Table of contents

We aim to clarify confusions in the literature as to whether or not dynamical density functional theories for the one-body density of a classical Brownian fluid should contain a stochastic noise term. We point out that a stochastic as well as a deterministic equation of motion for the density distribution can be justified, depending on how the fluid one-body density is defined—i.e. whether it is an ensemble averaged density distribution or a spatially and/or temporally coarse grained density distribution.

We calculate the asymptotic behaviour of the one-body density matrix of one-dimensional impenetrable bosons in finite size geometries. Our approach is based on a modification of the replica method from the theory of disordered systems. We obtain explicit expressions for oscillating terms, similar to fermionic Friedel oscillations. These terms are universal and originate from the strong short-range correlations between bosons in one dimension.

We study the spectral and transport properties of magnons in a model of a disordered magnet called Mattis glass, at vanishing average magnetization. We find that in two-dimensional space, the magnons are localized with the localization length which diverges as a power of frequency at small frequencies. In three-dimensional space, the long wavelength magnons are delocalized. In the delocalized regime in 3d (and also in 2d in a box whose size is smaller than the relevant localization length scale) the magnons move diffusively. The diffusion constant diverges at small frequencies. However, the divergence is slow enough so that the thermal conductivity of a Mattis glass is finite, and we evaluate it in this paper. This situation can be contrasted with that of phonons in structural glasses whose contribution to thermal conductivity is known to diverge (when inelastic scattering is neglected).

I present a hybrid method for the labelling of clusters in two-dimensional lattices, which combines the recursive approach with iterative scanning to reduce the stack size required by the pure recursive technique, while keeping its benefits: single pass and straightforward cluster characterization and percolation detection parallel to the labelling. While the capacity to hold the entire lattice in memory is usually regarded as the major constraint for the applicability of the recursive technique, the required stack size is the real limiting factor. Resorting to recursion only for the transverse direction greatly reduces the recursion depth and therefore the required stack. It also enhances the overall performance of the recursive technique, as is shown by results on a set of uniform random binary lattices and on a set of samples of the Ising model. I also show how this technique may replace the recursive technique in Wolff's cluster algorithm, decreasing the risk of stack overflow and increasing its speed, and the Hoshen–Kopelman algorithm in the Swendsen–Wang cluster algorithm, allowing effortless characterization during generation of the samples and increasing its speed.

In the present study, the nonlinear modulation of transverse waves propagating in a cubically nonlinear dispersive elastic medium is studied using a multi-scale expansion of wave solutions. It is found that the propagation of quasi-monochromatic transverse waves is described by a pair of coupled nonlinear Schrödinger (CNLS) equations. In the process of deriving the amplitude equations, it is observed that for a specific choice of material constants and wavenumber, the coefficient of nonlinear terms becomes zero, and the CNLS equations are no longer valid for describing the behaviour of transverse waves. In order to balance the nonlinear effects with the dispersive effects, by intensifying the nonlinearity, a new perturbation expansion is used near the critical wavenumber. It is found that the long time behaviour of the transverse waves about the critical wavenumber is given by a pair of coupled quintic nonlinear Schrödinger (CQNLS) equations. In the absence of one of the transverse waves, the CQNLS equations reduce to the single quintic nonlinear Schrödinger (QNLS) equation which has already been obtained in the context of water waves. By using a modified form of the so-called tanh method, some travelling wave solutions of the CQNLS equations are presented.

We study in detail the dynamics of a hierarchy of piecewise maps generated by a one-parameter family of trigonometric chaotic maps and a one-parameter family of elliptic chaotic maps of the cn and sn type. We calculate the Lyapunov exponent and Kolmogorov–Sinai entropy of the these maps with respect to a control parameter. Non-ergodicity of these piecewise maps is proven analytically and investigated numerically. The invariant measure of these maps, which are not equal to 1 or 0, appears to be characteristic for non-ergodic behaviour. A quantity of interest is the Kolmogorov–Sinai entropy, which, for these maps, is smaller compared with the sum of the positive Lyapunov exponents, and it confirms the non-ergodicity of the maps.

In this work we plot loop configurations that minimize energy functionals depending on geometrical invariants of the loop itself. In particular, we consider a family of functionals including curvature and torsion terms, both linear and quadratic, such that their combinations produce geometrical invariants. In [1], Noether's theorem was advantageously used to identify the constants of integration of the Euler–Lagrange equations describing the equilibrium of loops for this family of energy functionals. Those authors demonstrated the integrability of these functionals by means of quadratures. In this work we follow that approach to realize numeric calculations and to show plots of numerical solutions of the relevant equations for equilibrium of loops, as these were presented but not studied in [1]. We then show as the main result of this work a representative catalogue of such solutions in Euclidean three-dimensional space.

