Phase transitions of quasistationary states in the Hamiltonian Mean Field model

The out-of equilibrium dynamics of the Hamiltonian Mean Field (HMF) model is studied in presence of an externally imposed magnetic field h. Lynden-Bell’s theory of violent relaxation is revisited and shown to adequately capture the system dynamics, as revealed by direct Vlasov based numerical simulations in the limit of vanishing field. This includes the existence of an out-of-equilibrium phase transition separating magnetized and non magnetized phases. We also monitor the fluctuations in time of the magnetization, which allows us to elaborate on the choice of the correct order parameter when challenging the performance of Lynden-Bell’s theory. The presence of the field h removes the phase transition, as it happens at equilibrium. Moreover, regions with negative susceptibility are numerically found to occur, in agreement with the predictions of the theory.     

Covariant Hamiltonian dynamics with negative-energy states

A relativistic quantum mechanics is studied for bound hadronic systems in the framework of the point form relativistic Hamiltonian dynamics. Negative-energy states are introduced taking into account the restrictions imposed by a correct definition of the Poincaré group generators. We obtain nonpathological, manifestly covariant wave equations that dynamically contain the contributions of the negative-energy states. Auxiliary negative-energy states are also introduced, specially for studying the interactions of the hadronic systems with external probes.     

On the Gibbs states for one-dimensional lattice Boson systems with a long-range interaction

We consider an infinite chain of interacting quantum (anharmonic) oscillators. The pair potential for the oscillators at lattice distanced is proportional to {d ²[log(d+1)]F(d)}^–1 where _rZ [rF(r)]^–1 < . We prove that for any value of the inverse temperature> 0 there exists a limiting Gibbs state which is translationally invariant and ergodic. Furthermore, it is analytic in a natural sense. This shows the absence of phase transitions in the systems under consideration for any value of the thermodynamic parameters.     

Interaction of phonons with quasistationary states

Decay of the states via this interaction is considered. The line width in ferromagnetic resonance as a function of temperature is calculated in the spin-wave approximation, and the results are compared with experiment.I am indebted to L. N. Ovander for direct!on in this work and for valuable advice.     

Quasi-stationary states in low-dimensional Hamiltonian systems

We address a simple connection between results of Hamiltonian non-linear dynamical theory and thermostatistics. Using a properly defined dynamical temperature in low-dimensional symplectic maps, we display and characterize long-standing quasi-stationary states that eventually cross over to a Boltzmann–Gibbs-like regime. As time evolves, the geometrical properties (e.g., fractal dimension) of the phase space change sensibly, and the duration of the anomalous regime diverges with decreasing chaoticity. The scenario that emerges is consistent with the non-extensive statistical mechanics one.     

On the theory of quasistationary states

Under the assumption of an exponential law of decay of quasistationary states, an integral equation is obtained for the wave function ⁺ of a quasistationary state together with an expression for the energy shift E. These are used to obtain an expression for the decay probability and an optical theorem for quasistationary states and also the connection between the decay amplitude and the amplitude of resonance scattering.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 43–46, September, 1974.     

Thermodynamics and dynamics of systems with long-range interactions

We review simple aspects of the thermodynamic and dynamical properties of systems with long-range pairwise interactions (LRI), which decay as 1/r^d+σ at large distances r in d dimensions. Two broad classes of such systems are discussed. (i) Systems with a slow decay of the interactions, termed “strong” LRI, where the energy is super-extensive. These systems are characterized by unusual properties such as inequivalence of ensembles, negative specific heat, slow decay of correlations, anomalous diffusion and ergodicity breaking. (ii) Systems with faster decay of the interaction potential, where the energy is additive, thus resulting in less dramatic effects. These interactions affect the thermodynamic behavior of systems near phase transitions, where long-range correlations are naturally present. Long-range correlations are often present in systems driven out of equilibrium when the dynamics involves conserved quantities. Steady state properties of driven systems with local dynamics are considered within the framework outlined above.     

Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

We study numerically statistical distributions of sums of chaotic orbit coordinates, viewed as independent random variables, in weakly chaotic regimes of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam (FPU-β) oscillator chains with different boundary conditions and numbers of particles and a microplasma of identical ions confined in a Penning trap and repelled by mutual Coulomb interactions. For the FPU systems we show that, when chaos is limited within “small size” phase space regions, statistical distributions of sums of chaotic variables are well approximated for surprisingly long times (typically up to t≈10⁶) by a q-Gaussian (1<q<3) distribution and tend to a Gaussian (q=1) for longer times, as the orbits eventually enter into “large size” chaotic domains. However, in agreement with other studies, we find in certain cases that the q-Gaussian is not the only possible distribution that can fit the data, as our sums may be better approximated by a different so-called “crossover” function attributed to finite-size effects. In the case of the microplasma Hamiltonian, we make use of these q-Gaussian distributions to identify two energy regimes of “weak chaos”—one where the system melts and one where it transforms from liquid to a gas state-by observing where the q-index of the distribution increases significantly above the q=1 value of strong chaos.     

