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Duccio Fanelli

Duccio Fanelli appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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cond-mat.stat-mechnlin.PSnlin.AOphysics.soc-phBiological Physicscond-mat.dis-nnnlin.CDcond-mat.softphysics.plasm-phPopulations and EvolutionCell Behaviorcond-mat.othermath-phmath.DSmath.MPphysics.chem-phSocial and Information Networksastro-phchao-dyncond-mat.mes-hallMachine Learningmath.CONeurons and Cognitionphysics.acc-phphysics.ao-phphysics.data-an

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Published work

69 published item(s)

preprint2026arXiv

Exact Fixed-Point Constraints in Neural-ODEs with Provable Universality

We introduce a technique that enables Neural-ODEs to approximate arbitrary velocity fields with a priori planted fixed-points. Specifically, a recipe is given to explicitly accommodate for a finite collection of points in the reference multi-dimensional space of the Neural-ODE where the velocity field is exactly equal to zero. In this way, the gradient-based training is rigorously constrained inside the prescribed hypothesis class while leaving the expressive power of the Neural-ODE unaltered. We rigorously prove the universality of the Neural-ODE under any local constraints in the velocity field and give a computationally convenient way of imposing the fixed points. Our method is then tested on two paradigmatic physical models.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Theory of synchronisation and pattern formation on time varying networks

Synchronisation and pattern formation have been intensely addressed for systems evolving on static networks. Extending the study to include the inherent ability of the network to adjust over time proved cumbersome and led to conclusions which lack of generality, as relying on peculiar assumptions. Here, the master stability formalism is extended to account, in a thoroughly general prospect, for the additional contributions as stemming from the time evolution of the underlying network. The theory is successfully challenged against two illustrative testbeds, which can be respectively ascribed to synchronisation and Turing settings.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Mobility-based prediction of SARS-CoV-2 spreading

The rapid spreading of SARS-CoV-2 and its dramatic consequences, are forcing policymakers to take strict measures in order to keep the population safe. At the same time, societal and economical interactions are to be safeguarded. A wide spectrum of containment measures have been hence devised and implemented, in different countries and at different stages of the pandemic evolution. Mobility towards workplace or retails, public transit usage and permanence in residential areas constitute reliable tools to indirectly photograph the actual grade of the imposed containment protocols. In this paper, taking Italy as an example, we will develop and test a deep learning model which can forecast various spreading scenarios based on different mobility indices, at a regional level. We will show that containment measures contribute to "flatten the curve" and quantify the minimum time frame necessary for the imposed restrictions to result in a perceptible impact, depending on their associated grade.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Prestrain-induced contraction in 1D random elastic chains

Prestrained elastic networks arise in a number of biological and technological systems ranging from the cytoskeleton of cells to tensegrity structures. To understand the response of such a network as a function of the prestrain, we consider a minimal model in one dimension. We do this by considering a chain (1D network) of elastic springs upon which a random, zero mean, finite variance prestrain is imposed. Numerical simulations and analytical predictions quantify the magnitude of the contraction as a function of the variance of the prestrain, and show that the chain always shrinks. To test these predictions, we vary the topology of the chain and consider more complex connectivity and show that our results are relatively robust to these changes.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Analysis and forecast of COVID-19 spreading in China, Italy and France

In this note we analyze the temporal dynamics of the coronavirus disease 2019 outbreak in China, Italy and France in the time window $22/01-09/03/2020$. A first analysis of simple day-lag maps points to some universality in the epidemic spreading, suggesting that simple mean-field models can be meaningfully used to gather a quantitative picture of the epidemic spreading, and notably the height and time of the peak of confirmed infected individuals. The analysis of the same data within a simple susceptible-(confirmed) infected-recovered-deaths model indicates that the kinetic parameter that describes the rate of recovery seems to be the same, irrespective of the country, while the infection and death rates appear to be more variable. The model places the peak in Italy around March 20$^{\rm th}$ 2020, with a maximum number of confirmed infected individuals of about 16,000.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

COVID-19: The unreasonable effectiveness of simple models

When the novel coronavirus disease SARS-CoV2 (COVID-19) was officially declared a pandemic by the WHO in March 2020, the scientific community had already braced up in the effort of making sense of the fast-growing wealth of data gathered by national authorities all over the world. However, despite the diversity of novel theoretical approaches and the comprehensiveness of many widely established models, the official figures that recount the course of the outbreak still sketch a largely elusive and intimidating picture. Here we show unambiguously that the dynamics of the COVID-19 outbreak belongs to the simple universality class of the SIR model and extensions thereof. Our analysis naturally leads us to establish that there exists a fundamental limitation to any theoretical approach, namely the unpredictable non-stationarity of the testing frames behind the reported figures. However, we show how such bias can be quantified self-consistently and employed to mine useful and accurate information from the data. In particular, we describe how the time evolution of the reporting rates controls the occurrence of the apparent epidemic peak, which typically follows the true one in countries that were not vigorous enough in their testing at the onset of the outbreak. The importance of testing early and resolutely appears as a natural corollary of our analysis, as countries that tested massively at the start clearly had their true peak earlier and less deaths overall.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Dynamical systems on Hypergraphs

Networks are a widely used and efficient paradigm to model real-world systems where basic units interact pairwise. Many body interactions are often at play, and cannot be modelled by resorting to binary exchanges. In this work, we consider a general class of dynamical systems anchored on hypergraphs. Hyperedges of arbitrary size ideally encircle individual units so as to account for multiple, simultaneous interactions. These latter are mediated by a combinatorial Laplacian, that is here introduced and characterised. The formalism of the Master Stability Function is adapted to the present setting. Turing patterns and the synchronisation of non linear (regular and chaotic) oscillators are studied, for a general class of systems evolving on hypergraphs. The response to externally imposed perturbations bears the imprint of the higher order nature of the interactions.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Generalized patterns from local and non local reactions

A class of systems is considered, where immobile species associated to distinct patches, the nodes of a network, interact both locally and at a long-range, as specified by an (interaction) adjacency matrix. Non local interactions are treated in a mean-field setting which enables the system to reach a homogeneous consensus state, either constant or time dependent. We provide analytical evidence that such homogeneous solution can turn unstable under externally imposed disturbances, following a symmetry breaking mechanism which anticipates the subsequent outbreak of the patterns. The onset of the instability can be traced back, via a linear stability analysis, to a dispersion relation that is shaped by the spectrum of an unconventional reactive Laplacian. The proposed mechanism prescinds from the classical Local Activation and Lateral Inhibition scheme, which sits at the core of the Turing recipe for diffusion driven instabilities. Examples of systems displaying a fixed-point or a limit cycle, in their uncoupled versions, are discussed. Taken together, our results pave the way for alternative mechanisms of pattern formation, opening new possibilities for modeling ecological, chemical and physical interacting systems.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Generating directed networks with prescribed Laplacian spectra

