arXiv subjects

Lapo Casetti

Publications and source records attributed to Lapo Casetti.

At least 37 records · Page 2Linked to original sources

arXiv2018

Concavity, Response Functions and Replica Energy

In nonadditive systems, like small systems or like long-range interacting systems even in the thermodynamic limit, ensemble inequivalence can be related to the occurrence of negative response functions, this in turn being connected with anomalous concavity properties of the thermodynamic potentials associated to the various ensembles. We show how the type and number of negative response functions depend on which of the quantities E, V and N (energy, volume and number of particles) are constrained in the ensemble. In particular, we consider the unconstrained ensemble in which E, V and N fluctuate, physically meaningful only for nonadditive systems. In fact, its partition function is associated to the replica energy, a thermodynamic function that identically vanishes when additivity holds, but that contains relevant information in nonadditive systems.

cond-mat.stat-mech↗

arXiv2017

Dynamical origin of non-thermal states in galactic filaments

Observations strongly suggest that filaments in galactic molecular clouds are in a non-thermal state. As a simple model of a filament we study a two-dimensional system of self-gravitating point particles by means of numerical simulations of the dynamics, with various methods: direct $N$-body integration of the equations of motion, particle-in-cell simulations and a recently developed numerical scheme that includes multiparticle collisions in a particle-in-cell approach. Studying the collapse of Gaussian overdensities we find that after the damping of virial oscillations the system settles in a non-thermal steady state whose radial density profile is similar to the observed ones, thus suggesting a dynamical origin of the non-thermal states observed in real filaments. Moreover, for sufficiently cold collapses the density profiles are anticorrelated with the kinetic temperature, i.e., exhibit temperature inversion, again a feature that has been found in some observations of filaments. The same happens in the state reached after a strong perturbation of an initially isothermal cylinder. Finally, we discuss our results in the light of recent findings in other contexts (including non-astrophysical ones) and argue that the same kind of non-thermal states may be observed in any physical system with long-range interactions.

astro-ph.GA↗

arXiv2017

Non-thermal states in models of filaments: a dynamical study

We study the origin of the non-thermal profiles observed in filamentary structures in galactic molecular clouds by means of numerical dynamical simulations. We find that such profiles are intrinsic features of the end products of dissipationless collapse in cylindrical symmetry. Moreover, for sufficiently cold initial conditions, we obtain end states characterized by markedly anticorrelated radial density and temperature profiles. Gravitational, dissipationless dynamics alone is thus sufficient to reproduce, at least qualitatively, many of the properties of the observed non-thermal structures.

astro-ph.GA↗

arXiv2017

Heat transport in oscillator chains with long-range interactions coupled to thermal reservoirs

We investigate thermal conduction in arrays of long-range interacting rotors and Fermi-Pasta-Ulam (FPU) oscillators coupled to two reservoirs at different temperatures. The strength of the interaction between two lattice sites decays as a power $α$ of the inverse of their distance. We point out the necessity of distinguishing between energy flows towards/from the reservoirs and those within the system. We show that energy flow between the reservoirs occurs via a direct transfer induced by long-range couplings and a diffusive process through the chain. To this aim, we introduce a decomposition of the steady-state heat current that explicitly accounts for such direct transfer of energy between the reservoir. For $0\leq α<1$, the direct transfer term dominates, meaning that the system can be effectively described as a set of oscillators each interacting with the thermal baths. Also, the heat current exchanged with the reservoirs depends on the size of the thermalised regions: in the case in which such size is proportional to the system size $N$, the stationary current is independent on $N$. For $α> 1$, heat transport mostly occurs through diffusion along the chain: for the rotors transport is normal, while for FPU the data are compatible with an anomalous diffusion, possibly with an $α$ -dependent characteristic exponent.

