Meccanica statistica di non equilibrio: trasporto e diffusione in sistemi complessi
Relatore: Stefano Lepri
Gli argomenti di tesi sono di tipo teorico e riguardano il comportamento di sistemi fuori equilibrio. Una preparazione di base in meccanica statistica e’ essenziale. La conoscenza di un linguaggio di programmazione per lo sviluppo di codici di simulazione e’ auspicabile.
Il moto di una particella soggetta a forze aleatorie e’ descritto dalle leggi del moto browniano (diffusione).… Read the rest
Roberto Livi and Paolo Politi
Nonequilibrium Statistical Physics – A modern perspective
(Cambridge University Press, 2017)
Statistical mechanics has been proven to be successful at describing physical systems at thermodynamic equilibrium. Since most natural phenomena occur in nonequilibrium conditions, the present challenge is to find suitable physical approaches for such conditions: this book provides a pedagogical pathway that explores various perspectives.… Read the rest
In all languages, rules have exceptions in the form of irregularities. Since rules make a language efficient, the persistence of irregularity is an anomaly. How language systems become rule governed, how and why they sustain exceptions to rules? Frequent words are unlikely to change over time (e.g., frequent verbs tend to maintain an irregular past tense form). What is the role of frequency in maintaining exceptions to rules?… Read the rest
Social networks have empirically been found to be assortative (i.e., the degree of neighboring nodes are positively correlated), while other networks (e.g., technological, biological) show the opposite pattern (disassortative). Why is that so?
How do these patterns change in signed networks, where relations indicate trust/distrust, friendship/enmity? Do individuals who dislike many others tend to dislike each other, or do they dislike those who dislike only very few others?… Read the rest
The intermittent and self-similar fluctuations displayed by a slow crack during the propagation in a heterogeneous medium can be quantitatively described by an extension of a classical statistical model for fracture. The model yields the correct dynamical and morphological scaling, and allows to demonstrate that the scale invariance originates from the presence of a non-equilibrium, reversible, critical transition which, in the presence of dissipation, gives rise to self-organized critical behaviour.… Read the rest
Synchronization is a long known phenomenon dating back to Huygens experiments who observed that suspending two pendula “…in the same wooden beam, the motions of each pendulum in opposite swings were so much in agreement that they never receded the least bit from each other and the sound of each was always heard simultaneously“. In spite of the early discovery, the phenomenon was fully understood much later with the experiments and theoretical analysis of E.… Read the rest
In recent years it has become widely recognized that many large-scale phenomena observed in social systems are the “macroscopic” complex effect of the “microscopic” simple behavior of a large number of interacting agents. This has led social scientists to the introduction of elementary models of social behavior (cellular automata, agent-based models). Many of these models are somehow relatives of models that have been introduced in modern traditional statistical physics, and it is natural to approach them using the same concepts and tools that have been successfully applied in physics.… Read the rest
In social phenomena every individual interacts with a limited number of peers, usually negligible as compared with the total number of people in the system. In spite of that, human societies are characterized by stunning global regularities. There are transitions from disorder to order, like the spontaneous emergence of a common language/culture or the creation of consensus about a specific topic.… Read the rest
There exists an overwhelming diversity of landscapes on Earth. A cornerstone of modern geomorphology came with the realization that all the different features of the terrestrial surface result from the accumulated effect of current geological agents [Lyell, 1830]. This principle established for the first time a qualitative relationship between pattern and process in geology.
More than one century later, fractal geometry gave a theoretical framework able to provide quantitative measures for the patterns of landscapes, which were identified in a first approximation as self-similar, and triggered the research on mechanistic and theoretical models to identify the underlying constructive rules responsible for their appearance.… Read the rest
Even before the identification of galaxies as stellar systems, astronomers have classified them based on their visual appearance. Galaxies in the local universe can organized in a sequence of morphologies (e.g. the Hubble sequence) which must be the result of the specific processes that originated them.
The relative roles over cosmic time of processes such as the merging of dark matter haloes, dissipation, starburst, feedback, active galactic nuclei (AGN) activity, etc.,… Read the rest
A globular cluster (GC) is a spherical collection of stars that orbits a galactic core as a satellite.
