arXiv subjects

Renato Pakter

Publications and source records attributed to Renato Pakter.

At least 19 recordsLinked to original sources

arXiv2026

Comment on: Discontinuous codimension-two bifurcation in a Vlasov equation (arXiv:2212.01250)

We comment on the recent work by Yamaguchi and Barr\'e [Phys. Rev. E 107, 054203 (2023)], which uses linear stability analysis of the Vlasov equation to characterize phase transitions in a generalized Hamiltonian Mean Field (gHMF) model. By performing extensive molecular dynamics simulations with $N=10^8$ particles, we demonstrate that the bifurcation analysis of the initial stationary distribution is insufficient to predict either the location or the nature of the phase transition to a quasi-stationary state (qSS). Specifically, we show that for bimodal momentum distributions, the instability threshold identified by the authors does not correspond to a ferromagnetic transition; instead, the system remains in a paramagnetic state characterized by magnetization oscillations with a zero time-average. We find that the true paramagnetic-ferromagnetic transition is discontinuous (first-order) and occurs at significantly larger coupling strengths, characterized by a clear coexistence of states. These results indicate that linear bifurcation and symmetry-breaking phase transitions are distinct phenomena in long-range interacting systems, and that the former lacks the predictive power to describe the long-time fate of the system.

cond-mat.stat-mech↗

arXiv2025

Mixing and Ergodicity in Systems with Long-Range Interactions

We present a theory of collisionless relaxation in systems with long-range interactions. Contrary to Lynden-Bell's theory of violent relaxation, which assumes global ergodicity and mixing, we show that quasi-stationary states (qSS) observed in these systems exhibit broken global ergodicity. We propose that relaxation towards equilibrium occurs through a process of local mixing, where particles spread over energy shells defined by the manifold to which their trajectories are confined. To demonstrate our theory, we study the Hamiltonian Mean Field (HMF) model, a paradigmatic system with long-range interactions. Our theory accurately predicts the particle distribution functions in qSS observed in molecular dynamics simulations without any adjustable parameters. Additionally, it precisely forecasts the phase transitions observed in the HMF model.

cond-mat.stat-mech↗

arXiv2020

Linear and non-linear instabilities of Kirchhoff's Elliptical Vortices

We investigate the stability of a uniform elliptical vortex in a two-dimensional incompressible Euler fluid. It's demonstrated that for small eccentricities, the vortex relaxes to a core-halo structure that undergoes rigid rotation with the central core remaining elliptical. For large eccentricities, the vortex splits into two quasi-circular vortices that revolve around the center of mass. Independent of the aspect ratio, the steady-state displays a low-density halo. We present a theory that qualitatively explains the transition between the two states. All theoretical results are compared with extensive molecular dynamics simulations based on the vortex-in-cell algorithm.

physics.flu-dyn↗

arXiv2020

Entropy production and Vlasov equation for self-gravitating systems

The evolution of a self-gravitating system to a non-equilibrium steady state occurs through a process of violent relaxation. In the thermodynamic limit the dynamics of a many body system should be governed by the Vlasov equation. Recently, however, a question was raised regarding the validity of Vlasov equation during the process of violent relaxation. In this paper we will explore the entropy production during the relaxation process using N-body molecular dynamics simulations. We will show that the entropy production time grows as $N^α$, with $α> 0$ and in the limit $N \rightarrow \infty$, entropy will remain constant, consistent with the Vlasov equation. Furthermore, we will show that the mean field dynamics constructed on the basis of the Vlasov equation is in excellent agreement with the full molecular dynamics simulations, justifying the applicability of Vlasov equation during the violent relaxation phase of evolution.

cond-mat.stat-mech↗

arXiv2018

Non-equilibrium Statistical Mechanics of Two-dimensional Vortices

It has been observed empirically that two dimensional vortices tend to cluster forming a giant vortex. To account for this observation Onsager introduced a concept of negative absolute temperature in equilibrium statistical mechanics. In this Letter we will show that in the thermodynamic limit a system of interacting vortices does not relax to the thermodynamic equilibrium, but becomes trapped in a non-equilibrium stationary state. We will show that the vortex distribution in this non-equilibrium stationary state has a characteristic core-halo structure, which can be predicted {\it a priori}. All the theoretical results are compared with explicit molecular dynamics simulations.

