On the fluctuations spectrum of inhomogeneous plasmas and fluids

A general formula for the fluctuations spectrum of inhomogeneous plasmas and fluids at statistical equilibrium is found. It is valid for linearized equations of conservative systems and uses a rigorous treatment within Gibbs statistics. It opens the way for quantitative calculations of two-dimensional systems.     

A non-equilibrium Ising model of turbulence

We introduce a model of interacting lattices at different resolutions driven by the two-dimensional Ising dynamics with a nearest-neighbor interaction. We study this model both with tools borrowed from equilibrium statistical mechanics as well as non-equilibrium thermodynamics. Our findings show that this model keeps the signature of the equilibrium phase transition. The critical temperature of the equilibrium models corresponds to the state maximizing the entropy and delimits two out-of-equilibrium regimes, one satisfying the Onsager relations for systems close to equilibrium and one resembling convective turbulent states. Since the model preserves the entropy and energy fluxes in the scale space, it seems a good candidate for parametric studies of out-of-equilibrium turbulent systems.     

Surface properties of a classical two-dimensional one-component plasma: Exact results

At one special temperature, the equilibrium statistical mechanics of a classical two-dimensional one-component plasma can be worked out exactly. This model is used for computing the density profile and the two-body correlation function for three kinds of electrified interfaces: charge particles attracted by a charged plate, charged particles near the permeable boundary of a semiinfinite background, charged particles near the interface between two backgrounds of different densities. Sum rules are discussed.     

Magnetic properties of a nearly classical one-component plasma in three or two dimensions: II. Strong field

The equilibrium statistical mechanics of a nearly classical one-component plasma, submitted to a strong magnetic field, is studied, in three or two dimensions, by a suitable expansion of the Wigner distribution function. A strong magnetic field quenches the quantum fluctuations transverse to the field. The situation is especially simple for a two-dimensional plasma, which has a classical behaviour in the strong-field limit; as a consequence, a classical Wigner crystallization can be induced by the magnetic field.     

Statistical mechanics results in the P(φ)2 quantum field theory

We apply ideas and methods from classical statistical mechanics to study the P(φ)₂ self-coupled two-dimensional Boson field theory in the Euclidean region. In particular, we consider correlation inequalities of Griffiths type; the thermodynamic limit for the pressure, the average interaction and the entropy; and the equilibrium equations for states associated with a given interaction.     

A low diffusion particle method for simulating compressible inviscid flows

A new particle method is presented for the numerical simulation of compressible inviscid gas flows, through procedures which involve relatively small modifications to an existing direct simulation Monte Carlo (DSMC) algorithm. Implementation steps are outlined for simulations involving various grid geometries and for gas mixtures comprising an arbitrary number of species. The proposed method is compared with other numerical schemes through a series of one-dimensional and two-dimensional test cases, and is shown to provide a significant reduction in both artificial diffusion and statistical scatter effects relative to existing DSMC-based equilibrium particle methods.     

Nonequilibrium relaxation method — an alternative simulation strategy

One well-established simulation strategy to study the thermal phases and transitions of a given microscopic model system is the so-called equilibrium method, in which one first realizes the equilibrium ensemble of a finite system and then extrapolates the results to infinite system. This equilibrium method traces over the standard theory of the thermal statistical mechanics, and over the idea of the thermodynamic limit. Recently, an alternative simulation strategy has been developed, which analyzes the nonequilibrium relaxation (NER) process. It is called theNER method. NER method has some advantages over the equilibrium method. The NER method provides a simpler analyzing procedure. This implies less systematic error which is inevitable in the simulation and provides efficient resource usage. The NER method easily treats not only the thermodynamic limit but also other limits, for example, non-Gibbsian nonequilibrium steady states. So the NER method is also relevant for new fields of the statistical physics. Application of the NER method have been expanding to various problems: from basic first- and second-order transitions to advanced and exotic phases like chiral, KT spin-glass and quantum phases. These studies have provided, not only better estimations of transition point and exponents, but also qualitative developments. For example, the universality class of a random system, the nature of the two-dimensional melting and the scaling behavior of spin-glass aging phenomena have been clarified.     

