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1.
《Physica A》2006,365(1):184-189
We briefly discuss the state of the art on the anomalous dynamics of the Hamiltonian mean field (HMF) model. We stress the important role of the initial conditions for understanding the microscopic nature of the intriguing metastable quasi-stationary states (QSS) observed in the model and the connections to Tsallis statistics and glassy dynamics. We also present new results on the existence of metastable states in the Kuramoto model and discuss the similarities with those found in the HMF model. The existence of metastability seems to be quite a common phenomenon in fully coupled systems, whose origin could be also interpreted as a dynamical mechanism preventing or hindering synchronization. 相似文献
2.
The effect of Jeans term in a multicomponent self-gravitating quantum magnetoplasma is investigated employing the quantum hydrodynamic (QHD) model. The effects of quantum Bohm potential and statistical terms as well as the ambient magnetic field are also investigated on both dust and ion dynamics driven waves in this Letter. We state the conditions that can drive the system unstable in the presence of Jeans term. The limiting cases are also presented. The present work may have relevance in the dense astrophysical environments where the self-gravitating effects are expected to play a pivotal role. 相似文献
3.
P. H. Chavanis 《The European Physical Journal B - Condensed Matter and Complex Systems》2006,52(1):47-59
We present first elements of kinetic theory appropriate
to the inhomogeneous phase of the Hamiltonian Mean Field (HMF)
model. In particular, we investigate the case of strongly
inhomogeneous distributions for T→0 and exhibit
curious behaviour of the force auto-correlation function and
friction coefficient. The temporal correlation function of the
force has an oscillatory behaviour which averages to zero over a
period. By contrast, the effects of friction accumulate with time
and the friction coefficient does not satisfy the Einstein
relation. On the contrary, it presents the peculiarity to increase
linearly with time. Motivated by this result, we provide analytical
solutions of a simplified kinetic equation with a time dependent
friction. Analogies with self-gravitating systems and other systems
with long-range interactions are also mentioned. 相似文献
4.
We study a paradigmatic system with long-range interactions: the Hamiltonian mean-field (HMF) model. 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 nonequilibrium first-order phase transition between paramagnetic and ferromagnetic states. 相似文献
5.
P. H. Chavanis 《The European Physical Journal B - Condensed Matter and Complex Systems》2011,80(3):275-306
We study the thermodynamics of the Hamiltonian mean field
(HMF) model with an external potential playing the role of a
“magnetic field”. If we consider only fully stable states,
the caloric curve does not present any phase
transition. However, if we take into account metastable states (for
a restricted class of perturbations), we find a very rich
phenomenology. In particular, the caloric curve
displays a region of negative specific heat in the microcanonical
ensemble in which the temperature decreases as the energy
increases. This leads to ensembles inequivalence and to zeroth
order phase transitions similar to the “gravothermal catastrophe”
and to the “isothermal collapse” of self-gravitating systems. In
the present case, they correspond to the reorganization of the
system from an “anti-aligned” phase (magnetization pointing in
the direction opposite to the magnetic field) to an “aligned”
phase (magnetization pointing in the same direction as the magnetic
field). We also find that the magnetic susceptibility can be
negative in the microcanonical ensemble so that the magnetization
decreases as the magnetic field increases. The magnetic curves can
take various shapes depending on the values of energy or temperature. We
describe first order phase
transitions and hysteretic cycles involving positive or negative
susceptibilities. We also show that this model exhibits gaps in the
magnetization at fixed energy, resulting in ergodicity breaking. 相似文献
6.
7.
《Physics letters. A》2019,383(26):125829
We investigate the quasi-equilibrium state of one-dimensional self-gravitating systems. If the null virial condition is satisfied at initial time, it is found that the number density around the center of the system at the quasi-equilibrium state has the universality similar to two- and three-dimensional self-gravitating systems reported in Refs. [1], [2]. The reason why the null virial condition is sufficient for the universality is unveiled by the envelope equation. We present a phenomenological model to describe the universal structure by using a special Langevin equation with a distinctive random noise to self-gravitating systems. Additionally, we unveil a mechanism which decides the radius of the system. 相似文献
8.
