Phase separation transition in a driven diffusive system with anti-ferromagnetic interaction

One-dimensional non-equilibrium systems with short-range interaction can undergo phase transitions from homogeneous states to phase separated states as interaction (?) among particles is increased. One of the model systems where such a transition has been observed is the extended Katz-Lebowitz-Spohn (KLS) model with ferromagnetically interacting particles at ?=4/5. Here, the system remains homogeneous for small interaction strength (?<4/5), and for anti-ferromagnetic interactions (?<0). We show that the phase separation transitions can also occur in anti-ferromagnetic systems if interaction among particles depends explicitly on the size of the block (n) they belong to. We study this transition in detail for a specific case ?=δ/n, where phase separation occurs for δ<−1.     

Collapse of systems of relativistic particles

We prove in a very simple way that if a system of N non-relativistic particles interacting by Coulomb or gravitational forces has a (negative) binding energy increasing faster than N, the corresponding system necessarily collapses for N big enough if particles are given relativistic kinetic energy. At the same time our method allows to improve considerably a recently obtained sufficient condition on coupling constants for the collapse of ordinary matter.     

Dynamical Formation of Correlations in a Bose-Einstein Condensate

We consider the evolution of N bosons interacting with a repulsive short range pair potential in three dimensions. The potential is scaled according to the Gross-Pitaevskii scaling, i.e. it is given by N ² V(N(x _i ? x _j )). We monitor the behaviour of the solution to the N-particle Schrödinger equation in a spatial window where two particles are close to each other. We prove that within this window a short-scale interparticle structure emerges dynamically. The local correlation between the particles is given by the two-body zero energy scattering mode. This is the characteristic structure that was expected to form within a very short initial time layer and to persist for all later times, on the basis of the validity of the Gross-Pitaevskii equation for the evolution of the Bose-Einstein condensate. The zero energy scattering mode emerges after an initial time layer where all higher energy modes disperse out of the spatial window. We can prove the persistence of this structure up to sufficiently small times before three-particle correlations could develop.     

The Vlasov equation and the Hamiltonian mean-field model

We show that the quasi-stationary states of homogeneous (zero magnetization) states observed in the N-particle dynamics of the Hamiltonian mean-field (HMF) model are nothing but Vlasov stable homogeneous states. There is an infinity of Vlasov stable homogeneous states corresponding to different initial momentum distributions. Tsallis q-exponentials in momentum, homogeneous in angle, distribution functions are possible, however, they are not special in any respect, among an infinity of others. All Vlasov stable homogeneous states lose their stability because of finite N effects and, after a relaxation time diverging with a power-law of the number of particles, the system converges to the Boltzmann–Gibbs equilibrium.     

On the complete synchronization of the Kuramoto phase model

We discuss the asymptotic complete phase-frequency synchronization for the Kuramoto phase model with a finite size N. We present sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states. Our new sufficient conditions and decay rate depend only on the coupling strength and the diameter of initial phase and natural frequency configurations. But they are independent of the system size N, hence they can be used for the mean-field limit. For the complete synchronization estimates, we estimate the time evolution of the phase and frequency diameters for configurations. The initial phase configurations for identical oscillators located on the half circle will converge to the complete synchronized states exponentially fast. In contrast, for the non-identical oscillators, the complete frequency synchronization will occur exponentially fast for some restricted class of initial phase configurations. Our estimates are based on the monotonicity arguments of extremal phase and frequencies, which do not employ any linearization procedure of nonlinear coupling terms and detailed information on the eigenvalue of the linearized system around the complete synchronized states. We compare our analytical results with numerical simulations.     

Correlation Effects in the Moshinsky Model

We investigate quantum correlations in the ground state of the Moshinsky model formed by N harmonically interacting particles confined in a harmonic potential. The model is solvable which allows an exact determination of entanglement between the subset of p particles and the remaining N ? p particles. We study linear entropies and von Neumann entropies of the bipartitions and compare their behavior with that of the relative correlation energy and of the statistical Kutzelnigg coefficient.     