Complex real-world phenomena are often modeled as dynamical systems on networks. In many cases of interest, the spectrum of the underlying graph Laplacian sets the system stability and ultimately shapes the matter or information flow. This motivates devising suitable strategies, with rigorous mathematical foundation, to generate Laplacian that possess prescribed spectra. In this paper, we show that a weighted Laplacians can be constructed so as to exactly realize a desired complex spectrum. The method configures as a non trivial generalization of existing recipes which assume the spectra to be real. Applications of the proposed technique to (i) a network of Stuart-Landau oscillators and (ii) to the Kuramoto model are discussed. Synchronization can be enforced by assuming a properly engineered, signed and weighted, adjacency matrix to rule the pattern of pairing interactions.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Nonlinear walkers and efficient exploration of congested networks

Random walks are the simplest way to explore or search a graph, and have revealed a very useful tool to investigate and characterize the structural properties of complex networks from the real world, e.g. they have been used to identify the modules of a given network, its most central nodes and paths, or to determine the typical times to reach a target. Although various types of random walks whose motion is node biased have been proposed, which are still amenable to analytical solution, most if not all of them rely on the assumption of linearity and independence of the walkers. We introduce a novel class of nonlinear stochastic processes describing a system of interacting random walkers moving over networks with finite node capacities. The transition probabilities are modulated by nonlinear functions of the available space at the destination node, with a bias parameter that allows to tune the tendency of the walkers to avoid nodes occupied by other walkers. Firstly, we derive the master equation governing the dynamics of the system, and we determine an analytical expression for the occupation probability of the walkers at equilibrium in the most general case, and under different level of network congestions. Then, we study different type of synthetic and real-world networks, presenting numerical and analytical results for the entropy rate, a proxy for the network exploration capacities of the walkers.We find that, for each level of the nonlinear bias, there is an optimal crowding that maximises the entropy rate in a given network topology. The analysis suggests that a large fraction of real-world networks are organised in such a way as to favour exploration under congested conditions. Our work provides a general and versatile framework to model nonlinear stochastic processes whose transition probabilities vary in time depending on the current state of the system.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Random walks on hypergraphs

In the last twenty years network science has proven its strength in modelling many real-world interacting systems as generic agents, the nodes, connected by pairwise edges. Yet, in many relevant cases, interactions are not pairwise but involve larger sets of nodes, at a time. These systems are thus better described in the framework of hypergraphs, whose hyperedges effectively account for multi-body interactions. We hereby propose a new class of random walks defined on such higher-order structures, and grounded on a microscopic physical model where multi-body proximity is associated to highly probable exchanges among agents belonging to the same hyperedge. We provide an analytical characterisation of the process, deriving a general solution for the stationary distribution of the walkers. The dynamics is ultimately driven by a generalised random walk Laplace operator that reduces to the standard random walk Laplacian when all the hyperedges have size 2 and are thus meant to describe pairwise couplings. We illustrate our results on synthetic models for which we have a full control of the high-order structures, and real-world networks where higher-order interactions are at play. As a first application of the method, we compare the behaviour of random walkers on hypergraphs to that of traditional random walkers on the corresponding projected networks, drawing interesting conclusions on node rankings in collaboration networks. As a second application, we show how information derived from the random walk on hypergraphs can be successfully used for classification tasks involving objects with several features, each one represented by a hyperedge. Taken together, our work contributes to unveiling the effect of higher-order interactions on diffusive processes in higher-order networks, shading light on mechanisms at the hearth of biased information spreading in complex networked systems.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Resilience for stochastic systems interacting via a quasi-degenerate network

A stochastic reaction-diffusion model is studied on a networked support. In each patch of the network two species are assumed to interact following a non-normal reaction scheme. When the interaction unit is replicated on a directed linear lattice, noise gets amplified via a self-consistent process which we trace back to the degenerate spectrum of the embedding support. The same phenomenon holds when the system is bound to explore a quasi degenerate network. In this case, the eigenvalues of the Laplacian operator, which governs species diffusion, accumulate over a limited portion of the complex plane. The larger the network, the more pronounced the amplification. Beyond a critical network size, a system deemed deterministically stable, hence resilient, may turn unstable, yielding seemingly regular patterns in the concentration amount. Non-normality and quasi-degenerate networks may therefore amplify the inherent stochasticity, and so contribute to altering the perception of resilience, as quantified via conventional deterministic methods.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Stabilizing Stuart-Landau oscillators via time-varying networks

A procedure is developed and tested to enforce synchronicity in a family of Stuart-Landau oscillators, coupled through a symmetric network. The proposed method exploits network plasticity, as an inherent non autonomous drive. More specifically, we assume that the system is initially confined on a network which turns the underlying homogeneous synchronous state unstable. A properly engineered network can be always generated, which links the same set of nodes, and allows for synchronicity to be eventually restored, upon performing continuously swappings, at a sufficient rate, between the two aforementioned networks. The result is cast in rigorous terms, as follows an application of the average theorem and the critical swapping rate determined analytically.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Absence of thermalization for systems with long-range interactions coupled to a thermal bath

We investigate the dynamics of a small long-range interacting system, in contact with a large long-range thermal bath. Our analysis reveals the existence of striking anomalies in the energy flux between the bath and the system. In particular, we find that the evolution of the system is not influenced by the kinetic temperature of the bath, as opposed to what happens for short-range collisional systems. As a consequence, the system may get hotter also when its initial temperature is larger than the bath temperature. This observation is explained quantitatively in the framework of the collisionless Vlasov description of toy models with long-range interactions and shown to be valid whenever the Vlasov picture applies, from cosmology to plasma physics.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Drift-induced Benjamin-Feir instabilities

A modified version of the Ginzburg-Landau equation is introduced which accounts for asymmetric couplings between neighbors sites on a one-dimensional lattice, with periodic boundary conditions. The drift term which reflects the imposed microscopic asymmetry seeds a generalized class of instabilities, reminiscent of the Benjamin-Feir type. The uniformly synchronized solution is spontaneously destabilized outside the region of parameters classically associated to the Benjamin-Feir instability, upon injection of a non homogeneous perturbation. The ensuing patterns can be of the traveling wave type or display a patchy, colorful mosaic for the modulus of the complex oscillators amplitude.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Fluctuating hydrodynamics approximation of the stochastic Cowan-Wilson model

We consider a stochastic version of the Wilson-Cowan model which accommodates for discrete populations of excitatory and inhibitory neurons. The model assumes a finite carrying capacity with the two populations being constant in size. The master equation that governs the dynamics of the stochastic model is analyzed by an expansion in powers of the inverse population size, yielding a coupled pair of non-linear Langevin equations with multiplicative noise. Gillespie simulations show the validity of the obtained approximation, for the parameter region where the system exhibits dynamical bistability. We report analytical progress by silencing the retroaction of the activators on the inhibitors, while still assigning the parameters so to fall in the region of deterministic bistability for the excitatory species. The proposed approach forms the basis of a perturbative generalization which applies to the case where a modest degree of coupling is restored.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Pattern formation in a two-component reaction-diffusion system with delayed processes on a network

