Table of contents

A model of state reduction in relativistic quantum field theory involving a nonlinear stochastic extension of Schrödinger's equation is outlined. The eigenstates of the annihilation operator are chosen as the preferred basis onto which reduction occurs. These are the coherent states which saturate the bound of the Heisenberg uncertainty relation, exhibiting classical-like behaviour. The quantum harmonic oscillator is studied in detail before generalizing to relativistic scalar quantum field theory. The infinite rates of increase in energy density which have plagued recent relativistic proposals of dynamical state reduction are absent in this model. This is because the state evolution equation does not drive particle creation from the vacuum. The model requires the specification of a preferred sequence of space-like hyper-surfaces supporting the time-like state evolution. However, it is shown that the choice of preferred surfaces has no effect on perturbative results to second order in the coupling parameter. It is demonstrated how state reduction to a charge density basis can be induced in fermionic matter via an appropriate coupling to a bosonic field undergoing this mechanism.

A continuous complex rotation of time is shown to smooth out the huge fluctuations that characterize chaotic tunnelling. This is illustrated in the kicked rotor model (quantum standard map) where the period of the map is complexified: the associated chaotic classical dynamics, if significant for θ = 0, is blurred out long before the Wick rotation is completed (θ = π/2). The influence of resonances on tunnelling rates weakens exponentially as θ increases from zero, all the more rapidly the sharper the fluctuations. The long-range fluctuations can therefore be identified in a deterministic way without ambiguity. When the last ones have been washed out, tunnelling recovers the (quasi-)integrable exponential behaviour governed by the action of a regular instanton.

We give a simple derivation of all n-point densities for the eigenvalues of the real Ginibre ensemble with even dimension N as quaternion determinants. A very simple symplectic kernel governs both the real and complex correlations. One- and two-point correlations are discussed in more detail. Scaling forms for large dimension N are derived.

In this paper, an initially unmagnetized pair plasma with asymmetric velocity distributions is investigated where any unstable Weibel mode must be isolated, with discrete values for the growth rates and the unstable wavenumbers. For both a non-relativistic distribution with thermal spread and a high-relativistic two-stream distribution it is shown that isolated modes are excited and that, as the asymmetry tends to zero, the growth rate remains finite, as long as the distribution function is not precisely symmetric.

Regarding the quantum mechanical transmission–reflection problem in one dimension with a potential of the form of the derivative of the Dirac delta function δ'(x) = dδ(x)/dx, Christiansen et al recently found that, depending on how δ'(x) is interpreted, there can be a resonance which leads to partial transmission. This is in contrast to the earlier consensus that such a potential allows no transmission. The δ'(x) can be regarded as the narrow-width limit of a certain function Δ'(x) of a finite range. Christiansen et al assumed a rectangular function for Δ'(x). We examine various other forms and how the resonance depends on the shape of Δ'(x). We also present some general observations related to the 'threshold anomaly'.

We study models for surface growth with a wetting and a roughening transition using simple and pair mean-field approximations. The simple mean-field equations are solved exactly and they predict the roughening transition and the correct growth exponents in a region of the phase diagram. The pair mean-field equations, which are solved numerically, show a better accordance with numerical simulation and correctly predict a growing interface with a constant velocity at the moving phase. Also, when detailed balance is fulfilled, the pair mean field becomes the exact solution of the model.

We use the recently derived form factor expansions of the diagonal two-point correlation function of the square Ising model to study the susceptibility for a magnetic field applied only to one diagonal of the square lattice, for the isotropic Ising model. We exactly evaluate the one and two particle contributions χ⁽¹⁾_d and χ⁽²⁾_d of the corresponding susceptibility, and obtain linear differential equations for the three and four particle contributions, as well as the five particle contribution χ⁽⁵⁾_d, but only modulo a given prime. We use these exact linear differential equations to show that not only the Russian-doll structure but also the direct sum structure on the linear differential operators for the n-particle contributions χ⁽ⁿ⁾_d are quite directly inherited from the direct sum structure on the form factors f⁽ⁿ⁾. We show that the nth particle contributions χ⁽ⁿ⁾_d have their singularities at roots of unity. These singularities become dense on the unit circle |sinh 2E_v/kTsinh 2E_h/kT| = 1 as n → ∞.