We discuss an effective spin-glass Hamiltonian which can be used to study the glassy-like dynamics observed in the metastable states of the Hamiltonian mean field (HMF) model. By means of the Replica formalism, we were able to find a self-consistent equation for the glassy order parameter which reproduces, in a restricted energy region below the phase transition, the microcanonical simulations for the polarization order parameter recently introduced in the HMF model. 相似文献
9.
P. H. Chavanis L. Delfini 《The European Physical Journal B - Condensed Matter and Complex Systems》2009,69(3):389-429
We apply the Nyquist method to the Hamiltonian mean field (HMF) model in order to settle the linear dynamical stability of
a spatially homogeneous distribution function with respect to the
Vlasov equation. We consider the case of Maxwell (isothermal) and Tsallis (polytropic) distributions and show that the system
is stable above a critical kinetic temperature Tc and unstable below it. Then, we consider a symmetric double-humped distribution, made of the superposition of two decentered
Maxwellians, and show the existence of a re-entrant phase in the stability diagram. When we consider an asymmetric double-humped
distribution, the re-entrant phase disappears above a
critical value of the asymmetry factor Δ > 1.09. We also consider the HMF model with a repulsive interaction. In that case,
single-humped distributions are always stable. For asymmetric double-humped distributions, there is a re-entrant phase for
1 ≤ Δ < 25.6, a double re-entrant phase for 25.6 < Δ < 43.9 and no re-entrant phase for Δ > 43.9. Finally, we extend our results
to arbitrary potentials of interaction and mention the connexion between the HMF model, Coulombian plasmas and gravitational
systems. We discuss the relation between linear dynamical stability and formal nonlinear dynamical stability and show their
equivalence for spatially
homogeneous distributions. We also provide a criterion of dynamical stability for spatially inhomogeneous systems. 相似文献
10.
P. H. Chavanis 《The European Physical Journal B - Condensed Matter and Complex Systems》2006,53(4):487-501
Systems with long-range interactions can reach a Quasi
Stationary State (QSS) as a result of a violent collisionless
relaxation. If the system mixes well (ergodicity), the QSS can be
predicted by the statistical theory of Lynden-Bell (1967) based on
the Vlasov equation. When the initial condition takes only two
values, the Lynden-Bell distribution is similar to the Fermi-Dirac
statistics. Such distributions have recently been observed in
direct numerical simulations of the HMF model (Antoniazzi et al. 2006). In this paper, we determine the caloric curve
corresponding to the Lynden-Bell statistics in relation with the
HMF model and analyze the dynamical and thermodynamical stability
of spatially homogeneous solutions by using two general criteria
previously introduced in the literature. We express the critical
energy and the critical temperature as a function of a degeneracy
parameter fixed by the initial condition. Below these critical
values, the homogeneous Lynden-Bell distribution is not a maximum
entropy state but an unstable saddle point. Known stability
criteria corresponding to the Maxwellian distribution and the
water-bag distribution are recovered as particular limits of our
study. In addition, we find a critical point below which the
homogeneous Lynden-Bell distribution is always stable. We apply
these results to the situation considered in Antoniazzi et
al. For a given energy, we find a critical initial
magnetization above which the homogeneous Lynden-Bell distribution
ceases to be a maximum entropy state. For an energy U=0.69, this
transition occurs above an initial magnetization Mx=0.897. In
that case, the system should reach an inhomogeneous Lynden-Bell
distribution (most mixed) or an incompletely mixed state (possibly
fitted by a Tsallis distribution). Thus, our theoretical study
proves that the dynamics is different for small and large initial
magnetizations, in agreement with numerical results of Pluchino et
al. (2004). This new dynamical phase transition may reconcile the
two communities by showing that they study different regimes. 相似文献
11.
P. H. Chavanis 《The European Physical Journal B - Condensed Matter and Complex Systems》2006,52(1):61-82
We study the relaxation of a test particle immersed
in a bath of field particles interacting via weak long-range
forces. To order 1/N in the N→+∞ limit, the
velocity distribution of the test particle satisfies a
Fokker-Planck equation whose form is related to the Landau and
Lenard-Balescu equations in plasma physics. We provide explicit
expressions for the diffusion coefficient and friction force in the
case where the velocity distribution of the field particles is
isotropic. We consider (i) various dimensions of space d=3,2 and
1; (ii) a discret spectrum of masses among the particles; (iii)
different distributions of the bath including the Maxwell
distribution of statistical equilibrium (thermal bath) and the step
function (water bag). Specific applications are given for
self-gravitating systems in three dimensions, Coulombian systems in
two dimensions and for the HMF model in one dimension. 相似文献
12.