Reaction-diffusion systems with time-delay defined on complex networks have been studied in the framework of the emergence of Turing instabilities. The use of the Lambert W-function allowed us get explicit analytic conditions for the onset of patterns as a function of the main involved parameters, the time-delay, the network topology and the diffusion coefficients. Depending on these parameters, the analysis predicts whether the system will evolve towards a stationary Turing pattern or rather to a wave pattern associated to a Hopf bifurcation. The possible outcomes of the linear analysis overcome the respective limitations of the single-species case with delay, and that of the classical activator-inhibitor variant without delay. Numerical results gained from the Mimura-Murray model support the theoretical approach.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Quantifying the degree of average contraction of Collatz orbits

We here elaborate on a quantitative argument to support the validity of the Collatz conjecture, also known as the (3x + 1) or Syracuse conjecture. The analysis is structured as follows. First, three distinct fixed points are found for the third iterate of the Collatz map, which hence organise in a period 3 orbit of the original map. These are 1, 2 and 4, the elements which define the unique attracting cycle, as hypothesised by Collatz. To carry out the calculation we write the positive integers in modulo 8 (mod8 ), obtain a closed analytical form for the associated map and determine the transitions that yield contracting or expanding iterates in the original, infinite-dimensional, space of positive integers. Then, we consider a Markov chain which runs on the reduced space of mod8 congruence classes of integers. The transition probabilities of the Markov chain are computed from the deterministic map, by employing a measure that is invariant for the map itself. Working in this setting, we demonstrate that the stationary distribution sampled by the stochastic system induces a contracting behaviour for the orbits of the deterministic map on the original space of the positive integers. Sampling the equilibrium distribution on the congruence classes mod8^m for any m, which amounts to arbitrarily reducing the degree of imposed coarse graining, returns an identical conclusion.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Reaction rate of a composite core-shell nanoreactor with multiple, spatially distributed embedded nano-catalysts

We present a detailed theory for the total reaction rate constant of a composite core-shell nanoreactor, consisting of a central solid core surrounded by a hydrogel layer of variable thickness, where a given number of small catalytic nanoparticles are embedded at prescribed positions and are endowed with a prescribed surface reaction rate constant. Besides the precise geometry of the assembly, our theory accounts explicitly for the diffusion coefficients of the reactants in the hydrogel and in the bulk as well as for their transfer free energy jump upon entering the hydrogel shell. Moreover, we work out an approximate analytical formula for the overall rate constant, which is valid in the physically relevant range of geometrical and chemical parameters. We discuss in depth how the diffusion-controlled part of the rate depends on the essential variables, including the size of the central core. In particular, we derive some simple rules for estimating the number of nanocatalysts per nanoreactor for an efficient catalytic performance in the case of small to intermediate core sizes. Our theoretical treatment promises to provide a very useful and flexible tool for the design of superior performing nanoreactor geometries and with optimized nanoparticle load.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Suppressing escape events in maps of the unit interval with demographic noise

We explore the properties of discrete-time stochastic processes with a bounded state space, whose deterministic limit is given by a map of the unit interval. We find that, in the mesoscopic description of the system, the large jumps between successive iterates of the process can result in probability leaking out of the unit interval, despite the fact that the noise is multiplicative and vanishes at the boundaries. By including higher-order terms in the mesoscopic expansion, we are able to capture the non-Gaussian nature of the noise distribution near the boundaries, but this does not preclude the possibility of a trajectory leaving the interval. We propose a number of prescriptions for treating these escape events, and we compare the results with those obtained for the metastable behavior of the microscopic model, where escape events are not possible. We find that, rather than truncating the noise distribution, censoring this distribution to prevent escape events leads to results which are more consistent with the microscopic model. The addition of higher moments to the noise distribution does not increase the accuracy of the final results, and it can be replaced by the simpler Gaussian noise.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Tune the topology to create or destroy patterns

We consider the dynamics of a reaction-diffusion system on a multigraph. The species share the same set of nodes but can access different links to explore the embedding spatial support. By acting on the topology of the networks we can control the ability of the system to self-organise in macroscopic patterns, emerging as a symmetry breaking instability of an homogeneous fixed point. Two different cases study are considered: on the one side, we produce a global modification of the networks, starting from the limiting setting where species are hosted on the same graph. On the other, we consider the effect of inserting just one additional single link to differentiate the two graphs. In both cases, patterns can be generated or destroyed, as follows the imposed, small, topological perturbation. Approximate analytical formulae allows to grasp the essence of the phenomenon and can potentially inspire innovative control strategies to shape the macroscopic dynamics on multigraph networks

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Delay induced Turing-like waves for one species reaction-diffusion model on a network

A one species time-delay reaction-diffusion system defined on a complex networks is studied. Travelling waves are predicted to occur as follows a symmetry breaking instability of an homogenous stationary stable solution, subject to an external non homogenous perturbation. These are generalized Turing-like waves that materialize in a single species populations dynamics model, as the unexpected byproduct of the imposed delay in the diffusion part. Sufficient conditions for the onset of the instability are mathematically provided by performing a linear stability analysis adapted to time delayed differential equation. The method here developed exploits the properties of the Lambert W-function. The prediction of the theory are confirmed by direct numerical simulation carried out for a modified version of the classical Fisher model, defined on a Watts-Strogatz networks and with the inclusion of the delay.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Emergence of a collective crystal in a classical system with long-range interactions

A one-dimensional long-range model of classical rotators with an extended degree of complexity, as compared to paradigmatic long-range systems, is introduced and studied. Working at constant density, in the thermodynamic limit one can prove the statistical equivalence with the Hamiltonian Mean Field model (HMF) and $α$-HMF: a second order phase transition is indeed observed at the critical energy threshold $\varepsilon_c=0.75$. Conversely, when the thermodynamic limit is performed at infinite density (while keeping the length of the hosting interval $L$ constant), the critical energy $\varepsilon_c$ is modulated as a function of $L$. At low energy, a self-organized collective crystal phase is reported to emerge, which converges to a perfect crystal in the limit $ε\rightarrow 0$. To analyze the phenomenon, the equilibrium one particle density function is analytically computed by maximizing the entropy. The transition and the associated critical energy between the gaseous and the crystal phase is computed. Molecular dynamics show that the crystal phase is apparently split into two distinct regimes, depending on the the energy per particle $\varepsilon$. For small $\varepsilon$, particles are exactly located on the lattice sites; above an energy threshold $\varepsilon{*}$, particles can travel from one site to another. However, $\varepsilon{*}$ does not signal a phase transition but reflects the finite time of observation: the perfect crystal observed for $\varepsilon >0$ corresponds to a long lasting dynamical transient, whose life time increases when the $\varepsilon >0$ approaches zero.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Generalized maximum entropy approach to quasi-stationary states in long range systems

