The setting of a theory of complexity is greatly facilitated if it is carried out within a discrete framework. Most physical and mathematical problems, however, find their natural formulation in the real or complex field. Since the transformation of continuous quantities into a symbolic form is much more straightforward than the converse, it is convenient to adopt a common representation for complex systems based on integer arithmetics. This choice, in fact, does not restrict the generality of the approach, as this chapter will show. Moreover, discrete patterns actually occur in relevant physical systems and in mathematical models: consider, for example, magnets, alloys, crystals, DNA chains, and cellular automata. We recall, however, that a proposal for a theory of computational complexity over the real and complex fields has been recently advanced (Blum, 1990).

The symbolic representation of continuous systems also helps to elucidate the relationship between chaotic phenomena and random processes, although it is by no means restricted to nonlinear dynamics. Indeed, von Neumann's discrete automaton (von Neumann, 1966) was introduced to model natural organisms, which are mixed, “analogue–digital” systems: the genes are discrete information units, whereas the enzymes they control function analogically. Fluid configurations of the kind reproduced in Fig. 2.2 also lend themselves to discretization: owing to the constancy of the wavelength (complexity being associated with the orientation of the subdomains), a one-dimensional cut through the pattern yields a binary signal (high-low).

Most of the physical processes illustrated in the previous chapter are conveniently described by a set of partial differential equations (PDEs) for a vector field Ψ(x, t) which represents the state of the system in phase space X. The coordinates of Ψ are the values of observables measured at position x and time t: the corresponding field theory involves an infinity of degrees of freedom and is, in general, nonlinear.

A fundamental distinction must be made between conservative and dissipative systems: in the former, volumes in phase space are left invariant by the flow; in the latter, they contract to lower dimensional sets, thus suggesting that fewer variables may be sufficient to describe the asymptotic dynamics. Although this is often the case, it is by no means true that a dissipative model can be reduced to a conservative one acting in a lower-dimensional space, since the asymptotic trajectories may wander in the whole phase space without filling it (see, e.g., the definition of a fractal measure in Chapter 5).

A system is conceptually simple if its evolution can be reduced to the superposition of independent oscillations. This is the integrable case, in which a suitable nonlinear coordinate change permits expression of the equations of motion as a system of oscillators each having its own frequency.

When the metal complex [Ru(phen)2(dppz)]2+ is bound to DNA it can luminesce. If the metal complex [Rh(phi)2(phen)]3+ is nearby on the strand, the luminescence is quenched by electron transfer. By varying concentrations and by varying the DNA it is possible to probe the distribution of complexes in this one-dimensional (1D) system, and to gather information about the electron transfer length and interparticle forces. Our model assumes random deposition with allowance for interactions among the complexes. Long strands of calf thymus (CT) DNA and short strands of a synthetic 28-mer were used in the experiments and, for fixed [Ru(phen)2(dppz)]2+ concentration, quenching was measured as a function of [Rh(phi)2(phen)]3+ concentration. In previous work, to be cited later, we reported an electron transfer length based on the CT-DNA data. However, the short-strand (28-mer) experiments show a remarkable difference from the previously analyzed data. In particular, the electron-transfer quenching upon irradiation is enhanced by a factor of approximately four. This requires the consideration of new physical effects on the short strands. Our proposal is to introduce complexcomplex repulsion as an additional feature. This allows a reasonable fit within the context of the random-deposition model, although it does not take into account changes in the structure of the 28-mer introduced by the metal complexes during the loading process.

Editor's note

Kinetic Ising models in 1D provide a gallery of exactly solvable systems with nontrivial dynamics. The emphasis has traditionally been on their exact solvability, although much attention has also been devoted to models with conservation laws that have to be treated by numerical and approximation methods.

Chapter 4 reviews these models with emphasis on steady states and the approach to steady-state behavior. Chapter 5 puts the simplest 1D kinetic Ising models into a wider framework of the evaluation of dynamical critical behavior, analytically, in 1D, and numerically, for general dimension. Finally, Ch. 6 describes low-temperature nonequilibrium properties such as domain growth and freezing.

For a general description of dynamical critical behavior, not limited to 1D, as well as an excellent review and classification of various types of dynamics, the reader is directed to the classical work [1]. Certain probabilistic cellular automata are equivalent to kinetic Ising models.

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.