Aurelio Patelli Stefano Ruffo 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2014,68(11):1-13
Long-range interacting N-particle systems get trapped into long-living out-of-equilibrium stationary states called quasi-stationary states (QSS). We study here the response to a small external perturbation when such systems are settled into a QSS. In the N → ∞ limit the system is described by the Vlasov equation and QSS are mapped into stable stationary solutions of such equation. We consider this problem in the context of a model that has recently attracted considerable attention, the Hamiltonian mean field (HMF) model. For such a model, stationary inhomogeneous and homogeneous states determine an integrable dynamics in the mean-field effective potential and an action-angle transformation allows one to derive an exact linear response formula. However, such a result would be of limited interest if restricted to the integrable case. In this paper, we show how to derive a general linear response formula which does not use integrability as a requirement. The presence of conservation laws (mass, energy, momentum, etc.) and of further Casimir invariants can be imposed a posteriori. We perform an analysis of the infinite time asymptotics of the response formula for a specific observable, the magnetization in the HMF model, as a result of the application of an external magnetic field, for two stationary stable distributions: the Boltzmann-Gibbs equilibrium distribution and the Fermi-Dirac one. When compared with numerical simulations the predictions of the theory are very good away from the transition energy from inhomogeneous to homogeneous states. 相似文献
13.
14.
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. 相似文献
15.
We investigate the fundamental characteristics of numerical irreversibility appearing in self-gravitating small N-body systems by means of a molecular dynamics method from the viewpoint of time-reversible dynamics. We reconsider a closed spherical system consisting of 250 point-particles interacting through the Plummer softened potential. To investigate the characteristics of numerical irreversibility, we examine the influence of the instability affected by the softening parameter for the softened potential (the instability considered here is the instability of a dynamical system in chaos theory, e.g., a separation rate of the distance between two nearby trajectories in phase space or speed space). To this end, under the restriction of constant initial energy, the softening parameter for the Plummer softened potential is varied from 0.005R to 0.050R, where R is the radius of the spherical container. We first confirm that the size of the softening parameter, i.e., the deviation of the potential from a pure gravitational potential, influences the virial ratio, the density, the pressure on the spherical container, etc., during an early stage of the relaxation process. Through a time-reversible simulation based on a velocity inversion technique, we demonstrate that numerical irreversibility due to round-off errors appears more rapidly with decreasing softening parameter. This means that the higher the instability of the system or the higher the separation rate of the distance between two nearby trajectories, the earlier the memory of the initial conditions is lost. We show that the memory loss time , when the simulated trajectory completely forgets its initial conditions, increases approximately linearly with the timescale of the chaotic system, i.e., the Lyapunov time tλ. In a small self-gravitating system, propagation of numerical irreversibility or loss of reversibility depends on both the energy state of the system and the instability affected by the softening parameter. 相似文献
16.
Jiulin Du 《Central European Journal of Physics》2005,3(3):376-381
Self-gravitating systems are generally thought to behavior non-extensively due to the long-range nature of gravitational forces.
We discuss a relation between the nonextensive parameter q of Tsallis statistics, the temperature gradient and the gravitational potential based on the equation of hydrostatic equilibrium
for self-gravitating systems. It is suggested that the nonextensive parameter in Tsallis statistics has a clear physical meaning
with regard to the non-isothermal nature of the systems with long-range interactions. Tsallis’ equilibrium distribution for
the self-gravitating systems describes the property of hydrostatic equilibrium of the systems. 相似文献
17.
Angelani L Parisi G Ruocco G Viliani G 《Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics》2000,61(2):1681-1691
We analyze the properties of a Lennard-Jones system at the level of the potential energy landscape. After an exhaustive investigation of the topological features of the landscape of the systems, obtained by studying small size samples, we describe the dynamics of the systems in multidimensional configurational space by means of a simple model. This considers the configurational space as a connected network of minima where the dynamics proceeds by jumps described by an appropriate master equation. Using this model we are able to reproduce the long-time dynamics and the low temperature regime. We investigate both the equilibrium regime and the off-equilibrium one, finding those typical glassy behaviors usually observed in the experiments such as (i) a stretched exponential relaxation, (ii) a temperature-dependent stretching parameter, (iii) a breakdown of the Stokes-Einstein relation, and (iv) the appearance of a critical temperature below which one observes a deviation from the fluctuation-dissipation relation as a consequence of the lack of equilibrium in the system. 相似文献
18.
