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.     

The coupled dynamics of two particles with different limit sets

We consider a system of two coupled particles evolving in a periodic and spatially symmetric potential under the influence of external driving and damping. The particles are driven individually in such a way that in the uncoupled regime, one particle evolves on a chaotic attractor, while the other evolves on regular periodic attractors. Notably, only the latter supports coherent particle transport. The influence of the coupling between the particles is explored, and in particular how it relates to the emergence of a directed current. We show that increasing the (weak) coupling strength subdues the current in a process, which in phase-space, is related to a merging crisis of attractors forming one large chaotic attractor in phase-space. Further, we demonstrate that complete current suppression coincides with a chaos-hyperchaos transition.     

Directed transport of two interacting particles in a washboard potential

We study the conservative and deterministic dynamics of two nonlinearly interacting particles evolving in a one-dimensional spatially periodic washboard potential. A weak tilt of the washboard potential is applied biasing one direction for particle transport. However, the tilt vanishes asymptotically in the direction of bias. Moreover, the total energy content is not enough for both particles to be able to escape simultaneously from an initial potential well; to achieve transport the coupled particles need to interact cooperatively. For low coupling strength the two particles remain trapped inside the starting potential well permanently. For increased coupling strength there exists a regime in which one of the particles transfers the majority of its energy to the other one, as a consequence of which the latter escapes from the potential well and the bond between them breaks. Finally, for suitably large couplings, coordinated energy exchange between the particles allows them to achieve escapes — one particle followed by the other — from consecutive potential wells resulting in directed collective motion. The key mechanism of transport rectification is based on the asymptotically vanishing tilt causing a symmetry breaking of the non-chaotic fraction of the dynamics in the mixed phase space. That is, after a chaotic transient, only at one of the boundaries of the chaotic layer do resonance islands appear. The settling of trajectories in the ballistic channels associated with transporting islands provides long-range directed transport dynamics of the escaping dimer.     

Diffusive Limit of a Two Dimensional Kinetic System of Partially Quantized Particles

A quantum-classical coupled system which models the diffusive transport of electrons partially confined in semiconductors nanostructures was presented in Ben Abdallah and Méhats (Proc. Edinb. Math. Soc. 49:513–549, 2006). In this model, electrons are assumed to behave like wave in the confinement direction and to have a classical behaviour in a diffusive regime in the transport direction parallel to the electron gas. It was formally derived from a kinetic system for partially quantized particles thanks to a diffusive limit when the mean free path becomes small with respect to the macroscopic length scale. This paper is devoted to the rigorous study of this limit for a transport in one dimension. In the transport direction, the motion of particles is described by a 1D Boltzmann equation. A Boltzmann-Schrödinger-Poisson system is then considered. Existence of renormalized solutions relying on the study of a quasistatic Schrödinger-Poisson system and on an entropy estimate is established. Its diffusive limit is then considered.     

Bistability effects of the brownian motion in periodic potentials

In the stationary state the equation of motion for particles moving in a periodic potential has two solutions, a locked one and a running one, for low and intermediate damping constants and for suitable external forces. The effect of an additional Langevin force to this bistable behaviour is investigated. For finite noise strength, the mobility depends continuously on the external force, whereas in the limit of vanishing strength of the noise force one gets a sharp transition between the locked and the running solution at a critical external force. This critical force is calculated exactly in the low friction limit and approximately for intermediate friction constants. Furthermore the temperature dependence for various forces including the critical one is shown in the low friction limit.     

