arXiv subjects

Alessandro Campa

Publications and source records attributed to Alessandro Campa.

36 records · Page 2Linked to original sources

arXiv2015

Nonequilibrium inhomogeneous steady state distribution in disordered, mean-field rotator systems

We present a novel method to compute the phase space distribution in the nonequilibrium stationary state of a wide class of mean-field systems involving rotators subject to quenched disordered external drive and dissipation. The method involves a series expansion of the stationary distribution in inverse of the damping coefficient; the expansion coefficients satisfy recursion relations whose solution requires computing a sparse matrix, making numerical evaluation simple and efficient. We illustrate our method for the paradigmatic Kuramoto model of spontaneous collective synchronization and for its two mode generalization, in presence of noise and inertia, and demonstrate an excellent agreement between simulations and theory for the phase space distribution.

cond-mat.stat-mech↗

arXiv2015

Thermodynamics of nonadditive systems

The usual formulation of thermodynamics is based on the additivity of macroscopic systems. However, there are numerous examples of macroscopic systems that are not additive, due to the long-range character of the interaction among the constituents. We present here an approach in which nonadditive systems can be described within a purely thermodynamics formalism. The basic concept is to consider a large ensemble of replicas of the system where the standard formulation of thermodynamics can be naturally applied and the properties of a single system can be consequently inferred. After presenting the approach, we show its implementation in systems where the interaction decays as $1/r^α$ in the interparticle distance $r$, with $α$ smaller than the embedding dimension $d$, and in the Thirring model for gravitational systems.

cond-mat.stat-mech↗

arXiv2014

Kuramoto model of synchronization: Equilibrium and nonequilibrium aspects

Recently, there has been considerable interest in the study of spontaneous synchronization, particularly within the framework of the Kuramoto model. The model comprises oscillators with distributed natural frequencies interacting through a mean-field coupling, and serves as a paradigm to study synchronization. In this review, we put forward a general framework in which we discuss in a unified way known results with more recent developments obtained for a generalized Kuramoto model that includes inertial effects and noise. We describe the model from a different perspective, highlighting the long-range nature of the interaction between the oscillators, and emphasizing the equilibrium and out-of-equilibrium aspects of its dynamics from a statistical physics point of view. In the first main section, we introduce the model and discuss for the noiseless and noisy dynamics and unimodal frequency distributions the synchronization transition that occurs in the stationary state. In the second section, we introduce the generalized model, and discuss its synchronization phase diagram for unimodal frequency distributions. In the third section, we describe deviations from the mean-field setting of the Kuramoto model by considering the generalized dynamics on a one-dimensional periodic lattice on the sites of which the oscillators reside and interact with a coupling that decays as an inverse power-law of their separation. For two specific cases, namely, in the absence of noise and inertia, and in the case when the natural frequencies are the same for all the oscillators, we discuss how the long-time transition to synchrony is governed by the dynamics of the mean-field mode (zero Fourier mode) of the spatial distribution of the oscillator phases.

cond-mat.stat-mech↗

arXiv2014

Nonequilibrium first-order transition in coupled oscillator systems with inertia and noise

We study the dynamics of a system of coupled oscillators of distributed natural frequencies, by including the features of both thermal noise, parametrized by a temperature, and inertial terms, parametrized by a moment of inertia. For a general unimodal frequency distribution, we report here the complete phase diagram of the model in the space of dimensionless moment of inertia, temperature, and width of the frequency distribution. We demonstrate that the system undergoes a nonequilibrium first-order phase transition from a synchronized phase at low parameter values to an incoherent phase at high values. We provide strong numerical evidence for the existence of both the synchronized and the incoherent phase, treating the latter analytically to obtain the corresponding linear stability threshold that bounds the first-order transition point from below. In the limit of zero noise and inertia, when the dynamics reduces to the one of the Kuramoto model, we recover the associated known continuous transition. At finite noise and inertia but in the absence of natural frequencies, the dynamics becomes that of a well-studied model of long-range interactions, the Hamiltonian mean-field model. Close to the first-order phase transition, we show that the escape time out of metastable states scales exponentially with the number of oscillators, which we explain to be stemming from the long-range nature of the interaction between the oscillators.

cond-mat.stat-mech↗

arXiv2012

Overdamped dynamics of long-range systems on a one-dimensional lattice: Dominance of the mean-field mode and phase transition

We consider the overdamped dynamics of a paradigmatic long-range system of particles residing on the sites of a one-dimensional lattice, in the presence of thermal noise. The internal degree of freedom of each particle is a periodic variable which is coupled to those of other particles with an attractive XY-like interaction. The coupling strength decays with the interparticle separation $r$ in space as $1/r^α$; ~$0 < α< 1$. We study the dynamics of the model in the continuum limit by considering the Fokker-Planck equation for the evolution of the spatial density of particles. We show that the equation allows a linearly stable stationary state which is always uniform in space, being non-uniform in the internal degrees below a critical temperature $T=1/2$ and uniform above, with a phase transition between the two at $T=1/2$. The state is the same as the equilibrium state of the mean-field version of the model, obtained by considering $α=0$. Our analysis also lets us to compute the growth and decay rates of spatial Fourier modes of density fluctuations. The growth rates compare very well with numerical simulations.

