequences of our duality. 
Jie Ren, V. Y. Chernyak, N. A. Sinitsyn
cond-mat.stat-mech physics.chem-ph