Ergodic systems ofn balls in a billiard table

We consider the motion ofn balls in billiard tables of a special form and we prove that the resulting dynamical systems are ergodic on a constant energy surface; in fact, they enjoy theK-property. These are the first systems of interacting particles proven to be ergodic for an arbitrary number of particles.     

Three particles on a ring

We study the ergodic properties of three classical particles of unequal mass moving on a ring and colliding like hard points.Although it is generally believed that this system is ergodic, we show that extensive numerical calculations do not support this belief.This may be connected with our discovery that for vanishing total momentum, any initial state with one particle at rest is periodic, independent of the mass values.The analogous periodicity for two unequal mass particles bouncing in a one-dimensional box can be understood on the basis of a remarkable property of a billiard in the form of a rectangular triangle.     

Mean square relaxation times for evolution of random fields

We consider the problem of relaxation times for Markov evolution of systems composed of a countable number of locally interacting particles, each one of which has a finite phase space. We give a theorem for comparison of mean square relaxation times of evolutions possessing the same ergodic stationary state. We give a reduction theorem for “attractive” evolutions. The results are applied to a generalization of the Glauber evolution of the one dimensional Ising chain.     

On the solutions of 3-particle spin elliptic Calogero-Moser systems

We describe a class of the singular solutions to the multicomponent analogs of the Lamé equation, arising as equations of motion of the elliptic Calogero-Moser systems of particles carrying spin 1/2. At special value of the coupling constant we propose the ansatz which allows one to get meromorphic solutions with two arbitrary parameters. They are quantized upon the requirement of the regularity of the wave function on the hyperplanes at which particles meet and imposing periodic boundary conditions. We find also the extra integrals of motion for three-particle systems which commute with the Hamiltonian for arbitrary values of the coupling constant.     

Relativistic quantum mechanics of spin-zero and spin-half particles

We consider the quantum mechanics of directly interacting relativistic particles of spin-zero and spin-half. We introduce a scalar product in the vector space of physical states which is finite, positive definite and relativistically invariant and keeps orthogonal eigenstates of total four momentum belonging to different eigenvalues. This allows us to show that the vector space of physical states is, in fact, a Hilbert space. The case of two particles is explicitly considered and the Cauchy problem of physical wave function illustrated. The problem of a spin-1/2 particle interacting with a spin-zero particle is considered and a new equation is proposed for two spin-1/2 particles interacting via the most general form of interaction possible. The restrictions due to Hermiticity, space inversion and time reversal invariance are also considered.     

Freezing by heating in a driven mesoscopic system

We investigate a simple model corresponding to particles driven in opposite directions and interacting via a repulsive potential. The particles move off-lattice on a periodic strip and are subject to random forces as well. We show that this model-which can be considered as a continuum version of some driven diffusive systems-exhibits a paradoxical, new kind of transition called here "freezing by heating." One interesting feature of this transition is that a crystallized state with a higher total energy is obtained from a fluid state by increasing the amount of fluctuations.     

Interaction induced delocalization for two particles in a periodic potential

We consider two interacting particles evolving in a one-dimensional periodic structure embedded in a magnetic field. We show that the strong localization induced by the magnetic field for particular values of the flux per unit cell is destroyed as soon as the particles interact. We study the spectral and dynamical aspects of this transition.     

Analytic proof of the Sutherland conjecturet

Using the integral representation of the inverse of the logarithmic derivative of the elliptic theta function, the spectrum of the Lax matrix for the 1D system of particles interacting via an inverse sinh-squared potential is shown to be given by the asymptotic Bethe ansatz in the thermodynamic limit.     

Helix self-assembly from anisotropic molecules

We explore the potential energy landscape for clusters composed of disklike ellipsoidal particles interacting via an anisotropic potential based on the elliptic contact function. Over a wide range of parameter space we find global potential energy minima consisting of helices composed of one or more strands. Characterizing the potential energy surface in the region of helical global minima reveals a topology associated with "structure-seeking" systems. This result indicates that the helices will self-assemble over a wide range of temperature.     

