Introduction

One-dimensional (1D) kinetic Ising models are arguably the simplest stochastic systems that display collective behavior. Their simplicity permits detailed calculations of dynamical behavior both at and away from equilibrium, and they are therefore ideal testbeds for theories and approximation schemes that may be applied to more complex systems. Moreover, they are useful as models of relaxation in real 1D systems, such as biopolymers.

This chapter reviews the behavior of 1D kinetic Ising models at low, but not necessarily constant, temperatures. We shall concentrate on systems whose steady states correspond to thermodynamic equilibrium, and in particular on Glauber and Kawasaki dynamics. The case of nonequilibrium competition between these two kind of dynamics is covered in Ch. 4. We have also limited the discussion to the case of nearest-neighbor interactions, and zero applied magnetic field. The unifying factor in our approach is a consideration of the effect of microscopic processes on behavior at slow time scales and long length scales. It is appropriate to consider separately the cases of constant temperature, instantaneous cooling, and slow cooling, corresponding respectively to the phenomena of critical dynamics, domain growth, and freezing.

As zero temperature is approached, the phenomenon of critical dynamics (‘critical slowing-down’) is observed in 1D Ising models. In the exactly solvable cases of uniform chains with Glauber or Kawasaki dynamics, the critical properties are simply related both to the internal microscopic processes and to the conventional Van Hove theory of critical dynamics.

A challenge of modern science has been to understand complex, highly correlated systems, from many-body problems in physics to living organisms in biology. Such systems are studied by all the classical sciences, and in fact the boundaries between scientific disciplines have been disappearing; ‘interdisciplinary’ has become synonymous with ‘timely’. Many general theoretical advances have been made, for instance the renormalization group theory of correlated many-body systems. However, in complex situations the value of analytical results obtained for simple, usually one-dimensional (1D) or effectively infinite-dimensional (mean-field), models has grown in importance. Indeed, exact and analytical calculations deepen understanding, provide a guide to the general behavior, and can be used to test the accuracy of numerical procedures.

A generation of physicists have enjoyed the book Mathematical Physics in One Dimension …, edited by Lieb and Mattis, which has recently been re-edited. But what about mathematical chemistry or mathematical biology in 1D? Since statistical mechanics plays a key role in complex, many-body systems, it is natural to use it to define topical coverage spanning diverse disciplines. Of course, there is already literature devoted to 1D models in selected fields, for instance, or to analytically tractable models in statistical mechanics, e.g.,. However, in recent years there has been a tremendous surge of research activity in 1D reactions, dynamics, diffusion, and adsorption. These developments are reviewed in this book.

There are several reasons for the flourishing of studies of 1D many-body systems with stochastic time evolution.

We present some rigorous and computer-simulation results for a simple microscopic model, the asymmetric simple exclusion process, as it relates to the structure of shocks.

Introduction

In this chapter our concern is the underlying microscopic structure of hydrodynamic fields, such as the density, velocity and temperature of a fluid, that are evolving according to some deterministic autonomous equations, e.g., the Euler or Navier-Stokes equations. When the macroscopic fields described by these generally nonlinear equations are smooth we can assume that on the microscopic level the system is essentially in local thermodynamic equilibrium. What is less clear, however, and is of particular interest, both theoretical and practical, is the case where the evolution is not smooth—as in the occurrence of shocks. Looked at from the point of view of the hydrodynamical equations these correspond to mathematical singularities—at least at the compressible Euler level—possibly smoothed out a bit by the viscosity, at the Navier-Stokes level. But what about the microscopic structure of these shocks? Is there really a discontinuity, or at least a dramatic change in the density, at the microscopic scale or does it look smooth at that scale?

It is clear that this question cannot be answered by the macroscopic equations.

The experimental verification of models for one-dimensional (1D) reaction kinetics requires well-defined systems obeying pure 1D behavior. There is a number of such systems that can be interpreted in terms of 1D reaction kinetics. Many of them are based on the diffusive or coherent motion of excitons along well-defined chains or channels in the material. In this chapter they will be briefly reviewed.

We also present results of an experimental study on the reaction kinetics of a 1D diffusion-reaction system, on a picosecond-to-millisecond time scale. Tetramethylammonium manganese trichloride (TMMC) is a perfect model system for the study of this problem. The time-resolved luminescence of TMMC has been measured over nine decades in time. The nonexponential shape of the luminescence decay curves depends strongly on the exciting laser's power. This is shown to result from a fusion reaction (A+A → A) between photogenerated excitons, which for initial exciton densities ≲ 2xl0-4 as a fraction of the number of sites is very well described by the diffusionlimited single-species fusion model. At higher initial exciton densities the diffusion process, and thus the reaction rate, is significantly influenced by the heat produced in the fusion reaction. This is supported by Monte Carlo simulations.