By using theoretical analysis and computer simulations, we develop the Watts–Strogatz network model by including degree distribution, in an attempt to improve the comparison between characteristic path lengths and clustering coefficients predicted by the original Watts–Strogatz network model and those of the real networks with the small-world property. Good agreement between the predictions of the theoretical analysis and those of the computer simulations has been shown. It is found that the developed Watts–Strogatz network model can fit the real small-world networks more satisfactorily. Some other interesting results are also reported by adjusting the parameters in a model degree-distribution function. The developed Watts–Strogatz network model is expected to help in the future analysis of various social problems as well as financial markets with the small-world property.

We study the relaxational critical dynamics of the three-dimensional random anisotropy magnets with the non-conserved n-component order parameter coupled to a conserved scalar density. In the random anisotropy magnets, the structural disorder is present in the form of local quenched anisotropy axes of random orientation. When the anisotropy axes are randomly distributed along the edges of the n-dimensional hypercube, asymptotical dynamical critical properties coincide with those of the random-site Ising model. However the structural disorder gives rise to considerable effects for non-asymptotic critical dynamics. We investigate this phenomenon by a field-theoretical renormalization group analysis in the two-loop order. We study critical slowing down and obtain quantitative estimates for the effective and asymptotic critical exponents of the order parameter and scalar density. The results predict complex scenarios for the effective critical exponent approaching the asymptotic regime.

We develop rigorous, analytic techniques to study the behaviour of biased random walks on combs. This enables us to calculate exactly the spectral dimension of random comb ensembles for any bias scenario in the teeth or spine. Two specific examples of random comb ensembles are discussed: the random comb with nonzero probability of an infinitely long tooth at each vertex on the spine and the random comb with a power-law distribution of tooth lengths. We also analyse transport properties along the spine for these probability measures.

We have developed a modified Nagel–Schreckenberg cellular automata model for describing a conflicting vehicular traffic flow at the intersection of two streets. No traffic lights control the traffic flow. The approaching cars to the intersection yield to each other to avoid collision. Closed boundary condition is applied to the streets. Extensive Monte Carlo simulation is taken into account to find the model characteristics. In particular, we obtain the fundamental diagrams and show that the effect of the interaction of two streets can be regarded as a dynamic impurity located at the intersection point. Our results suggest that yielding mechanism gives rise to a high total flow throughout the intersection especially in the low density regime.

Lévy flights, which are Markovian continuous time random walks possibly accounting for extreme events, serve frequently as small-scale models for the spreading of matter in heterogeneous media. Among them, Brownian motion is a particular case where Fick's law holds: for a cloud of walkers, the flux is proportional to the gradient of the probability density of finding a particle at some place. Lévy flights resemble Brownian motion, except that jump lengths are distributed according to an α-stable Lévy law, possibly showing heavy tails and skewness. For α between 1 and 2, a fractional form of Fick's law is known to hold in infinite media: that the flux is proportional to a combination of fractional derivatives or the order of α − 1 of the density of walkers was obtained as a consequence of a fractional dispersion equation. We present a direct and natural proof of this result, based upon a novel definition of usual fractional derivatives, involving a convolution and a limiting process. Taking account of the thus obtained fractional Fick's law yields fractional dispersion equation for smooth densities. The method adapts to domains, limited by boundaries possibly implying non-trivial modifications to this equation.