From collective periodic running states to completely chaotic synchronised states in coupled particle dynamics

We consider the damped and driven dynamics of two interacting particles evolving in a symmetric and spatially periodic potential. The latter is exerted to a time-periodic modulation of its inclination. Our interest is twofold: First, we deal with the issue of chaotic motion in the higher-dimensional phase space. To this end, a homoclinic Melnikov analysis is utilised assuring the presence of transverse homoclinic orbits and homoclinic bifurcations for weak coupling allowing also for the emergence of hyperchaos. In contrast, we also prove that the time evolution of the two coupled particles attains a completely synchronised (chaotic) state for strong enough coupling between them. The resulting "freezing of dimensionality" rules out the occurrence of hyperchaos. Second, we address coherent collective particle transport provided by regular periodic motion. A subharmonic Melnikov analysis is utilised to investigate persistence of periodic orbits. For directed particle transport mediated by rotating periodic motion, we present exact results regarding the collective character of the running solutions entailing the emergence of a current. We show that coordinated energy exchange between the particles takes place in such a manner that they are enabled to overcome--one particle followed by the other--consecutive barriers of the periodic potential resulting in collective directed motion.     

One-particle and few-particle billiards

We study the dynamics of one-particle and few-particle billiard systems in containers of various shapes. In few-particle systems, the particles collide elastically both against the boundary and against each other. In the one-particle case, we investigate the formation and destruction of resonance islands in (generalized) mushroom billiards, which are a recently discovered class of Hamiltonian systems with mixed regular-chaotic dynamics. In the few-particle case, we compare the dynamics in container geometries whose counterpart one-particle billiards are integrable, chaotic, and mixed. One of our findings is that two-, three-, and four-particle billiards confined to containers with integrable one-particle counterparts inherit some integrals of motion and exhibit a regular partition of phase space into ergodic components of positive measure. Therefore, the shape of a container matters not only for noninteracting particles but also for interacting particles.     

Chaotic delocalization of two interacting particles in the classical Harper model

We study the problem of two interacting particles in the classical Harper model in the regime when one-particle motion is absolutely bounded inside one cell of periodic potential. The interaction between particles breaks integrability of classical motion leading to emergence of Hamiltonian dynamical chaos. At moderate interactions and certain energies above the mobility edge this chaos leads to a chaotic propulsion of two particles with their diffusive spreading over the whole space both in one and two dimensions. At the same time the distance between particles remains bounded by one or two periodic cells demonstrating appearance of new composite quasi-particles called chaons. The effect of chaotic delocalization of chaons is shown to be rather general being present for Coulomb and short range interactions. It is argued that such delocalized chaons can be observed in experiments with cold atoms and ions in optical lattices.     

Ergodicity of a Single Particle Confined in a Nanopore

We analyze the dynamics of a gas particle moving through a nanopore of adjustable width with particular emphasis on ergodicity. We give a measure of the portion of phase space that is characterized by quasiperiodic trajectories which break ergodicity. The interactions between particle and wall atoms are mediated by a Lennard-Jones potential, so that an analytical treatment of the dynamics is not feasible, but making the system more physically realistic. In view of recent studies, which proved non-ergodicity for systems with scatterers interacting via smooth potentials, we find that the non-ergodic component of the phase space for energy levels typical of experiments, is surprisingly small, i.e. we conclude that the ergodic hypothesis is a reasonable approximation even for a single particle trapped in a nanopore. Due to the numerical scope of this work, our focus will be the onset of ergodic behavior which is evident on time scales accessible to simulations and experimental observations rather than ergodicity in the infinite time limit.     

Periodic composites: quasi-uniform heat conduction,Janus thermal illusion,and illusion thermal diodes

Manipulating thermal conductivities at will plays a crucial role in controlling heat flow. By developing an effective medium theory including periodicity, here we experimentally show that nonuniform media can exhibit quasi-uniform heat conduction. This provides capabilities in proposing Janus thermal illusion and illusion thermal rectification. For the former, we study, via experiment and theory, a big periodic composite containing a small periodic composite with circular or elliptic particles. As a result, we reveal the Janus thermal illusion that describes the whole periodic system with both invisibility illusion along one direction and visibility illusion along the perpendicular direction, which is fundamentally different from the existing thermal illusions for misleading thermal detection. Further, the Janus illusion helps to design two different periodic systems that both work as thermal diodes but with nearly the same temperature distribution, heat fluxes and rectification ratios, thus being called illusion thermal diodes. Such thermal diodes differ from those extensively studied in the literature, and are useful for the areas that require both thermal rectification and thermal camouflage. This work not only opens a door for designing novel periodic composites in thermal camouflage and heat rectification, but also holds for achieving similar composites in other disciplines like electrostatics, magnetostatics, and particle dynamics.     