The stability analysis of rolling motion of hypersonic vehicles and its validations

The stability of the rolling motion of near space hypersonic vehicles with rudder control is studied using method of qualitative analysis of nonlinear differential equations, and the stability criteria of the deflected rolling motions are improved. The outcomes can serve as the basis for further study regarding the influence of pitching and lateral motion on the stability of rolling motion. To validate the theoretical results, numerical simulations were done for the rolling motion of two hypersonic vehicles with typical configurations. Also, wind tunnel experiments for four aircraft models with typical configurations have been done. The results show that: 1) there exist two dynamic patterns of the rolling motion under statically stable condition. The first one is point attractor, for which the motion of aircraft returns to the original state. The second is periodic attractor, for which the aircraft rolls periodically. 2) Under statically unstable condition, there exist three dynamic patterns of rolling motion, namely, the point attractor, periodic attractor around deflected state of rolling motion, and double periodic attractors or chaotic attractors.     

分数阶对数耦合系统在非周期外力作用下的定向输运现象

本文讨论了分数阶对数耦合系统在非周期外力作用情况下, 耦合粒子链的定向输运现象. 由于粒子在黏性介质中的运动具有“记忆性”, 所以本文通过将系统建模为分数阶对数耦合模型来研究各个系统参数对粒子链运动状态的影响. 数值仿真表明: 1)对于此类系统, 只有在存在外力作用的情况下粒子链才能够产生定向输运现象, 并且粒子链平均流速随着外力的增大而增大. 2)对于分数阶阶数较小的系统, 阻尼记忆性对粒子链的运动状态有显著的影响, 具体表现为: 粒子链的平均流速存在上界(这个上界非常小), 无论外力、耦合力以及噪声强度如何变化, 粒子链的平均流速都不会超过这个上界. 当系统的阻尼力很大且外力为零时, 粒子链不会产生定向输运现象. 3) 当系统的阶数与外力较大时, 虽然粒子链能够产生定向流, 但是此时系统对耦合力与噪声具有免疫性. 4) 耦合力与噪声强度对粒子链运动的影响只在外力较小的情况下有所表现. 在这种情况下, 当系统阶数充分大时, 粒子链的平均流速随着耦合力与噪声强度的变化而变化, 并且伴随着定向流的产生.     

Effective Dynamics of a Tracer Particle Interacting with an Ideal Bose Gas

We study a system consisting of a heavy quantum particle, called the tracer particle, coupled to an ideal gas of light Bose particles, the ratio of masses of the tracer particle and a gas particle being proportional to the gas density. All particles have non-relativistic kinematics. The tracer particle is driven by an external potential and couples to the gas particles through a pair potential. We compare the quantum dynamics of this system to an effective dynamics given by a Newtonian equation of motion for the tracer particle coupled to a classical wave equation for the Bose gas. We quantify the closeness of these two dynamics as the mean-field limit is approached (gas density ${\to \infty}$ ). Our estimates allow us to interchange the thermodynamic with the mean-field limit.     

Superdiffusion and viscoelastic vortex flows in a two-dimensional complex plasma

Viscoelastic vortical fluid motion in a strongly coupled particle system has been observed experimentally. Optical tracking of particle motion in a complex plasma monolayer reveals high grain mobility and large scale vortex flows coexistent with partial preservation of the global hexagonal lattice structure. The transport of particles is superdiffusive and ascribed to Lévy statistics on short time scales and to memory effects on the longer scales influenced by cooperative motion. At these longer time scales, the transport is governed by vortex flows covering a wide spectrum of temporal and spatial scales.     

Twister quintessence scenario

We study generic solutions in a non-minimally coupled to gravity scalar field cosmology. It is shown that dynamics for both canonical and phantoms scalar fields with the potential can be reduced to the dynamical system from which the exact forms for an equation of the state parameter can be derived. We have found the stationary solutions of the system and discussed their stability. Within the large class of admissible solutions we have found a non-degenerate critical points and we pointed out multiple attractor type of trajectory travelling in neighborhood of three critical points at which we have the radiation dominating universe, the barotropic matter dominating state and finally the de Sitter attractor. We have demonstrated the stability of this trajectory which we call the twister solution. Discovered evolutional path is only realized if there exist the non-minimal coupling constant. We have found simple duality relations between twister solutions in phantom and canonical scalar fields in the radiation domination phase. For the twister trajectory we have found an oscillating regime of approaching the de Sitter attractor.     

