arXiv subjects

Alessandro Campa

Publications and source records attributed to Alessandro Campa.

At least 19 recordsLinked to original sources

arXiv2026

Restoration of Ensemble Equivalence by Quantum Fluctuations

We study the thermodynamic phase diagram of a one-dimensional quantum spin chain subjected to both mean-field and nearest-neighbor interactions, and to a transverse magnetic field $h$. The purpose is to determine the effect of the quantum fluctuations, due to the transverse field, on the phase diagram, in particular with respect to the occurrence of ensemble inequivalence. We denote our model as a quantum Nagle-Kardar model. To perform the calculation of the canonical partition function, we show that, due to the presence of the mean-field term, in the thermodynamic limit one can use the Hubbard-Stratonovich transformation in spite of the non-commutativity of the different operators appearing in the Hamiltonian, and we adopt a procedure of successive approximations that lead to the determination of the phase diagram thanks to a scaling property of the phase transition lines. The results show that the ensemble inequivalence, present in the classical Nagle-Kardar model, is removed above a threshold value $h_c$ for the transverse field. For $h$ larger than $h_c$ the phase diagram exhibits only second-order phase transition lines, implying therefore restoration of ensemble equivalence.

cond-mat.stat-mech↗

arXiv2026

Phase diagram of the one-dimensional three-state Potts model with an additional mean-field interaction

We derive the phase diagram of the one-dimensional three-state Potts model with an additional mean-field interaction in the canonical ensemble. The free energy is obtained by mapping the model onto the spin-$1$ Blume-Emery-Griffiths model and solving it by using an Hubbard-Stratonovich transformation combined with the transfer matrix method. A complex structure with lines of first-order transitions, two triple points and a critical point appears at finite temperature. The phase diagram is two-dimensional, since there are two adjustable parameters, the nearest-neighbour coupling $K$ and the temperature $T$. We show that the phase diagram does not present second-order phase transition lines, due to the fact that the order parameter is not a symmetry-breaking one. Quite remarkably, we are able to determine analytically one of the first-order phase-transition lines. We also prove that, when the nearest-neighbour coupling $K$ is large and negative, the first-order transition temperature becomes asymptotically independent of $K$.

cond-mat.stat-mech↗

arXiv2025

Ensemble Inequivalence in Ising Chains with Competing Interactions

We study the effect of competing interactions on ensemble inequivalence. We consider a one-dimensional Ising model with ferromagnetic mean-field interactions and short-range nearest-neighbor and next-nearest-neighbor couplings which can be either ferromagnetic or antiferromagnetic. Despite the relative simplicity of the model, our calculations in the microcanonical ensemble reveal a rich phase diagram. The comparison with the corresponding phase diagram in the canonical ensemble shows the presence of phase transition points and lines which are different in the two ensembles. As an example, in a region of the phase diagram where the canonical ensemble shows a critical point and a critical end point, the microcanonical ensemble has an additional critical point and also a triple point. The regions of ensemble inequivalence typically occur at lower temperatures and at larger absolute values of the competing couplings. The presence of two free parameters in the model allows us to obtain a fourth-order critical point, which can be fully characterized by deriving its Landau normal form.

cond-mat.stat-mech↗

arXiv2023

Synchronization in a System of Kuramoto Oscillators with Distributed Gaussian Noise

We consider a system of globally-coupled phase-only oscillators with distributed intrinsic frequencies and evolving in presence of distributed Gaussian, white noise, namely, a Gaussian, white noise whose strength for every oscillator is a specified function of its intrinsic frequency. In the absence of noise, the model reduces to the celebrated Kuramoto model of spontaneous synchronization. For two specific forms of the mentioned functional dependence and for a symmetric and unimodal distribution of the intrinsic frequencies, we unveil the rich long-time behavior that the system exhibits, which stands in stark contrast to the case in which the noise strength is the same for all the oscillators. Namely, in the studied dynamics, the system may exist in either a synchronized or an incoherent or a time-periodic state; interestingly, all of these states also appear as long-time solutions of the Kuramoto dynamics for the case of bimodal frequency distributions, but in the absence of any noise in the dynamics.

cond-mat.stat-mech↗

arXiv2023

Modified Thirring model beyond the excluded-volume approximation

Long-range interacting systems may exhibit ensemble inequivalence and can possibly attain equilibrium states under completely open conditions, for which energy, volume and number of particles simultaneously fluctuate. Here we consider a modified version of the Thirring model for self-gravitating systems with attractive and repulsive long-range interactions in which particles are treated as hard spheres in dimension d=1,2,3. Equilibrium states of the model are studied under completely open conditions, in the unconstrained ensemble, by means of both Monte Carlo simulations and analytical methods and are compared with the corresponding states at fixed number of particles, in the isothermal-isobaric ensemble. Our theoretical description is performed for an arbitrary local equation of state, which allows us to examine the system beyond the excluded-volume approximation. The simulations confirm the theoretical prediction of the possible occurrence of first-order phase transitions in the unconstrained ensemble. This work contributes to the understanding of long-range interacting systems exchanging heat, work and matter with the environment.

