Transport process and local thermal reservoirs

A random walk of $N$ particles on a lattice with $M$ sites is studied under the constraint that each lattice site is coupled to its own mesoscopic heat bath. Such a situation can be conveniently described by using the master equation in a quantized Hamiltonian formulation where the exclusion principle is included by using Pauli operators. If all reservoirs are mutually in contact, giving rise to a temperature gradient, an evolution equation for the particle density with two different currents already results in the mean-field approximation. One is the conventional diffusive current, driven by the density gradient, whereas the other includes a coupling between the local density and the temperature gradient. Due to the competitive currents, the system exhibits a stationary solution, where the local density is determined by the local temperature field and depends on the filling factor $M∕N$. The stability of the solution is related to the eigenvalues of a Schrödinger-like equation. In the case of a fixed temperature gradient the stationary density distribution remains stable. The approach used is totally different from and an alternative to the conventional Onsager ansatz.

Figure 1

Reduced particle density $n_s∕\rho$ as function of $x∕L$ according to Eq. (18) for different temperature gradients $\mu \mathrm{\Delta }T=0$, 0.1, 0.2, 0.5, 1.0, 2.0, 5.0, 10. The arrow points in the direction of increasing temperature gradients.

Figure 2

Reduced particle density $n_s∕\rho$ as function of $x∕L$ according to Eq. (18) for a fixed temperature gradient $\mu \mathrm{\Delta }T=10$, but different filling factors $M∕N=0.01$, 0.05, 0.2, 0.5, 0.8, 0.95, 0.99 in the direction of the arrow.