cond-mat.stat-mech↗

arXiv2010

Dynamical stability criterion for inhomogeneous quasi-stationary states in long-range systems

We derive a necessary and sufficient condition of linear dynamical stability for inhomogeneous Vlasov stationary states of the Hamiltonian Mean Field (HMF) model. The condition is expressed by an explicit disequality that has to be satisfied by the stationary state, and it generalizes the known disequality for homogeneous stationary states. In addition, we derive analogous disequalities that express necessary and sufficient conditions of formal stability for the stationary states. Their usefulness, from the point of view of linear dynamical stability, is that they are simpler, although they provide only sufficient criteria of linear stability. We show that for homogeneous stationary states the relations become equal, and therefore linear dynamical stability and formal stability become equivalent.

cond-mat.stat-mech↗

arXiv2008

Dynamical phase transitions in long-range Hamiltonian systems and Tsallis distributions with a time-dependent index

We study dynamical phase transitions in systems with long-range interactions, using the Hamiltonian Mean Field (HMF) model as a simple example. These systems generically undergo a violent relaxation to a quasi-stationary state (QSS) before relaxing towards Boltzmann equilibrium. In the collisional regime, the out-of-equilibrium one-particle distribution function (DF) is a quasi-stationary solution of the Vlasov equation, slowly evolving in time due to finite $N$ effects. For subcritical energies $7/12<U<3/4$, we exhibit cases where the DF is well-fitted by a Tsallis $q$-distribution with an index $q(t)$ slowly decreasing in time from $q\simeq 3$ (semi-ellipse) to $q=1$ (Boltzmann). When the index $q(t)$ reaches a critical value $q_{crit}(U)$, the non-magnetized (homogeneous) phase becomes Vlasov unstable and a dynamical phase transition is triggered, leading to a magnetized (inhomogeneous) state. While Tsallis distributions play an important role in our study, we explain this dynamical phase transition by using only conventional statistical mechanics. For supercritical energies, we report for the first time the existence of a magnetized QSS with a very long lifetime.

cond-mat.stat-mech↗

arXiv2007

Long time behavior of quasi-stationary states of the Hamiltonian Mean-Field model

The Hamiltonian Mean-Field model has been investigated, since its introduction about a decade ago, to study the equilibrium and dynamical properties of long-range interacting systems. Here we study the long-time behavior of long-lived, out-of-equilibrium, quasi-stationary dynamical states, whose lifetime diverges in the thermodynamic limit. The nature of these states has been the object of a lively debate, in the recent past. We introduce a new numerical tool, based on the fluctuations of the phase of the instantaneous magnetization of the system. Using this tool, we study the quasi-stationary states that arise when the system is started from different classes of initial conditions, showing that the new observable can be exploited to compute the lifetime of these states. We also show that quasi-stationary states are present not only below, but also above the critical temperature of the second order magnetic phase transition of the model. We find that at supercritical temperatures the lifetime is much larger than at subcritical temperatures.

cond-mat.stat-mech↗

arXiv2006

Long-Range Effects in Layered Spin Structures

We study theoretically layered spin systems where long-range dipolar interactions play a relevant role. By choosing a specific sample shape, we are able to reduce the complex Hamiltonian of the system to that of a much simpler coupled rotator model with short-range and mean-field interactions. This latter model has been studied in the past because of its interesting dynamical and statistical properties related to exotic features of long-range interactions. It is suggested that experiments could be conducted such that within a specific temperature range the presence of long-range interactions crucially affect the behavior of the system.

cond-mat.stat-mech↗

arXiv2006

On the computation of the entropy in the microcanonical ensemble for mean-field-like systems

Two recently proposed expressions for the computation of the entropy in the microcanonical ensemble are compared, and their equivalence is proved. These expressions are valid for a certain class of statistical mechanics systems, that can be called mean-field-like systems. Among these, this work considers only the systems with the most usual hamiltonian structure, given by a kinetic energy term plus interaction terms depending only on the configurational coordinates.

cond-mat.stat-mech↗

arXiv2005

Dynamics and thermodynamics of rotators interacting with both long and short range couplings

The effect of nearest-neighbor coupling on the thermodynamic and dynamical properties of the ferromagnetic Hamiltonian Mean Field model (HMF) is studied. For a range of antiferromagnetic nearest-neighbor coupling, a canonical first order transition is observed, and the canonical and microcanonical ensembles are non-equivalent. In studying the relaxation time of non-equilibrium states it is found that as in the HMF model, a class of non-magnetic states is quasi-stationary, with an algebraic divergence of their lifetime with the number of degrees of freedom $N$. The lifetime of metastable states is found to increase exponentially with $N$ as expected.

cond-mat.stat-mech↗

arXiv2005

Microcanonical solution of the mean-field $ϕ^4$ model: comparison with time averages at finite size