cond-mat.stat-mech↗

arXiv2017

Long-range interacting systems in the unconstrained ensemble

Completely open systems can exchange heat, work, and matter with the environment. While energy, volume, and number of particles fluctuate under completely open conditions, the equilibrium states of the system, if they exist, can be specified using the temperature, pressure, and chemical potential as control parameters. The unconstrained ensemble is the statistical ensemble describing completely open systems and the replica energy is the appropriate free energy for these control parameters from which the thermodynamics must be derived. It turns out that macroscopic systems with short-range interactions cannot attain equilibrium configurations in the unconstrained ensemble, since temperature, pressure, and chemical potential cannot be taken as a set of independent variables in this case. In contrast, we show that systems with long-range interactions can reach states of thermodynamic equilibrium in the unconstrained ensemble. To illustrate this fact, we consider a modification of the Thirring model and compare the unconstrained ensemble with the canonical and grand canonical ones: the more the ensemble is constrained by fixing the volume or number of particles, the larger the space of parameters defining the equilibrium configurations.

cond-mat.stat-mech↗

arXiv2016

Surprises from quenches in long-range interacting systems: Temperature inversion and cooling

What happens when one of the parameters governing the dynamics of a long-range interacting system of particles in thermal equilibrium is abruptly changed (quenched) to a different value? While a short-range system, under the same conditions, will relax in time to a new thermal equilibrium with a uniform temperature across the system, a long-range system shows a fast relaxation to a nonequilibrium {\em quasistationary state} (QSS). The lifetime of such an off-equilibrium state diverges with the system size, and the temperature is non-uniform across the system. Quite surprisingly, the density profile in the QSS obtained after the quench is anticorrelated with the temperature profile in space, thus exhibiting the phenomenon of {\em temperature inversion}: denser regions are colder than sparser ones. We illustrate with extensive molecular dynamics simulations the ubiquity of this scenario in a prototypical long-range interacting system subject to a variety of quenching protocols, and in a model that mimics an experimental setup of atoms interacting with light in an optical cavity. We further demonstrate how a procedure of iterative quenching combined with filtering out the high-energy particles in the system may be employed to cool the system. Temperature inversion is observed in nature in some astrophysical settings; our results imply that such a phenomenon should be observable, and could even be exploitable to advantage, also in controlled laboratory experiments.

cond-mat.stat-mech↗

arXiv2016

Phase transitions in Thirring's model

In his pioneering work on negative specific heat, Walter Thirring in\-tro\-duced a model that is solvable in the microcanonical ensemble. Here, we give a complete description of the phase-diagram of this model in both the microcanonical and the canonical ensemble, highlighting the main features of ensemble inequivalence. In both ensembles, we find a line of first-order phase transitions which ends in a critical point. However, neither the line nor the point have the same location in the phase-diagram of the two ensembles. We also show that the microcanonical and canonical critical points can be analytically related to each other using a Landau expansion of entropy and free energy, respectively, in analogy with what has been done in [O. Cohen, D. Mukamel, J. Stat. Mech., P12017 (2012)]. Examples of systems with certain symmetries restricting the Landau expansion have been considered in this reference, while no such restrictions are present in Thirring's model. This leads to a phase diagram that can be seen as a prototype for what happens in systems of particles with kinematic degrees of freedom dominated by long-range interactions.

cond-mat.stat-mech↗

arXiv2015

Temperature inversion in long-range interacting systems

Temperature inversions occur in nature, e.g., in the solar corona and in interstellar molecular clouds: somewhat counterintuitively, denser parts of the system are colder than dilute ones. We propose a simple and appealing way to spontaneously generate temperature inversions in systems with long-range interactions, by preparing them in inhomogeneous thermal equilibrium states and then applying an impulsive perturbation. In similar situations, short-range systems would typically relax to another thermal equilibrium, with uniform temperature profile. By contrast, in long-range systems, the interplay between wave-particle interaction and spatial inhomogeneity drives the system to nonequilibrium stationary states that generically exhibit temperature inversion. We demonstrate this mechanism in a simple mean-field model and in a two-dimensional self-gravitating system. Our work underlines the crucial role the range of interparticle interaction plays in determining the nature of steady states out of thermal equilibrium.

