Introduction

In many experiments on the adhesion of colloidal particles and proteins on substrates, the relaxation time scales are much longer than the times for the formation of the deposit. Owing to its relevance for the theoretical study of such systems, much attention has been devoted to the problem of irreversible monolayer particle deposition, termed random sequential adsorption (RSA) or the car parking problem; for reviews see. In RSA studies the depositing particles (on randomly chosen sites) are represented by hard-core extended objects; they are not allowed to overlap.

In this chapter, numerical Monte Carlo studies and analytical considerations are reported for 1D and 2D models of multilayer adsorption processes. Deposition without screening is investigated; in certain models the density may actually increase away from the substrate. Analytical studies of the RSA late stage coverage behavior show the crossover from exponential time dependence for the lattice case to the power-law behavior in continuum deposition. In 2D, lattice and continuum simulations rule out some ‘exact’ conjectures for the jamming coverage. For the deposition of dimers on a 1D lattice with diffusional relaxation the limiting coverage (100%) is approached according to the power law; this is preceded, for fast diffusion, by the mean-field crossover regime with intermediate, ∼ 1/t, behavior. In the case of k-mer deposition (k>> 3) with diffusion the void fraction decreases according to the power law t-1/(k-1).

Editor's note

Much interest has been devoted recently to various systems described in the continuum limit by variants of nonlinear diffusion equations. These include versions of the KPZ equation, Burgers’ equation, etc. Chapter 13 surveys nonlinear effects associated with shock formation in hard-core particle systems. Exact solution methods and results for such systems are then presented in Ch. 14.

Selected nonlinear effects in surface growth are reviewed in Ch. 15. Their relation to kinetic Ising models and a survey of some results were also presented in Ch. 4 (Sec. 4.6). This is a vast field with many recent results; see (and Chs. 4, 15) for review-type literature. Some surface-growth effects were also reviewed in Ch. 11.

The nonequilibrium ID systems covered in this book are effectively (1 + 1)- dimensional, where the second ‘dimension’ is time. For stochastic dynamics, the latter is frequently viewed as ‘Euclidean time’ in the field-theory nomenclature. Certain directed-walk models of surface fluctuations associated with wetting transitions, etc., as well as related models of polymer adsorption at surfaces, are effectively (0 + l)-dimensional in this classification, where the spatial dimension along the surface is effectively the Euclidean-time dimension. This property is shared by 1D quantum mechanics, to which the solution of many surface models reduces in the continuum limit. These models share simplicity, the availability of exact solutions, and the importance of fluctuations with the (l + l)-dimensional systems.

It has been well established by theory and simulations that the reaction kinetics of diffusion-limited reactions can be affected by the spatial dimension in which they occur. The types of reactions A + B → C, A + A → A. and A + C → C have been shown, theoretically and/or by simulation, to exhibit nonclassical reaction kinetics in 1D. We present here experimental results that have been collected for effectively 1D systems.

An A + B → C type reaction has been experimentally investigated in a long, thin capillary tube in which the reactants, A and B, are initially segregated. This initial segregation of reactants means that the net diffusion is along the length of the capillary only, making the system effectively 1D and allowing some of the properties of the resulting reaction front to be studied. The reaction rates of molecular coagulation and excitonic fusion reactions, A + A → A, well as trapping reactions, A + C → C, were observed via the phosphorescence (P) and delayed fluorescence (F) of naphthalene within the channels of Nuclepore membranes and Vycor glass and in the isolated chains of dilute polymer blends. In these experiments, the nonclassical kinetics is measured in terms of the heterogeneity exponent, h, from the equation rate ∼ F = kt-hPn, which gives the time dependence of the rate coefficient. Classically h = 0, while h = 1/2 in ID for A + A → A as well as A + C → C type reactions.

A generalized aggregation model of charged particles that diffuse and coalesce randomly in discrete space-time is studied, numerically and analytically. A statistically invariant steady state is established when randomly charged particles are uniformly and continuously injected. The exact steadystate size distribution obeys a power law whose exponent depends on the type of injection. The stability of the power-law size distribution is proved. The spatial correlations of the system are analyzed by a powerful new method, the interval distribution of a level set, and a scaling relation is obtained.

Introduction

The study of far-from-equilibrium systems has attracted much attention in the last two decades. Though many macroscopic phenomena in nature, such as turbulence, lightning, earthquakes, fracture, erosion, the formation of clouds, aerosols, and interstellar dusts, are typical far-from-equilibrium problems, no unified view has yet been established. The substantial difficulties in studying such systems are the following. First, far-from-equilibrium systems satisfy neither detailed balance nor, at the macroscopic level, the equipartition principle. Second, the system is usually open to an outside source. A common method to describe such systems is by abstracting the macroscopic essential features of the observed system and constructing a model in macroscopic terms irrespective of the microscopic (molecular) dynamics. In other words, we make a far-from-equilibrium model by assuming appropriate irreversible rules for the macroscopic dynamics.

Two recent developments involving activation and transport processes in simple stochastic nonlinear systems are reviewed in this chapter. The first is the idea of ‘resonant activation’ in which the mean first-passage time for escape over a fluctuating barrier passes through a minimum as the characteristic time scale of the fluctuating barrier is varied. The other is the notion of active transport in a fluctuating environment by so-called ‘ratchet’ mechanisms. Here, nonequilibrium fluctuations combined with spatial anisotropy conspire to generate systematic motion. The fundamental principles of these phenomena are covered, and some motivations for their study are described.

Introduction

The study of the interplay of noise and nonlinear dynamics presents many challenges, and interesting phenomena and insights appear even in onedimensional (1D) systems. Examples include Kramers’ fundamental theory of the Arrhenius temperature dependence of activated rate processes, Landauer's further insights into the role of multiplicative noise, and the theory of noise-induced transitions. This chapter reviews more recent developments which go beyond those studies in that the characteristic time scale of the fluctuations plays a major role in the dynamics of the system, whereas the phenomena in are fundamentally white-noise effects. Specifically, the two effects to be described in this chapter are the phenomena of ‘resonant activation’ and transport in ‘stochastic ratchets’.

Resonant activation is a generalization of Kramers’ model of activation over a potential barrier to the situation where the barrier itself is fluctuating randomly.

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.