cond-mat.stat-mech↗

arXiv2017

Entropy production in systems with long range interactions

On a fine grained scale the Gibbs entropy of an isolated system remains constant throughout its dynamical evolution. This is a consequence of Liouville's theorem for Hamiltonian systems and appears to contradict the second law of thermodynamics. In reality, however, there is no problem since the thermodynamic entropy should be associated with the Boltzmann entropy, which for non-equilibrium systems is different from Gibbs entropy. The Boltzmann entropy accounts for the microstates which are not accessible from a given initial condition, but are compatible with a given macrostate. In a sense the Boltzmann entropy is a coarse grained version of the Gibbs entropy and will not decrease during the dynamical evolution of a macroscopic system. In this paper we will explore the entropy production for systems with long range interactions. Unlike for short range systems, in the thermodynamic limit, the probability density function for these systems decouples into a product of one particle distribution functions and the coarse grained entropy can be calculated explicitly. We find that the characteristic time for the entropy production scales with the number of particles as $N^α$, with $α> 0$, so that in the thermodynamic limit entropy production takes an infinite amount of time.

cond-mat.stat-mech↗

arXiv2014

Ensemble inequivalence in a mean-field XY model with ferromagnetic and nematic couplings

We explore ensemble inequivalence in long-range interacting systems by studying an XY model of classical spins with ferromagnetic and nematic coupling. We demonstrate the inequivalence by mapping the microcanonical phase diagram onto the canonical one, and also by doing the inverse mapping. We show that the equilibrium phase diagrams within the two ensembles strongly disagree within the regions of first-order transitions, exhibiting interesting features like temperature jumps. In particular, we discuss the coexistence and forbidden regions of different macroscopic states in both the phase diagrams.

cond-mat.stat-mech↗

arXiv2014

Reply to 'Comment on "Vortex distribution in a confining potential"

We argue that contrary to recent suggestions, non-extensive statistical mechanics has no relevance for inhomogeneous systems of particles interacting by short-range potentials. We show that these systems are perfectly well described by the usual Boltzmann-Gibbs statistical mechanics.

cond-mat.stat-mech↗

arXiv2014

Nonequilibrium stationary states of 3D self-gravitating systems

Three dimensional self-gravitating systems do not evolve to thermodynamic equilibrium, but become trapped in nonequilibrium quasistationary states. In this Letter we present a theory which allows us to a priori predict the particle distribution in a final quasistationary state to which a self-gravitating system will evolve from an initial condition which is isotropic in particle velocities and satisfies a virial constraint 2K=-U, where K is the total kinetic energy and U is the potential energy of the system.

cond-mat.stat-mech↗

arXiv2013

Symmetry Breaking in d-Dimensional Self-gravitating Systems

Systems with long-range interactions, such as self-gravitating clusters and magnetically confined plasmas, do not relax to the usual Boltzmann-Gibbs thermodynamic equilibrium, but become trapped in quasi-stationary states (QSS) the life time of which diverges with the number of particles. The QSS are characterized by the lack of ergodicity which can result in a symmetry broken QSS starting from a spherically symmetric particle distribution. We will present a theory which allows us to quantitatively predict the instability threshold for spontaneous symmetry breaking for a class of d-dimensional self-gravitating systems.

cond-mat.stat-mech↗

arXiv2013

Nonequilibrium Statistical Mechanics of Systems with Long-Range Interactions: Ubiquity of Core-Halo Distributions

Systems with long-range (LR) forces, for which the interaction potential decays with the interparticle distance with an exponent smaller than the dimensionality of the embedding space, remain an outstanding challenge to statistical physics. The internal energy of such systems lacks extensivity and additivity. Although the extensivity can be restored by scaling the interaction potential with the number of particles, the non-additivity still remains. Lack of additivity leads to inequivalence of statistical ensembles. Before relaxing to thermodynamic equilibrium, isolated systems with LR forces become trapped in out-of-equilibrium quasi-stationary state (qSS), the lifetime of which diverges with the number of particles. Therefore, in thermodynamic limit LR systems will not relax to equilibrium. The qSSs are attained through the process of collisionless relaxation. Density oscillations lead to particle-wave interactions and excitation of parametric resonances. The resonant particles escape from the main cluster to form a tenuous halo. Simultaneously, this cools down the core of the distribution and dampens out the oscillations. When all the oscillations die out the ergodicity is broken and a qSS is born. In this report, we will review a theory which allows us to quantitatively predict the particle distribution in the qSS. The theory is applied to various LR interacting systems, ranging from plasmas to self-gravitating clusters and kinetic spin models.