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

Critical energy density of O$(n)$ models in $d=3$

A relation between O$(n)$ models and Ising models has been recently conjectured [L. Casetti, C. Nardini, and R. Nerattini, Phys. Rev. Lett. 106, 057208 (2011)]. Such a relation, inspired by an energy landscape analysis, implies that the microcanonical density of states of an O$(n)$ spin model on a lattice can be effectively approximated in terms of the density of states of an Ising model defined on the same lattice and with the same interactions. Were this relation exact, it would imply that the critical energy densities of all the O$(n)$ models (i.e., the average values per spin of the O$(n)$ Hamiltonians at their respective critical temperatures) should be equal to that of the corresponding Ising model; it is therefore worth investigating how different the critical energies are and how this difference depends on $n$. We compare the critical energy densities of O$(n)$ models in three dimensions in some specific cases: the O$(1)$ or Ising model, the O$(2)$ or $XY$ model, the O$(3)$ or Heisenberg model, the O$(4)$ model and the O$(\infty)$ or spherical model, all defined on regular cubic lattices and with ferromagnetic nearest-neighbor interactions. The values of the critical energy density in the $n=2$, $n=3$, and $n=4$ cases are derived through a finite-size scaling analysis of data produced by means of Monte Carlo simulations on lattices with up to $128^3$ sites. For $n=2$ and $n=3$ the accuracy of previously known results has been improved. We also derive an interpolation formula showing that the difference between the critical energy densities of O$(n)$ models and that of the Ising model is smaller than $1\%$ if $n<8$ and never exceeds $3\%$ for any $n$.

cond-mat.stat-mech↗

arXiv2014

Velocity filtration and temperature inversion in a system with long-range interactions

Temperature inversion due to velocity filtration, a mechanism originally proposed to explain the heating of the solar corona, is demonstrated to occur also in a simple paradigmatic model with long-range interactions, the Hamiltonian mean-field model. Using molecular dynamics simulations, we show that when the system settles into an inhomogeneous quasi-stationary state in which the velocity distribution has suprathermal tails, the temperature and density profiles are anticorrelated: denser parts of the system are colder than dilute ones. We argue that this may be a generic property of long-range interacting systems.

cond-mat.stat-mech↗

arXiv2013

Density of states of the $XY$ model: an energy landscape approach

Among the stationary configurations of the Hamiltonian of a classical O$(n)$ lattice spin model, a class can be identified which is in one-to-one correspondence with all the the configurations of an Ising model defined on the same lattice and with the same interactions. Starting from this observation it has been recently proposed that the microcanonical density of states of an O$(n)$ model could be written in terms of the density of states of the corresponding Ising model. Later, it has been shown that a relation of this kind holds exactly for two solvable models, the mean-field and the one-dimensional $XY$ model, respectively. We apply the same strategy to derive explicit, albeit approximate, expressions for the density of states of the two-dimensional $XY$ model with nearest-neighbor interactions on a square lattice. The caloric curve and the specific heat as a function of the energy density are calculated and compared against simulation data, yielding a very good agreement over the entire energy density range. The concepts and methods involved in the approximations presented here are valid in principle for any O$(n)$ model.

cond-mat.stat-mech↗

arXiv2012

Exploring the energy landscape of XY models

We investigate the energy landscape of two- and three-dimensional XY models with nearest-neighbor interactions by analytically constructing several classes of stationary points of the Hamiltonian. These classes are analyzed, in particular with respect to possible signatures of the thermodynamic phase transitions of the models. We find that, even after explicitly breaking the global O(2) symmetry of the XY spins, an exponentially large class of stationary points are singular and occur in continuous one-parameter families. This property may complicate the use of theoretical tools developed for the investigation of phase transitions based on stationary points of the energy landscape, and we discuss strategies to avoid these difficulties.

cond-mat.stat-mech↗

arXiv2012

Geometry of the energy landscape of the self-gravitating ring

We study the global geometry of the energy landscape of a simple model of a self-gravitating system, the self-gravitating ring (SGR). This is done by endowing the configuration space with a metric such that the dynamical trajectories are identified with geodesics. The average curvature and curvature fluctuations of the energy landscape are computed by means of Monte Carlo simulations and, when possible, of a mean-field method, showing that these global geometric quantities provide a clear geometric characterization of the collapse phase transition occurring in the SGR as the transition from a flat landscape at high energies to a landscape with mainly positive but fluctuating curvature in the collapsed phase. Moreover, curvature fluctuations show a maximum in correspondence with the energy of a possible further transition, occurring at lower energies than the collapse one, whose existence had been previously conjectured on the basis of a local analysis of the energy landscape and whose effect on the usual thermodynamic quantities, if any, is extremely weak. We also estimate the largest Lyapunov exponent $λ$ of the SGR using the geometric observables. The geometric estimate always gives the correct order of magnitude of $λ$ and is also quantitatively correct at small energy densities and, in the limit $N\to\infty$, in the whole homogeneous phase.

