Nonlinear excitations and electric transport in dissipative Morse-Toda lattices

We investigate the onset and maintenance of nonlinear soliton-like excitations in chains of atoms with Morse interactions at rather high densities, where the exponential repulsion dominates. First we discuss the atomic interactions and approximate the Morse potential by an effective Toda potential with adapted density-dependent parameters. Then we study several mechanisms to generate and stabilize the soliton-like excitations: (i) External forcing: we shake the masses periodically, mimicking a piezoelectric-like excitation, and delay subsequent damping by thermal excitation; (ii) heating, quenching and active friction: we heat up the system to a relatively high temperature Gaussian distribution, then quench to a low temperature, and subsequently stabilize by active friction. Finally, we assume that the atoms in the chain are ionized with free electrons able to move along the lattice. We show that the nonlinear soliton-like excitations running on the chain interact with the electrons. They influence their motion in the presence of an external field creating dynamic bound states (“solectrons”, etc.). We show that these bound states can move very fast and create extra current. The soliton-induced contribution to the current is constant, field-independent for a significant range of values when approaching the zero-field value.     

Phase behavior and collective excitations of the Morse ring chain

Using primarily numerical methods we study clustering processes and collective excitations in a one-dimensional ring chain. The ring chain is constituted by N identical point particles with next neighbors interacting via nonlinear Morse springs. If the system is coupled to a heat bath (Gaussian white noise and viscous friction), then depending on the particle density and the bath temperature different phase-like states can be distinguished. This will be illustrated by means of numerically calculated phase diagrams. In order to identify collective excitations activated by the heat bath we calculate the spectrum of the normalized dynamical structure factor (SDF). Our numerical results show that the transition regions between different phase-like states are typically characterized by a 1/f-type SDF spectrum, reflecting the fact that near critical points correlations on all length and time scales become important. In the last part of the paper we also discuss a non-equilibrium effect, which occurs if an additional nonlinearly velocity-dependent force is included in the equations of motions. In particular it will be shown that such additional dissipative effects may stabilize cluster configurations.Received: 27 June 2003, Published online: 2 October 2003PACS: 05.70.Fh Phase transitions: general studies - 05.70.Ln Non-equilibrium and irreversible processes - 05.40.-a Fluctuation phenomena, random processes, noise and Brownian motion     

Coulomb Friction Driving Brownian Motors

We review a family of models recently introduced to describe Brownian motors under the influence of Coulomb friction, or more general non-linear friction laws. It is known that, if the heat bath is modeled as the usual Langevin equation (linear viscosity plus white noise), additional non-linear friction forces are not sufficient to break detailed balance, i.e. cannot produce a motor effect. We discuss two possibile mechanisms to elude this problem. A first possibility, exploited in several models inspired to recent experiments, is to replace the heat bath's white noise by a "collisional noise", that is the effect of random collisions with an external equilibrium gas of particles. A second possibility is enlarging the phase space, e.g. by adding an external potential which couples velocity to position, as in a Klein-Kramers equation. In both cases, non-linear friction becomes sufficient to achieve a non-equilibrium steady state and, in the presence of an even small spatial asymmetry, a motor effect is produced.     

Thermodynamics and transport in an active Morse ring chain

We investigate the stochastic dynamics of an one-dimensional ring with N self-driven Brownian particles. In this model neighboring particles interact via conservative Morse potentials. The influence of the surrounding heat bath is modeled by Langevin-forces (white noise) and a constant viscous friction coefficient γ. The Brownian particles are provided with internal energy depots which may lead to active motions of the particles. The depots are realized by an additional nonlinearly velocity-dependent friction coefficient γ ₁(v) in the equations of motions. In the first part of the paper we study the partition functions of time averages and thermodynamical quantities (e.g. pressure) characterizing the stationary physical system. Numerically calculated non-equilibrium phase diagrams are represented. The last part is dedicated to transport phenomena by including a homogeneous external force field that breaks the symmetry of the model. Here we find enhanced mobility of the particles at low temperatures. Received 21 July 2001     

Phase synchronization in unidirectionally coupled chaotic ratchets

We study chaotic phase synchronization of unidirectionally coupled deterministic chaotic ratchets. The coupled ratchets were simulated in their chaotic states and perfect phase locking was observed as the coupling was gradually increased. We identified the region of phase synchronization for the ratchets and show that the transition to chaotic phase synchronization is via an interior crisis transition to strange attractor in the phase space.     

均匀湍流中的奇怪吸引子

本文对于实测的湍流信号采用延迟坐标重建相空间技术计算了它的关联维数,熵和最大Лялнов指数,从而指出这是一种受随机噪声干扰的确定性的混沌现象,它在相空间的吸引子是一个随机噪声背景上的奇怪吸引子。     

New Rational Solutions for Relativistic Discrete Toda Lattice System

In the present work, we examine the soliton excitations in the relativistic Toda lattice model using the rotational expansion method, where the coupling between the lattice sites is varied. For specific choices of the coupling strength we proceed to analyze the nonlinear wave excitations arising in the model which are found to be dark, singular and periodic solitary wave profiles. These solitary wave profiles are admitted to show possible modulation in its amplitude.     

