Scaling limits of interacting particle systems.

*(English)*Zbl 0927.60002
Grundlehren der Mathematischen Wissenschaften. 320. Berlin: Springer. xvi, 442 p. (1999).

The subject of “interacting particle systems” started to emerge as a new field of probability theory roughly about 1970, and received its first comprehensive textbook treaty in the now classical Grundlehren volume “Interacting particles” by T. M. Liggett (1985; Zbl 0559.60078). In the mean time, one other major book, H. Spohn’s “Large scale dynamics of interacting particles” has appeared (1991; Zbl 0742.76002). The present “Scaling limits of interacting particle systems”, another eight years later, reflects on the enormous progress in the field and in particular on the theme that has become the central focus of interest, the hydrodynamic behaviour of a particle system then observed at macroscopic scales of space and time.

An interacting particle system is a stochastic model for the time evolution of a set of particles, in the context of the present book, placed on a lattice \(Z^d\), or the torus \(T^d_N=(Z/NZ)^d\), respectively. The dynamics consists on each particle making a step to some other (mostly neighboring) site at a random time, while taking into account the distribution of particles at that time: in the simple exclusion process, no steps to sites that are already occupied are allowed, while in the zero range process, the rate at which a particle tries to leave a site \(x\) depends on the number of particles occupying that same site. These two processes furnish the main examples treated in the text. The question one now wants to ask is how to describe the time evolution of the configuration on a macroscopic scale; that is, rescaling in such a way that the \(N^d\) sites of the torus \(T^d_N\) are embedded in the unit torus \(T^D=[0,1)^d\), one considers the distribution of the density of particles on that torus, and seeks to derive the equations governing its time evolution, after appropriate rescaling, in the limit as \(N\uparrow\infty\). What one finds are, surprising or not, partial differential equations known from hydrodynamics.

The precise way of formalizing such a result is exposed in Chapter 1 in the simplest possible setting, that of independent random walks. Already here the important difference between symmetric and asymmetric processes can be observed, and in particular the dependence of the appropriate time rescaling on these features is explained. The formalism of local equilibrium and its conservation are given. This chapter is a self-contained elementary introduction that is very accessible and requires no hard tools whatsoever. The following two chapters provide general background on the two types of interacting particles mainly studied here, and the appropriate weak formulation of a local equilibrium with a given density profile. In Chapter 4 the first hydrodynamic limit is derived, namely the heat equation systems for the symmetric simple exclusion process. In Chapters 5 and 6 the entropy method and the relative entropy method are introduced and applied to the zero range process. Chapter 7 returns to the entropy method and extends it to so-called non-gradient systems, which for the first time require sharp estimates on the spectral gap of the generator of the process. Chapters 8 and 9 deal with attractive processes, and a main result is the proof of conservation of local equilibrium for attractive processes. Chapter 10 takes on large deviation principles for density fields, and Chapter 11 studies the equilibrium fluctuations of the density field. The bulk of the text is supplemented by two appendices that provide the main technical tools used in the book.

Although the presentation of all the major achievements in the field of hydrodynamic limits centered around only two types of processes, a rich collection of comments and references at the end of each chapter gives a comprehensive overview of related results on other examples. Due to these notes, the book also can serve as a reference on the state of the art today. Overall the book is extremely well written and a pleasure to read. Great care is taken not only to present proofs, but to explain the underlying ideas and problems. It will be an indispensable reference for researchers in the field, and at the same time an introduction for the advanced student.

An interacting particle system is a stochastic model for the time evolution of a set of particles, in the context of the present book, placed on a lattice \(Z^d\), or the torus \(T^d_N=(Z/NZ)^d\), respectively. The dynamics consists on each particle making a step to some other (mostly neighboring) site at a random time, while taking into account the distribution of particles at that time: in the simple exclusion process, no steps to sites that are already occupied are allowed, while in the zero range process, the rate at which a particle tries to leave a site \(x\) depends on the number of particles occupying that same site. These two processes furnish the main examples treated in the text. The question one now wants to ask is how to describe the time evolution of the configuration on a macroscopic scale; that is, rescaling in such a way that the \(N^d\) sites of the torus \(T^d_N\) are embedded in the unit torus \(T^D=[0,1)^d\), one considers the distribution of the density of particles on that torus, and seeks to derive the equations governing its time evolution, after appropriate rescaling, in the limit as \(N\uparrow\infty\). What one finds are, surprising or not, partial differential equations known from hydrodynamics.

The precise way of formalizing such a result is exposed in Chapter 1 in the simplest possible setting, that of independent random walks. Already here the important difference between symmetric and asymmetric processes can be observed, and in particular the dependence of the appropriate time rescaling on these features is explained. The formalism of local equilibrium and its conservation are given. This chapter is a self-contained elementary introduction that is very accessible and requires no hard tools whatsoever. The following two chapters provide general background on the two types of interacting particles mainly studied here, and the appropriate weak formulation of a local equilibrium with a given density profile. In Chapter 4 the first hydrodynamic limit is derived, namely the heat equation systems for the symmetric simple exclusion process. In Chapters 5 and 6 the entropy method and the relative entropy method are introduced and applied to the zero range process. Chapter 7 returns to the entropy method and extends it to so-called non-gradient systems, which for the first time require sharp estimates on the spectral gap of the generator of the process. Chapters 8 and 9 deal with attractive processes, and a main result is the proof of conservation of local equilibrium for attractive processes. Chapter 10 takes on large deviation principles for density fields, and Chapter 11 studies the equilibrium fluctuations of the density field. The bulk of the text is supplemented by two appendices that provide the main technical tools used in the book.

Although the presentation of all the major achievements in the field of hydrodynamic limits centered around only two types of processes, a rich collection of comments and references at the end of each chapter gives a comprehensive overview of related results on other examples. Due to these notes, the book also can serve as a reference on the state of the art today. Overall the book is extremely well written and a pleasure to read. Great care is taken not only to present proofs, but to explain the underlying ideas and problems. It will be an indispensable reference for researchers in the field, and at the same time an introduction for the advanced student.

Reviewer: A.Bovier (Berlin)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

82-02 | Research exposition (monographs, survey articles) pertaining to statistical mechanics |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82C43 | Time-dependent percolation in statistical mechanics |