Modelling Quasicrystals at Positive Temperature

We consider a two-dimensional lattice model of equilibrium statistical mechanics, using nearest neighbor interactions based on the matching conditions for an aperiodic set of 16 Wang tiles. This model has uncountably many ground state configurations, all of which are nonperiodic. The question addressed in this paper is whether nonperiodicity persists at low but positive temperature. We present arguments, mostly numerical, that this is indeed the case. In particular, we define an appropriate order parameter, prove that it is identically zero at high temperatures, and show by Monte Carlo simulation that it is nonzero at low temperatures.     

Mixing Times for the Mean-Field Blume-Capel Model via Aggregate Path Coupling

We investigate the relationship between the mixing times of the Glauber dynamics of a statistical mechanical system with its thermodynamic equilibrium structure. For this we consider the mean-field Blume-Capel model, one of the simplest statistical mechanical models that exhibits the following intricate phase transition structure: within a two-dimensional parameter space there exists a curve at which the model undergoes a second-order, continuous phase transition, a curve where the model undergoes a first-order, discontinuous phase transition, and a tricritical point which separates the two curves. We determine the interface between the regions of slow and rapid mixing. In order to completely determine the region of rapid mixing, we employ a novel extension of the path coupling method, successfully proving rapid mixing even in the absence of contraction between neighboring states.     

SCDA for 3D lattice gases with repulsive interaction

The selfconsistent diagram approximation (SCDA) is generalized for three-dimensional lattice gases with nearest neighbor repulsive interactions. The free energy is represented in a closed form through elementary functions. Thermodynamical (phase diagrams, chemical potential and mean square fluctuations), structural (order parameter, distribution functions) as well as diffusional characteristics are investigated. The calculation results are compared with the Monte Carlo simulation data to demonstrate high precision of the SCDA in reproducing the equilibrium lattice gas characteristics. It is shown that similarly to two-dimensional systems the specific statistical memory effects strongly influence the lattice gas diffusion in the ordered states. Received 7 August 2002 / Received in final form 22 January 2003 Published online 24 April 2003     

Pre-formed Cooper pairs and Bose-Einstein condensation in cuprate superconductors

A two-dimensional (2D) assembly of noninteracting, temperature-dependent, pre-formed Cooper pairs in chemical/thermal equilibrium with unpaired fermions is examined in a binary boson-fermion statistical model as the Bose-Einstein condensation (BEC) singularity temperature T_c is approached from above. Compared with BCS theory (which is not a BEC theory) substantially higher T_cs are obtained without any adjustable parameters, that fall roughly within the range of empirical T_cs for quasi-2D cuprate superconductors.     

Phase mixing and island saturation in Hamiltonian reconnection

The nonlinear evolution of a Hamiltonian magnetic field line reconnection in a two-dimensional fluid plasma leads to a macroscopic equilibrium with a finite-size island and fine-scale spatial structures. The latter arise from the phase mixing of the Lagrangian invariant fields. This equilibrium is the analog of the Bernstein-Greene-Kruskal equilibrium solution for electrostatic Langmuir waves.     

Statistical equilibrium states and long-time dynamics for a transport equation

Statistical equilibrium states for a linear transport equation were defined in a previous work. We consider here the two-dimensional case: we show that under some mild assumptions these equilibrium states actually describe the long-time dynamics of the system.     

The two-dimensional Coulomb gas on a sphere: Exact results

At the special value of the reduced inverse temperature=2, the equilibrium statistical mechanics of a two-dimensional Coulomb gas confined to the surface of a sphere is an exactly solvable problem, just as it was for the Coulomb gas in a plane. The thermodynamic quantities and all the correlation functions can be calculated. Use is made of an isomorphism between the classical Coulomb gas and the free Fermi field theory associated with the Dirac operator on the sphere.Laboratory associated with the Centre National de la Recherche Scientifique.     