Introduction

Reactions between (quasi-)particles in low-dimensional systems is an important topic in such diverse fields as heterogeneous catalysis, solid state physics, and biochemistry.

The kinetics of the diffusion-limited coalescence process, A + A → A, can be solved exactly in several ways. In this chapter we focus on the particular technique of interparticle distribution functions (IPDFs), which enables the exact solution of some nontrivial generalizations of the basic coalescence process. These models display unexpectedly rich kinetic behavior, including instances of anomalous kinetics, self-ordering, and a dynamic phase transition. They also reveal interesting finite-size effects and shed light on the combined effects of internal and external fluctuations. An approximation based on the IPDF method is employed for analysis of the crossover between the reaction-controlled and diffusion-controlled regimes in coalescence when the reaction rate is finite.

Introduction

Reaction-diffusion systems are those in which the reactants are transported by diffusion. Two fundamental time scales characterize these systems: (a) the diffusion time—the typical time between collisions of reacting particles, and (b) the reaction time—the time that particles take to react when in proximity. When the reaction time is much larger than the diffusion time, the process is reaction-limited. In this case the law of mass action holds and the kinetics is well described by classical rate equations. In recent years there has been a surge of interest in the less tractable case of diffusion-limited processes, where the reaction time may be neglected.

Editor's note

The next three chapters cover the topics of monolayer adsorption and, to a limited extent, multilayer adsorption, in those systems where finite particle dimensions provide the main interparticle interaction mechanism. Furthermore, the particles are larger than the unit cells of the underlying lattice (for lattice models). As a result, deposition without relaxation leads to interesting random jammed states where small vacant areas can no longer be covered. This basic process of random sequential adsorption, and its generalizations, are described in Ch. 10.

Added relaxation processes lead to the formation of denser deposits, yielding ordered states (full coverage in 1D). Chapter 11 is devoted to diffusional relaxation. The detachment of recombined particles is another relaxation mechanism, reviewed in Ch. 12. The detachment of originally deposited particles, although modeling an important experimental process, has been studied much less extensively.

While several exact results are available, as well as extensive Monte Carlo studies, it is interesting to note that many theoretical advances in deposition models with added relaxation have been derived by exploring relations to other 1D systems. These range from Heisenberg spin models to reactiondiffusion systems (Part I). However, most of these relations are limited to 1D.

Besides their theoretical interest, 1D deposition models find applications in characterizing certain reactions on polymer chains, in modeling traffic flow, and in describing the attachment of small molecules on DNA. The latter application is described in Ch. 22.

The dynamics of a grand-canonical ensemble of hard-core particles in a onedimensional (1D) random environment is considered. Two types of randomness are studied: static and dynamic. The equivalence of a grand-canonical ensemble of hard-core particles and a system of noninteracting fermions is used to evaluate the average number of particles per site and the density of creation and annihilation processes. Exact solutions are obtained for Cauchy distributions of the random environment. It is shown that a new physical state is spontaneously created by dynamic randomness.

Introduction

The Brownian motion of a particle in a realistic system may be affected by fluctuations of the environment. One can distinguish these fluctuations according to their time scales. There are fluctuations with time scales large compared to the Brownian motion of the particle. Those are usually considered as impurities and can be described by static randomness. On the other hand, there are also dynamic stochastic processes that occur on time scales equal to or even shorter than the time scale of the Brownian particle. They can be described by dynamic randomness. Mainly for technical reasons it will be assumed that both types of randomness are statistically independent with respect to space and, for the dynamic randomness, also with respect to time.

The purpose of this chapter is to discuss methods for analysis of the dynamics of a 1D ensemble of hard-core particles in a static or dynamic random environment.

Basic features of the kinetics of diffusion-controlled two-species annihilation, A + B → 0, as well as that of single-species annihilation, A + A→ 0, and coalescence, A + A → A, under diffusion-controlled and ballistically controlled conditions, are reviewed in this chapter. For two-species annihilation, the basic mechanism that leads to the formation of a coarsening mosaic of A- and B-domains is described. Implications for the distribution of reactants are also discussed. For single-species annihilation, intriguing phenomena arise for ‘heterogeneous’ systems, where the mobilities (in the diffusion-controlled case) or the velocities (in the ballistically controlled case) of each ‘species’ are drawn from a distribution. For such systems, the concentrations of the different ‘species’ decay with time at different power-law rates. Scaling approaches account for many aspects of the kinetics. New phenomena associated with discrete initial velocity distributions and with mixed ballistic and diffusive reactant motion are discussed. A scaling approach is outlined to describe the kinetics of a ballistic coalescence process which models traffic on a single-lane road with no passing allowed.