Globular clusters are very tightly bound by gravity, which gives them their spherical shapes and relatively high stellar densities toward their centers. The name of this category of star cluster is derived from the Latin globulus—a small sphere. Globular clusters are fairly common; there are about 158 currently known globular clusters in the Milky Way, with perhaps 10–20 more undiscovered.… Read the rest
The wide applicability of the random walks (RW) to natural phenomena relies just on the possibility to introduce appropriate generalizations on the probabilistic nature of displacements. A straightforward generalization is realized by assuming correlations in displacements to obtain the so called correlated random walks (CRW).
This possibility extends also to a set of particles distributed in space leading to the definition of spatially correlated random walks.… Read the rest
A System with long-range interactions is characterized by an inter-particle potential which decays at large distances with a power law exponent which is smaller than the dimension of the embedding space. Classical examples include for instance: self-gravitating systems, unscreened Coulomb systems, ion beams, wave-particle systems of relevance to plasma physics and others.
The behaviour of the above mentioned systems is interesting both from the point of view of stable (or metastable) states, because equilibrium statistical mechanics shows new types of phase transitions and cases of ensemble inequivalence, and from the dynamical point of view, because they display peculiar fast relaxation followed by the formation of quasi-stationary states that are related to the underlying Vlasov-like equation.… Read the rest
The most prominent feature of the initial conditions of the matter spatial distribution in the early universe, in standard theoretical models, derived from inflationary mechanisms, is that matter density field presents on large scale super-homogeneous features. This means the following. If one considers the paradigm of uniform distributions, the Poisson process where particles are placed completely randomly in space, the mass fluctuations in a sphere of radius R growths as the volume of the sphere.… Read the rest
In equilibrium statistical mechanics the distinction between short and long range interactions is given by the integrability or not of the pair potential. However for what concerns only the clustering dynamics of a particle distribution under the effect of an attractive pair interaction, it seems by recent works that the distinction is given by the integrability of the pair force instead of the potential.… Read the rest
The discovery that simple deterministic nonlinear systems could display dynamical evolution characterized by a randomness similar to stochastic processes changed very much researchers’ attitude toward determinism and predictability of natural phenomena. Determinstic chaos has been successfully invoked to interpret several irregular behaviors, however its role to the fundaments of statistical physics still remains debated in modern statistical mechanics. In other terms, one is tempted to think that a macroscopic system with chaotic microscopic interactions is more robust with respect to statistical mechanical principles thant the same system with non-chaotic interactions.… Read the rest
In hard magnetic materials, the domain walls movement or even creation is suppressed, and other mechanisms, like domains nucleation and coherent spin rotation enter in the process of magnetization reversal. For these kind of materials a description in terms of spin models is more appropriate. We focused on the non-equilibrium properties of the random field Ising model (RFIM), to describe the competition between quenched disorder and exchange interactions and their effect on the hysteretic behavior.… Read the rest
Problems susceptible to be mathematically represented by stochastic Langevin equations including a multiplicative noise abound not only in physics, but also in biology, ecology, economy, or social sciences. In a broad sense a Langevin equation is said to be multiplicative if the noise amplitude depends on the state variables themselves. In this sense, problems exhibiting absorbing states, i.e. fluctuation-less states in which the system can be trapped, are described by equations whose noise amplitude is proportional to the square-root of the (space and time dependent) activity density, vanishing at the absorbing state.… Read the rest
Both the static and the dynamical behaviour occurring in mean field spin glass models models can be interpreted as consequences of the complex (free) energy landscape that spin glasses have, with many minima, valleys and saddles. Traditionally, much attention has been devoted in the past to the analysis of absolute minima, i.e. equilibrium states. More recently, we have understood that also metastable states, i.e.… Read the rest
Many features predicted by mean field spin glass models, such as the behaviour of susceptibilities and correlation functions or the occurrence of aging and off-equilibrium dynamics, are qualitatively observed in experiments, suggesting that the mean field scenario may hold for finite dimensional systems also. To investigate this hypothesis a field theory for the fluctuations around the mean field solution has been developed.… Read the rest
Close to the glass transition supercooled liquids display an impressive increase of the relaxation time, without any clear sign of growing thermodynamic order, nor correlation length. This is at variance with physical intuition, which suggests that a large relaxation time is always associated to a large correlation length. Even though dynamical length scales were introduced and measured, nothing similar was thought to be possible for thermodynamic lengths.… Read the rest