Direct transition from a stable equilibrium to quasiperiodicity in non-smooth systems

The purpose of this Letter is to show how a border-collision bifurcation in a piecewise-smooth dynamical system can produce a direct transition from a stable equilibrium point to a two-dimensional invariant torus. Considering a system of nonautonomous differential equations describing the behavior of a power electronic DC/DC converter, we first determine the chart of dynamical modes and show that there is a region of parameter space in which the system has a single stable equilibrium point. Under variation of the parameters, this equilibrium may collide with a discontinuity boundary between two smooth regions in phase space. When this happens, one can observe a number of different bifurcation scenarios. One scenario is the continuous transformation of the stable equilibrium into a stable period-1 cycle. Another is the transformation of the stable equilibrium into an unstable period-1 cycle with complex conjugate multipliers, and the associated formation of a two-dimensional (ergodic or resonant) torus.     

Large Deviation Principles and Complete Equivalence and Nonequivalence Results for Pure and Mixed Ensembles

We consider a general class of statistical mechanical models of coherent structures in turbulence, which includes models of two-dimensional fluid motion, quasi-geostrophic flows, and dispersive waves. First, large deviation principles are proved for the canonical ensemble and the microcanonical ensemble. For each ensemble the set of equilibrium macrostates is defined as the set on which the corresponding rate function attains its minimum of 0. We then present complete equivalence and nonequivalence results at the level of equilibrium macrostates for the two ensembles. Microcanonical equilibrium macrostates are characterized as the solutions of a certain constrained minimization problem, while canonical equilibrium macrostates are characterized as the solutions of an unconstrained minimization problem in which the constraint in the first problem is replaced by a Lagrange multiplier. The analysis of equivalence and nonequivalence of ensembles reduces to the following question in global optimization. What are the relationships between the set of solutions of the constrained minimization problem that characterizes microcanonical equilibrium macrostates and the set of solutions of the unconstrained minimization problem that characterizes canonical equilibrium macrostates? In general terms, our main result is that a necessary and sufficient condition for equivalence of ensembles to hold at the level of equilibrium macrostates is that it holds at the level of thermodynamic functions, which is the case if and only if the microcanonical entropy is concave. The necessity of this condition is new and has the following striking formulation. If the microcanonical entropy is not concave at some value of its argument, then the ensembles are nonequivalent in the sense that the corresponding set of microcanonical equilibrium macrostates is disjoint from any set of canonical equilibrium macrostates. We point out a number of models of physical interest in which nonconcave microcanonical entropies arise. We also introduce a new class of ensembles called mixed ensembles, obtained by treating a subset of the dynamical invariants canonically and the complementary set microcanonically. Such ensembles arise naturally in applications where there are several independent dynamical invariants, including models of dispersive waves for the nonlinear Schrödinger equation. Complete equivalence and nonequivalence results are presented at the level of equilibrium macrostates for the pure canonical, the pure microcanonical, and the mixed ensembles.     

Mathematical Modeling of the Magnetization of a Two-Dimensional System of Magnetic Moments

Using a system that reaches its minimum energy of interaction at equilibrium, the magnetization of a discrete two-dimensional system of interacting magnetic dipoles by an external magnetic field is modeled mathematically. Magnetization curves for rectangular two-dimensional clusters of dipoles and the region of the magnetic domain are calculated.     

Pressures for a One-Component Plasma on a Pseudosphere

The classical (i.e., non-quantum) equilibrium statistical mechanics of a two-dimensional one-component plasma (a system of charged point-particles embedded in a neutralizing background) living on a pseudosphere (an infinite surface of constant negative curvature) is considered. In the case of a flat space, it is known that, for a one-component plasma, there are several reasonable definitions of the pressure, and that some of them are not equivalent to each other. In the present paper, this problem is revisited in the case of a pseudosphere. General relations between the different pressures are given. At one special temperature, the model is exactly solvable in the grand canonical ensemble. The grand potential and the one-body density are calculated in a disk, and the thermodynamic limit is investigated. The general relations between the different pressures are checked on the solvable model.