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.

Random sequential adsorption (RSA) and cooperative sequential adsorption (CSA) on 1D lattices provide a remarkably broad class of far-fromequilibrium processes that are amenable to exact analysis. We examine some basic models, discussing both kinetics and spatial correlations. We also examine certain continuum limits obtained by increasing the characteristic size in the model (e.g., the size of the adsorbing species in RSA, or the mean island size in CSA models having a propensity for clustering). We indicate that the analogous 2D processes display similar behavior, although no exact treatment is possible here.

Introduction

In the most general scenario for chemisorption or epitaxial growth at single crystal surfaces, species adsorb at a periodic array of adsorption sites, hop between adjacent sites, and possibly desorb from the surface. Such processes can be naturally described within a lattice-gas formalism. The microscopic rates for different processes in general depend on the local environment and satisfy detailed-balance constraints. The net adsorption rate is determined by the difference in chemical potential between the gas phase and the adsorbed phase. In many cases, thermal desorption can be ignored for a broad range of typical surface temperatures, T. Furthermore, for sufficiently low T, thermally activated surface diffusion is also inoperative, so then species are irreversibly (i.e., permanently) bound at their adsorption sites. Henceforth, we consider the latter regime exclusively. Clearly the resultant adlayer is in a far-from-equilibrium state determined by the kinetics of the adsorption process.

Continuous phase transitions from an absorbing to an active state arise in diverse areas of physics, chemistry and biology. This chapter reviews the current understanding of phase diagrams and scaling behavior at such transitions, and recent developments bearing on universality.

Introduction

Stochastic processes often possess one or more absorbing states—configurations with arrested dynamics, admitting no escape. Phase transitions between an absorbing state and an active regime have been of interest in physics since the late 1950s, when Broadbent and Hammersley introduced directed percolation (DP). Subsequent incarnations include Reggeon field theory, a high-energy model of peripheral interest to most condensed matter physicists, and a host of more familiar problems such as autocatalytic chemical reactions, epidemics, and transport in disordered media. For the simpler examples—Schlögl's models, the contact process, and directed percolation itself—many aspects of critical behavior are well in hand. In the mid-1980s absorbing-state transitions found renewed interest due to the catalysis models devised by Ziff and others, and to a proposed connection with the transition to turbulence. A further impetus has been the ongoing quest to characterize universality classes for these transitions. Parallel to these developments, probabilists studying interacting particle systems have established a number of fundamental theorems for models with absorbing states.

Interest in the influence of kinetic rules on the phase diagram has spawned many models over the last decade; the majority must go unmentioned here.

Recent results for the Glauber-type kinetic Ising models are reviewed in this chapter. Exact solutions for chains and simulational results for the dynamical exponents for square and cubic lattices are given.

Introduction

A study on the dynamical behavior of the Ising model must begin with the introduction of a temporal evolution rule, because the Ising model itself does not have any a priori dynamics naturally introduced from the kinetic theory. Various kinds of dynamics are possible and some are useful to describe and predict physical phenomena or to make simulation studies of the equilibrium state. The Ising model with an appropriately defined temporal evolution rule is called the kinetic Ising model.

The statistical mechanical studies of the dynamical behavior in and around the equilibrium state started in the 1950s. During that decade, theoretical and computational developments provided a breakthrough and advanced such studies. The Kubo theory and its successful application established the linearly perturbed regime around the equilibrium state generally treated by methods of statistical mechanics. It gave a means of investigating the dynamic behavior of macroscopic systems. Another great advance in that decade was the application of computing machines to statistical physics. Dynamical Monte Carlo (MC) simulation on computers gave rise to the problem of computational efficiency, which is related to the dynamical behavior of the system, although this aspect became clear rather recently, in the 1980s.