Systems with long-range interactions display a short-time relaxation towards Quasi Stationary States (QSS) whose lifetime increases with the system size. In the paradigmatic Hamiltonian Mean-field Model (HMF) out-of-equilibrium phase transitions are predicted and numerically detected which separate homogeneous (zero magnetization) and inhomogeneous (nonzero magnetization) QSS. In the former regime, the velocity distribution presents (at least) two large, symmetric, bumps, which cannot be self-consistently explained by resorting to the conventional Lynden-Bell maximum entropy approach. We propose a generalized maximum entropy scheme which accounts for the pseudo-conservation of additional charges, the even momenta of the single particle distribution. These latter are set to the asymptotic values, as estimated by direct integration of the underlying Vlasov equation, which formally holds in the thermodynamic limit. Methodologically, we operate in the framework of a generalized Gibbs ensemble, as sometimes defined in statistical quantum mechanics, which contains an infinite number of conserved charges. The agreement between theory and simulations is satisfying, both above and below the out of equilibrium transition threshold. A precedently unaccessible feature of the QSS, the multiple bumps in the velocity profile, is resolved by our new approach.

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preprint2015arXiv

Multiple scale theory of topology driven pattern on directed networks

Dynamical processes on networks are currently being considered in different domains of cross-disciplinary interest. Reaction-diffusion systems hosted on directed graphs are in particular relevant for their widespread applications, from neuroscience, to computer networks and traffic systems. Due to the peculiar spectrum of the discrete Laplacian operator, homogeneous fixed points can turn unstable, on a directed support, because of the topology of the network, a phenomenon which cannot be induced on undirected graphs. A linear analysis can be performed to single out the conditions that underly the instability. The complete characterization of the patterns, which are eventually attained beyond the linear regime of exponential growth, calls instead for a full non linear treatment. By performing a multiple time scale perturbative calculation, we here derive an effective equation for the non linear evolution of the amplitude of the most unstable mode, close to the threshold of criticality. This is a Stuart-Landau equation whose complex coefficients appear to depend on the topological features of the embedding directed graph. The theory proves adequate versus simulations, as confirmed by operating with a paradigmatic reaction-diffusion model.

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preprint2015arXiv

Pattern formation for reactive species undergoing anisotropic diffusion

Turing instabilities for a two species reaction-diffusion systems is studied under anisotropic diffusion. More specifically, the diffusion constants which characterize the ability of the species to relocate in space are direction sensitive. Under this working hypothesis, the conditions for the onset of the instability are mathematically derived and numerically validated. Patterns which closely resemble those obtained in the classical context of isotropic diffusion, develop when the usual Turing condition is violated, along one of the two accessible directions of migration. Remarkably, the instability can also set in when the activator diffuses faster than the inhibitor, along the direction for which the usual Turing conditions are not matched.

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preprint2015arXiv

The second will be first: competition on directed networks

Multiple sinks competition is investigated for a walker diffusing on directed complex networks. The asymmetry of the imposed spatial support makes the system non transitive. As a consequence, it is always possible to identify a suitable location for the second absorbing sink that screens at most the flux of agents directed against the first trap, whose position has been preliminarily assigned. The degree of mutual competition between pairs of nodes is analytically quantified through apt indicators that build on the topological characteristics of the hosting graph. Moreover, the positioning of the second trap can be chosen so as to minimize, at the same time the probability of being in turn shaded by a thirdly added trap. Supervised placing of absorbing traps on a asymmetric disordered and complex graph is hence possible, as follows a robust optimization protocol. This latter is here discussed and successfully tested against synthetic data.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Theory of diffusion-influenced reactions in complex geometries

Chemical reactions involving diffusion of reactants and subsequent chemical fixation steps are generally termed "diffusion-influenced" (DI). Virtually all biochemical processes in living media can be counted among them, together with those occurring in an ever-growing number of emerging nano-technologies. The role of the environment's geometry (obstacles, compartmentalization) and distributed reactivity (competitive reactants, traps) is key in modulating the rate constants of DI reactions, and is therefore a prime design parameter. Yet, it is a formidable challenge to build a comprehensive theory able to describe the environment's "reactive geometry". Here we show that such a theory can be built by unfolding this many-body problem through addition theorems for special functions. Our method is powerful and general and allows one to study a given DI reaction occurring in arbitrary "reactive landscapes", made of multiple spherical boundaries of given size and reactivity. Importantly, ready-to-use analytical formulas can be derived easily in most cases.

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preprint2015arXiv

WKB versus generalized van Kampen system-size expansion: the stochastic logistic equation

Stochastic fluctuations are central to the understanding of extinction dynamics. In the context of population models they allow for the description of the transition from the vicinity of a non-trivial fixed point of the deterministic dynamics to a trivial fixed point, where the population has become extinct. To characterize analytically the fluctuations of a given stochastic population model, one can operate within the so-called linear-noise approximation. Here the fluctuations are taken to be Gaussian, and the phenomenon of extinction is so rare as to be negligible, for all practical purposes. When the size of the population becomes small, non-Gaussian fluctuations are instead found. Two analytical schemes are in principle available to determine the nature of the distribution of associated fluctuations: the generalized van Kampen system-size expansion beyond the conventional order of approximation and a WKB-like method. Here we investigate the accuracy of these two different approximation schemes, with reference to a simple stochastic process that converges to the logistic equation in the deterministic limit.

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preprint2014arXiv

Beam-Plasma Instability and Fast Particles: the Lynden-Bell Approach

The beam-plasma instability, i.e. the response of the plasma bulk to the injection of supra thermal charged-particle beams, results to be appropriately characterized by a long-range interaction system. This physical system hosts a number of very interesting phenomena and, in particular, the emergence of long-lived quasi-stationary states. We characterize the self-consistent distribution functions of such out-of-equilibrium states by means of the Lynden-Bell's theory. The prediction of this theory, based on the statistical mechanics of the Vlasov equation, are checked against the outcomes of numerical simulations of the discrete system. Moreover, a phenomenological study of the effective resonance band for the system response is also addressed. A threshold value has been found in the initial spread of beam-particle momenta. This threshold allows to discriminate between the resonant and non-resonant regimes. The analysis of the thermalization of a few percents of the beam population characterized by large initial momenta (with respect to the main part of the beam itself) is also performed and it confirms and deepens the understanding of the physical meaning of the mentioned threshold.

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preprint2014arXiv

Diffusion of tagged particles in a crowded medium

The influence of crowding on the diffusion of tagged particles in a dense medium is investigated in the framework of a mean-field model, derived in the continuum limit from a microscopic stochastic process with exclusion. The probability distribution function of the tagged particles obeys to a nonlinear Fokker-Planck equation, where the drift and diffusion terms are determined self-consistently by the concentration of crowders in the medium. Transient sub-diffusive or super-diffusive behaviours are observed, depending on the selected initial conditions, that bridge normal diffusion regimes characterized by different diffusion coefficients. These anomalous crossovers originate from the microscopic competition for space and reflect the peculiar form of the non-homogeneous advection term in the governing Fokker-Planck equation. Our results strongly warn against the overly simplistic identification of crowding with anomalous transport {\em tout court}.