Quantum Statistics of the Toda Oscillator in the Wigner Function Formalism

Classical and quantum mechanical Toda systems (Toda molecules, Toda lattices, Toda quantum fields) recently found growing interest as nonlinear systems showing solitons and chaos. In this paper the statistical thermodynamics of a system of quantum mechanical Toda oscillators characterized by a potential energy V(q) = V_o cos h q is treated within the Wigner function formalism (phase space formalism of quantum statistics). The partition function is given as a Wigner- Kirkwood series expansion in terms of powers of h² (semiclassical expansion). The partition function and all thermodynamic functions are written, with considerable exactness, as simple closed expressions containing only the modified Hankel functions K_o and K₁ of the purely imaginary argument iζ with ζ = V_o/kT.     

New Rational Solutions for Relativistic Discrete Toda Lattice System

In the present work, we examine the soliton excitations in the relativistic Toda lattice model using the rotational expansion method, where the coupling between the lattice sites is varied. For specific choices of the coupling strength we proceed to analyze the nonlinear wave excitations arising in the model which are found to be dark, singular and periodic solitary wave profiles. These solitary wave profiles are admitted to show possible modulation in its amplitude.     

Activation by nonlinear oscillations and solitonic excitations

Local excitations in molecular systems are studied taking into account the influence of soft impurities. The dynamics of activation processes (high-energy events) due to nonlinear mechanisms is studied. The following examples of classical macroscopic systems with strong nonlinear interaction are investigated: 1D Toda chains, 1D Morse rings, and 3D systems of hard spheres including impurities. It is shown that solitonlike excitations may lead to the concentration of energy at definite sites (weak springs or soft spheres). The accumulation of energy is mainly due to soliton-fusion effects. In thermal equilibrium an optimum temperature exists, where the thermally averaged potential energy is preferably partitioned to the soft springs embedded into a hard-spring solvent. Further, we show that the effect of thermal energy localization and the temperature dependence also persists for solutions of soft spheres in hard-sphere solvents.     

Phase transitions in the adsorbed molecular chains

Rotational excitations of molecular adsorbed layers are studied theoretically. Nonlinear dynamical equations are obtained with accounting of quadrupolar interactions between molecules and freezing of translational degrees of freedom. The equilibrium positions of the molecules are found to be experimentally observed structures with alternating rotational ordering of planar rotors along the direction to the nearest neighbor (for linear or square structures) under low temperature. Dynamical analysis gives an integral of motion (energy) of the chain that in the long-wave limit leads consequently to the existence of four phases. The first one corresponds to oscillations near equilibrium ordered states. The second phase corresponds to low-energy rotational excitations along ‘valleys’ (easy directions in the effective potential) that do not destroy strong correlations between molecules while structural data can show rotational disorder (melting). The third phase corresponds to an energy that is enough to travel between ‘valleys’; only some ‘islands’ in the angle space are forbidden. Complete destruction of correlation when the energy is over the peaks of the effective potential corresponds to the fourth phase. Therefore rotational melting is a complex phenomenon that has several stages. Presented at the 2nd International Conference “Physics of Liquid Matter: Modern Problems” (September 2003, Kyiv, Ukraine).     

An integrable model in general relativity

Melnikov-method-based theoretical results are demonstrated concerning the relative effectiveness of any two weak excitations in suppressing homoclinic/heteroclinic chaos of a relevant class of dissipative, low-dimensional and non-autonomous systems for the main resonance between the chaos-inducing and chaos-suppressing excitations. General analytical expressions are derived from the analysis of generic Melnikov functions providing the boundaries of the regions as well as the enclosed area in the amplitude/initial phase plane of the chaos-suppressing excitation where homoclinic/heteroclinic chaos is inhibited. The relevance of the theoretical results on chaotic attractor elimination is confirmed by means of Lyapunov exponent calculations for a two-well Duffing oscillator. Received 21 May 2002 / Received in final form 13 September 2002 Published online 29 November 2002     

Detecting variation in chaotic attractors

If the output of an experiment is a chaotic signal, it may be useful to detect small changes in the signal, but there are a limited number of ways to compare signals from chaotic systems, and most known methods are not robust in the presence of noise. One may calculate dimension or Lyapunov exponents from the signal, or construct a synchronizing model, but all of these are only useful in low noise situations. I introduce a method for detecting small variations in a chaotic attractor based on directly calculating the difference between vector fields in phase space. The differences are found by comparing close strands in phase space, rather than close neighbors. The use of strands makes the method more robust to noise and more sensitive to small attractor differences.     