We solve the mean-field $ϕ^4$ model in an external magnetic field in the microcanonical ensemble using two different methods. The first one is based on Rugh's microcanonical formalism and leads to express macroscopic observables, such as temperature, specific heat, magnetization and susceptibility, as time averages of convenient functions of the phase-space. The approach is applicable for any finite number of particles $N$. The second method uses large deviation techniques and allows us to derive explicit expressions for microcanonical entropy and for macroscopic observables in the $N \to\infty$ limit. Assuming ergodicity, we evaluate time averages in molecular dynamics simulations and, using Rugh's approach, we determine the value of macroscopic observables at finite $N$. These averages are affected by a slow time evolution, often observed in systems with long-range interactions. We then show how the finite $N$ time averages of macroscopic observables converge to their corresponding $N\to\infty$ values as $N$ is increased. As expected, finite size effects scale as $N^{-1}$.

cond-mat.stat-mech↗

arXiv2001

Metastable states in a class of long-range Hamiltonian systems

We numerically show that metastable states, similar to the Quasi Stationary States found in the so called Hamiltonian Mean Field Model, are also present in a generalized model in which $N$ classical spins (rotators) interact through ferromagnetic couplings decaying as $r^{-α}$, where $r$ is their distance over a regular lattice. Scaling laws with $N$ are briefly discussed.

cond-mat.stat-mech↗

arXiv2000

Universal behavior in the static and dynamic properties of the $α$-XY model

The $α$-XY model generalizes, through the introduction of a power-law decaying potential, a well studied mean-field hamiltonian model with attractive long-range interactions. In the $α$-model, the interaction between classical rotators on a lattice is gauged by the exponent $α$ in the couplings decaying as $r^α$, where $r$ are distances between sites. We review and comment here a few recent results on the static and dynamic properties of the $α$-model. We discuss the appropriate $α$ dependent rescalings that map the canonical thermodynamics of the $α$-model into that of the mean field model. We also show that the chaotic properties of the model, studied as a function of $α$ display an universal behaviour.

cond-mat.stat-mech↗

arXiv2000

Bubble propagation in a helicoidal molecular chain

We study the propagation of very large amplitude localized excitations in a model of DNA that takes explicitly into account the helicoidal structure. These excitations represent the ``transcription bubble'', where the hydrogen bonds between complementary bases are disrupted, allowing access to the genetic code. We propose these kind of excitations in alternative to kinks and breathers. The model has been introduced by Barbi et al. [Phys. Lett. A 253, 358 (1999)], and up to now it has been used to study on the one hand low amplitude breather solutions, and on the other hand the DNA melting transition. We extend the model to include the case of heterogeneous chains, in order to get closer to a description of real DNA; in fact, the Morse potential representing the interaction between complementary bases has two possible depths, one for A-T and one for G-C base pairs. We first compute the equilibrium configurations of a chain with a degree of uncoiling, and we find that a static bubble is among them; then we show, by molecular dynamics simulations, that these bubbles, once generated, can move along the chain. We find that also in the most unfavourable case, that of a heterogeneous DNA in the presence of thermal noise, the excitation can travel for well more 1000 base pairs.

cond-mat.soft↗

arXiv2000

Long-range interacting classical systems: universality in mixing weakening

Through molecular dynamics, we study the $d=2,3$ classical model of $N$ coupled rotators (inertial XY model) assuming a coupling constant which decays with distance as $r_{ij}^{-α}$ ($α\ge 0$). The total energy $ $ is asymptotically $\propto N {\tilde N}$ with ${\tilde N} \equiv [N^{1-α/d}-(α/d)]/[1-α/d]$, hence the model is thermodynamically extensive if $α/d>1$ and nonextensive otherwise. We numerically show that, for energies above some threshold, the (appropriately scaled) maximum Lyapunov exponent is $\propto N^{-κ}$ where $κ$ is an {\it universal} (one and the same for $d=1,2$ and 3, and all energies) function of $α/d$, which monotonically decreases from 1/3 to zero when $α/d$ increases from 0 to 1, and identically vanishes above 1. These features are consistent with the nonextensive statistical mechanics scenario, where thermodynamic extensivity is associated with {\it exponential} mixing in phase space, whereas {\it weaker} (possibly {\it power-law} in the present case) mixing emerges at the $N \to \infty$ limit whenever nonextensivity is observed.

cond-mat.stat-mech↗

arXiv2000

Canonical solution of a system of long-range interacting rotators on a lattice

The canonical partition function of a system of rotators (classical X-Y spins) on a lattice, coupled by terms decaying as the inverse of their distance to the power alpha, is analytically computed. It is also shown how to compute a rescaling function that allows to reduce the model, for any d-dimensional lattice and for any alpha<d, to the mean field (alpha=0) model.

cond-mat.stat-mech↗

arXiv1998

First experimental tests of the Peyrard-Bishop model applied to the melting of very short DNAs

The melting curves of short heterogeneous DNA chains in solution are calculated on the basis of statistical thermodynamics and compared to experiments. The computation of the partition function is based on the Peyrard-Bishop hamiltonian, which has already been adopted in the theoretical description of the melting of long DNA chains. In the case of short chains it is necessary to consider not only the breaking of the hydrogen bonds between single base pairs, but also the complete dissociation of the two strands forming the double helix.

physics.bio-ph↗