A Monte Carlo Method for Calculating Lynden-Bell Equilibrium in Self-Gravitating Systems.

We present a Monte Carlo approach that allows us to easily implement Lynden-Bell (LB) entropy maximization for an arbitrary initial particle distribution. The direct maximization of LB entropy for an arbitrary initial distribution requires an infinite number of Lagrange multipliers to account for the Casimir invariants. This has restricted studies of Lynden-Bell's violent relaxation theory to only a very small class of initial conditions of a very simple waterbag form, for which the entropy maximization can be performed numerically. In the present approach, an arbitrary initial distribution is discretized into density levels which are then evolved using an efficient Monte Carlo algorithm towards the final equilibrium state. A comparison is also made between the LB equilibrium and explicit Molecular Dynamics simulations. We find that for most initial distributions, relaxation is incomplete and the system is not able to reach the state of maximum LB entropy. In particular, we see that the tail of the stationary particle distribution is very different from the one predicted by the theory of violent relaxation, with a hard cutoff instead of an algebraic decay predicted by LB's theory.

Euler fluid in two dimensions: Statistical approach.

We use Kirchhoff's vortex formulation of 2D Euler fluid equations to explore the equilibrium state to which a 2D incompressible fluid relaxes from an arbitrary initial flow. The vortex dynamics obeys Hamilton's equations of motion with x and y coordinates of the vortex position forming a conjugate pair. A state of fluid can, therefore, be expressed in terms of an infinite number of infinitesimal vortices. If the vortex dynamics is mixing, the final equilibrium state of the fluid should correspond to the maximum of Boltzmann entropy, with the constraint that all the Casimir invariants of the fluid must be preserved. This is the fundamental assumption of Lynden-Bell's theory of collisionless relaxation. In this paper, we will present a Monte Carlo method which allows us to find the maximum entropy state of the fluid starting from an arbitrary initial condition. We will then compare this prediction with the results of molecular dynamics simulation and demonstrate that the final state to which the fluid evolves is, actually, very different from that corresponding to the maximum of entropy. This indicates that the mixing assumption is not correct. We will then present a different approach based on core-halo distribution which allows us to accurately predict the final state to which the fluid will relax, starting from an arbitrary initial condition.

Nonequilibrium 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 the concept of negative absolute temperature in equilibrium statistical mechanics. In this Letter, we show that in the thermodynamic limit a system of interacting vortices does not relax to the thermodynamic equilibrium but becomes trapped in a nonequilibrium stationary state. We show that the vortex distribution in this nonequilibrium stationary state has a characteristic core-halo structure, which can be predicted a priori. All the theoretical results are compared with explicit molecular dynamics simulations.

Dynamics, thermodynamics, and phase transitions of classical spins interacting through the magnetic field.

We introduce and study a one dimensional model of classical planar spins interacting self-consistently through magnetic field. The spins and the magnetic field evolve in time according to the Hamiltonian dynamics which mimics that of a free electron laser. We show that by rescaling the energy due to magnetic field inhomogeneity, in equilibrium, this system can be mapped onto a model very similar to the paradigmatic globally coupled Hamiltonian mean-field (HMF) model. The system exhibits a continuous equilibrium phase transition from paramagnetic to ferromagnetic phase, however unlike HMF, we do not see any magnetized quasistationary states.

Stability and self-organization of planetary systems.

We show that stability of planetary systems is intimately connected with their internal order. An arbitrary initial distribution of planets is susceptible to catastrophic events in which planets either collide or are ejected from the planetary system. These instabilities are a fundamental consequence of chaotic dynamics and of Arnold diffusion characteristic of many body gravitational interactions. To ensure stability over astronomical time scale of a realistic planetary system-in which planets have masses comparable to those of planets in the solar system-the motion must be quasiperiodic. A dynamical mechanism is proposed which naturally evolves a planetary system to a quasiperiodic state from an arbitrary initial condition. A planetary self-organization predicted by the theory is similar to the one found in our solar system.

Chaos and relaxation to equilibrium in systems with long-range interactions.

