This book offers an informal, easy-to-understand account of topics in modern physics and mathematics. The focus is, in particular, on statistical mechanics, soft matter, probability, chaos, complexity, and models, as well as their interplay. The book features 28 key entries and it is carefully structured so as to allow readers to pursue different paths that reflect their interests and priorities, thereby avoiding an excessively systematic presentation that might stifle interest. While the majority of the entries concern specific topics and arguments, some relate to important protagonists of science, highlighting and explaining their contributions. Advanced mathematics is avoided, and formulas are introduced in only a few cases. The book is a user-friendly tool that nevertheless avoids scientific compromise. It is of interest to all who seek a better grasp of the world that surrounds us and of the ideas that have changed our perceptions. Il libro prova a colmare almeno in parte quel vuoto lasciato dalla divulgazione scientifica mainstream, interessata principalmente a dare risalto agli aspetti più sensazionalistici e bizzarri delle scoperte scientifiche, svelando il fascino presente in argomenti che non vengono solitamente discussi nei libri di divulgazione e nelle vite di scienziati poco conosciuti al grande pubblico, ma che hanno posto le basi per la scienza come la conosciamo ora.

Standard reaction-diffusion systems are characterized by infinite velocities and no persistence in the movement of individuals, two conditions that are violated when considering living organisms. Here we consider a discrete particle model in which individuals move following a persistent random walk with finite speed and grow with logistic dynamics. We show that, when the number of individuals is very large, the individual-based model is well described by the continuous reactive Cattaneo equation (RCE), but for smaller values of the carrying capacity important finite-population effects arise. The effects of fluctuations on the propagation speed are investigated both considering the RCE with a cutoff in the reaction term and by means of numerical simulations of the individual-based model. Finally, a more general Lévy walk process for the transport of individuals is examined and an expression for the front speed of the resulting traveling wave is proposed.

Understanding the conditions ensuring the persistence of a population is an issue of primary importance in population biology. The first theoretical approach to the problem dates back to the 1950s with the Kierstead, Slobodkin, and Skellam (KiSS) model, namely a continuous reaction-diffusion equation for a population growing on a patch of finite size L surrounded by a deadly environment with infinite mortality, i.e., an oasis in a desert. The main outcome of the model is that only patches above a critical size allow for population persistence. Here we introduce an individual-based analog of the KiSS model to investigate the effects of discreteness and demographic stochasticity. In particular, we study the average time to extinction both above and below the critical patch size of the continuous model and investigate the quasistationary distribution of the number of individuals for patch sizes above the critical threshold.

This book reviews the basic ideas of the Law of Large Numbers with its consequences to the deterministic world and the issue of ergodicity. Applications of Large Deviations and their outcomes to Physics are surveyed. The book covers topics encompassing ergodicity and its breaking and the modern applications of Large deviations to equilibrium and non-equilibrium statistical physics, disordered and chaotic systems, and turbulence.

This contribution aims at introducing the topics of this book. We start with a brief historical excursion on the developments from the law of large numbers to the central limit theorem and large deviations theory. The same topics are then presented using the language of probability theory. Finally, some applications of large deviations theory in physics are briefly discussed through examples taken from statistical mechanics, dynamical and disordered systems.

We investigate invasions from a biological reservoir to an initially empty, heterogeneous habitat in the presence of advection. The habitat consists of a periodic alternation of favorable and unfavorable patches. In the latter the population dies at fixed rate. In the former it grows either with the logistic or with an Allee effect type dynamics, where the population has to overcome a threshold to glow. We study the conditions for successful invasions and the speed of the invasion process, which is numerically and analytically investigated in several limits. Generically advection enhances the downstream invasion speed but decreases the population size of the invading species, and can even inhibit the invasion process. Remarkably, however, the rate of population increase, which quantifies the invasion efficiency, is maximized by an optimal advection velocity. In models with Allee effect, differently from the logistic case, above a critical unfavorable patch size the population localizes in a favorable patch, being unable to invade the habitat. However, we show that advection, when intense enough, may activate the invasion process.