System of interacting harmonic oscillators in rotationally invariant noncommutative phase space

Rotationally invariant space with noncommutativity of coordinates and noncommutativity of momenta of canonical type is considered. A system of N interacting harmonic oscillators in uniform field and a system of N particles with harmonic oscillator interaction are studied. We analyze effect of noncommutativity on the energy levels of these systems. It is found that influence of coordinates noncommutativity on the energy levels of the systems increases with increasing of the number of particles. The spectrum of N free particles in uniform field in rotationally invariant noncommutative phase space is also analyzed. It is shown that the spectrum corresponds to the spectrum of a system of N harmonic oscillators with frequency determined by the parameter of momentum noncommutativity.     

On the Asymptotic Dynamics of a Quantum System Composed by Heavy and Light Particles

We consider a non-relativistic quantum system consisting of K heavy and N light particles in dimension three, where each heavy particle interacts with the light ones via a two-body potential α V. No interaction is assumed among particles of the same kind. Choosing an initial state in a product form and assuming α sufficiently small we characterize the asymptotic dynamics of the system in the limit of small mass ratio, with an explicit control of the error. In the case K = 1 the result is extended to arbitrary α. The proof relies on a perturbative analysis and exploits a generalized version of the standard dispersive estimates for the Schrödinger group. Exploiting the asymptotic formula, an application to the problem of the decoherence effect produced on a heavy particle by the interaction with the light ones is also outlined.     

New super-Calogero models and OSp(4|2) superconformal mechanics

We report on the new approach to constructing superconformal extensions of the Calogerotype systems with an arbitrary number of involved particles. It is based upon the superfield gauging of non-Abelian isometries of some supersymmetric matrix models. Among its applications, we focus on the new N = 4 superconformal system yielding the U(2)-spin Calogero model in the bosonic sector, and the one-particle case of this system, which is a new OSp(4|2) superconformal mechanics with nondynamical U(2) spin variables. The characteristic feature of thesemodels is that the strength of the conformal inverse-square potential is quantized.     

Ground States of the 2D Sticky Disc Model: Fine Properties and N 3/4 Law for the Deviation from the Asymptotic Wulff Shape

We investigate ground state configurations for a general finite number N of particles of the Heitmann-Radin sticky disc pair potential model in two dimensions. Exact energy minimizers are shown to exhibit large microscopic fluctuations about the asymptotic Wulff shape which is a regular hexagon: There are arbitrarily large N with ground state configurations deviating from the nearest regular hexagon by a number of ～N ^3/4 particles. We also prove that for any N and any ground state configuration this deviation is bounded above by ～N ^3/4. As a consequence we obtain an exact scaling law for the fluctuations about the asymptotic Wulff shape. In particular, our results give a sharp rate of convergence to the limiting Wulff shape.     

Renormalization and phase transition in the quantum ofCPN−1 model (D=2,3)

We prove in the framework of the 1/N expansion the ultraviolet renormalizibility of the ofCP^N?1 model when D = 2, 3. It is shown that when D = 2 the model gains infrared divergences in higher orders of 1/N. When D = 3, the phase transition of second order occurs: above the critical points there exist N massive scalar charged particles and one particle corresponding to the massless isoscalar vector field, but below it there exist only N ? 1 massless particles.     

Scattering of a Light Particle by a System of Harmonic Oscillators

We discuss the asymptotic wave function of a quantum system in ?³ composed by heavy and light particles, in the case where the light particles are in scattering states and no interaction is assumed among particles of the same kind. We first review a recent result concerning the case of K heavy and N light particles, where the one-particle potential acting on each heavy particle decays at infinity. Then we consider the case of one light particle interacting with a system of harmonic oscillators and prove the same kind of result following, with some modification, the proof of the previous case. A possible application to the analysis of the scattering of a light particle from condensed matter is also outlined.     