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arXiv2022

Relaxation dynamics in a long-range system with mixed Hamiltonian and non-Hamiltonian interactions

Sometimes the dynamics of a physical system is described by non-Hamiltonian equations of motion, and additionally, the system is characterized by long-range interactions. A concrete example is that of particles interacting with light as encountered in free-electron laser and cold-atom experiments. In this work, we study the relaxation dynamics to non-Hamiltonian systems, more precisely, to systems with interactions of both Hamiltonian and non-Hamiltonian origin. Our model consists of $N$ globally-coupled particles moving on a circle of unit radius; the model is one-dimensional. We show that in the infinite-size limit, the dynamics, similarly to the Hamiltonian case, is described by the Vlasov equation. In the Hamiltonian case, the system eventually reaches an equilibrium state, even though one has to wait for a long time diverging with $N$ for this to happen. By contrast, in the non-Hamiltonian case, there is no equilibrium state that the system is expected to reach eventually. We characterize this state with its average magnetization. We find that the relaxation dynamics depends strongly on the relative weight of the Hamiltonian and non-Hamiltonian contributions to the interaction. When the non-Hamiltonian part is predominant, the magnetization attains a vanishing value, suggesting that the system does not sustain states with constant magnetization, either stationary or rotating. On the other hand, when the Hamiltonian part is predominant, the magnetization presents long-lived strong oscillations, for which we provide a heuristic explanation. Furthermore, we find that the finite-size corrections are much more pronounced than those in the Hamiltonian case; we justify this by showing that the Lenard-Balescu equation, which gives leading-order corrections to the Vlasov equation, does not vanish, contrary to what occurs in one-dimensional Hamiltonian long-range systems.

cond-mat.stat-mech↗

arXiv2022

The study of the dynamics of the order parameter of coupled oscillators in the Ott-Antonsen scheme for generic frequency distributions

The Ott-Antonsen ansatz shows that, for certain classes of distribution of the natural frequencies in systems of $N$ globally coupled Kuramoto oscillators, the dynamics of the order parameter, in the limit $N\to \infty$, evolves, under suitable initial conditions, in a manifold of low dimension. This is not possible when the frequency distribution, continued in the complex plane, has an essential singularity at infinite; this is the case for example, of a Gaussian distribution. In this work we propose a simple approximation scheme that allows to extend also to this case the representation of the dynamics of the order parameter in a low dimensional manifold. Using as a working example the Gaussian frequency distribution, we compare the dynamical evolution of the order parameter of the system of oscillators, obtained by the numerical integration of the $N$ equations of motion, with the analogous dynamics in the low dimensional manifold obtained with the application of the approximation scheme. The results confirm the validity of the approximation. The method could be employed for general frequency distributions, allowing the determination of the corresponding phase diagram of the oscillator system.

cond-mat.stat-mech↗

arXiv2021

Ensemble inequivalence in the Blume-Emery-Griffiths model near a fourth order critical point

The canonical phase diagram of the Blume-Emery-Griffiths (BEG) model with infinite-range interactions is known to exhibit a fourth order critical point at some negative value of the bi-quadratic interaction $K<0$. Here we study the microcanonical phase diagram of this model for $K<0$, extending previous studies which were restricted to positive $K$. A fourth order critical point is found to exist at coupling parameters which are different from those of the canonical ensemble. The microcanonical phase diagram of the model close to the fourth order critical point is studied in detail revealing some distinct features from the canonical counterpart.

cond-mat.stat-mech↗

arXiv2021

Computation of microcanonical entropy at fixed magnetization without direct counting

We discuss a method to compute the microcanonical entropy at fixed magnetization without direct counting. Our approach is based on the evaluation of a saddle-point leading to an optimization problem. The method is applied to a benchmark Ising model with simultaneous presence of mean-field and nearest-neighbour interactions for which direct counting is indeed possible, thus allowing a comparison. Moreover, we apply the method to an Ising model with mean-field, nearest-neighbour and next-nearest-neighbour interactions, for which direct counting is not straightforward. For this model, we compare the solution obtained by our method with the one obtained from the formula for the entropy in terms of all correlation functions. This example shows that for general couplings our method is much more convenient than direct counting methods to compute the microcanonical entropy at fixed magnetization.

cond-mat.stat-mech↗

arXiv2021

Monte Carlo simulations in the unconstrained ensemble

The unconstrained ensemble describes completely open systems whose control parameters are chemical potential, pressure, and temperature. For macroscopic systems with short-range interactions, thermodynamics prevents the simultaneous use of these intensive variables as control parameters, because they are not independent and cannot account for the system size. When the range of the interactions is comparable with the size of the system, however, these variables are not truly intensive and may become independent, so equilibrium states defined by the values of these parameters may exist. Here, we derive a Monte Carlo algorithm for the unconstrained ensemble and show that simulations can be performed using chemical potential, pressure, and temperature as control parameters. We illustrate the algorithm by applying it to physical systems where either the system has long-range interactions or is confined by external conditions. The method opens up a new avenue for the simulation of completely open systems exchanging heat, work, and matter with the environment.