The cycle expansion of the thermodynamical zeta function for the Lyapunov exponent of a product of random matrices typically converges exponentially with the maximal cycle length (Mainieri 1992 Phys. Rev. Lett.68 1965). In this paper we show that the convergent exponents are given by the spectrum of a properly defined evolution operator, which describes how a steady distribution of vector direction is established under the action of random matrices. The exponential decay terms are automatically eliminated in the cycle expansion of the spectral determinant, which greatly accelerates the convergence provided all matrix elements are positive numbers. As a marginal case, the random Fibonacci series is studied in detail, and it is shown that this method is helpful.

The simple cubic lattice Green function is investigated, where {ℓ₁, ℓ₂, ℓ₃} is a set of integers, w = w₁ + iw₂ is a complex variable and α is a real parameter in the interval (0, ∞). In particular, a new and direct method is used to prove that G(2n, n, n; α, w) and G(n, n, n; 1, w) can be expressed in terms of a product of two ₂F₁ hypergeometric functions, where n = 0, 1, 2, .... In earlier work, Delves and Joyce obtained these ₂F₁ product forms by solving complicated fourth-order linear differential equations of the Fuchsian type. In this paper Fourier generating functions and a known Lie group addition theorem play crucial roles in the derivation of the product forms. Many-term recurrence relations are also derived for G(2n, n, n; α, w) and G(n, n, n; α, w).

The position and momentum information entropies of D-dimensional quantum systems with central potentials, such as the isotropic harmonic oscillator and the hydrogen atom, depend on the entropies of the (hyper)spherical harmonics. In turn, these entropies are expressed in terms of the entropies of the Gegenbauer (ultraspherical) polynomials C^(λ)_n(x), the parameter λ being either an integer or a half-integer number. Up till now, however, the exact analytical expression of the entropy of Gegenbauer polynomials of arbitrary degree n has only been obtained for the particular values of the parameter λ = 0, 1, 2. Here we present a novel approach to the evaluation of the information entropy of Gegenbauer polynomials, which makes use of trigonometric representations for these polynomials and complex integration techniques. Using this method, we are able to find the analytical expression of the entropy for arbitrary values of both n and .

Symmetries have been used with great success to determine solutions of differential equations with which they are associated. In addition to reducing the order of the equation, one can also use the Lie algebra of the symmetries to transform the equation into 'canonical form'. The canon, in this case, is determined by the 'standard' realization of the Lie algebra. Following a comment in Olver P J (1995 Equivalence, Invariants and Symmetry (Cambridge: Cambridge University Press)), we conjecture that while Lie algebras may have non-equivalent realizations in the usual (point transformation) sense, all realizations of the same Lie algebra are equivalent when considered on the appropriate ordered jet space. We show how this result can have useful implications for ordinary differential equations, including linearization for equations thought to be inherently nonlinear.

In this paper we consider embedded eigenvalues of a Schrödinger Hamiltonian in a waveguide induced by a symmetric perturbation. It is shown that these eigenvalues become unstable and turn into resonances after twisting of the waveguide. The perturbative expansion of the resonance width is calculated for weakly twisted waveguides and the influence of the twist on resonances in a concrete model is discussed in detail.

We deal with polynomial vector fields of the form ∑^d_k=1P_k(x₁, ..., x_d)∂/∂x_k with d ⩾ 2. Let m_k be the degree of P_k. We call (m₁, ..., m_d) the degree of . We provide the best upper bounds for the polynomial vector field in the function of its degree (m₁, ..., m_d) of (1) the maximal number of invariant hyperplanes, (2) the maximal number of parallel invariant hyperplanes, and (3) the maximal number of invariant hyperplanes that pass through a single point. Moreover, if m_i = m, i = 1, ..., d, we show that these best upper bounds are reached taking into account the multiplicity of the invariant hyperplanes.

In this paper, combinatorial aspects of normal ordering of arbitrary words in the creation and annihilation operators of the q-deformed boson are discussed. In particular, it is shown how, by introducing appropriate q-weights for the associated 'Feynman diagrams', the normally ordered form of a general expression in the creation and annihilation operators can be written as a sum over all q-weighted Feynman diagrams, representing Wick's theorem in the present context.

