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.     

Impossibility of reducing the motion of a system of N particles with pair interaction to m-particle (2 < m < N) motions in relative coordinates

It was shown that in classical mechanics, if there exists the complete integral, the problem of motion of a system of N particles with pair interaction cannot be reduced to problems of motion of 2 < m < N particles in relative coordinates.     

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.     

The many-body problem with an energy-dependent confining potential

We consider systems of N bosons bound by two-body harmonic interactions, whose frequency depends on the total energy of the system. Such energy dependent confining interactions between the bosons yield remarkable properties of the many-body system. As the quantum numbers increase, the total energy cannot exceed a saturation energy, which is independent of the number of particles N. Moreover, the ground state energy increases with N. As a result, the density of states tends rapidly to infinity as N and/or the quantum numbers increase.     

Chaotic motion at the emergence of the time averaged energy decay

A system plus environment conservative model is used to characterize the nonlinear dynamics when the time averaged energy for the system particle starts to decay. The system particle dynamics is regular for low values of the N environment oscillators and becomes chaotic in the interval 13≤N≤15, where the system time averaged energy starts to decay. To characterize the nonlinear motion we estimate the Lyapunov exponent (LE), determine the power spectrum and the Kaplan-Yorke dimension. For much larger values of N the energy of the system particle is completely transferred to the environment and the corresponding LEs decrease. Numerical evidence shows the connection between the variations of the amplitude of the particles energy time oscillation with the time averaged energy decay and trapped trajectories.     

Ground state of an impurity in a homogeneous system in the hypernetted chain approximation

The hypernetted chain theory of the ground state of a homogeneous N-particle medium NM with an impurity particle is presented. The N identical particles are fermions with spin-isospin degeneracy ν, or bosons (in the limit of ν → ∞). The ground-state wave-function of the system is assumed in the Jastrow form with central, state-independent correlation functions. Central, spin-isospin-dependent two-body interactions both in NM and between the impurity particle and the particles of NM are considered. Expressions for the ground-state energy of the system and for the separation energy of the impurity particle are derived. The simplified case of the chain approximation is also considered.     

Brunnian and Efimov N-Body States

The Efimov effect was originally formulated for three particles. The underlying principle of model independence is extended in this article in several directions. We present our definitions of the concepts of universality and scale independence. In this context we review briefly the scaling relations established for two- and three-body structures of nuclear halos. We emphasize the difference between the two extremes of weak binding named Efimov and Brunnian states. They arise respectively for two-body interactions at threshold of binding either two or N particles. We restrict the Hilbert space to include no more than two-body correlations, and discuss the derived excited N-body Efimov states both for zero- and finite-range two-body interactions. Then we investigate the relation between radius and binding energy extremely close to threshold of binding the Brunnian N-body system. Radii of both ground and first excited states for N?=?4, 5, 6 remain finite as the binding energy vanishes, and the distances between pairs of particles are substantially larger than the range of the two-body potential. The radii decrease with N and increase with excitation energy. The computed radii are larger for the complete than for the restricted Hilbert space. Model independence at the Brunnian threshold is strongly indicated.     

Dynamics of collective decoherence and thermalization

We analyze the dynamics of N interacting spins (quantum register) collectively coupled to a thermal environment. Each spin experiences the same environment interaction, consisting of an energy conserving and an energy exchange part.We find the decay rates of the reduced density matrix elements in the energy basis. We show that if the spins do not interact among each other, then the fastest decay rates of off-diagonal matrix elements induced by the energy conserving interaction is of order N², while that one induced by the energy exchange interaction is of the order N only. Moreover, the diagonal matrix elements approach their limiting values at a rate independent of N. For a general spin system the decay rates depend in a rather complicated (but explicit) way on the size N and the interaction between the spins.Our method is based on a dynamical quantum resonance theory valid for small, fixed values of the couplings. We do not make Markov-, Born- or weak coupling (van Hove) approximations.     

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.     

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.     

Emission of helium nuclei in heavy ion interactions at energies ≧ 100 MeV/nucleon

The energy and angular distributions of helium particles emitted in interactions between nuclei in the cosmic radiation and nuclei in photoemulsions at energies ≧ 100 MeV/nucleon have been studied. The data obtained is impossible to interpret on the basis of a statistical decay of excited nuclei. For example, it is found that more than 28% of the helium nuclei are emitted in processes different from simple evaporation. The differential energy distribution of the helium nuclei in the energy interval (40–200) MeV can be represented by the relationN(E)dE=constE ^?a dE, wherea≈1.2. The large spread in angles and energies of the fast helium particles emitted in heavy ion interactions can to a certain degree be understood, if it is assumed that interactions between nucleons and clusters of nucleons occur.     

