Entropy analysis of a nearest-neighbor attractive/repulsive exclusion process on one-dimensional lattices. (English) Zbl 0705.60093

Yaguchi, Hirotake

Ann. Probab. 18, No. 2, 556-580 (1990).

The author studies a class of nearest neighbour exclusion processes with state space \(X=\{0,1\}^ Z\). The particles move on Z according to the following rules. Consider four subsequent sites x-1, x, \(x+1\) and \(x+2\). Assume that there is a particle at x and that \(x+1\) is vacant. Then the particle at x jumps to \(x+1\) with rate \(\alpha\) (\(\beta\)) if x-1 is occupied (vacant) (similar remarks apply to the situation where \(x+1\) is occupied and x vacant). Here, \(\alpha\) and \(\beta\) are positive real numbers. If \(\alpha <\beta\) \((\alpha >\beta)\), then the exclusion process is called attractive (repulsive). On the other hand, if \(\alpha =\beta\), the process is the well-known simple exclusion process.
Let \(\Omega\) be the generator of the above process. The author investigates the structure of stationary measures by employing the method of relative entropy which was used e.g. by R. Holley [Commun. Math. Phys. 23, 87-99 (1971; Zbl 0241.60096)] in order to show that a probability measure on X is stationary for Markov processes with generator \({\bar \Omega}\) iff it has the regular clustering property (RCP) with index \(\beta\) /\(\alpha\). The RCP which is introduced in the present paper is a kind of generalization of the exchangeability property of measures. The author then shows that the extremal points of the set of measures having the RCP with a given index are renewal measures.

Reviewer: K.Schürger

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60G50	Sums of independent random variables; random walks
60J25	Continuous-time Markov processes on general state spaces

Keywords:

nearest neighbour exclusion processes; simple exclusion process; stationary measures; regular clustering property; exchangeability property of measures

Citations:

Zbl 0241.60096