We consider the problem of two-dimensional projectile motion in which the resistance acting on an object moving in air is proportional to the square of the velocity of the object (quadratic resistance law). It is well known that the quadratic resistance law is valid in the range of the Reynolds number: 1 × 10³ ∼ 2 × 10⁵ (for instance, a sphere) for practical situations, such as throwing a ball. It has been considered that the equations of motion of this case are unsolvable for a general projectile angle, although some solutions have been obtained for a small projectile angle using perturbation techniques. To obtain a general analytic solution, we apply Liao's homotopy analysis method to this problem. The homotopy analysis method, which is different from a perturbation technique, can be applied to a problem which does not include small parameters. We apply the homotopy analysis method for not only governing differential equations, but also an algebraic equation of a velocity vector to extend the radius of convergence. Ultimately, we obtain the analytic solution to this problem and investigate the validation of the solution.

A class of bound-state problems which represents the coupling of a three-level atom with a two-dimensional system involving two shape-invariant potentials was introduced in a previous paper. In this paper, we considered second-order parasupersymmetric quantum-mechanical models and two possible kinds for the coupling Hamiltonian (linear and nonlinear in the potential ladder operators). In the present paper, using an algebraic formulation for shape-invariant potential systems, we study the quantum dynamics of these coupled systems and obtain the temporal behaviour of some dynamical variables related with the atom and the coupling potentials. An application is given for a couple of shape-invariant potentials (Pöschl–Teller + self-similar potentials).

In this work, we exploit the Grover operator for the weight analysis of a Boolean function, specifically to solve the weight-decision problem. The weight w is the fraction of all possible inputs for which the output is 1. The goal of the weight-decision problem is to find the exact weight w from the given two weights w₁ and w₂ satisfying a general weight condition as w₁ + w₂ = 1 and 0 < w₁ < w₂ < 1. First, we propose a limited weight-decision algorithm where the function has another constraint: a weight is in for integer k. Second, by changing the phases in the last two Grover iterations, we propose a general weight-decision algorithm which is free from the above constraint. Finally, we show that when our algorithm requires O(k) queries to find w with a unit success probability, any classical algorithm requires at least Ω(k²) queries for a unit success probability. In addition, we show that our algorithm requires fewer queries to solve this problem compared with the quantum counting algorithm.

We show that the state with the highest known average two-particle von Neumann entanglement entropy proposed by Sudbery and one of the authors gives a local maximum of this entropy. We also show that this is not the case for an alternative highly entangled state proposed by Brown et al.

Entanglement in the ground state of the XY model on the infinite chain can be measured by the von Neumann entropy of a block of neighbouring spins. We study a double scaling limit: the size of the block is much larger than 1 but much smaller than the length of the whole chain. The entropy of the block has an asymptotic limit in the gapped regimes. We study this limiting entropy as a function of the anisotropy and of the magnetic field. We identify its minima at product states and its divergencies at the quantum phase transitions. We find that the curves of constant entropy are ellipses and hyperbolas, and that they all meet at one point (essential critical point). Depending on the approach to the essential critical point, the entropy can take any value between 0 and ∞. In the vicinity of this point, small changes in the parameters cause large change of the entropy.

The factorization approach for complex Hamiltonians has been used to obtain the exactly solvable nonlocal variant of -symmetric local potentials. The formalism is used to obtain exact eigenvalues and eigenfunctions of the nonlocal -symmetric Scarf potential.

