349 research outputs found
Equilibrium and out of equilibrium phase transitions in systems with long range interactions and in 2D flows
In self-gravitating stars, two dimensional or geophysical flows and in
plasmas, long range interactions imply a lack of additivity for the energy; as
a consequence, the usual thermodynamic limit is not appropriate. However, by
contrast with many claims, the equilibrium statistical mechanics of such
systems is a well understood subject. In this proceeding, we explain briefly
the classical approach to equilibrium and non equilibrium statistical mechanics
for these systems, starting from first principles. We emphasize recent and new
results, mainly a classification of equilibrium phase transitions, new
unobserved equilibrium phase transition, and out of equilibrium phase
transitions. We briefly discuss what we consider as challenges in this field
Vlasov stability of the Hamiltonian Mean Field model
We investigate the dynamical stability of a fully-coupled system of
inertial rotators, the so-called Hamiltonian Mean Field model. In the limit , and after proper scaling of the interactions, the -space
dynamics is governed by a Vlasov equation. We apply a nonlinear stability test
to (i) a selected set of spatially homogeneous solutions of Vlasov equation,
qualitatively similar to those observed in the quasi-stationary states arising
from fully magnetized initial conditions, and (ii) numerical coarse-grained
distributions of the finite- dynamics. Our results are consistent with
previous numerical evidence of the disappearance of the homogenous
quasi-stationary family below a certain energy.Comment: 11 pages, 5 figures. Submitted as a contribution to the proceedings
of the International Workshop on Trends and Perspectives on Extensive and
Non-Extensive Statistical Mechanics, November, 19-21, 2003, Angra dos Reis,
Brazi
Statistical mechanics and dynamics of solvable models with long-range interactions
The two-body potential of systems with long-range interactions decays at
large distances as , with , where is the
space dimension. Examples are: gravitational systems, two-dimensional
hydrodynamics, two-dimensional elasticity, charged and dipolar systems.
Although such systems can be made extensive, they are intrinsically non
additive. Moreover, the space of accessible macroscopic thermodynamic
parameters might be non convex. The violation of these two basic properties is
at the origin of ensemble inequivalence, which implies that specific heat can
be negative in the microcanonical ensemble and temperature jumps can appear at
microcanonical first order phase transitions. The lack of convexity implies
that ergodicity may be generically broken. We present here a comprehensive
review of the recent advances on the statistical mechanics and
out-of-equilibrium dynamics of systems with long-range interactions. The core
of the review consists in the detailed presentation of the concept of ensemble
inequivalence, as exemplified by the exact solution, in the microcanonical and
canonical ensembles, of mean-field type models. Relaxation towards
thermodynamic equilibrium can be extremely slow and quasi-stationary states may
be present. The understanding of such unusual relaxation process is obtained by
the introduction of an appropriate kinetic theory based on the Vlasov equation.Comment: 118 pages, review paper, added references, slight change of conten
Recommended from our members
On study of deterministic conservative solvers for the nonlinear boltzmann and landau transport equations
textThe Boltzmann Transport Equation (BTE) has been the keystone of the kinetic theory, which is at the center of Statistical Mechanics bridging the gap between the atomic structures and the continuum-like behaviors. The existence of solutions has been a great mathematical challenge and still remains elusive. As a grazing limit of the Boltzmann operator, the Fokker-Planck-Landau (FPL) operator is of primary importance for collisional plasmas. We have worked on the following three different projects regarding the most important kinetic models, the BTE and the FPL Equations. (1). A Discontinuous Galerkin Solver for Nonlinear BTE. We propose a deterministic numerical solver based on Discontinuous Galerkin (DG) methods, which has been rarely studied. As the key part, the weak form of the collision operator is approximated within subspaces of piecewise polynomials. To save the tremendous computational cost with increasing order of polynomials and number of mesh nodes, as well as to resolve loss of conservations due to domain truncations, the following combined procedures are applied. First, the collision operator is projected onto a subspace of basis polynomials up to first order. Then, at every time step, a conservation routine is employed to enforce the preservation of desired moments (mass, momentum and/or energy), with only linear complexity. The asymptotic error analysis shows the validity and guarantees the accuracy of these two procedures. We applied the property of ``shifting symmetries" in the weight matrix, which consists in finding a minimal set of basis matrices that can exactly reconstruct the complete family of collision weight matrix. This procedure, together with showing the sparsity of the weight matrix, reduces the computation and storage of the collision matrix from O(N3) down to O(N^2). (2). Spectral Gap for Linearized Boltzmann Operator. Spectral gaps provide information on the relaxation to equilibrium. This is a pioneer field currently unexplored form the computational viewpoint. This work, for the first time, provides numerical evidence on the existence of spectral gaps and corresponding approximate values. The linearized Boltzmann operator is projected onto a Discontinuous Galerkin mesh, resulting in a ``collision matrix". The original spectral gap problem is then approximated by a constrained minimization problem, with objective function the Rayleigh quotient of the "collision matrix" and with constraints the conservation laws. A conservation correction then applies. We also study the convergence of the approximate Rayleigh quotient to the real spectral gap. (3). A Conservative Scheme for Approximating Collisional Plasmas. We have developed a deterministic conservative solver for the inhomogeneous Fokker-Planck-Landau equations coupled with Poisson equations. The original problem is splitted into two subproblems: collisonless Vlasov problem and collisonal homogeneous Fokker-Planck-Landau problem. They are handled with different numerical schemes. The former is approximated using Runge-Kutta Discontinuous Galerkin (RKDG) scheme with a piecewise polynomial basis subspace covering all collision invariants; while the latter is solved by a conservative spectral method. To link the two different computing grids, a special conservation routine is also developed. All the projects are implemented with hybrid MPI and OpenMP. Numerical results and applications are provided.Computational Science, Engineering, and Mathematic
Relaxation of a test particle in systems with long-range interactions: diffusion coefficient and dynamical friction
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 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 explict 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
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
Green's Function and Pointwise Behaviors of the One-Dimensional modified Vlasov-Poisson-Boltzmann System
The pointwise space-time behaviors of the Green's function and the global
solution to the modified Vlasov- Poisson-Boltzmann (mVPB) system in
one-dimensional space are studied in this paper. It is shown that, the Green's
function admits the diffusion wave, the Huygens's type sound wave, the singular
kinetic wave and the remainder term decaying exponentially in space-time. These
behaviors are similar to the Boltzmann equation (Liu and Yu in Comm. Pure Appl.
Math. 57: 1543-1608, 2004). Furthermore, we establish the pointwise space-time
nonlinear diffusive behaviors of the global solution to the nonlinear mVPB
system in terms of the Green's function
- …