Spatial and velocity statistics of heavy point-like particles in incompressible, homogeneous, and isotropic three-dimensional turbulence is studied by means of direct numerical simulations at two values of the Taylor-scale Reynolds number Re-lambda similar to 200 and Re-lambda similar to 400, corresponding to resolutions of 512(3) and 2048(3) grid points, respectively. Particles Stokes number values range from St approximate to 0.2 to 70. Stationary small-scale particle distribution is shown to display a singular -multifractal- measure, characterized by a set of generalized fractal dimensions with a strong sensitivity on the Stokes number and a possible, small Reynolds number dependency. Velocity increments between two inertial particles depend on the relative weight between smooth events - where particle velocity is approximately the same of the fluid velocity-, and caustic contributions - when two close particles have very different velocities. The latter events lead to a non-differentiable small-scale behaviour for the relative velocity. The relative weight of these two contributions changes at varying the importance of inertia. We show that moments of the velocity difference display a quasi bi-fractal-behavior and that the scaling properties of velocity increments for not too small Stokes number are in good agreement with a recent theoretical prediction made by K. Gustavsson and B. Mehlig arXiv:1012.1789v1 [physics.fludyn], connecting the saturation of velocity scaling exponents with the fractal dimension of particle clustering.

Coalescence growth of droplets is a fundamental process for liquid cloud evolution. The initiation of collisions and coalescence occurs when a few droplets become large enough to fall. Gravitational collisions represent the most efficient mechanism for multi-disperse solutions, when droplets span a large variety of sizes. However, turbulence provides another mechanism for droplets coalescence, taking place also in the case of uniform condensational growth leading to narrow droplet-size spectra. We consider the problem of estimating the rate of collisions of small droplets dispersed in a highly turbulent medium. The problem is investigated by means of high-resolution direct numerical simulations of a three-dimensional turbulent flow, seeded with inertial particles, up to resolutions of 2048^3 grid points. Rate of collision is estimated in terms of the probability to find particles at close positions, and of the statistics of particles velocity. In particular, we show that the statistics of velocity differences between inertial particles suspended in an incompressible turbulent flow is extremely intermittent. When particles are separated by distances of the order of their diameter, the competition between quiet regular regions and multivalued caustics leads to a quasi bi-fractal behavior of the particle velocity statistics, with high-order moments bringing the signature of caustics. This results in large probabilities that close particles have important velocity differences. Together with preferential concentration of particles in low-vorticity regions, caustics contribute to speed-up collisions between inertial particles. Implications for the early stage of rain droplets formation are discussed.

The statistics of velocity differences between pairs of heavy inertial point particles suspended in an incompressible turbulent flow is studied and found to be extremely intermittent. The problem is particularly relevant to the estimation of the efficiency of collisions among heavy particles in turbulence. We found that when particles are separated by distances within the dissipative subrange, the competition between regions with quiet regular velocity distributions and regions where very close particles have very different velocities (caustics) leads to a quasi bi-fractal behaviour of the particle velocity structure functions. Contrastingly, we show that for particles separated by inertial-range distances, the velocity-difference statistics can be characterized in terms of a local roughness exponent, which is a function of the scale-dependent particle Stokes number only. Results are obtained from high-resolution direct numerical simulations up to 2048 3 collocation points and with millions of particles for each Stokes number.

Relative dispersion of tracers - i.e. very small, neutrally buoyant particles-, is particularly efficient in incompressible turbulent flows. Due to the non smooth behaviour of velocity differences in the inertial range, the separation distance between two trajectories, R(t)=X1(t)-X2(t) , grows as a power of time superdiffusively, R2(t)t3 , as first observed by L.F. Richardson [1]. This now well established result has no counterpart in the theory of heavy particle suspensions, namely finite-size particles with a mass density much larger that of the carrier fluid. The complete knowledge of particle properties of mixing in turbulent flows -yet an open problem-, is of great importance in cloud physics, or in estimating pollutant dispersion for hazardous safety purposes.

We present results from recent direct numerical simulations of heavy particle transport in homogeneous, isotropic, fully developed turbulence, with grid resolution up to 5123 and R? ? 185. By following the trajectories of millions of particles with different Stokes numbers, St ? [0.16 : 3.5], we are able to characterize in full detail the statistics of particle acceleration. We focus on the probability density function of the normalised acceleration a/arms and on the behaviour of their rootmean-squared acceleration arms as a function of both St and R?. We explain our findings in terms of two concurrent mechanisms: particle clustering, very effective for small St, and filtering induced by finite particle response time, taking over at larger St.