Exact solutions of a new elliptic Calogero–Sutherland model

A quantum Hamiltonian describing N particles on a line interacting pairwise via an elliptic function potential in the presence of an external field is introduced. For a discrete set of values of the strength of the external potential, it is shown that a finite number of eigenfunctions and eigenvalues of the model can be exactly computed in an algebraic way.     

Dynamical Windings of Random Walks and Exclusion Models. Part I: Thermodynamic Limit in ℤ2

We consider a system consisting of a planar random walk on a square lattice, subjected to stochastic elementary local deformations. Both numerical and theoretical results are reported. Depending on the deformation transition rates, and specifically on a parameter η which breaks the symmetry between the left and right orientation, the winding distribution of the walk is modified, and the system can be in three different phases: folded, stretched and glassy. An explicit mapping is found, leading to consider the system as a coupling of two exclusion processes: particles of the first one move in a landscape defined by particles of the second one, and vice-versa. This can be viewed as an inhomogeneous exclusion process. For all closed or periodic initial sample paths, a convenient scaling permits to show a convergence in law (or almost surely on a modified probability space) to a continuous curve, the equation of which is given by a system of two non linear stochastic differential equations. The deterministic part of this system is explicitly analyzed via elliptic functions. In a similar way, by using a formal fluid limit approach, the dynamics of the system is shown to be equivalent to a system of two coupled Burgers equations.     

Ergodic properties of an infinite system of particles moving independently in a periodic field

We investigate the ergodic properties of a general class of infinite systems of independent particles which undergo nontrivial collisions with an external field, e.g. fixed convex barriers (the Lorentz gas). We relate the ergodic properties of these systems to the ergodic properties for a single particle moving in a finite box (with periodic boundary conditions) with the same dynamics. We prove that when the one particle system is mixing or aK-system for a sequence of boxes approaching infinity so is the infinite particle system with an equilibrium measure obtained as a Poisson construction over the one particle phase space.Research sponsored in part by the Air Force Office of Scientific Research Grant No. 73-2430 A and The National Science Foundation Grant No. GP-16147 A, No. 1.     

Dynamics of a pair of spherical gravitating shells

The dynamical N body problem for a system of mass points interacting solely through gravitational forces is not integrable. The difficulties which arise in constructing accurate numerical codes for simulating the motion over long time scales are legend. Thus, in order to test their theories, astronomers and astrophysicists resort to simpler, one-dimensional models which avoid the problems of binary formation, escape, and the singularity of the inverse square force law. To date, the most frequently employed "test" model consists of a system of parallel mass sheets moving perpendicular to their surface. While this system avoids all of the above problems, the time scale for reaching equilibrium is extremely long and probably arises from the system's weak ergodic properties, which become manifest even in the three sheet system. Here we consider a different one-dimensional gravitating system consisting of nonrotating concentric mass shells. For the case of two shells we investigate the structure of the phase space by studying the stability of periodic trajectories. By employing an event driven algorithm, we are able to directly investigate the influence of the singularity without having to resort to regularization of the force. Although stable structures are present at every energy, we find that the ergodic properties of this system are more robust than its planar counterpart. (c) 1997 American Institute of Physics.     

Transitions in two sinusoidally coupled Josephson junction rotators

We investigate the dynamics of two sinusoidally coupled Josephson junction rotators to provide a clear knowledge of the behaviors in different regions of the parameter space. The dynamical states are identified, and the transitions among these states are studied. The properties of the current–voltage curves are investigated. Further more, we observed the chaotic states in some regions of parameter space. We conjecture it may caused by the competition of two periodic potentials: one is the external field, another is the interacting of two particles.     

Entropy-driven formation of the gyroid cubic phase

We show, by computer simulation, that tapered or pear-shaped particles, interacting through purely repulsive interactions, can freely self-assemble to form the three-dimensionally periodic, gyroid cubic phase. The Ia3d gyroid cubic phase is formed by these particles on both compression of an isotropic configuration and expansion of a smectic A bilayer arrangement. For the latter case, it is possible to identify the steps by which the topological transformation from nonintersecting planes to fully interpenetrating, periodic networks takes place.