Introduction

There are a number of interesting kinetic and geometric features associated with diffusion-controlled two-species annihilation, A + B → 0, and with single-species reactions, A + A → 0 and A + A → A, under diffusion-controlled and ballistically controlled conditions.

In two-species annihilation, there is a spontaneous symmetry breaking in which large-scale single-species heterogeneities form when the initial concentrations of the two species are equal and spatially uniform.

Editor's note

The first three chapters of the book cover topics in reactions and catalysis. Chemical reactions comprise a vast field of study. The recent interest in models in low dimension has been due to the importance of two-dimensional surface geometry, appropriate, for instance, in heterogeneous catalysis. In addition, several experimental systems realize 1D reactions (Part VII).

The classical theory of chemical reactions, based on rate equations and, for nonuniform densities, diffusion-like differential equations, frequently breaks down in low dimension. Recent advances have included the elucidation of this effect in terms of fluctuation-dominated dynamics. Numerous models have been developed and modern methods in the theory of critical phenomena applied. The techniques employed range from exact solutions to renormalization-group, numerical, and scaling methods.

Models of reactions in 1D are also interrelated with many other 1D systems ranging from kinetic Ising models (Part II) and deposition (Part IV) to nucleation (Part III). Chapter 1 reviews the scaling theory of basic reactions and summarizes numerous results. One of the methods of obtaining exact solutions in 1D, the interparticle-distribution approach, is reviewed in Ch. 2. Other methods for deriving exact results in 1D are not considered in this Part. Instead, closely related systems and solution techniques based on kinetic Ising models and cellular automata are presented in Chs. 4, 6, 8. Coagulation models in Ch. 9 employ methods that have also been applied to reactions.

More complicated models of catalysis, directed percolation, and kinetic phase transitions, are treated in Ch. 3.

Exact solutions for the phase-ordering dynamics of three one-dimensional models are reviewed in this chapter. These are the lattice Ising model with Glauber dynamics, a nonconserved scalar field governed by time-dependent Ginzburg-Landau (TDGL) dynamics, and a nonconserved 0(2) model (or XY model) with TDGL dynamics. The first two models satisfy conventional dynamic scaling. The scaling functions are derived, together with the (in general nontrivial) exponent describing the decay of autocorrelations. The 0(2) model has an unconventional scaling behavior associated with the existence of two characteristic length scales—the ‘phase coherence length’ and the ‘phase winding length’.

Introduction

The theory of phase-ordering dynamics, or ‘domain coarsening’, following a temperature quench from a homogeneous phase to a two-phase region has a history going back more than three decades to the pioneering work of Lifshitz, Lifshitz and Slyozov, and Wagner. The current status of the field has been recently reviewed.

The simplest scenario can be illustrated using the ferromagnetic Ising model. Consider a temperature quench, at time t = 0, from an initial temperature TI, which is above the critical temperature TC to a final temperature TF, which is below TC-At TF there are two equilibrium phases, with magnetization ±M0. Immediately after the quench, however, the system is in an unstable disordered state corresponding to equilibrium at temperature TI. The theory of phase-ordering kinetics is concerned with the dynamical evolution of the system from the initial disordered state to the final equilibrium state.

Editor's note

Nucleation, phase separation, cluster growth and coarsening, ordering, and spinodal decomposition are interrelated topics of great practical importance. While most experimental realizations of these phenomena are in three (bulk) and two (surface) dimensions, there has been much interest in lattice and continuum (off-lattice) 1D stochastic dynamical systems modeling these irreversible processes.

The main applications of 1D models have been in testing various scaling theories such as cluster-size-distribution scaling and scaling forms of orderparameter correlation functions. Exact solutions are particularly useful in this regard, and the focus of all three chapters in this Part is on exactly solvable models. Additional literary sources are cited in the chapters, including general review- articles as well as other studies in 1D.

Chapter 7 reviews exact solutions of three different models of phaseordering dynamics, including results based on the Glauber-Ising model introduced in Part II. Chapter 8 review's a model with synchronous (cellularautomaton) dynamics and relations to reactions (Part I). In both chapters exact results for scaling of the two-point correlation function are obtained. Finally, Ch. 9 describes models of coagulating particles and associated results for cluster-size-distribution scaling.

The aim of this chapter is to summarize briefly recent results on directed walks and provide a guide to the literature. We shall restrict consideration to the equilibrium properties of directed interfaces and polymers, focusing particularly on their collapse and binding transitions. The walks will lie in a nonrandom environment.