Core-softened potentials,multiple liquid–liquid critical points,and density anomaly regions: An exact solution

The pressure versus temperature phase diagram of a system of particles interacting through a multiscale shoulder-like potential is exactly computed in one dimension. The N-shoulder potential exhibits N density anomaly regions in the phase diagram if the length scales can be connected by a convex curve. The result is analyzed in terms of the convexity of the Gibbs free energy.     

Infinite Range Interaction Model of a Structural Glass

In this paper a simple mean-field model for the liquid-glass phase transition is proposed. This is the low density D-dimensional system of N particles interacting via infinite-range oscillating potential. In the framework of the replica approach it is shown that such a system exhibits the phase transition between the high-temperature liquid phase and the low-temperature glass phase. This phase transition is described in terms of the standard one-step replica symmetry breaking scheme.     

Relativistic and separable classical hamiltonian particle dynamics

We show within the Hamiltonian formalism the existence of classical relativistic mechanics of N scalar particles interacting at a distance which satisfies the requirements of Poincaré invariance, separability, world-line invariance and Einstein causality. The line of approach which is adopted here uses the methods of the theory of systems with constraints applied to manifestly covariant systems of particles. The study is limited to the case of scalar interactions remaining weak in the whole phase space and vanishing at large space-like separation distances of the particles. Poincaré invariance requires the inclusion of many-body, up to N-body, potentials. Separability requires the use of individual or two-body variables and the construction of the total interaction from basic two-body interactions. Position variables of the particles are constructed in terms of the canonical variables of the theory according to the world-line invariance condition and the subsidiary conditions of the non-relativistic limit and separability. Positivity constraints on the interaction masses squared of the particles ensure that the velocities of the latter remain always smaller than the velocity of light.     

On bound states in the continuum of N-body systems and the virial theorem

We prove generalized versions of the quantum mechanical virial theorem and apply them to the investigation of the spectrum of N body Hamiltonians. We show, in particular, that for N particles interacting through 2-body potentials which may have singularities but “don't wiggle too much,” no positive energy bound state can exist. We also prove results on the absence of bound states with energy bigger than some value E⁰ ? − ∞ and extend them to the case of N particles interacting through ν-body forces (ν = 1, 2,…, N) and with an external electromagnetic field. Also some remarks for the case of a Dirac electron in an external potential are given as well as for some problems with boundary conditions. A by-product of this investigation is the unitarity of the S matrix and the strong asymptotic completeness for systems of N particles interacting by 2-body forces which are not restricted to be purely repulsive.     

Drift velocity in the necklace model for reptation in a two-dimensional square lattice

We report the chain dynamics in the necklace model that mimics the reptation of a chain of N particles in a two-dimensional square lattice. We focus on the drift velocity under an applied static field. The characteristics of the model allow us to determine the effects of the forces on the chains and the resulting mechanisms that affect the drift velocity. Results obtained through Monte Carlo simulations were analyzed and discussed and distinct regimes as a function of the force strength and N were identified. We found that for small total applied forces, the drift velocity scales as 1/N. When the applied force to every particle is small but the total applied force is not, the tube deforms in such a way that the drift velocity does not depend on N. Large forces, applied to every particle, can straight chains such that the distance between the chain ends increases faster than the number of particles. Also, large forces can deform the chain within the tube what is directly related to a decrease of the drift velocity.     

Quantum Dynamics with Mean Field Interactions: a New Approach

We propose a new approach for the study of the time evolution of a factorized N-particle bosonic wave function with respect to a mean-field dynamics with a bounded interaction potential. The new technique, which is based on the control of the growth of the correlations among the particles, leads to quantitative bounds on the difference between the many-particle Schrödinger dynamics and the one-particle nonlinear Hartree dynamics. In particular the one-particle density matrix associated with the solution to the N-particle Schrödinger equation is shown to converge to the projection onto the one-dimensional subspace spanned by the solution to the Hartree equation with a speed of convergence of order 1/N for all fixed times.