Ergodic properties of plane billiards with symmetric potentials

We study chaotic behaviour of the motion of a particle moving like in a billiard table outside some disks where a symmetric potential acts. Quadratic forms introduced in (Markarian, 1988) to study non-vanishing Lyapunov exponents are used.     

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.     

Ergodic properties of equilibrium states

Possible ergodic properties of Gibbs states are discussed by constructing a number of examples. In particular existence of Gibbs states which are mixing but not extremal is shown.     

Supersymmetry and solvable periodic potentials

We use the formalism of supersymmetric quantum mechanics to enlarge considerably the limited class of analytically solvable one-dimensional periodic potentials. In particular, we derive and discuss the energy-band structure of the Lamé potentials pm sn² (x, m) and associated Lamé potentials pm sn² (x, m) + qm cn²(x, m)/dn² (x, m), both of which involve Jacobi elliptic functions with modulus parameter m. We find several new analytic expressions for band-edge energies and wave functions. The supersymmetric partners of Lamé and associated Lamé potentials constitute even more new solvable potentials with exactly the same energy-band structure.     

拓扑材料周期表

人们曾经认为,外部导电但内部绝缘的材料是不常见的。然而,德国德累斯顿马克斯·普朗克固体化学物理研究所的计算化学家MaiaVergniory及其同事最近证明了事实并非如此。他们发现了数以万计的此类拓扑绝缘体以及其他具有值得关注的拓扑特性的材料,并将这些结果创建成拓扑材料数据库(www.topologicalquantumchemistry.com)。Vergniory对Margaret Harris讲述了她所在的团队如何进行这种搜索以及数据库对其领域的意义。     

Diffusion in periodic potentials

We discuss the static and dynamic phenomena connected with the Brownian motion of a particle in a periodic potential. Specifically we consider the dynamic mobility and the dynamic structure factor. We show with numerical examples how translational and oscillatory motion show up in these quantities. The calculations are done for the onedimensional case. A main tool is the continued fraction expansion of the relevant correlation functions. It enables us, for example, to obtain numerically accurate results for the mobility in a cosine potential. Possible applications of this model lie in the fields of superionic conductors and molecular crystals with rotational diffusion.     

Ergodic properties of the Lozi mappings

In this paper, we construct the Bowen-Ruelle measure for the Lozi mapping, an almost everywhere hyperbolic diffeomorphism of the plane. We also derive some of its properties which are similar to those of an axiomA system.     

Ergodic properties of infinite-dimensional stochastic systems

The ergodic properties of two stochastic models _I and _II are investigated. Each model is described by a fieldx(t),t > 0, on the lattice =Z ^d,d < . For _I,x(t) evolves according to the equations wherex _s (t) R for eachs eF. Here the {w_s(t): s } are independent, one-dimensional Wiener processes, ₂ is a bounded interaction between adjacent lattice sites, and the potentials ₁ and ₂ satisfy appropriate regularity conditions. It is shown that for each model,x(t) is a Markov process on an infinite-dimensional phase spaceX. The probability measures onX that satisfy the Dobrushin-Lanford-Ruelle (DLR) conditions are stationary for this process and have a mixing property. Moreover, for _I any stationary, time-reversal-invariant probability measure that has certain regularity properties must satisfy the DLR conditions.This paper is based on a portion of the author's Ph.D. thesis.⁽²⁾     

Ergodic properties of anomalous diffusion processes

In this paper we study ergodic properties of some classes of anomalous diffusion processes. Using the recently developed measure of dependence called the Correlation Cascade, we derive a generalization of the classical Khinchin theorem. This result allows us to determine ergodic properties of Lévy-driven stochastic processes. Moreover, we analyze the asymptotic behavior of two different fractional Ornstein–Uhlenbeck processes, both originating from subdiffusive dynamics. We show that only one of them is ergodic.     

Stochastic resonance in periodic potentials

We have studied the motion of a particle in a periodic potential plus a bias, driven by a noise and a coherent forcing. The response (power spectrum) of the particle at the driving forcing frequency is considered for different values of the noise intensity and of the bias. It is shown via direct simulation that the response displays the phenomenon of stochastic resonance, although the phenomenology is somehow different from the one observed in the standard bistable system.     

Diffusion in flashing periodic potentials

The one-dimensional overdamped Brownian motion in a symmetric periodic potential modulated by external time-reversible noise is analyzed. The calculation of the effective diffusion coefficient is reduced to the mean first passage time problem. We derive general equations to calculate the effective diffusion coefficient of Brownian particles moving in arbitrary supersymmetric potential modulated by: (i) an external white Gaussian noise and (ii) a Markovian dichotomous noise. For both cases the exact expressions for the effective diffusion coefficient are derived. We obtain acceleration of diffusion in comparison with the free diffusion case for fast fluctuating potentials with arbitrary profile and for sawtooth potential in case (ii). In this case the parameter region where this effect can be observed is given. We obtain also a finite net diffusion in the absence of thermal noise. For rectangular potential the diffusion slows down, for all parameters of noise and of potential, in comparison with the case when particles diffuse freely.     