Out-of-equilibrium states as statistical equilibria of an effective dynamics in a system with long-range interactions

We study the formation of coherent structures in a system with long-range interactions where particles moving on a circle interact through a repulsive cosine potential. Nonequilibrium structures are shown to correspond to statistical equilibria of an effective dynamics, which is derived using averaging techniques. This simple behavior might be a prototype of others observed in more complicated systems with long-range interactions, such as two-dimensional incompressible fluids and wave-particle interaction in a plasma.     

Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems

We study the growth of correlations in systems with weak long-range interactions. Starting from the BBGKY hierarchy, we determine the evolution of the two-body correlation function by using an expansion of the solutions of the hierarchy in powers of 1/N in a proper thermodynamic limit N→+∞, where N is the number of particles. These correlations are responsible for the “collisional” evolution of the system beyond the Vlasov regime due to finite N effects. We obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. These peculiarities are specific to systems with unshielded long-range interactions. For spatially homogeneous systems with short memory time like plasmas, we recover the classical Landau (or Lenard-Balescu) equations. An interest of our approach is to develop a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems. This enlightens the basic physics and provides novel kinetic equations with a clear physical interpretation. However, unless we restrict ourselves to spatially homogeneous systems, closed kinetic equations can be obtained only if we ignore some collective effects between particles. General exact coupled equations taking into account collective effects are also given. We use this kinetic theory to discuss the processes of violent collisionless relaxation and slow collisional relaxation in systems with weak long-range interactions. In particular, we investigate the dependence of the relaxation time with the system size N and try to provide a coherent discussion of all the numerical results obtained for these systems.     

Quasi-stationary states and incomplete violent relaxation in systems with long-range interactions

We discuss the nature of quasi-stationary states (QSS) with non-Boltzmannian distribution in systems with long-range interactions in relation with a process of incomplete violent relaxation based on the Vlasov equation. We discuss several attempts to characterize these QSS. We show that their distribution is non-universal and explain why their prediction is difficult in general.     

Hamiltonian approach to the dynamics of Darwinian systems

It is shown that equations describing the dynamics of Darwinian systems (DS) far from the bifurcation points may be expressed in Hamiltonian form. The cases of DS with constant organization and DS with a constant flux through the system are considered. The configurational part of phase space is formed by variables containing information on the structure of the system. Momentum variables may be regarded as specific rates of multiplication. The evolution of DS with constant organization in this phase space is expressed as uniform rectilinear motion. In the case of DS with a constant flux, the motion occurs in some effective constant and uniform field. The meaning of the elements of the Hamiltonian structure is described in terms of theoretical biology. Tomsk State University. Scientific-Research Institute of Biological Systems, Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 23–28, July, 1997.     

Zeldovich's regularization method in the theory of quasistationary states

The complex quasienergy, including the level width Γ, is calculated for a loosely bound atomic state in an intense monochromatic laser field of circular polarization. The method proposed by Zeldovich for regularizing divergent integrals that involve the Gamow wave function is employed in this calculation. The convergence of the method is demonstrated, and the conditions of its applicability are indicated.     

Hamiltonian dynamics of vortex lines in hydrodynamic-type systems

It is shown that the degeneracy of the noncanonical Poisson bracket operating on the space of solenoidal vector fields that arises due to the freezing-in of the curl of the velocity [E. A. Kuznetsov and A. V. Mikhailov, Phys. Lett. A 77, 37 (1980)] is lifted when the vorticity Ω is represented in terms of vortex lines. This representation makes it possible to integrate the equation of motion of the vorticity for a system with the Hamiltonian H=∫∣Ω∣d r. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 12, 1015–1020 (25 June 1998)     

Fractional dynamics of systems with long-range space interaction and temporal memory

Field equations with time and coordinate derivatives of noninteger order are derived from a stationary action principle for the cases of power-law memory function and long-range interaction in systems. The method is applied to obtain a fractional generalization of the Ginzburg-Landau and nonlinear Schrödinger equations. As another example, dynamical equations for particle chains with power-law interaction and memory are considered in the continuous limit. The obtained fractional equations can be applied to complex media with/without random parameters or processes.