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preprint2014arXiv

Ensemble Inequivalence in Systems with Wave-Particle Interaction

The classical wave-particle Hamiltonian is considered in its generalized version, where two modes are assumed to interact with the co-evolving charged particles. The equilibrium statistical mechanics solution of the model is worked out analytically, both in the canonical and the microcanonical ensembles. The competition between the two modes is shown to yield ensemble inequivalence, at variance with the standard scenario where just one wave is allowed to develop. As a consequence, both temperature jumps and negative specific heat can show up in the microcanonical ensemble. The relevance of these findings for both plasma physics and Free Electron Laser applications is discussed.

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preprint2014arXiv

Intrinsic noise and two-dimensional maps: Quasicycles, quasiperiodicity, and chaos

We develop a formalism to describe the discrete-time dynamics of systems containing an arbitrary number of interacting species. The individual-based model, which forms our starting point, is described by a Markov chain, which in the limit of large system sizes is shown to be very well-approximated by a Fokker-Planck-like equation, or equivalently by a set of stochastic difference equations. This formalism is applied to the specific case of two species: one predator species and its prey species. Quasi-cycles --- stochastic cycles sustained and amplified by the demographic noise --- previously found in continuous-time predator-prey models are shown to exist, and their behavior predicted from a linear noise analysis is shown to be in very good agreement with simulations. The effects of the noise on other attractors in the corresponding deterministic map, such as periodic cycles, quasiperiodicity and chaos, are also investigated.

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preprint2014arXiv

Optimal search strategies on complex networks

Complex networks are ubiquitous in nature and play a role of paramount importance in many contexts. Internet and the cyberworld, which permeate our everyday life, are self-organized hierarchical graphs. Urban traffic flows on intricate road networks, which impact both transportation design and epidemic control. In the brain, neurons are cabled through heterogeneous connections, which support the propagation of electric signals. In all these cases, the true challenge is to unveil the mechanisms through which specific dynamical features are modulated by the underlying topology of the network. Here, we consider agents randomly hopping along the links of a graph, with the additional possibility of performing long-range hops to randomly chosen disconnected nodes with a given probability. We show that an optimal combination of the two jump rules exists that maximises the efficiency of target search, the optimum reflecting the topology of the network.

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preprint2014arXiv

The theory of individual based discrete-time processes

A general theory is developed to study individual based models which are discrete in time. We begin by constructing a Markov chain model that converges to a one-dimensional map in the infinite population limit. Stochastic fluctuations are hence intrinsic to the system and can induce qualitative changes to the dynamics predicted from the deterministic map. From the Chapman-Kolmogorov equation for the discrete-time Markov process, we derive the analogues of the Fokker-Planck equation and the Langevin equation, which are routinely employed for continuous time processes. In particular, a stochastic difference equation is derived which accurately reproduces the results found from the Markov chain model. Stochastic corrections to the deterministic map can be quantified by linearizing the fluctuations around the attractor of the map. The proposed scheme is tested on stochastic models which have the logistic and Ricker maps as their deterministic limits.

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preprint2014arXiv

Topology-driven instabilities: the theory of pattern formation on directed networks

The theory of pattern formation in reaction-diffusion systems is extended to the case of a directed network. Due to the structure of the network Laplacian of the scrutinised system, the dispersion relation has both real and imaginary parts, at variance with the conventional case for a symmetric network. It is found that the homogeneous fixed point can become unstable due to the topology of the network, resulting in a new class of instabilities which cannot be induced on undirected graphs. Results from a linear stability analysis allow the instability region to be analytically traced. Numerical simulations show that the instability can lead to travelling waves, or quasi-stationary patterns, depending on the characteristics of the underlying graph. The results presented here could impact on the diverse range of disciplines where directed networks are found, such as neuroscience, computer networks and traffic systems.

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preprint2014arXiv

Turing instabilities from a limit cycle

The Turing instability is a paradigmatic route to patterns formation in reaction-diffusion systems. Following a diffusion-driven instability, homogeneous fixed points can become unstable when subject to external perturbation. As a consequence, the system evolves towards a stationary, nonhomogeneous attractor. Stable patterns can be also obtained via oscillation quenching of an initially synchronous state of diffusively coupled oscillators. In the literature this is known as the oscillation death phenomenon. Here we show that oscillation death is nothing but a Turing instability for the first return map associated to the excitable system in its synchronous periodic state. In particular we obtain a set of closed conditions for identifying the domain in the parameters space that yields the instability. This is a natural generalisation of the original Turing relations, to the case where the homogeneous solution of the examined system is a periodic function of time. The obtained framework applies to systems embedded in continuum space, as well as those defined on a network-like support. The predictive ability of the theory is tested numerically, using different reaction schemes.

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preprint2014arXiv

Turing instabilities on Cartesian product networks

The problem of Turing instabilities for a reaction-diffusion system defined on a complex Cartesian product networks is considered. To this end we operate in the linear regime and expand the time dependent perturbation on a basis formed by the tensor product of the eigenvectors of the discrete Laplacian operators, associated to each of the individual networks that build the Cartesian product. The dispersion relation which controls the onset of the instability depends on a set of discrete wave- lenghts, the eigenvalues of the aforementioned Laplacians. Patterns can develop on the Cartesian network, if they are supported on at least one of its constituive sub-graphs. Multiplex networks are also obtained under specific prescriptions. In this case, the criteria for the instability reduce to compact explicit formulae. Numerical simulations carried out for the Mimura-Murray reaction kinetics confirm the adequacy of the proposed theory.

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preprint2014arXiv

Turing patterns in multiplex networks

The theory of patterns formation for a reaction-diffusion system defined on a multiplex is developed by means of a perturbative approach. The intra-layer diffusion constants act as small parameter in the expansion and the unperturbed state coincides with the limiting setting where the multiplex layers are decoupled. The interaction between adjacent layers can seed the instability of an homogeneous fixed point, yielding self-organized patterns which are instead impeded in the limit of decoupled layers. Patterns on individual layers can also fade away due to cross-talking between layers. Analytical results are compared to direct simulations.

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preprint2013arXiv

Intrinsic noise and discrete-time processes

A general formalism is developed to construct a Markov chain model that converges to a one-dimensional map in the infinite population limit. Stochastic fluctuations are therefore internal to the system and not externally specified. For finite populations an approximate Gaussian scheme is devised to describe the stochastic fluctuations in the non-chaotic regime. More generally, the stochastic dynamics can be captured using a stochastic difference equation, derived through an approximation to the Markov chain. The scheme is demonstrated using the logistic map as a case study.