For time-periodic perturbations of the wave equation in given by a potential q(t, x), we obtain an upper bound of the number of the resonances . We establish for m N large enough a trace formula relating the iterations of the monodromy operator U(mT, 0), T > 0, and the sum ∑_jz^m_j of all resonances counted with their multiplicities.

We show that the Casimir operators of the semidirect products of the exceptional Lie algebra G₂ and a Heisenberg algebra can be constructed explicitly from the Casimir operators of G₂.

We give a geometrical interpretation of the notion of μ-prolongations of vector fields and of the related concept of μ-symmetry for partial differential equations (extending to PDEs the notion of λ-symmetry for ODEs). We give in particular a result concerning the relationship between μ-symmetries and standard exact symmetries. The notion is also extended to the case of conditional and partial symmetries, and we analyse the relation between local μ-symmetries and nonlocal standard symmetries.

Any eigenfunction of the Laplacian on a sphere is given in terms of a unique set of directions: these are Maxwell's multipoles, their existence and uniqueness being known as Sylvester's theorem. Here, the theorem is proved by realizing the multipoles are pairs of opposite vectors in Majorana's sphere representation of quantum spins. The proof involves a physicist's standard tools of quantum angular momentum algebra, integral kernels and Gaussian integration. Various other proofs are compared, including an alternative using the calculus of spacetime spinors.

A practical method of inverse scattering is proposed to determine a finite-range scattering potential from phase-shift data evaluated at a set of discrete energies. By using the asymptotic behaviour of the phase shifts and the connection between short- and long-wavelength scales, a relatively few input data are found to be adequate. The method works well for repulsive and moderately attractive potentials, useful in many applications.

Tensor operators in graded representations of Z₂-graded Hopf algebras are defined and their elementary properties are derived. The Wigner–Eckart theorem for irreducible tensor operators for U_q[osp(1|2)] is proven. Examples of tensor operators in the irreducible representation space of Hopf algebra U_q[osp(1|2)] are considered. The reduced matrix elements for the irreducible tensor operators are calculated. A construction of some elements of the centre of U_q[osp(1|2)] is given.

Canonical coherent states can be written as infinite series in powers of a single complex number z and a positive integer ρ(m). The requirement that these states realize a resolution of the identity typically results in a moment problem, where the moments form the positive sequence of real numbers {ρ(m)}^∞_m=0. In this paper we obtain new classes of vector coherent states by simultaneously replacing the complex number z and the moments ρ(m) of the canonical coherent states by n × n matrices. Associated oscillator algebras are discussed with the aid of a generalized matrix factorial. Two physical examples are discussed. In the first example coherent states are obtained for the Jaynes–Cummings model in the weak coupling limit and some physical properties are discussed in terms of the constructed coherent states. In the second example coherent states are obtained for a conditionally exactly solvable supersymmetric radial harmonic oscillator.

Trajectory-based approaches to quantum mechanics include the de Broglie–Bohm interpretation and Nelson's stochastic interpretation. It is shown that the usual route to establishing the validity of such interpretations, via a decomposition of the Schrödinger equation into a continuity equation and a modified Hamilton–Jacobi equation, fails for some quantum states. A very simple example is provided by a quantum particle in a box, described by a wavefunction that is initially uniform over the interior of the box. For this example, there is no corresponding continuity or modified Hamilton–Jacobi equation, and the space-time dependence of the wavefunction has a known fractal structure. Examples with finite average energies are also constructed.

Witten's the non-relativistic formalism of supersymmetric quantum mechanics was based on a factorization and partnership between Schrödinger equations. We show how it accommodates a transition to the partnership between relativistic Klein–Gordon equations.

We clarify the relation among canonical, metric and Belinfante's energy–momentum tensors for general tensor field theories. For any tensor field T, we define a new tensor field in terms of which metric and Belinfante's energy–momentum tensors are readily computed. We show that the latter is the one that arises naturally from Noether's theorem for an arbitrary spacetime and it coincides on-shell with the metric one.

This paper focuses on upscaling of the transport equation for heterogeneous porous media with random flow. We consider the local flow field being a stationary random field and develop an upscaling by the recently developed coarse graining method which is based on filtering procedures in Fourier space. The coarse graining method is used to obtain an upscaled dispersion tensor which depends on the given length scale of the upscaling. We give explicit results for the scale-dependent dispersion coefficient in lowest-order perturbation theory. For finite length scales the upscaled dispersion models the effect of the unresolved subscale flow fluctuations, and for a global upscaling the upscaled value agrees with the well-known macrodispersion coefficient, which is, however, nearly approached for length scales larger than tenfold of the correlation length.