We consider general quantum random walks in a d-dimensional half-space. We first obtain a path integral formula for general quantum random walks in a d-dimensional space. Our path integral formula is valid for general quantum random walks on Cayley graphs as well. Then the path integral formula is applied to obtain the scaling limit of the exit distribution, the expectation of exit time and the asymptotic behaviour of the exit probabilities, for general quantum random walks in a half-space under some conditions on amplitude functions. The conditions are shown to be satisfied by both the Hadamard and Grover quantum random walks in two-dimensional half-spaces. For the two-dimensional case, we show that the critical exponent for the scaling limit of the hitting distribution is 1 as the lattice spacing tends to zero, i.e. the natural magnitude of the hitting position is of order O(1) if the lattice spacing is set to be 1/n. We also show that the rate of convergence of the total hitting probability has lower bound n⁻² and upper bound n⁻²⁺ for any > 0. For a quantum random walk with a fixed starting point, we show that the probability of hitting times at the hyperplane decays faster than that of the classical random walk. In both one and two dimensions, given the event of a hit, the conditional expectation of hitting times is finite, in contrast to being infinite for the classical case. In the one-dimensional case, we also obtain an exact order of the probability distribution of the hitting time at 0.

We propose a mechanism for fixing the velocity of relativistic solitons based on the breaking of the Lorentz symmetry of the sine-Gordon (SG) model. The proposal is first elaborated for a molecular chain model as the simple pendulum limit of a double pendulums chain. It is then generalized to a full class of two-dimensional field theories of the sine-Gordon type. From a phenomenological point of view, the mechanism allows one to select the speed of a SG soliton just by tuning elastic couplings constants and kinematical parameters. From a fundamental, field-theoretical point of view we show that the characterizing features of relativistic SG solitons (existence of conserved topological charges and stability) may be still preserved even if the Lorentz symmetry is broken and a soliton of a given speed is selected.

We derive a formula connecting in any dimension the Green function on the (D + 1)-dimensional Euclidean Rindler space and the one for a minimally coupled scalar field with a mass m in the D-dimensional hyperbolic space. The relation takes a simple form in the momentum space where the Green functions are equal at the momenta (p₀, p) for Rindler and for hyperbolic space with a simple additive relation between the squares of the mass and the momenta. The formula has applications to finite temperature Green functions, Green functions on the cone and on the (compactified) Milne spacetime. Analytic continuations and interacting quantum fields are briefly discussed.

We consider the stochastic quantization method for scalar fields defined in a curved manifold and also in a flat spacetime with event horizon. The two-point function associated with a massive self-interacting scalar field is evaluated, up to the first-order level in the coupling constant λ, for the case of an Einstein and also a Rindler Euclidean metric, respectively. Its value for the asymptotic limit of the Markov parameter τ → ∞ is exhibited. The divergences therein are taken care of by employing a covariant stochastic regularization, where all the symmetries of the original theory are preserved.

The known extended algebras associated with p-branes are shown to be generated as topological charge algebras of the standard p-brane actions. A representation of the charges in terms of superspace forms is constructed. The charges are shown to be the same in standard/extended superspace formulations of the action.

The MTK equations (Martyna G J, Tobias D J and Klein M L 1994 J. Chem. Phys.101 4177–89), which simulate the constant-pressure, constant-temperature NPT ensemble, have been modified to simulate an anisotropic pressure along a single coordinate axis, thus rendering the NP_zzT ensemble. The necessary theory of non-Hamiltonian systems is briefly reviewed in order to analytically prove that the proposed equations indeed sample the desired ensemble. A previously derived geometric integrator for the MTK equations is modified to take into account the anisotropic pressure and volume fluctuations. We choose a Lennard-Jones fluid as an illustrative example. The density distribution function, as well as various thermodynamic and interfacial properties of the model system in a liquid–vapour coexistence state, was computed to test the robustness of the proposed equations of motion to simulate the NP_zzT ensemble.

The asymptotic evaluation and expansion of the Keesom integral, K(a), is discussed at some length in Battezzati and Magnasco (2004 J. Phys. A: Math. Gen.37 9677; 2005 J. Phys. A: Math. Gen.38 6715). Here, using standard identities, it is shown that this triple integral can be reduced to a single integral from which the asymptotic behaviour is readily obtained using Laplace's method.

Reference [15] should read:

[15] Mihalache D, Bertolotti M and Sibilia C 1989 Progress in Optics XXVII ed E Wolf (Amsterdam: Elsevier) pp 229--313.