cond-mat.stat-mech↗

arXiv2012

Caloric curve of star clusters

Self-gravitating systems, like globular clusters or elliptical galaxies, are the prototypes of many-body systems with long-range interactions, and should be the natural arena where to test theoretical predictions on the statistical behaviour of long-range-interacting systems. Systems of classical self-gravitating particles can be studied with the standard tools of equilibrium statistical mechanics, provided the potential is regularized at small length scales and the system is confined in a box. The confinement condition looks rather unphysical in general, so that it is natural to ask whether what we learn with these studies is relevant to real self-gravitating systems. In order to provide a first answer to this question we consider a basic, simple, yet effective model of globular clusters, the King model. This model describes a self-consistently confined system, without the need of any external box, but the stationary state is a non-thermal one. In particular, we consider the King model with a short-distance cutoff on the interactions and we discuss how such a cutoff affects the caloric curve, i.e. the relation between temperature and energy. We find that the cutoff stabilizes a low-energy phase which is absent in the King model without cutoff; the caloric curve of the model with cutoff turns out to be very similar to that of previously studied confined and regularized models, but for the absence of a high-energy gas-like phase. We briefly discuss the possible phenomenological as well as theoretical implications of these results.

cond-mat.stat-mech↗

arXiv2012

Density of states of continuous and discrete spin models: a case study

A relation between O(n) lattice spin models and Ising models defined on the same lattice was recently put forward [L. Casetti, C. Nardini, and R. Nerattini, Phys. Rev. Lett. 106, 057208 (2011)]. Such a relation, inspired by an energy landscape analysis, implies that the density of states of an O(n) spin model on a lattice can be effectively approximated, at least close to the phase transition, in terms of the density of states of an Ising model defined on the same lattice and with the same interactions. In the present paper we show that such a relation exactly holds, albeit in a slightly modified form, in the special cases of the mean-field XY model and of the one-dimensional XY model. We also discuss the possible consequences of this result for the general case.

cond-mat.stat-mech↗

arXiv2010

On a microcanonical relation between continuous and discrete spin models

A relation between a class of stationary points of the energy landscape of continuous spin models on a lattice and the configurations of a Ising model defined on the same lattice suggests an approximate expression for the microcanonical density of states. Based on this approximation we conjecture that if a O(n) model with ferromagnetic interactions on a lattice has a phase transition, its critical energy density is equal to that of the n = 1 case, i.e., a system of Ising spins with the same interactions. The conjecture holds true in the case of long-range interactions. For nearest-neighbor interactions, numerical results are consistent with the conjecture for n=2 and n=3 in three dimensions. For n=2 in two dimensions (XY model) the conjecture yields a prediction for the critical energy of the Berezinskij-Kosterlitz-Thouless transition, which would be equal to that of the two-dimensional Ising model. We discuss available numerical data in this respect.

cond-mat.stat-mech↗

arXiv2010

Solvable model of a self-gravitating system

We introduce and discuss an effective model of a self-gravitating system whose equilibrium thermodynamics can be solved in both the microcanonical and the canonical ensemble, up to a maximization with respect to a single variable. Such a model can be derived from a model of self-gravitating particles confined on a ring, referred to as the self-gravitating ring (SGR) model, allowing a quantitative comparison between the thermodynamics of the two models. Despite the rather crude approximations involved in its derivation, the effective model compares quite well with the SGR model. Moreover, we discuss the relation between the effective model presented here and another model introduced by Thirring forty years ago. The two models are very similar and can be considered as examples of a class of minimal models of self-gravitating systems.

cond-mat.stat-mech↗