Brownian motors arranged on nontrivial networks to achieve fast transport

A method of calculating velocity, efficiency as well as coefficient of performance of the refrigerator in terms of complex generating functions is presented for a Brownian motor that operates between two heat baths. Based on the analytic method developed by Goldhirsch and Gefen (Phys Rev A 33:2583, 1986, Phys Rev A 35:1317, 1987), first the dependence for the velocity V, the efficiency $$\eta $$ η as well as the coefficient of performance of the refrigerator $$P_{\text {ref}}$$ P ref on model parameters is explored. Not only the previously known results are reconfirmed for a single Brownian engine but also the thermodynamic feature of M Brownian ratchets that are arranged in a complex network is investigated. The results that are shown in this paper depict that a proper arrangement of the motors on the network may achieve fast transport. It is shown that at a quasistatic limit, Carnot efficiency or Carnot refrigerator is attainable as long as the heat exchange via kinetic energy is omitted. For all cases, far from the quasistatic limit, the efficiency and the coefficient of performance of the refrigerator are far lower than Carnot efficiency or Carnot refrigerator. Moreover, a way of transporting a Brownian particle to a desired location in a network with many dangling bonds and loops is presented. Graphic abstract