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preprint2013arXiv

Linear noise approximation for stochastic oscillations of intracellular calcium

A stochastic model of intracellular calcium oscillations is analytically studied. The governing master equation is expanded under the linear noise approximation and a closed prediction for the power spectrum of fluctuations analytically derived. A peak in the obtained power spectrum profile signals the presence of stochastic, noise induced, oscillations which extend also outside the region where a deterministic limit cycle is predicted to occur.

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preprint2013arXiv

Nonlinear macroscopic transport equations in many-body systems from microscopic exclusion processes

Describing particle transport at the macroscopic or mesoscopic level in non-ideal environments poses fundamental theoretical challenges in domains ranging from inter and intra-cellular transport in biology to diffusion in porous media. Yet, often the nature of the constraints coming from many-body interactions or reflecting a complex and confining environment are better understood and modeled at the microscopic level. In this paper we investigate the subtle link between microscopic exclusion processes and the mean-field equations that ensue from them in the continuum limit. We derive a generalized nonlinear advection diffusion equation suitable for describing transport in a inhomogeneous medium in the presence of an external field. Furthermore, taking inspiration from a recently introduced exclusion process involving agents with non-zero size, we introduce a modified diffusion equation appropriate for describing transport in a non-ideal fluid of $d$-dimensional hard spheres. We consider applications of our equations to the problem of diffusion to an absorbing sphere in a non-ideal self-crowded fluid and to the problem of gravitational sedimentation. We show that our formalism allows one to recover known results. Moreover, we introduce the notions of point-like and extended crowding, which specify distinct routes for obtaining macroscopic transport equations from microscopic exclusion processes.

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preprint2013arXiv

Persistent random walk with exclusion

Modelling the propagation of a pulse in a dense {\em milieu} poses fundamental challenges at the theoretical and applied levels. To this aim, in this paper we generalize the telegraph equation to non-ideal conditions by extending the concept of persistent random walk to account for spatial exclusion effects. This is achieved by introducing an explicit constraint in the hopping rates, that weights the occupancy of the target sites. We derive the mean-field equations, which display nonlinear terms that are important at high density. We compute the evolution of the mean square displacement (MSD) for pulses belonging to a specific class of spatially symmetric initial conditions. The MSD still displays a transition from ballistic to diffusive behaviour. We derive an analytical formula for the effective velocity of the ballistic stage, which is shown to depend in a nontrivial fashion upon both the density (area) and the shape of the initial pulse. After a density-dependent crossover time, nonlinear terms become negligible and normal diffusive behaviour is recovered at long times.

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preprint2013arXiv

Stochastic Turing Patterns for systems with one diffusing species

The problem of pattern formation in a generic two species reaction--diffusion model is studied, under the hypothesis that only one species can diffuse. For such a system, the classical Turing instability cannot take place. At variance, by working in the generalized setting of a stochastic formulation to the inspected problem, Turing like patterns can develop, seeded by finite size corrections. General conditions are given for the stochastic Turing patterns to occur. The predictions of the theory are tested for a specific case study.

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preprint2013arXiv

The linear noise approximation for reaction-diffusion systems on networks

Stochastic reaction-diffusion models can be analytically studied on complex networks using the linear noise approximation. This is illustrated through the use of a specific stochastic model, which displays traveling waves in its deterministic limit. The role of stochastic fluctuations is investigated and shown to drive the emergence of stochastic waves beyond the region of the instability predicted from the deterministic theory. Simulations are performed to test the theoretical results and are analyzed via a generalized Fourier transform algorithm. This transform is defined using the eigenvectors of the discrete Laplacian defined on the network. A peak in the numerical power spectrum of the fluctuations is observed in quantitative agreement with the theoretical predictions.

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preprint2012arXiv

Multispecies reaction diffusion models and the Turing instability revisited

The Turing instability paradigm is revisited in the context of a multispecies diffusion scheme derived from a self-consistent microscopic formulation. The analysis is developed with reference to the case of two species. These latter share the same spatial reservoir and experience a degree of mutual interference due to the competition for the available resources. Turing instability can set in for all ratios of the main diffusivities, also when the (isolated) activator diffuses faster then the (isolated) inhibitor. This conclusion, at odd with the conventional vision, is here exemplified for the Brusselator model and ultimately stems from having assumed a generalized model of multispecies diffusion, fully anchored to first principles, which also holds under crowded conditions.

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preprint2012arXiv

Stochastic Turing Patterns on a Network

The process of stochastic Turing instability on a network is discussed for a specific case study, the stochastic Brusselator model. The system is shown to spontaneously differentiate into activator-rich and activator-poor nodes, outside the region of parameters classically deputed to the deterministic Turing instability. This phenomenon, as revealed by direct stochastic simulations, is explained analytically, and eventually traced back to the finite size corrections stemming from the inherent graininess of the scrutinized medium.

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preprint2011arXiv

Analytical study of non Gaussian fluctuations in a stochastic scheme of autocatalytic reactions

A stochastic model of autocatalytic chemical reactions is studied both numerically and analytically. The van Kampen perturbative scheme is implemented, beyond the second order approximation, so to capture the non Gaussianity traits as displayed by the simulations. The method is targeted to the characterization of the third moments of the distribution of fluctuations, originating from a system of four populations in mutual interaction. The theory predictions agree well with the simulations, pointing to the validity of the van Kampen expansion beyond the conventional Gaussian solution.

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preprint2011arXiv

Efficiency of cyclic devices working with non-Boltzmannian fluids: challenging the second principle of thermodynamics

According to classical Boltzmannian thermodynamics, the efficiency of a cyclic machine is strictly lower than one. Such a result is a straightforward consequence of the second principle of thermodynamics. Recent advances in the study of the thermodynamics of long-range interacting system report however on a rather intricate zoology of peculiar behaviors, which are occasionally in contrast with customarily accepted scenarios, dueling with intuition and common sense. In this paper, a thermodynamical cycle is assembled for an ideal device working with non-Boltzmaniann long-range fluid and operating in contact with two thermal reservoirs. The system is analytically shown to violate the second principle of thermodynamics, a phenomenon that ultimately relates to the existence of regions with negative kinetic specific heat, in the canonical ensemble for the system under scrutiny. We argue that the validity of the second principle of thermodynamics can be possibly restored, by revisiting the definition of canonical ensemble, as well as the Fourier law of heat transport, and consequently relaxing the constraint on the maximal efficiency as imposed by the Carnot theorem.

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preprint2011arXiv

Non-Gaussian fluctuations in stochastic models with absorbing barriers

The dynamics of a one-dimensional stochastic model is studied in presence of an absorbing boundary. The distribution of fluctuations is analytically characterized within the generalized van Kampen expansion, accounting for higher order corrections beyond the conventional Gaussian approximation. The theory is shown to successfully capture the non Gaussian traits of the sought distribution returning an excellent agreement with the simulations, for {\it all times} and arbitrarily {\it close} to the absorbing barrier. At large times, a compact analytical solution for the distribution of fluctuations is also obtained, bridging the gap with previous investigations, within the van Kampen picture and without resorting to alternative strategies, as elsewhere hypothesized.