Orthofermions versus Gutzwiller projection operator approach for the infinite U Hubbard model

A comparison of two well-known approaches for strongly correlated electron systems, namely, nested Bethe ansatz implemented through orthofermion algebra and Gutzwiller projection operator formalism, is made by calculating the energy spectrum of 1D infinite U Hubbard model for a finite system consisting of three particles on a four site anisotropic closed chain. It is shown that orthofermion algebra always leads to at least an eight hold degeneracy in the energy spectrum corresponding to all 2³ spin configurations, consistent with the nested Bethe ansatz solution leading to a N²-fold degeneracy of energy levels of an N electron system. Such a degeneracy is absent in the Gutzwiller projection operator approach. This finding shows the limitations of the Gutzwiller projection method and at the same time the relevance of orthofermion approach for the infinite U Hubbard model.     

Relativistic versus nonrelativistic solution of the N-fermion problem in a hyperradius-confining potential

We investigate the quantum system of N identical fermions in the relativistic limit. In this article the considered potential is a combination of Coulombic, linear confining and harmonic oscillator terms. By using Jacobi coordinates and introducing the hyperradius quantity we obtain the wave functions of the system as well as the corresponding energy eigenvalues. Assuming that all particles are confined within a hypersphere we calculate the corresponding x _bag . In particular we consider the case N = 3 which corresponds to baryonic systems. By using the experimental values of the charge radius of each baryon we calculate the potential coefficients. Within our treatment the results of the MIT bag model are recovered for N = 1. Finally we compare the results obtained by the Dirac equation with the corresponding results of the Schrödinger equation and we find that the energy spectra obtained by the former are much closer to experimental values.     

On Collisions Times of ‘Self-Sorting’ Interacting Particles in One-Dimension with Random Initial Positions and Velocities

We investigate a one-dimensional system of N particles, initially distributed with random positions and velocities, interacting through binary collisions. The collision rule is such that there is a time after which the N particles do not interact and become sorted according to their velocities. When the collisions are elastic, we derive asymptotic distributions for the final collision time of a single particle and the final collision time of the system as the number of particles approaches infinity, under different assumptions for the initial distributions of the particles’ positions and velocities. For comparison, a numerical investigation is carried out to determine how a non-elastic collision rule, which conserves neither momentum nor energy, affects the median collision time of a particle and the median final collision time of the system.     

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.     

Statistics of partitions of the kinetic energy of small nanoclusters

In Aquilanti, Lombardi, and Sevryuk, J. Chem. Phys. 2004, V. 121, No. 12, P. 5579 and Sevryuk, Lombardi, and Aquilanti, Phys. Rev. A. 2005, V. 72, No. 3, P. 033201, we defined several partitions of the total kinetic energy of a system of classical particles into terms corresponding to various motion modes. In this work, we study the statistics of these terms for clusters with the number of particles N from 3 to 100 (at randomly selected particle coordinates and velocities). Some new kinetic energy components are defined and studied. Two limiting situations are considered, those of particles of equal masses and particles whose masses vary randomly. With equal masses, the mean values of almost all cluster kinetic energy components are expressed in terms of N with the use of very simple equations.     

Continuous Particles in the Canonical Ensemble as an Abstract Polymer Gas

We revisit the expansion recently proposed by Pulvirenti and Tsagkarogiannis for a system of N continuous particles in the Canonical Ensemble. Under the sole assumption that the particles interact via a tempered and stable pair potential and are subjected to the usual free boundary conditions, we show the analyticity of the Helmholtz free energy at low densities and, using the Penrose tree graph identity, we establish a lower bound for the convergence radius which happens to be identical to the lower bound of the convergence radius of the virial series in the Grand Canonical ensemble established by Lebowitz and Penrose in 1964. We also show that the free energy can be written as a series in powers of the density whose k-th order coefficient coincides, modulo terms o(N)/N, with the k-th order virial coefficient divided by k+1, according to its expression in terms of the m-th order (with m≤k+1) simply connected cluster integrals first given by Mayer in 1942. We finally give an upper bound for the k-th order virial coefficient which slightly improves, at high temperatures, the bound obtained by Lebowitz and Penrose.     

Energy fluctuation and correlation in Tsallis statistics

We investigate the fluctuation of the energy in the framework of Tsallis statistics and find the correlation plays an important role in energy fluctuations. In Tsallis statistics, the correlation is induced by the nonextensivity of Tsallis entropy and exists between particles even if the particles are dynamically independent. By taking the generalized ideal gas as an example, we get that when the particle number N is large enough, the relative fluctuation of the energy is proportional to 1/N instead of in Boltzmann statistics. Thus, the relative energy fluctuation is much smaller in Tsallis statistics than that in Boltzmann statistics. Besides, we demonstrate that the introduction of correlation between particle energies leads to smaller energy fluctuations in Tsallis statistics.     

Deconfining and chiral phase transitions in quantum chromodynamics at finite temperature

We give further arguments to support the claim of Svetitsky and Yaffe that the finite-temperature transition in 4-dimensional SU(N) gauge theories is in the universality class of 3-dimensional Z_N spin models. We show that this implies a smoothing out of the transition when quarks are added to the system as long as N ≠ 3. For N = 3 the pure gauge transition is expected to be first order and will be smoothed by quarks only if the quark contribution to the internal energy is larger than the latent heat of transition.