This book by Nino Boccara presents a compilation of model systems commonly termed as `complex'. It starts with a definition of the systems under consideration and how to build up a model to describe the complex dynamics. The subsequent chapters are devoted to various categories of mean-field type models (differential and recurrence equations, chaos) and of agent-based models (cellular automata, networks and power-law distributions). Each chapter is supplemented by a number of exercises and their solutions.

The table of contents looks a little arbitrary but the author took the most prominent model systems investigated over the years (and up until now there has been no unified theory covering the various aspects of complex dynamics). The model systems are explained by looking at a number of applications in various fields.

The book is written as a textbook for interested students as well as serving as a compehensive reference for experts. It is an ideal source for topics to be presented in a lecture on dynamics of complex systems. This is the first book on this `wide' topic and I have long awaited such a book (in fact I planned to write it myself but this is much better than I could ever have written it!). Only section 6 on cellular automata is a little too limited to the author's point of view and one would have expected more about the famous Domany--Kinzel model (and more accurate citation!).

In my opinion this is one of the best textbooks published during the last decade and even experts can learn a lot from it. Hopefully there will be an actualization after, say, five years since this field is growing so quickly.

The price is too high for students but this, unfortunately, is the normal case today. Nevertheless I think it will be a great success!

This is a handbook for a computational approach to reacting flows, including background material on statistical mechanics. In this sense, the title is somewhat misleading with respect to other books dedicated to the statistical theory of turbulence (e.g. Monin and Yaglom). In the present book, emphasis is placed on modelling (engineering closures) for computational fluid dynamics. The probabilistic (pdf) approach is applied to the local scalar field, motivated first by the nonlinearity of chemical source terms which appear in the transport equations of reacting species. The probabilistic and stochastic approaches are also used for the velocity field and particle position; nevertheless they are essentially limited to Lagrangian models for a local vector, with only single-point statistics, as for the scalar. Accordingly, conventional techniques, such as single-point closures for RANS (Reynolds-averaged Navier-Stokes) and subgrid-scale models for LES (large-eddy simulations), are described and in some cases reformulated using underlying Langevin models and filtered pdfs. Even if the theoretical approach to turbulence is not discussed in general, the essentials of probabilistic and stochastic-processes methods are described, with a useful reminder concerning statistics at the molecular level.

The book comprises 7 chapters. Chapter 1 briefly states the goals and contents, with a very clear synoptic scheme on page 2.

Chapter 2 presents definitions and examples of pdfs and related statistical moments.

Chapter 3 deals with stochastic processes, pdf transport equations, from Kramer-Moyal to Fokker-Planck (for Markov processes), and moments equations. Stochastic differential equations are introduced and their relationship to pdfs described. This chapter ends with a discussion of stochastic modelling.

The equations of fluid mechanics and thermodynamics are addressed in chapter 4. Classical conservation equations (mass, velocity, internal energy) are derived from their counterparts at the molecular level. In addition, equations are given for multicomponent reacting systems. The chapter ends with miscellaneous topics, including DNS, (idea of) the energy cascade, and RANS.

Chapter 5 is devoted to stochastic models for the large scales of turbulence. Langevin-type models for velocity (and particle position) are presented, and their various consequences for second-order single-point corelations (Reynolds stress components, Kolmogorov constant) are discussed. These models are then presented for the scalar. The chapter ends with compressible high-speed flows and various models, ranging from k- to hybrid RANS-pdf.

Stochastic models for small-scale turbulence are addressed in chapter 6. These models are based on the concept of a filter density function (FDF) for the scalar, and a more conventional SGS (sub-grid-scale model) for the velocity in LES.

The final chapter, chapter 7, is entitled `The unification of turbulence models' and aims at reconciling large-scale and small-scale modelling.

This book offers a timely survey of techniques in modern computational fluid mechanics for turbulent flows with reacting scalars. It should be of interest to engineers, while the discussion of the underlying tools, namely pdfs, stochastic and statistical equations should also be attractive to applied mathematicians and physicists. The book's emphasis on local pdfs and stochastic Langevin models gives a consistent structure to the book and allows the author to cover almost the whole spectrum of practical modelling in turbulent CFD. On the other hand, one might regret that non-local issues are not mentioned explicitly, or even briefly. These problems range from the presence of pressure-strain correlations in the Reynolds stress transport equations to the presence of two-point pdfs in the single-point pdf equation derived from the Navier--Stokes equations.