cond-mat.stat-mech↗

arXiv2013

Topology of Collisionless Relaxation

Using extensive molecular dynamics simulations we explore the fine-grained phase space structure of systems with long-range interactions. We find that if the initial phase space particle distribution has no holes, the final stationary distribution will also contain a compact simply connected region. The microscopic holes created by the filamentation of the initial distribution function are always restricted to the outer regions of the phase space. In general, for complex multilevel distributions it is very difficult to a priori predict the final stationary state without solving the full dynamical evolution. However, we show that for multilevel initial distributions satisfying a generalized virial condition, it is possible to predict the particle distribution in the final stationary state using Casimir invariants of the Vlasov dynamics.

cond-mat.stat-mech↗

arXiv2013

Nonequilibrium phase transitions in systems with long-range interactions

We introduce a generalized Hamiltonian Mean Field Model (gHMF)-XY model with both linear and quadratic coupling between spins and explicit Hamiltonian dynamics. In addition to the usual paramagnetic and ferromagnetic phases, this model also possesses a nematic phase. The gHMF can be solved explicitly using Boltzmann-Gibbs (BG) statistical mechanics, in both canonical and microcanonical ensembles. However, when the resulting microcanonical phase diagram is compared with the one obtained using molecular dynamics simulations, it is found that the two are very different. We will present a dynamical theory which allows us to explicitly calculate the phase diagram obtained using molecular dynamics simulations without any adjustable parameters. The model illustrates the fundamental role played by dynamics as well the inadequacy of BG statistics for systems with long-range forces in the thermodynamic limit.

cond-mat.stat-mech↗

arXiv2012

Ergodicity Breaking and Parametric Resonances in Systems with Long-Range Interactions

We explore the mechanism responsible for the ergodicity breaking in systems with long-range forces. In thermodynamic limit such systems do not evolve to the Boltzmann-Gibbs equilibrium, but become trapped in an out-of-equilibrium quasi-stationary-state. Nevertheless, we show that if the initial distribution satisfies a specific constraint - a generalized virial condition - the quasi- stationary-state is very close to ergodic and can be described by Lynden-Bell statistics. On the other hand if the generalized virial condition is violated, parametric resonances are excited, leading to chaos and ergodicity breaking.

cond-mat.stat-mech↗

arXiv2011

Statistical Mechanics of 1d Self-Gravitating Systems: The Core-Halo Distribution

We study, using both theory and simulations, a system of self-gravitating sheets. A new statistical mechanics theory - free of any adjustable parameters - is derived to quantitatively predict the final stationary state achieved by this system after the process of collisionless relaxation is completed. The theory shows a very good agreement with the numerical simulations. The model sheds new light on the general mechanism of relaxation of self-gravitating systems and may help us to understand cold matter distribution in the Universe.

astro-ph.GA↗

arXiv2011

Core-halo distribution in the Hamiltonian Mean-Field Model

We study a paradigmatic system with long-range interactions: the Hamiltonian Mean-Field Model (HMF). It is shown that in the thermodynamic limit this model does not relax to the usual equilibrium Maxwell-Boltzmann distribution. Instead, the final stationary state has a peculiar core-halo structure. In the thermodynamic limit, HMF is neither ergodic nor mixing. Nevertheless, we find that using dynamical properties of Hamiltonian systems, it is possible to quantitatively predict both the spin distribution and the velocity distribution functions in the final stationary state, without any adjustable parameters. We also show that HMF undergoes a non-equilibrium first-order phase transition between paramagnetic and ferromagnetic states.

cond-mat.stat-mech↗

arXiv2011

Rejoinder on: Thermostatistics of Overdamped Motion of Interacting Particles

In their Reply [arXiv:1104.5036] to our Comment [arXiv:1104.0697] Andrade et al. state that we have "chosen to categorically dismiss their elaborate and solid conceptual approach without employing any concepts or tools from Statistical Mechanics". Because of the one page restriction imposed by PRL, we could not address all of the flaws of the original paper. Therefore, we are grateful to Andrade et al., for giving us an opportunity to further elaborate on our Comment. In this Rejoinder we show that the model studied by Andrade et al. has nothing to do with the Tsallis entropy either at T=0 or at any other temperature.

cond-mat.stat-mech↗

arXiv2011

Comment on: Thermostatistics of Overdamped Motion of Interacting Particles [arXiv:1008.1421]

In a recent paper, Phys. Rev. Lett. 105 260601 (2010) [arXiv:1008.1421], Andrade et al., argued that classical particles confined in a parabolic trap at T=0 distribute themselves in accordance with the Tsallis statistics. To prove their point the authors performed molecular dynamics simulations. Here we show that the model of Andrade et al. can be solved exactly. The distribution of particles at T=0 has nothing to do with the Tsallis entropy and is determined simply by the force balance.

cond-mat.stat-mech↗