Understanding the function and control of channel transport is of paramount importance for cell physiology and nanotechnology. In particular, if several species are involved, the mechanisms of selectivity, competition, cooperation, pumping, and its modulation need to be understood. What lacks is a rigorous mathematical approach within the framework of stochastic thermodynamics, which explains the impact of interparticle in-channel interactions on the transport properties of the respective species. To achieve this, stochastic channel transport of two species is considered in a model, which different from mean field approaches, explicitly conserves the spatial correlation of the species within the channel by analysis of the stochastic dynamics within a state space, the elements of which are the channel’s spatial occupation states. The interparticle interactions determine the stochastic transitions between these states. Local flow and entropy production in this state space reveal the respective particle flows through the channel and the intensity of the Brownian ratchet like rectifying forces, which these species exert mutually on each other, together with its thermodynamic effectiveness and costs. Perfect coupling of transport of the two species is realized by an attractive empty channel and strong repulsive forces between particles of the same species. This confines the state space to a subspace with circular topology, in which the concentration gradients as thermodynamic driving forces act in series, and channel flow of both species becomes equivalent. For opposing concentration gradients, this makes the species with the stronger gradient the driving, positive entropy producing one; the other is driven and produces negative entropy. Gradients equal in magnitude make all flows vanish, and thermodynamic equilibrium occurs. A differential interparticle interaction with less repulsive forces within particles of one species but maintenance of this interaction for the other species adds a bypass path to this circular subspace. On this path, which is not involved in coupling of the two species, a leak flow of the species with less repulsive interparticle interaction emerges, which is directed parallel to its concentration gradient and, hence, produces positive entropy here. Different from the situation with perfect coupling, appropriate strong opposing concentration gradients may simultaneously parallelize the flow of their respective species, which makes each species produce positive entropy. The rectifying potential of the species with the bypass option is diminished. This implies the existence of a gradient of the other species, above which its flow and gradient are parallel for any gradient of the less coupled species. The opposite holds for the less coupled species. Its flow may always be rectified and turned anti-parallel to its gradient by a sufficiently strong opposing gradient of the other one.
The set of points with Markovian symbolic dynamics for non-uniformly hyperbolic diffeomorphisms