Thermodynamics and phase transitions in dissipative and active Morse chains

We study the evolution of a simple one-dimensional chain of N=4 particles with Morse interactions and periodic boundary conditions which are imbedded into a heat bath creating dissipation and noise. The investigation is concentrated on thermodynamic properties for equilibrium, near-equilibrium and far-equilibrium conditions. For the thermodynamic equilibrium, created by white noise and passive friction obeying Einsteins fluctuation dissipation relation, we find a standard phase diagram. By applying active friction forces the system is driven to stationary non-equilibrium states, creating conditions where various self-sustained oscillations are excited. Thermodynamic quantities like energy, pressure and entropy are calculated near equilibrium, around a critical distance from equilibrium and far from equilibrium. We observe maximal order (minimum entropy) in certain region of the noise temperature, a phenomenon which is reminiscent of stochastic resonance. With increasing distance from equilibrium new phases corresponding to the existence of several attractors of the dynamical stem appear.     

Topological Landau-Ginzburg theory with a rational potential and the dispersionless KP hierarchy

Based on the dispersionless KP (dKP) theory, we study a topological Landau-Ginzburg (LG) theory characterized by a rational potential. Writing the dKP hierarchy in a general form treating all the primaries in an equal basis, we find that the hierarchy naturally includes the dispersionless (continuous) limit of Toda hierarchy and its generalizations having a finite number of primaries. Several flat solutions of the topological LG theory are obtained in this formulation, and are identified with those discussed by Dubrovin. We explicitly construct gravitational descendants for all the primary fields. Giving a residue formulae for the 3-point functions of the fields, we show that these 3-point functions satisfy the topological recursion relation. The string equation is obtained as the generalized hodograph solutions of the dKP hierarchy, which show that all the gravitational effects to the constitutive equations (2-point functions) can be renormalized into the coupling constants in the small phase space. Supported in part by NSF grant DMS-9403597.     

Relaxation of a test particle in systems with long-range interactions: diffusion coefficient and dynamical friction

We study the relaxation of a test particle immersed in a bath of field particles interacting via weak long-range forces. To order 1/N in the N→+∞ limit, the velocity distribution of the test particle satisfies a Fokker-Planck equation whose form is related to the Landau and Lenard-Balescu equations in plasma physics. We provide explicit expressions for the diffusion coefficient and friction force in the case where the velocity distribution of the field particles is isotropic. We consider (i) various dimensions of space d=3,2 and 1; (ii) a discret spectrum of masses among the particles; (iii) different distributions of the bath including the Maxwell distribution of statistical equilibrium (thermal bath) and the step function (water bag). Specific applications are given for self-gravitating systems in three dimensions, Coulombian systems in two dimensions and for the HMF model in one dimension.     

Remarks on the moduli space of SU(2) monopoles and Toda flow

Rational functions of degreek which parametrize the moduli space of SU(2)k-monopoles, can be regarded as transfer functions for certain linear dynamical systems. It is shown that a time shift for linear systems induces a finite nonperiodic complexk×k Toda flow on the parameter space of generic SU(2)k-monopoles. Thus, there exists an integrable flow of the Toda type over the moduli space of SU(2) monopoles.This research was supported in part by National Science Foundation under Grant ECS-8718026.     

高维微分-差分模型的Virasoro对称子代数，多线性变量分离解和局域激发模式

寻找高维可积模型是非线性科学中的重要课题.利用无穷维Virasoro对称子代数［σ(f₁),σ(f₂)］=σ(f′₁f₂-f′₂f₁)和向量场的延拓结构理论，能够得到各种高维模型.选取一些特殊的实现，可以给出具有无穷维Virasoro对称子代数意义下的高维微分可积模型.把该方法推广到微分-差分模型上，构造出具有弱多线性变量分离可解性的(3+1)维类Toda晶格.另外，该模型的一个约化方程为具有多线性变量分离可解性的(2+1)维特殊Toda晶格.连续运用对称约化方法可以得到此特殊Toda晶格的一个(1+1)维约化方程具有多线性变量分离可解性.因为得到的精确解里含有低维任意函数，从而可以构造出丰富地局域激发模式，如dromion解，lump解，环孤子解，呼吸子解，瞬子解，混沌斑图和分形斑图等等. 关键词： Virasoro代数 微分-差分模型 变量分离 局域激发模式     

Statistical mechanics of confined systems with rotational excitations

New experimental and numerical investigations of confined systems of particles demonstrate the existence of rotational excitations. We develop here a statistical theory of finite systems, including rotational modes, by introducing the angular momentum into the formalism and constructing the relevant distributions. As special applications we study systems driven to a prescribed kinetic energy by negative friction or special isokinetic thermostats. Several distribution functions which are solutions of the Liouville or Fokker–Planck equations are given. The theory is applied to Coulomb clusters confined by parabolic forces.     

On the phase space approach to complexity

We elaborate in some detail on a new phase space approach to complexity, due to Y.-C. Zhang. We show in particular that the connection between maximal complexity and power law noise or correlations can be derived from a simple variational principle. For a 1D signal we find 1/f noise, in accordance with Zhang.