Marc Chemtob 《Nuclear Physics A》1984,420(3):461-495
The violations of isospin symmetry induced in the two-nucleon system at the quark level by the mass difference between up and down quarks are studied in a quark cluster model. Quark dynamics are treated by means of the standard non-relativistic quark model with a quark hamiltonian consisting of a confining harmonic potential, eventually corrected for anharmonicities, and a spin-dependent potential truncated to the contact-gluon-exchange hyperfine interaction. The resonating group method is adopted to treat the six-quark system and we restrict ourselves to configurations of two three-quark clusters with nucléon quantum numbers. π- and σ-meson-mediated quark interactions are tentatively considered in an attempt to achieve a good matching to the empirical strong NN potentials. We supply explicit formulas for the various kernels. Equivalent adiabatic potentials are calculated for the pp, np and nn systems in low partial waves. We also solve the resonating group scattering equations for these systems and give predictions for phase observables and low-energy parameters. 相似文献
19.
Pierre-Henri Chavanis 《Physica A》2011,390(9):1546-1574
We develop the kinetic theory of Brownian particles with long- and short-range interactions. Since the particles are in contact with a thermal bath fixing the temperature T, they are described by the canonical ensemble. We consider both overdamped and inertial models. In the overdamped limit, the evolution of the spatial density is governed by the generalized mean field Smoluchowski equation including a mean field potential due to long-range interactions and a generically nonlinear barotropic pressure due to short-range interactions. This equation describes various physical systems such as self-gravitating Brownian particles (Smoluchowski-Poisson system), bacterial populations experiencing chemotaxis (Keller-Segel model) and colloidal particles with capillary interactions. We also take into account the inertia of the particles and derive corresponding kinetic and hydrodynamic equations generalizing the usual Kramers, Jeans, Euler and Cattaneo equations. For each model, we provide the corresponding form of free energy and establish the H-theorem and the virial theorem. Finally, we show that the same hydrodynamic equations are obtained in the context of nonlinear mean field Fokker-Planck equations associated with generalized thermodynamics. However, in that case, the nonlinear pressure is due to the bias in the transition probabilities from one state to the other leading to non-Boltzmannian distributions while in the former case the distribution is Boltzmannian but the nonlinear pressure arises from the two-body correlation function induced by the short-range potential of interaction. As a whole, our paper develops connections between the topics of long-range interactions, short-range interactions, nonlinear mean field Fokker-Planck equations and generalized thermodynamics. It also justifies from a kinetic theory based on microscopic processes, the basic equations that were introduced phenomenologically to describe self-gravitating Brownian particles, chemotaxis and colloidal suspensions with attractive interactions. 相似文献
20.
We perform a linear dynamical stability analysis of a general hydrodynamic model of chemotactic aggregation [P.H. Chavanis, C. Sire, Physica A 384 (2007) 199]. Specifically, we study the stability of an infinite and homogeneous distribution of cells against “chemotactic collapse”. We discuss the analogy between the chemotactic collapse of biological populations and the gravitational collapse (Jeans instability) of self-gravitating systems. Our hydrodynamic model involves a pressure force which can take into account several effects like anomalous diffusion or the fact that the organisms cannot interpenetrate. We also take into account the degradation of the chemical which leads to a shielding of the interaction like for a Yukawa potential. Finally, our hydrodynamic model involves a friction force which quantifies the importance of inertial effects. In the strong friction limit, we obtain a generalized Keller–Segel model similar to the generalized Smoluchowski–Poisson system describing self-gravitating Langevin particles. For small frictions, we obtain a hydrodynamic model of chemotaxis similar to the Euler–Poisson system describing a self-gravitating barotropic gas. We show that an infinite and homogeneous distribution of cells is unstable against chemotactic collapse when the “velocity of sound” in the medium is smaller than a critical value. We study in detail the linear development of the instability and determine the range of unstable wavelengths, the growth rate of unstable modes and the damping rate, or the pulsation frequency, of the stable modes as a function of the friction parameter and shielding length. For specific equations of state, we express the stability criterion in terms of cell density. 相似文献