(One may recall that, even without scalar transport, a general closure problem for turbulence statistics results from both non-linearity and non-locality of Navier-Stokes equations, the latter coming from, e.g., the nonlocal relationship of velocity and pressure in the quasi-incompressible case. These two aspects are often intricately linked. It is well known that non-linearity alone is not responsible for the `problem', as evidenced by 1D turbulence without pressure (`Burgulence' from the Burgers equation) and probably 3D (cosmological gas). A local description in terms of pdf for the velocity can resolve the `non-linear' problem, which instead yields an infinite hierarchy of equations in terms of moments. On the other hand, non-locality yields a hierarchy of unclosed equations, with the single-point pdf equation for velocity derived from NS incompressible equations involving a two-point pdf, and so on. The general relationship was given by Lundgren (1967,Phys. Fluids10 (5), 969-975), with the equation for pdf at n points involving the pdf at n+1 points. The nonlocal problem appears in various statistical models which are not discussed in the book. The simplest example is full RST or ASM models, in which the closure of pressure-strain correlations is pivotal (their counterpart ought to be identified and discussed in equations (5-21) and the following ones). The book does not address more sophisticated non-local approaches, such as two-point (or spectral) non-linear closure theories and models, `rapid distortion theory' for linear regimes, not to mention scaling and intermittency based on two-point structure functions, etc. The book sometimes mixes theoretical modelling and pure empirical relationships, the empirical character coming from the lack of a nonlocal (two-point) approach.)

In short, the book is orientated more towards applications than towards turbulence theory; it is written clearly and concisely and should be useful to a large community, interested either in the underlying stochastic formalism or in CFD applications.

Since the discovery of the renormalization group theory in statistical physics, the realm of applications of the concepts of scale invariance and criticality has pervaded several fields of natural and social sciences. This is the leitmotiv of Didier Sornette's book, who in Critical Phenomena in Natural Sciences reviews three decades of developments and applications of the concepts of criticality, scale invariance and power law behaviour from statistical physics, to earthquake prediction, ruptures, plate tectonics, modelling biological and economic systems and so on.

This strongly interdisciplinary book addresses students and researchers in disciplines where concepts of criticality and scale invariance are appropriate: mainly geology from which most of the examples are taken, but also engineering, biology, medicine, economics, etc. A good preparation in quantitative science is assumed but the presentation of statistical physics principles, tools and models is self-contained, so that little background in this field is needed. The book is written in a simple informal style encouraging intuitive comprehension rather than stressing formal derivations. Together with the discussion of the main conceptual results of the discipline, great effort is devoted to providing applied scientists with the tools of data analysis and modelling necessary to analyse, understand, make predictions and simulate systems undergoing complex collective behaviour.

The book starts from a purely descriptive approach, explaining basic probabilistic and geometrical tools to characterize power law behaviour and scale invariant sets. Probability theory is introduced by a detailed discussion of interpretative issues warning the reader on the use and misuse of probabilistic concepts when the emphasis is on prediction of low probability rare---and often catastrophic---events. Then, concepts that have proved useful in risk evaluation, extreme value statistics, large limit theorems for sums of independent variables with power law distribution, random walks, fractals and multifractal formalisms, etc, are discussed in an immediate and direct way so as to provide ready-to-use tools for analysing and representing power law behaviour in natural phenomena. The exposition then continues discussing the main developments, allowing the reader to understand theoretically and model strongly correlated behaviour. After a concise, but useful, introduction to the fundamentals of statistical physics a discussion of equilibrium critical phenomena and the renormalization group is proposed to the reader. With the centrality of the problem of non-equilibrium behaviour in mind, a discussion is devoted to tentative applications of the concept of temperature in the off-equilibrium context.

Particular emphasis is given to the development of long range correlation and of precursors of phase transitions, and their role in the prediction of catastrophic events. Then, basic models such as percolation and rupture models are described. A central position in the book is occupied by a chapter on mechanisms for power laws and a subsequent one on self-organized criticality as a general paradigm for critical behaviour as proposed by P Bak and collaborators. The book concludes with a chapter on the prediction of fields generated by a random distribution of sources.

The book maintains the promise of the title of providing concepts and tools to tackle criticality and self-organization. The second edition, while retaining the structure of the first edition, considerably extends the scope with new examples and applications of a research field which is constantly growing.