In the thermodynamic limit, systems with long-range interactions do not relax to equilibrium, but become trapped in nonequilibrium stationary states. For a finite number of particles a nonequilibrium state has a finite lifetime, so that eventually a system will relax to thermodynamic equilibrium. The time that a system remains trapped in a quasistationary state (QSS) scales with the number of particles as N(δ), with δ>0, and diverges in the thermodynamic limit. In this paper we will explore the role of chaotic dynamics on the time that a system remains trapped in a QSS. We discover that chaos, measured by the Lyapunov exponents, favors faster relaxation to equilibrium. Surprisingly, weak chaos favors faster relaxation than strong chaos.

Ergodicity breaking and quasistationary states in systems with long-range interactions.

In the thermodynamic limit, systems with long-range interactions do not relax to equilibrium, but become trapped in quasistationary states (qSS), the life time of which diverges with the number of particles. In this paper we will explore the relaxation of the Hamiltonian Mean-Field model to qSS for a class of initial conditions of the multilevel water-bag form. We will show that if the initial distribution satisfies the virial condition, thereby reducing mean field changes, the final distribution in the qSS can be predicted very accurately using a reduced exactly integrable model. The calculated distribution functions obtained using this approach are found to be more accurate than the ones predicted by the Lynden-Bell theory.

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

We argue that contrary to recent suggestions, nonextensive 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.

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.

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.

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 quasistationary states (QSS) the lifetime 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.

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.

Adiabatic-nonadiabatic transition in warm long-range interacting systems: the transport of intense inhomogeneous beams.

We investigate the role of the temperature in the onset of singularities and the consequent breakdown in a macroscopic fluid model for long-range interacting systems. In particular, we consider an adiabatic fluid description for the transport of intense inhomogeneous charged particle beams. We find that there exists a critical temperature below which the fluid model always develops a singularity and breaks down as the system evolves. As the critical temperature is approached, however, the time for the occurrence of the singularity diverges. Therefore, the critical temperature separates two distinct dynamical phases: a nonadiabatic transport at lower temperatures and a completely adiabatic evolution at higher temperatures. These findings are verified with the aid of self-consistent N-particle simulations.

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 quasistationary 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.

Nonequilibrium phase transitions in systems with long-range interactions.

We introduce a generalized Hamiltonian mean field model-an 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 generalized Hamiltonian mean field model can be solved explicitly using Boltzmann-Gibbs 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 Boltzmann-Gibbs statistics for systems with long-range forces in the thermodynamic limit.

Core-halo distribution in the Hamiltonian mean-field model.

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.

Driven one-component plasmas.

A statistical theory is presented that allows the calculation of the stationary state achieved by a driven one-component plasma after a process of collisionless relaxation. The stationary Vlasov equation with appropriate boundary conditions is reduced to an ordinary differential equation, which is then solved numerically. The solution is then compared with the molecular-dynamics simulation. A perfect agreement is found between the theory and the simulations. The full current-voltage phase diagram is constructed.

Collisionless relaxation in gravitational systems: from violent relaxation to gravothermal collapse.

Theory and simulations are used to study collisionless relaxation of a gravitational N -body system. It is shown that when the initial one-particle distribution function satisfies the virial condition--potential energy is minus twice the kinetic energy--the system quickly relaxes to a metastable state described quantitatively by the Lynden-Bell distribution with a cutoff. If the initial distribution function does not meet the virial requirement, the system undergoes violent oscillations, resulting in a partial evaporation of mass. The leftover particles phase-separate into a core-halo structure. The theory presented allows us to quantitatively predict the amount and the distribution of mass left in the central core, without any adjustable parameters. On a longer time scale tauG-N , collisionless relaxation leads to a gravothermal collapse.

Collisionless relaxation in non-neutral plasmas.

A theoretical framework is presented which allows us to quantitatively predict the final stationary state achieved by a non-neutral plasma during a process of collisionless relaxation. As a specific application, the theory is used to study relaxation of charged-particle beams. It is shown that a fully matched beam relaxes to the Lynden-Bell distribution. However, when a mismatch is present and the beam oscillates, parametric resonances lead to a core-halo phase separation. The approach developed accounts for both the density and the velocity distributions in the final stationary state.