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.

Two great intellectual triumphs occurred in the first quarter of the twentieth century: the invention of the theory of relativity, and the rather more labored arrival of quantum mechanics. The former largely embodies the ideas, and almost exclusively the vision, of Albert Einstein; the latter benefited from many creative minds, and from a constant interplay between theory and experiment. Probability plays no direct role in relativity, which is thus outside the purview and purpose of this book. However, Einstein will make an appearance in the following pages, in what some would claim was his most revolutionary role – as diviner of the logical and physical consequences of every idea that was put forward in the formative stages of quantum mechanics. It is one of the ironies of the history of science that Einstein, who contributed so importantly to the task of creating this subject, never accepted that the job was finished when a majority of physicists did. As a consequence, he participated only indirectly in the great adventure of the second quarter of the century: the use of quantum mechanics to explain the structure of atoms, nuclei, and matter.

This book has grown out of a course I have taught five times during the last 15 years at Cornell University. The College of Arts & Sciences at Cornell has a ‘distribution requirement in science,’ which can be fulfilled in a variety of ways. The Physics Department has for many years offered a series of ‘general education’ courses; any two of them satisfy the science requirement. The descriptions of these courses in the Cornell catalog begin with the words: ‘Intended for non-scientists; does not serve as a prerequisite for further science courses. Assumes no scientific background but will use high school algebra.’ This tradition was begun in the 1950s by two distinguished physicists, Robert R. Wilson and Philip Morrison, with a two-semester sequence ‘Aspects of the Physical World,’ which became known locally as ‘Physics for Poets.’ At the present time some three or four one-semester courses for non-scientists, ‘Reasoning about Luck’ sometimes among them, are offered each year.

What I try to do in this book and why is said in Chapter 1, but some words may be useful here. I started the enterprise lightheartedly hoping to do my bit to combat the widely perceived problems of scientific illiteracy and – to use a fashionable word – innumeracy, by teaching how to reason quantitatively about the uses of probability in descriptions of the natural world. I quickly discovered that the italicized word makes for great difficulties.

Isolated systems left to themselves, we have argued, evolve in such a way as to increase their entropy. The successes achieved by the use of this principle in the last two chapters cannot, however, hide the fact that it has up to now been pure assertion, more than reasonable to be sure but only vaguely connected to the motions of the microscopic constituents of matter. Let us now explore this connection, and the reasons why time and disorder flow in the same direction.

This is a curiously subtle question. From the point of view of common sense, it is not a puzzle at all. Shown a film of, for example, a lit match progressively returning to its unburnt condition, we instantly recognize a trick achieved by running a projector backwards. While it is by no means true that every macroscopically quiescent material object is in equilibrium, it is a general feature of common experience that left to themselves inanimate isolated systems become increasingly disordered. Even when the process is very slow, it is inexorable: cars eventually rust away. More often, it happens in front of your eyes: the ice cube in your drink melts, the sugar you add to your tea dissolves and never reassembles in your spoon.

It is time to correct and soften a striking difference in emphasis between the chapters on Mechanics (6) and Statistical Mechanics (8). Contrast the simple and regular motions in the former with the unpredictable jigglings in the latter. The message here will be that even in mechanics easy predictability is not by any means universal, and generally found only in carefully chosen simple examples. Sensitivity to precise ‘aiming’ is found in problems just slightly more complicated than those we considered in our discussion of mechanics. However, the time it takes for such sensitivity to manifest itself in perceptibly different trajectories depends on the particular situation. ‘Chaotic’ as a technical term has come to refer to trajectories with a sensitive dependence on initial conditions, and the general subject of their study is now called Chaos.

There is a long tradition of teaching mechanics via simple examples, such as the approximately circular motion of the moon under the gravitational influence of the earth. This is a special case of the ‘two body problem’ – two massive objects orbiting about each other – whose solution in terms of elliptical trajectories is one of the triumphs of Newton's Laws.

That a gas, air for example, is made up of a myriad tiny bodies, now called molecules, moving about randomly and occasionally colliding with each other and with the walls of a containing vessel, is today a commonplace fact that can be verified in many ways. Interestingly, this molecular view was widely though not universally accepted long before there were experimental methods for directly confirming it. The credit for the insight has to go to Chemistry, because a careful study of chemical reactions revealed regularities that could most easily be understood on the basis of the molecular hypothesis. These regularities were known to chemists by the end of the eighteenth century. By this time, the notion of distinct chemical species, or ‘elements,’ was well established, these being substances, like oxygen and sulfur, that resisted further chemical breakdown. It was discovered that when elements combine to make chemical compounds they do so in definite weight ratios. It was also found that when two elements combine to make more than one compound the weights of one of the elements, when referred to a definite weight of the second, stand one to another in the ratio of small integers – which sentence is perhaps too Byzantine to be made sense of without a definite example.

Probability enters theoretical physics in two important ways: in the theory of heat, which is a manifestation of the irregular motions of the microscopic constituents of matter; and, in quantum mechanics, where it plays the bizarre but, as far as we know, fundamental role already briefly mentioned in the discussion of radioactive decay.

Before we can understand heat, we have to understand motion. What makes objects move, and how do they move? Isaac Newton, in the course of explaining the motion of planets and of things around us that we can see and feel with our unaided senses, answered these questions for such motions three centuries ago. The science he founded has come to be called classical or Newtonian mechanics, to distinguish it from quantum mechanics, the theory of motion in the atomic and sub-atomic world.

Classical mechanics is summarized in Newton's ‘laws’ of motion. These will here be illustrated by an example involving the gravitational attraction, described by Newton's ‘law’ of gravitation.