Any scientific book has to solve the dichotomy between the depth of discussion, the pedagogical character of exposition and the quantity of material discussed. In general the book, which evolved from a graduate student course, favours these last two aspects at the expense of the first one. This makes the book very readable and means that, while complicated concepts are always explained by means of simple examples, important results are often mentioned but not derived or discussed in depth. Most of the time this style of exposition manages to successfully convey the essential information, other times unfortunately, e.g. in the case of the chapter on disordered systems, the presentation appears rather superficial. This is the price we pay for a book covering an impressively vast subject area and the huge bibliography (more than 1000 references) furnishes a necessary guide for acquiring the working knowledge of the subject covered. I would recommend it to teachers planning introductory courses on the field of complex systems and to researchers wanting to learn about an area of great contemporary interest.

This book contains 18 contributions from different authors. Its subtitle `Econophysics, Bioinformatics, and Pattern Recognition' says more precisely what it is about: not so much about central problems of conventional statistical physics like equilibrium phase transitions and critical phenomena, but about its interdisciplinary applications.

After a long period of specialization, physicists have, over the last few decades, found more and more satisfaction in breaking out of the limitations set by the traditional classification of sciences. Indeed, this classification had never been strict, and physicists in particular had always ventured into other fields. Helmholtz, in the middle of the 19th century, had considered himself a physicist when working on physiology, stressing that the physics of animate nature is as much a legitimate field of activity as the physics of inanimate nature. Later, Max Delbrück and Francis Crick did for experimental biology what Schrödinger did for its theoretical foundation. And many of the experimental techniques used in chemistry, biology, and medicine were developed by a steady stream of talented physicists who left their proper discipline to venture out into the wider world of science.

The development we have witnessed over the last thirty years or so is different. It started with neural networks where methods could be applied which had been developed for spin glasses, but todays list includes vehicular traffic (driven lattice gases), geology (self-organized criticality), economy (fractal stochastic processes and large scale simulations), engineering (dynamical chaos), and many others. By staying in the physics departments, these activities have transformed the physics curriculum and the view physicists have of themselves. In many departments there are now courses on econophysics or on biological physics, and some universities offer degrees in the physics of traffic or in econophysics.

In order to document this change of attitude, many former Institutes for Statistical Physics are now renamed as Institutes for Complex Systems Science, manifesting thereby the claim that studying the complexity of the world surrounding us is a legitimate branch of physics: after the science of the infinitely large and the science of the infinitely small, it is now the science of the infinitely complex.

The present book tries to give an overview of these developments. No volume of 360 pages can of course give a complete and balanced account. Therefore it is necessary to pick out representative problems, and to illustrate with them how concepts and methods from statistical physics can be made useful in circumstances which their creators never had in mind. This is essentially the goal that the book tries to attain, as also stated on its back cover: `This book provides a unique insight into the latest breakthroughs in a consistent manner, at a level accessible to undergraduates, yet with enough attention to the theory and computation to satisfy the professional researcher.'

Measured against these high goals, the book has failed. The articles are of very uneven quality. The only paper written manifestly for undergraduates is the one on first passage problems by Ding and Rangarajan. Others, like the articles on protein folding by Hansmann, on clustering by Steinbach et al, and on thermal convection by Rogers et al should still be very useful for students, but, for example, the excellent article by Y-K Yu on sequence alignment is written mainly for specialists.

While the above articles (and several other ones) are indeed well written and of sufficiently broad interest to be included in such a volume, I cannot say this of all the papers. Some seem more the outcome of a PhD thesis (Thomakos on predicting the direction of a time series, Jirsa on variability of timing), rather than a review of a more substantial piece of work. Others (for example Aspneset al on the variability of stock markets) are extremely technical, and some (Stanley et al on economy) are just sloppy and superficial. The article on competition in biological systems by Louzoun et al is somewhat annoying because the authors seem extremely surprised that reaction--diffusion systems in low dimension are dominated by fluctuations and do not obey mean field behaviour, neglecting completely a vast literature accumulated during the last 20 years. The one by Stramaglia et al is equally annoying since it was obviously motivated by the more interesting work by Domany et al (from which it departs mainly by replacing the very natural and well understood Potts model by a much less understood synchronization process), without giving much of a credit.

Finally, some of the authors missed the opportunity to point out direct and beautiful relationships with other problems in statistical physics. A striking example is the article by Yu which does not mention any relation with surface roughening or directed polymers, although he and his collaborators discovered it. It is this chance to apply results obtained in one context to a completely different one which underlies the unity of physics and which contributes much to its fascination.

After having made all these negative remarks, I should say that the book is nevertheless a rich source of inspiration and gives fascinating outlooks in a field which is sure to remain active over the coming decades. The fact that it is not perfect just reflects the chaotic and fast-moving state of an emerging field, and thus might indeed give the right impression of ongoing construction works which might induce newcomers to join it. Whether it is suitable as a source for beginners to learn systematically about the subject is doubtful.