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preprint2011arXiv

Phase transitions of quasistationary states in the Hamiltonian Mean Field model

The out-of-equilibrium dynamics of the Hamiltonian Mean Field (HMF) model is studied in presence of an externally imposed magnetic field h. Lynden-Bell's theory of violent relaxation is revisited and shown to adequately capture the system dynamics, as revealed by direct Vlasov based numerical simulations in the limit of vanishing field. This includes the existence of an out-of-equilibrium phase transition separating magnetized and non magnetized phases. We also monitor the fluctuations in time of the magnetization, which allows us to elaborate on the choice of the correct order parameter when challenging the performance of Lynden-Bell's theory. The presence of the field h removes the phase transition, as it happens at equilibrium. Moreover, regions with negative susceptibility are numerically found to occur, in agreement with the predictions of the theory.

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preprint2011arXiv

Statistical theory of quasi stationary states beyond the single water-bag case study

An analytical solution for the out-of-equilibrium quasi-stationary states of the paradigmatic Hamiltonian Mean Field (HMF) model can be obtained from a maximum entropy principle. The theory has been so far tested with reference to a specific class of initial condition, the so called (single-level) water-bag type. In this paper a step forward is taken by considering an arbitrary number of overlapping water bags. The theory is benchmarked to direct microcanonical simulations performed for the case of a two-levels water-bag. The comparison is shown to return an excellent agreement.

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preprint2010arXiv

A spatial model of autocatalytic reactions

Biological cells with all of their surface structure and complex interior stripped away are essentially vesicles - membranes composed of lipid bilayers which form closed sacs. Vesicles are thought to be relevant as models of primitive protocells, and they could have provided the ideal environment for pre-biotic reactions to occur. In this paper, we investigate the stochastic dynamics of a set of autocatalytic reactions, within a spatially bounded domain, so as to mimic a primordial cell. The discreteness of the constituents of the autocatalytic reactions gives rise to large sustained oscillations, even when the number of constituents is quite large. These oscillations are spatio-temporal in nature, unlike those found in previous studies, which consisted only of temporal oscillations. We speculate that these oscillations may have a role in seeding membrane instabilities which lead to vesicle division. In this way synchronization could be achieved between protocell growth and the reproduction rate of the constituents (the protogenetic material) in simple protocells.

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preprint2010arXiv

Deterministic and stochastic aspects of VEGF-A production and the cooperative behavior of tumoral cell colony

A model is proposed to study the process of hypoxia-induced angiogenesis in cancer cells. The model accounts for the role played by the vascular endothelial growth factor (VEGF)-A in regulating the oxygen intake. VEGF-A is dynamically controlled by the HIF-1alpha concentration. If not degraded, HIF-1alpha can bind to the subunit termed HIF-1beta and so experience translocation to the nucleus, to exert its proper transcriptional activity. The delicate balance between these opposing tendencies translates into the emergence of distinct macroscopic behaviors in terms of the associated molecular concentrations that we here trace back to normoxia, hypoxia and death regimes. These aspects are firstly analyzed with reference to the ideal mean-field scenario. Stochastic fluctuations are also briefly discussed and shown to seed a cooperative interaction among cellular units, competing for the same oxygen reservoir.

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preprint2010arXiv

Diffusion in a crowded environment

We analyze a pair of diffusion equations which are derived in the infinite system--size limit from a microscopic, individual--based, stochastic model. Deviations from the conventional Fickian picture are found which ultimately relate to the depletion of resources on which the particles rely. The macroscopic equations are studied both analytically and numerically, and are shown to yield anomalous diffusion which does not follow a power law with time, as is frequently assumed when fitting data for such phenomena. These anomalies are here understood within a consistent dynamical picture which applies to a wide range of physical and biological systems, underlining the need for clearly defined mechanisms which are systematically analyzed to give definite predictions.

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preprint2010arXiv

Dynamical description of vesicle growth and shape change

We systematize and extend the description of vesicle growth and shape change using linear nonequilibrium thermodynamics. By restricting the study to shape changes from spheres to axisymmetric ellipsoids, we are able to give a consistent formulation which includes the lateral tension of the vesicle membrane. This allows us to generalize and correct a previous calculation. Our present calculations suggest that, for small growing vesicles, a prolate ellipsoidal shape should be favored over oblate ellipsoids, whereas for large growing vesicles oblates should be favored over prolates. The validity of this prediction is examined in the light of the various assumptions made in its derivation.

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preprint2010arXiv

Existence of Quasi-stationary states at the Long Range threshold

In this paper the lifetime of quasi-stationary states (QSS) in the $α-$HMF model are investigated at the long range threshold ($α=1$). It is found that QSS exist and have a diverging lifetime $τ(N)$ with system size which scales as $\mbox{\ensuremathτ(N)\ensuremath{\sim}}\log N$, which contrast to the exhibited power law for $α<1$ and the observed finite lifetime for $α>1$. Another feature of the long range nature of the system beyond the threshold ($α>1$) namely a phase transition is displayed for $α=1.5$. The definition of a long range system is as well discussed.

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preprint2010arXiv

Experimental perspectives for systems based on long-range interactions

The possibility of observing phenomena peculiar to long-range interactions, and more specifically in the so-called Quasi-Stationary State (QSS) regime is investigated within the framework of two devices, namely the Free-Electron Laser (FEL) and the Collective Atomic Recoil Laser (CARL). The QSS dynamics has been mostly studied using the Hamiltonian Mean-Field (HMF) toy model, demonstrating in particular the presence of first versus second order phase transitions from magnetized to unmagnetized regimes in the case of HMF. Here, we give evidence of the strong connections between the HMF model and the dynamics of the two mentioned devices, and we discuss the perspectives to observe some specific QSS features experimentally. In particular, a dynamical analog of the phase transition is present in the FEL and in the CARL in its conservative regime. Regarding the dissipative CARL, a formal link is established with the HMF model. For both FEL and CARL, calculations are performed with reference to existing experimental devices, namely the FERMI@Elettra FEL under construction at Sincrotrone Trieste (Italy) and the CARL system at LENS in Florence (Italy).

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preprint2010arXiv

Out of equilibrium statistical ensemble inequivalence

We consider a paradigmatic model describing the one-dimensional motion of $N$ rotators coupled through a mean-field interaction, and subject to the perturbation of an external magnetic field. The latter is shown to significantly alter the system behaviour, driving the emergence of ensemble inequivalence in the out-of-equilibrium phase, as signalled by a negative (microcanonical) magnetic susceptibility. The thermodynamic of the system is analytically discussed, building on a maximum entropy scheme justified from first principles. Simulations confirm the adequacy of the theoretical picture. Ensemble inequivalence is shown to rely on a peculiar phenomenon, different from the one observed in previous works. As a result, the existence of a convex intruder in the micro-canonical energy is found to be a necessary but not sufficient condition for inequivalence to be (macroscopically) observed. Negative temperature states are also found to occur. These intriguing phenomena reflect the non-Boltzmanian nature of the scrutinized problem and, as such, bear traits of universality that embraces equilibrium as well out-of-equilibrium regimes.

