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Particle systems and reaction-diffusion equations. (English) Zbl 0799.60093
Consider the particle system with spin space \(\{0,1\}\) on \(S=\varepsilon{\mathbb{Z}}^ d\quad (\varepsilon>0)\). At each site \(x\), the death \((t\to 0)\) rate is fixed to be 1 and the birth \((0\to 1)\) rate depends on a finite range with parameter \(\beta>0\). For every pair \(x,y\in S\), the values at \(x\) and \(y\) are exchanged at rate \(\varepsilon^{-2}/2\). It is more or less known that under some mild assumption, as \(\varepsilon\to 0\), the density of the distribution satisfies a reaction-diffusion equation. The systems studied in the paper have the point mass \(\delta_ 0\) on the \(\equiv 0\) configuration as its trivial stationary distribution. Moreover, the systems are assumed to be attractive, hence we have the upper stationary distribution \(\nu^ \varepsilon(\beta)\). Define \(\beta_ c(\varepsilon)=\inf\{\beta: \nu^ \varepsilon(\beta)\neq\delta_ 0\}\). The paper describes the limit of \(\beta_ c(\varepsilon)\) as \(\varepsilon\to 0\). The authors show for five examples that there exist non-trivial limits, which coincide with the critical value at which the speed of the traveling wave changes its sign. This indicates the use of the reaction-diffusion equation to describe the microscopic systems, it is the main new idea of the present paper and from the reviewer’s point of view, it is a critical progress in the field. Certainly, one expects the further development on the interaction between these two subjects: the reaction-diffusion processes and the reaction-diffusion equations. At the moment, we do not know what will happen if we remove the absorbing condition. Does there still exist phase transitions? To study this problem, one may need to consider the limits from different region of \(\beta\), not only from the above. For the background of the study and for further information, it may be helpful to refer to the reviewer’s book “From Markov chains to non-equilibrium particle systems” (1992; Zbl 0753.60055). Part IV of the book is devoted to the subject.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
35K35 Initial-boundary value problems for higher-order parabolic equations
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