Quantization and Fractional Quantization of Currents in Periodically   Driven Stochastic Systems I: Average Currents
  This article studies Markovian stochastic motion of a particle on a graph with finite number of nodes and periodically time-dependent transition rates that satisfy the detailed balance condition at any time. We show that under general conditions, the currents in the system on average become quantized or fractionally quantized for adiabatic driving at sufficiently low temperature. We develop the quantitative theory of this quantization and interpret it in terms of topological invariants. By implementing the celebrated Kirchhoff theorem we derive a general and explicit formula for the average generated current that plays a role of an efficient tool for treating the current quantization effects. 
Vladimir Y. Chernyak, John R. Klein, Nikolai A. Sinitsyn
cond-mat.mes-hall cond-mat.soft math.AT math.ST stat.TH