Thibault Damour is a theoretical physicist, and a member of the French Academy of Sciences. This book is the translation, by Eric Novak, of the original French Si Einstein m'etait conté (Le Cherche Midi, 2005). It is neither a book of theoretical physics nor a biography of Einstein. It is not a book of history nor philosophy of science. In Damour's words it was written to encourage the reader to share with Einstein `those times when he understood some part of the hidden order of the universe'. It is a relatively short book, written in a very fluent style, but it deals with all the major problems and achievements of Einstein's works.

Starting from special relativity, it continues with general relativity, quantum theories, unified field theory and a brief overview of the actual research related to Einstein's legacy. It is essentially a popular science book with some related exploration in history and philosophy to interpret physical theories. The most important problem discussed by Damour is the nature of time. On this subject, there is a very interesting short paragraph (pp 33--35) dedicated to the reception of the relativity idea by the great writer Marcel Proust and its counterpart within À la Recherche du Temps Perdu. A correct discussion of the implications of a relativistic time should imply the distinction of the different possible interpretations of this concept. Damour seems to conclude that only one interpretation is possible: `time does not exist', flowing of time is an illusion. One has to know that Einstein's ideas on time were related to Spinoza's perspective of a knowledge sub specie aeternitatis. However, other interpretations are possible and are related to the idea of time as an actuality. Damour speaks about the controversy between Einstein and Bergson, but Bergson is considered as a philosopher who did not understand relativity.

This philosophical problem of relativistic time is indeed related to a historical problem briefly discussed by Damour (pp 17--21, 48--52 and related endnotes): had Henri Poincaré constructed a special relativistic dynamics before Einstein? There is a long debate on this subject in the literature. Damour's answer is negative and his conclusions seem related to the conservation of a myth of Einstein, that is, the rise of special relativity is considered as a creatio ex nihilo within Einstein's mind and Einstein is considered as the only genius able to conceive the relativity of time. Poincaré's texts are undervalued and misunderstood by Damour's cutting quotations from their context. Damour never quotes La Science et l'Hypothèse (1902): we know it was read by Einstein and here Poincaré first (within chapters already published as separate papers in 1900) stated the relativity of time and of simultaneity. Damour never quotes Poincaré's paper published on 5 June 1905, La dynamique de l'èlectron, which presents the first relativistic dynamics, invariant by Lorentz transformations. Poincaré's (July 1905) introduction of a quadrimensional space-time is considered by Damour only a mathematical artifice (p 51) and Damour never said that Minkowski took this idea from Poincaré! Poincaré's interpretation of relativistic time implies that it is not an illusion but a complex net of different real flows related to different processes.

Poincaré and Einstein had different conceptions of Nature at the root of special relativity: respectively an electromagnetic conception (Poincaré) and a semi-mechanist one (Einstein). Thus, the (philosophical) meaning of relativity can be very different from the one presented by Damour. Furthermore, Damour accepts Kantian philosophy as a key to understanding relativity and quantum theories. This perspective seems to me very anachronistic and based on a misunderstanding: an interpretation of 20th century physical theories (relativity and quantum physics) is given within the framework of an 18th century philosophical perspective, created to give a foundation to Newton's theory. Relativity and quantum physics imply a breakdown of Kantian philosophy (see, for instance, G Bachelard's La Philosophie du Non). Relativity of space and time was considered possible only by overcoming the epistemological obstacle of Kantian idealistic foundation of Euclidean geometry and of Newton's absolute space and time. Relativity and quantum theories turn up not only the hierarchy between mathematics and physics, but also between epistemology (and logic) and physics: quantum physics implies not only a new conception of an indeterminate and unpredictable Nature, but a quantum logic too, that is, it implies a change in our way of thinking and knowing. When will the revolutionary impact of 20th century physics be reduced (by physicists themselves) to an already given philosophical framework?

