Macroscopic properties of a stationary non-equilibrium distribution for a non-gradient interacting particle system
A one-dimensional generalized symmetric simple exclusion process is considered. At most two particles per site are permitted for this process. The state space of the process is \(\{0,1,2\}^{T_ N}\), \(T_ N = \{0,1,2,3, \dots\}\) for each integer \(N\). The process is defined as the Markov one generator of which is given by \[ (L_ Nf) (\eta) = \sum_{k=0}^{N-1} (L_{k, k+1} f)(\eta) + (L_ gf) (\eta) + (L_ df) (\eta). \] The elementary generators \(L_{k,k+1}\), \(L_ g\), \(L_ d\) are defined as follows \[ (L_{k,k+1}f) (\eta) = d_{k,k+1} (\eta) \bigl[ f(\eta^{k,k+1}) - f(\eta) \bigr] + g_{k,k+1} (\eta) \bigl[ f (\eta^{k+1,k}) - f(\eta) \bigr] \] with the rate functions \[ d_{k,k+1} (\eta) = \begin{cases} 1 \quad & \text{if } \quad \eta_ k \geq 1,\;\eta_{k+1} \leq 1,\\ 0 \quad & \text{otherwise}, \end{cases}\quad g_{k,k+1} (\eta)= \begin{cases} 1 & \text{if } \quad \eta_ k \leq 1,\eta_{k+1}\geq 1, \\ 0 & \text{otherwise}, \end{cases} \] \[ \eta^{k,i}_ j = \begin{cases} \eta_ j \quad & \text{if} \quad j \neq k,i, \\ \eta_{k-1} \quad & \text{if} \quad j = k,\\ \eta_{i+1} \quad & \text{ if } j=i.\end{cases} \] The generators \(L_ g\), \(L_ d\) are the creation-destruction operators taking into account the boundary conditions. Thus the system is open with the boundaries which are two infinite reservoirs of particles with different densities. The hydrodynamic limit \(N \to \infty\) is studied. It is proved that in the stationary state the nonrandom density of random field converges to the solution of a nonlinear elliptic equation. In particular, it is proved Fick's law of the transport current.
scientific article; zbMATH DE number 740324