We present a collection of eight data sets from state-of-the-art experiments and numerical simulations on turbulent velocity statistics along particle trajectories obtained in different flows with Reynolds numbers in the range Rlambda[is-an-element-of][120:740]. Lagrangian structure functions from all data sets are found to collapse onto each other on a wide range of time lags, pointing towards the existence of a universal behavior, within present statistical convergence, and calling for a unified theoretical description. Parisi-Frisch multifractal theory, suitably extended to the dissipative scales and to the Lagrangian domain, is found to capture the intermittency of velocity statistics over the whole three decades of temporal scales investigated here.

A detailed comparison between data from experimental measurements and numerical simulations of Lagrangian velocity structure functions in turbulence is presented. Experimental data, at Reynolds number ranging from R? = 350 to R? = 815, are obtained in a swirling water flow between counter-rotating baffled disks. Direct numerical simulations (DNS) data, up to R? = 284, are obtained from a statistically homogeneous and isotropic turbulent flow. By integrating information from experiments and numerics, a quantitative understanding of the velocity scaling properties over a wide range of time scales and Reynolds numbers is achieved. To this purpose, we discuss in detail the importance of statistical errors, anisotropy effects, and finite volume and filter effects, finite trajectory lengths. The local scaling properties of the Lagrangian velocity increments in the two data sets are in good quantitative agreement for all time lags, showing a degree of intermittency that changes if measured close to the Kolmogorov time scales or at larger time lags. This systematic study resolves apparent disagreement between observed experimental and numerical scaling properties.

Distributions of heavy particles suspended in incompressible turbulent flows are investigated by means of high-resolution direct numerical simulations. It is shown that particles form fractal clusters in the dissipative range, with properties independent of the Reynolds number. Conversely, in the inertial range, the particle distribution is not scale-invariant. It is however shown that deviations from uniformity depends only on a rescaled contraction rate, and not on the local Stokes number given by dimensional analysis. Particle distribution is characterized by voids spanning all scales of the turbulent flow; their signature on the coarse-grained mass probability distribution is an algebraic behavior at small densities.

Particles with different density from the advecting turbulent fluids cluster due to the different response of light and heavy particles to turbulent fluctuations. This study focuses on the quantitative characterization of the segregation of dilute polydisperse inertial particles evolving in turbulent flow, as obtained from direct numerical simulation of homogeneous isotropic turbulence. We introduce an indicator of segregation amongst particles of different inertia and/or size, from which a length scale r(seg), quantifying the segregation degree between two particle types, is deduced.

Spatial distributions of heavy particles suspended in an incompressible isotropic and homogeneous turbulent flow are investigated by means of high resolution direct numerical simulations. In the dissipative range, it is shown that particles form fractal clusters with properties independent of the Reynolds number. Clustering is there optimal when the particle response time is of the order of the Kolmogorov time scale tau(eta). In the inertial range, the particle distribution is no longer scale invariant. It is, however, shown that deviations from uniformity depend on a rescaled contraction rate, which is different from the local Stokes number given by dimensional analysis. Particle distribution is characterized by voids spanning all scales of the turbulent flow; their signature in the coarse-grained mass probability distribution is an algebraic behavior at small densities.

Lyapunov exponents of heavy particles and tracers advected by homogeneous and isotropic turbulent flows are investigated by means of direct numerical simulations. For large values of the Stokes number, the main effect of inertia is to reduce the chaoticity with respect to fluid tracers. Conversely, for small inertia, a counterintuitive increase of the first Lyapunov exponent is observed. The flow intermittency is found to induce a Reynolds number dependency for the statistics of the finite-time Lyapunov exponents of tracers. Such intermittency effects are found to persist at increasing inertia. (c) 2006 American Institute of Physics.

The Lagrangian statistics of heavy particles and of fluid tracers transported by a fully developed turbulent flow are investigated by means of high resolution direct numerical simulations. The Lagrangian velocity structure functions are measured in a time range spanning about three decades, from a tenth of the Kolmogorov time scale, tau(eta), up to a few large-scale eddy turnover times. Strong evidence is obtained that fluid tracer statistics are contaminated in the time range tau is an element of[1:10]tau(eta) by a bottleneck effect due to vortex filament. This effect is found to be significantly reduced for heavy particles which are expelled from vortices by inertia. These findings help in clarifying the results of a recent study by H. Xu [Phys. Rev. Lett. 96, 024503 (2006)], where differences between experimental and numerical results on scaling properties of fluid tracers were reported. (c) 2006 American Institute of Physics.