The term “crackling noise” refers to the signal that some disordered systems produce as a response to an external driving field smoothly changing in time. Due to the presence of disorder, crackling signals are extremely irregular, despite the steady increase of the external forcing. They are typically characterized by a sequence of pulses of very different sizes and durations, separated by quiescence intervals.… Read the rest
The physics of thin and ultra-thin magnetic films has been extensively studied in the recent past, because of its important implications for applications to high frequency devices. Power losses in ferromagnetic materials generally depend on the frequency of the applied field, a phenomenon referred to as dynamic hysteresis. The problem has great importance from a purely theoretical point of view, for the understanding of the dynamics of disordered magnetic systems, which represents a central issue in non–equilibrium statistical mechanics.… Read the rest
During the last decade it has become clear that the topology in many systems, ranging from technological to social to biological, is not well described by regular lattices nor by random graphs. Complex networks, characterized by small-world effects, large connectivity fluctuations, clustering, correlations and other nontrivial features are often a better description of many natural and man-made systems. Since many of such networks describe the topological patterns that mediate various sorts of interactions among nodes, it is natural and interesting to wonder what is the effect of complex topologies on dynamical processes taking place on them.… Read the rest
Until now, the study of human dynamics has been done only qualitatively. Actually, the present possibility to have quantitative data on the kind and nature of social relationships through social networks is driving a rapid change in the field. Thanks to the emergence of detailed datasets that capture human behavior, we can now follow specific human actions in ultimate detail. One of the first measurable quantity with which one can describe the relationship between humans is the timing and order with which we perform specific tasks.… Read the rest
In a nutshell, a random laser is the coherent emission from active stochastic resonators.
In a series of articles around 1966, a Russian scientist V. S. Letokhov, of the Lebedev Physics Institute in Dubna considered the generation of light in the interstellar medium. In the presence of scatterers, as for example dust particles, photons diffuse like neutrons and, if some mechanism (following Letokhov a “negative absorption”) is able to increase their number, a sort of photonic reactor can be realized.… Read the rest
Spin Glasses are dilute magnetic alloys where the interactions between spins are randomly ferromagnetic or anti-ferromagnetic, and are considered as paradigmatic examples of frozen disorder. The presence of disorder (the random interactions) induces frustration and a greater difficulty for the system to find optimal configurations. As a consequence, these systems exhibit non trivial thermodynamic and dynamic properties, different and richer than those observed in their non disordered counterpart.… Read the rest
Collective phenomena in economics, social sciences and ecology are very attractive for statistical physicists, especially in view of the empirical abundance of non-trivial fluctuation patterns and statistical regularities — think of returns in financial markets or of allometric scaling in ecosystems — which pose intriguing theoretical challenges. On an abstract level, the problems at stake are indeed not too different from, say, understanding how spontaneous magnetization may arise in a magnetic system, since what one wants in both cases is to understand how the effects of interactions at the microscopic scale can build up to the macroscopic scale.… Read the rest
Scale-Free Networks are present in a wide list of phenomena. Examples range from the structure of the Internet and that of the WWW (we shall see in the following that they are different systems) to the interconnections between financial agents or species predation in ecological food webs. Thanks to the simplicity of graph theory it is very easy to provide a network description for different systems.… Read the rest
When a binary fluid mixture at the critical concentration is cooled from a high temperature to a sufficiently low temperature (below a critical one), the original homogeneous phase becomes unstable and spontaneously evolves into two phases separated by an interface. As time advances, an out-of-equilibrium process of phase ordering takes place through the formation of domains of a single phase that grows algebraically in time as L(t)~t1/3.… Read the rest
Recently, part of the research activity on turbulence has focused on temporal properties of turbulent statistics which are much less known than the equivalent spatial properties, and are expected to bring information on some of the mechanisms responsible for intermittency in turbulence, for example lagrangian motion is strongly affected by the presence of vortical motion around vortex filaments (see Figure 1).… Read the rest
The ability of efficiently mixing transported substances is one of the most distinctive properties of turbulence. For instance, it is turbulence (induced by the spoon) that allows cream to rapidly invade a cup of coffee, indeed if only molecular diffusion would be at play in the coffee at rest the same process would require many hours! Given the statistical complexity of a turbulent velocity field, it is natural to wonder about the resulting complexity in the statistical features of the transported concentration field of a substance (e.g.… Read the rest
We already mentioned that enhanced mixing is probably one of the most distinguishing feature of turbulence. When a turbulent flow is seeded with particulate matter having a finite size and/or density different from that of the carrier fluid, new features appear. The figure on the left show the instantaneous position particles which are heavier (e.g. water drops in air) resp. lighter (e.g.… Read the rest