Spectral behavior of quasi periodic potentials

Spectral properties of Schrödinger operators of the typeH =–+V, where is the Laplacian,V a quasiperiodic potential and a coupling constant, are developed.V is taken to be finite sum of exponentials with generic frequencies. For small a strong stability is shown. On the other hand, examples (in the finite diffeence case) are given, for which a transition in the type of spectrum occurs, as is increased.Research supported in part by grant NSF MCS 7900813     

Ergodic properties of the discontinuous sawtooth map

We prove the ergodicity for the discontinuous sawtooth map, adapting a technique previously used in billiard theory. The core of the proof is the construction of a Hopf chain passing through a countable dense set of discontinuity lines.     

Ergodic properties of a kicked damped particle

We investigate a class of nonlinear dynamical systems describing the movement of a particle in a viscous medium under the influence of a kick force. These systems can be regarded as a generalization of the Langevin approach to Brownian motion in the sense that the fluctuating force on the particle is not Gaussian white noise but an arbitrary non-gaussian process generated by a nonlinear dynamical system. We investigate how certain properties of the force (periodicity, ergodicity, mixing property) transfer to the velocity of the particle. Moreover, the relaxation properties of the system are analysed.Address after October 1, 1989: Institut für Theoretische Physik, RWTH, D-5100, Aachen, FRG     

One-dimensionalXY model: Ergodic properties and hydrodynamic limit

We prove theorems on convergence to a stationary state in the course of time for the one-dimensionalXY model and its generalizations. The key point is the well-known Jordan-Wigner transformation, which maps theXY dynamics onto a group of Bogoliubov transformations on the CARC ^*-algebra overZ ¹. The role of stationary states for Bogoliubov transformations is played by quasifree states and for theXY model by their inverse images with respect to the Jordan-Wigner transformation. The hydrodynamic limit for the one-dimensionalXY model is also considered. By using the Jordan-Wigner transformation one reduces the problem to that of constructing the hydrodynamic limit for the group of Bogoliubov transformations. As a result, we obtain an independent motion of normal modes, which is described by a hyperbolic linear differential equation of second order. For theXX model this equation reduces to a first-order transfer equation.     

Conduction bands in classical periodic potentials

The energy of a quantum particle cannot be determined exactly unless there is an infinite amount of time to perform the measurement. This paper considers the possibility that ΔE, the uncertainty in the energy, may be complex. To understand the effect of a particle having a complex energy, the behaviour of a classical particle in a one-dimensional periodic potential V(x) = −cos(x) is studied. On the basis of detailed numerical simulations it is shown that if the energy of such a particle is allowed to be complex, the classical motion of the particle can exhibit two qualitatively different behaviours: (i) The particle may hop from classically allowed site to nearest-neighbour classically allowed site in the potential, behaving as if it were a quantum particle in an energy gap and undergoing repeated tunnelling processes or (ii) the particle may behave as a quantum particle in a conduction band and drift at a constant average velocity through the potential as if it were undergoing resonant tunnelling. The classical conduction bands for this potential are determined numerically with high precision.     

Brownian motion in fluctuating periodic potentials

This work deals with the overdamped motion of a particle in a fluctuating one-dimensional periodic potential. If the potential has no inversion symmetry and its fluctuations are asymmetric and correlated in time, a net flow can be generated at finite temperatures. We present results for the stationary current for the case of a piecewise linear potential, especially for potentials being close to the case with inversion symmetry. The aim is to study the stationary current as a function of the potential. Depending on the form of the potential, the current changes sign once or even twice as a function of the correlation time of the potential fluctuations. To explain these current reversals, several mechanisms are proposed. Finally, we discuss to what extent the model is useful to understand the motion of biomolecular motors.     

Collisionally induced transport in periodic potentials

We study the transport of ultracold atoms in a tight optical lattice. For identical fermions the system is insulating under an external force while for bosonic atoms it is conducting. This reflects the different collisional properties of the particles and reveals the role of interparticle collisions in establishing a macroscopic transport in a perfectly periodic potential. Also in the case of fermions we can induce a transport by creating a collisional regime through the addition of bosons. We investigate the transport as a function of the collisional rate and observe a transition from a regime in which the mobility increases with increasing collisional rate to one in which it decreases. We compare our data with a theoretical model for electron transport in solids introduced by Esaki and Tsu.     

Optical solitons in PT periodic potentials

We investigate the effect of nonlinearity on beam dynamics in parity-time (PT) symmetric potentials. We show that a novel class of one- and two-dimensional nonlinear self-trapped modes can exist in optical PT synthetic lattices. These solitons are shown to be stable over a wide range of potential parameters. The transverse power flow within these complex solitons is also examined.