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preprint2010arXiv

Out-of-equilibrium mean-field dynamics of a model for wave-particle interaction

The out-of-equilibrium mean-field dynamics of a model for wave-particle interaction is investigated. Such a model can be regarded as a general formulation for all those applications where the complex interplay between particles and fields is known to be central, e.g., electrostatic instabilities in plasma physics, particle acceleration and free-electron lasers (FELs). The latter case is here assumed as a paradigmatic example. A transition separating different macroscopic regimes is numerically identified and interpreted by making use of the so-called violent relaxation theory. In the context of free-electron lasers, such a theory is showed to be effective in predicting the saturated regime for energies below the transition. The transition is explained as a dynamical switch between two metastable regimes, and is related to the properties of a stationary point of an entropic functional.

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preprint2010arXiv

Stationary states and fractional dynamics in systems with long range interactions

Dynamics of many-body Hamiltonian systems with long range interactions is studied, in the context of the so called $α-$HMF model. Building on the analogy with the related mean field model, we construct stationary states of the $α-$HMF model for which the spatial organization satisfies a fractional equation. At variance, the microscopic dynamics turns out to be regular and explicitly known. As a consequence, dynamical regularity is achieved at the price of strong spatial complexity, namely a microscopic inhomogeneity which locally displays scale invariance.

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preprint2010arXiv

Stochastic Turing patterns in the Brusselator model

A stochastic version of the Brusselator model is proposed and studied via the system size expansion. The mean-field equations are derived and shown to yield to organized Turing patterns within a specific parameters region. When determining the Turing condition for instability, we pay particular attention to the role of cross diffusive terms, often neglected in the heuristic derivation of reaction diffusion schemes. Stochastic fluctuations are shown to give rise to spatially ordered solutions, sharing the same quantitative characteristic of the mean-field based Turing scenario, in term of excited wavelengths. Interestingly, the region of parameter yielding to the stochastic self-organization is wider than that determined via the conventional Turing approach, suggesting that the condition for spatial order to appear can be less stringent than customarily believed.

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preprint2010arXiv

Universal fluctuations in tropospheric radar measurements

Radar data collected at an experimental facility arranged on purpose suggest that the footprint of atmospheric turbulence might be encoded in the radar signal statistics. Radar data probability distributions are calculated and nicely fitted by a one parameter family of generalized Gumbel (GG) distributions, G(a). A relation between the wind strength and the measured shape parameter "a" is obtained. Strong wind fluctuations return pronounced asymmetric leptokurtic profiles, while Gaussian profiles are eventually recovered as the wind fluctuations decrease. Besides stressing the crucial impact of air turbulence for radar applications, we also confirm the adequacy of G(a) statistics for highly correlated complex systems.

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preprint2009arXiv

Finite Size Effects for the Ising Model on Random Graphs with Varying Dilution

We investigate the finite size corrections to the equilibrium magnetization of an Ising model on a random graph with $N$ nodes and $N^γ$ edges, with $1 < γ\leq 2$. By conveniently rescaling the coupling constant, the free energy is made extensive. As expected, the system displays a phase transition of the mean-field type for all the considered values of $γ$ at the transition temperature of the fully connected Curie-Weiss model. Finite size corrections are investigated for different values of the parameter $γ$, using two different approaches: a replica-based finite $N$ expansion, and a cavity method. Numerical simulations are compared with theoretical predictions. The cavity based analysis is shown to agree better with numerics.

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preprint2008arXiv

Enhanced stochastic oscillations in autocatalytic reactions

We study a simplified scheme of $k$ coupled autocatalytic reactions, previously introduced by Togashi and Kaneko. The role of stochastic fluctuations is elucidated through the use of the van Kampen system-size expansion and the results compared with direct stochastic simulations. Regular temporal oscillations are predicted to occur for the concentration of the various chemical constituents, with an enhanced amplitude resulting from a resonance which is induced by the intrinsic graininess of the system. The associated power spectra are determined and have a different form depending on the number of chemical constituents, $k$. We make detailed comparisons in the two cases $k=4$ and $k=8$. Agreement between the theoretical and numerical results for the power spectrum are good in both cases. The resulting spectrum is especially interesting in the $k=8$ system, since it has two peaks, which the system-size expansion is still able to reproduce accurately.

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preprint2007arXiv

Exploring the thermodynamic limit of Hamiltonian models: convergence to the Vlasov equation

We here discuss the emergence of Quasi Stationary States (QSS), a universal feature of systems with long-range interactions. With reference to the Hamiltonian Mean Field (HMF) model, numerical simulations are performed based on both the original $N$-body setting and the continuum Vlasov model which is supposed to hold in the thermodynamic limit. A detailed comparison unambiguously demonstrates that the Vlasov-wave system provides the correct framework to address the study of QSS. Further, analytical calculations based on Lynden-Bell's theory of violent relaxation are shown to result in accurate predictions. Finally, in specific regions of parameters space, Vlasov numerical solutions are shown to be affected by small scale fluctuations, a finding that points to the need for novel schemes able to account for particles correlations.

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preprint2007arXiv

Out of equilibrium phase transitions in mean field Hamiltonian dynamics

Systems with long-range interactions display a short-time relaxation towards Quasi-Stationary States (QSSs), whose lifetime increases with system size. With reference to the Hamiltonian Mean Field (HMF) model, we here review Lynden-Bell's theory of ``violent relaxation''. The latter results in a maximum entropy scheme for a water-bag initial profile which predicts the presence of out-of-equilibrium phase transitions} separating homogeneous (zero magnetization) from inhomogeneous (non-zero magnetization) QSSs. Two different parametric representations of the initial condition are analyzed and the features of the phase diagram are discussed. In both representations we find a second order and a first order line of phase transitions that merge at a tricritical point. Particular attention is payed to the condition of existence and stability of the homogenous phase.

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preprint2007arXiv

Short-time dynamics in presence of wave-particles interactions: a perturbative approach

The self-consistent interaction between a beam of charged particles and a wave is considered, within a Vlasov picture. The model is discussed with reference to the case of a Free Electron Laser. Starting with a spatially bunched waterbag distribution, we derive, via perturbative methods, closed analytical expressions for the time evolution of the main macroscopic observables. Predictions of the theory are shown to agree with direct numerical simulations.

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preprint2000arXiv

On the dynamics of a self-gravitating medium with random and non-random initial conditions

The dynamics of a one-dimensional self-gravitating medium, with initial density almost uniform is studied. Numerical experiments are performed with ordered and with Gaussian random initial conditions. The phase space portraits are shown to be qualitatively similar to shock waves, in particular with initial conditions of Brownian type. The PDF of the mass distribution is investigated.

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