cond-mat.stat-mech↗

arXiv2019

Phase diagram of noisy systems of coupled oscillators with a bimodal frequency distribution

We study the properties of large systems of globally coupled oscillators in the presence of noise. When the distribution of the natural frequencies of the oscillators is bimodal and its analytical continuation in the complex plane has only few poles in the lower half plane, the dynamics of the system, governed by a Fokker-Planck equation for the single particle distribution function, can be reduced to a system of ordinary differential equations describing the dynamics of suitably defined order parameters, the first ones of which are related to the usual synchronization order parameter. We obtain the full phase diagram of the oscillator system, that shows a very rich behaviour, with regions characterized by synchronized states, regions with periodic states, and others with bi-stability, associated to the presence of hysteresis. The latter phenomenon is confirmed by numerical simulations ot the full system of coupled oscillators. We compare our results with those previously obtained for noiseless systems, and we show that for increasing noise the phase diagram changes qualitatively, tending to the simple diagram that is found for systems with unimodal frequency distributions.

cond-mat.stat-mech↗

arXiv2019

Phase transitions in the unconstrained ensemble

The unconstrained ensemble describes completely open systems in which energy, volume and number of particles fluctuate. Here we show that not only equilibrium states can exist in this ensemble, but also that completely open systems can undergo first-order phase transitions. This is shown by studying a modified version of the Thirring model with attractive and repulsive interactions and with particles of finite size. The model exhibits first-order phase transitions in the unconstrained ensemble, at variance with the analogous model with point-like particles. While unconstrained and grand canonical ensembles are equivalent for this model, we found inequivalence between the unconstrained and isothermal-isobaric ensembles. By comparing the thermodynamic phase diagram in the unconstrained case with that obtained in the isothermal-isobaric ensemble, we show that phase transitions under completely open conditions for this model are different from those in which the number of particles is fixed, highlighting the inequivalence of ensembles.

cond-mat.stat-mech↗

arXiv2019

Ising chains with competing interactions in presence of long-range couplings

In this paper we study an Ising spin chain with short-range competing interactions in presence of long-range ferromagnetic interactions in the canonical ensemble. The simultaneous presence of the frustration induced by the short-range couplings together with their competition with the long-range term gives rise to a rich thermodynamic phase diagram. We compare our results with the limit in which one of two local interactions is turned off, which was previously studied in the literature. Eight regions of parameters with qualitatively distinct properties are featured, with different first- and second-order phase transition lines and critical points.

cond-mat.stat-mech↗

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

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

Dynamics of coupled oscillator systems in presence of a local potential

We consider a long-range model of coupled phase-only oscillators subject to a local potential and evolving in presence of thermal noise. The model is a non-trivial generalization of the celebrated Kuramoto model of collective synchronization. We demonstrate by exact results and numerics a surprisingly rich long-time behavior, in which the system settles into either a stationary state that could be in or out of equilibrium and supports either global synchrony or absence of it, or, in a time-periodic synchronized state. The system shows both continuous and discontinuous phase transitions, as well as an interesting reentrant transition in which the system successively loses and gains synchrony on steady increase of the relevant tuning parameter.

nlin.AO↗

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↗

arXiv2016

The quasilinear theory in the approach of long-range systems to quasi-stationary states

We develop a quasilinear theory of the Vlasov equation in order to describe the approach of systems with long-range interactions to quasi-stationary states. We derive a diffusion equation governing the evolution of the velocity distribution of the system towards a steady state. This steady state is expected to correspond to the angle-averaged quasi-stationary distribution function reached by the Vlasov equation as a result of a violent relaxation. We compare the prediction of the quasilinear theory to direct numerical simulations of the Hamiltonian Mean Field model, starting from an unstable spatially homogeneous distribution, either Gaussian or semi-elliptical. We find that the quasilinear theory works reasonably well for weakly unstable initial conditions and that it is able to predict the energy marking the out-of-equilibrium phase transition between unmagnetized and magnetized quasi-stationary states. At energies lower than the out-of-equilibrium transition the quasilinear theory works less well, the disagreement with the numerical simulations increasing by decreasing the energy. In that case, we observe, in agreement with our previous numerical study [A. Campa and P.-H. Chavanis, Eur. Phys. J. B 86, 170 (2013)], that the quasi-stationary states are remarkably well fitted by polytropic distributions (Tsallis distributions) with index $n=2$ (Gaussian case) or $n=1$ (semi-elliptical case). In particular, these polytropic distributions are able to account for the region of negative specific heats in the out-of-equilibrium caloric curve, unlike the Boltzmann and Lynden-Bell distributions.

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