Directed walks and polymers

A clear introduction to the physics of directed walks is given by Privman and Švrakić in a book published in 1989. This summarizes the work up to that time and therefore here we shall aim to describe more recent progress after a brief description of the relevant models.

Many of the interesting results for nonrandom systems have been obtained for walks that should strictly be labeled partially directed. In these movement is allowed along either the positive or negative x-direction but only along the positive t-direction, as shown in Fig. 16.1. Hence the position ratof the walk in column t= i is unique.

Also shown in Fig. 16.1 for comparison is a fully directed walk, each step of which must have a nonzero component in the positive t-direction. This is a simpler model, which has been very useful in studying the behavior of interfaces in a random environment (not reviewed here; see). The partially directed walk reduces to the fully directed one if the constraint is imposed.

An exact solution of a lattice spin model of ordering in one dimension is reviewed in this chapter. The model dynamics is synchronous, cellularautomaton- like, and involves interface diffusion and pairwise annihilation as well as spin flips due to an external field that favors one of the phases. At phase coexistence, structure-factor scaling applies, and the scaling function is obtained exactly. For field-driven, off-coexistence ordering, the scaling description breaks down for large enough times. The order parameter and the spin-spin correlation function are derived analytically, and several temporal and spatial scales associated with them analyzed.

Introduction

Phase separation, nucleation, ordering, and cluster growth are interrelated topics of great practical importance. One-dimensional (1D) phase separation, for which exact results can be derived, is discussed in this chapter. The emphasis will be on dynamical rules that involve simultaneous updating of the 1D-lattice ‘spin’ variables. Such models allow a particularly transparent formulation in terms of equations of motion the linearity of which yields exact solvability.

The results are also related to certain reaction-diffusion models of annihilating particles (see Part I of this book), and to deposition-with-relaxation processes (Part IV). Some of these connections will be reviewed here as well. While certain reaction and deposition processes have experimental realizations in 1D (see Part VII), 1D models of nucleation and cluster growth have been explored mainly as test cases for modern scaling theories of, for instance, structure-factor scaling, which will be reviewed in detail.

The dynamics of the deposition and evaporation of k adjacent particles at a time on a linear chain is studied. For the case k = 2 (reconstituting dimers), a mapping to the spin-½ Heisenberg model leads to an exact evaluation of the autocorrelation function C(t). For k ≥ 3, the dynamics is more complex. The phase space decomposes into many dynamically disconnected sectors, the number of sectors growing exponentially with size. Each sector is labeled by an irreducible string (IS), which is obtained from a configuration by a nonlocal deletion algorithm. The IS is shown to be a shorthand way of encoding an infinite number of conserved quantities. The large-t behavior of C(t) is very different from one sector to another. The asymptotic behavior in most sectors can be understood in terms of the diffusive, noncrossing movement of individual elements of the IS. Finally, a number of related models, including several that are many-sector decomposable, are discussed.

Introduction

Problems related to random sequential adsorption (RSA), initially studied several decades ago, have aroused renewed interest over the past few years. The reason for this is the growing realization that the basic process of deposition of extended objects, which is modeled by RSA, has diverse physical applications. In turn, this has led to the examination of a number of extensions, including the effect of interactions between atoms on adjacent sites, and the diffusion and desorption of single atoms.

In this chapter we give a brief review of one-dimensional (1D) kinetic Ising models that display nonequilibrium steady states. We describe how to construct such models, how to map them onto models of particle and surface dynamics, and how to derive and solve (in some cases) the equations of motion for the correlation functions. In the discussion of particular models, we focus on various problems characteristically occurring in studies of nonequilibrium systems such as the existence of phase transitions in 1D, the presence or absence of the fluctuation-dissipation theorem, and the derivation of the Langevin equations for mesoscopic degrees of freedom.

Introduction

The Ising model is a static, equilibrium, model. Its dynamical generalization was first considered by Glauber who introduced the single-spin-flip kinetic Ising model for describing relaxation towards equilibrium. Kawasaki then constructed a spin-exchange version of spin dynamics with the aim of studying such relaxational processes in the presence of conservation of magnetization. Other conservation laws were introduced soon afterwards by Kadanoff and Swift and thus the industry of kinetic Ising models wTas born.

The value of these models became apparent towards the end of the 1960s and the beginning of the 1970s when ideas of universality in static and dynamic critical phenomena emerged. Kinetic Ising models were simple enough to allow extensive analytical (series-expansion) and numerical (Monte Carlo) work, which was instrumental in determining critical exponents and checking universality.