The hallmark of a good book of problems is that it allows you to become acquainted with an unfamiliar topic quickly and efficiently. The Quantum Mechanics Solver fits this description admirably. The book contains 27 problems based mainly on recent experimental developments, including neutrino oscillations, tests of Bell's inequality, Bose–Einstein condensates, and laser cooling and trapping of atoms, to name a few. Unlike many collections, in which problems are designed around a particular mathematical method, here each problem is devoted to a small group of phenomena or experiments. Most problems contain experimental data from the literature, and readers are asked to estimate parameters from the data, or compare theory to experiment, or both. Standard techniques (e.g., degenerate perturbation theory, addition of angular momentum, asymptotics of special functions) are introduced only as they are needed. The style is closer to a non-specialist seminar rather than an undergraduate lecture. The physical models are kept simple; the emphasis is on cultivating conceptual and qualitative understanding (although in many of the problems, the simple models fit the data quite well). Some less familiar theoretical techniques are introduced, e.g. a variational method for lower (not upper) bounds on ground-state energies for many-body systems with two-body interactions, which is then used to derive a surprisingly accurate relation between baryon and meson masses. The exposition is succinct but clear; the solutions can be read as worked examples if you don't want to do the problems yourself. Many problems have additional discussion on limitations and extensions of the theory, or further applications outside physics (e.g., the accuracy of GPS positioning in connection with atomic clocks; proton and ion tumor therapies in connection with the Bethe–Bloch formula for charged particles in solids). The problems use mainly non-relativistic quantum mechanics and are organised into three sections: Elementary Particles, Nuclei and Atoms; Quantum Entanglement and Measurement; and Complex Systems. The coverage is not comprehensive; there is little on scattering theory, for example, and some areas of recent interest, such as topological aspects of quantum mechanics and semiclassics, are not included. The problems are based on examination questions given at the École Polytechnique in the last 15 years. The book is accessible to undergraduates, but working physicists should find it a delight.

This book is a new volume of a series designed to introduce the curious reader to anything from ancient Egypt and Indian philosophy to conceptual art and cosmology. Very handy in pocket size, Chaos promises an introduction to fundamental concepts of nonlinear science by using mathematics that is `no more complicated than X=2.

Anyone who ever tried to give a popular science account of research knows that this is a more challenging task than writing an ordinary research article. Lenny Smith brilliantly succeeds to explain in words, in pictures and by using intuitive models the essence of mathematical dynamical systems theory and time series analysis as it applies to the modern world. In a more technical part he introduces the basic terms of nonlinear theory by means of simple mappings. He masterly embeds this analysis into the social, historical and cultural context by using numerous examples, from poems and paintings over chess and rabbits to Olbers' paradox, card games and `phynance'.

Fundamental problems of the modelling of nonlinear systems like the weather, sun spots or golf balls falling through an array of nails are discussed from the point of view of mathematics, physics and statistics by touching upon philosophical issues. At variance with Laplace's demon, Smith's 21st century demon makes `real world' observations only with limited precision. This poses a severe problem to predictions derived from complex chaotic models, where small variations of initial conditions typically yield totally different outcomes. As Smith argues, this difficulty has direct implications on decision-making in everyday modern life. However, it also asks for an inherently probabilistic theory, which somewhat reminds us of what we are used to in the microworld.

There is little to criticise in this nice little book except that some figures are of poor quality thus not really reflecting the beauty of fractals and other wonderful objects in this field. I feel that occasionally the book is also getting a bit too intricate for the complete layman, and experts may not agree on all details of the more conceptual discussions.

Altogether I thoroughly enjoyed reading this book. It was a happy companion while travelling and a nice bedtime literature. It is furthermore an excellent reminder of the `big picture' underlying nonlinear science as it applies to the real world. I will gladly recommend this book as background literature for students in my introductory course on dynamical systems. However, the book will be of interest to anyone who is looking for a very short account on fundamental problems and principles in modern nonlinear science.