Phase segregation in driven diffusive systems

Driven diffusive models describe an array of atoms in an external periodic potential, when the motion is damped due to energy exchange with the substrate. The systems of this class have wide application in modeling of charge and mass transport in solids. Recently, the driven diffusive models have been used in tribology, where the driving force emerges due to motion of one of two substrates, which are separated by a thin atomic layer. When a dc force is applied to the atoms, the system exhibits the locked-to-sliding transition. During the transition the system may split in domains of two kinds, the running domains where the atoms move with almost maximum velocity, and the immobile domains (“traffic jams”). We discuss a new model for a 1D chain, where the particles have a complex structure treated in a mean-field fashion: particle collisions are inelastic and also each particle is considered as having its own thermostat. This model exhibits a hysteresis and the “traffic jams” state even at high temperatures due to the clustering of atoms with the same velocity.     

Statefinder Parameters for Coupled Quintessence Scenario in a Power Law Case

We investigate a coupled quintessence scenario, which can provide a natural solution to the cosmic coincidence problem. We assume that the mass of dark matter particles depends on a power law function of the scalar field associated to dark energy and meanwhile the scalar field evolves in a power law potential. Since the dynamics of this system is dominated by an attractor solution, the mass of dark matter particles is forced to change with time as to ensure that the ratio between the energy densities of dark matter and dark energy becomes a constant at late times,and one thus solves the cosmic coincidence problem naturally. We then apply a statefinder diagnostic to this coupled quintessence scenario. It is shown that the evolving trajectory of this scenario in the s-r diagram is quite different from those of other dark energy models.     

Statefinder Parameters for Coupled Quintessence Scenario in a Power Law Case

We investigate a coupled quintessence scenario, which can provide a natural solution to the cosmic coincidence problem. We assume that the mass of dark matter particles depends on a power law function of the scalar field associated to dark energy and meanwhile the scalar field evolves in a power law potential. Since the dynamics of this system is dominated by an attractor solution, the mass of dark matter particles is forced to change with time as to ensure that the ratio between the energy densities of dark matter and dark energy becomes a constant at late times, and one thus solves the cosmic coincidence problem naturally. We then apply a statefinder diagnostic to this coupled quintessence scenario. It is shown that the evolving trajectory of this scenario in the s-r diagram is quite different from those of other dark energy models.     

Persistent random walks in a one-dimensional random environment

Central limit theorems are obtained for persistent random walks in a onedimensional random environment. They also imply the central limit theorem for the motion of a test particle in an infinite equilibrium system of point particles where the free motion of particles is combined with a random collision mechanism and the velocities can take on three possible values.Work supported by the Central Research Fund of the Hungarian Academy of Sciences (grant No. 476/82).     

Memory Effects in Nonequilibrium Transport for Deterministic Hamiltonian Systems

We consider nonequilibrium transport in a simple chain of identical mechanical cells in which particles move around. In each cell, there is a rotating disc, with which these particles interact, and this is the only interaction in the model. It was shown in Ref. 1 that when the cells are weakly coupled, to a good approximation, the jump rates of particles and the energy-exchange rates from cell to cell follow linear profiles. Here, we refine that study by analyzing higher-order effects which are induced by the presence of external gradients for situations in which memory effects, typical of Hamiltonian dynamics, cannot be neglected. For the steady state we propose a set of balance equations for the particle number and energy in terms of the reflection probabilities of the cell and solve it phenomenologically. Using this approximate theory we explain how these asymmetries affect various aspects of heat and particle transport in systems of the general type described above and obtain in the infinite volume limit the deviation from the theory in Ref. 1 to first-order. We verify our assumptions with extensive numerical simulations.     

用连续法计算五维对流模型的定常解和周期解

利用连续算法(Continuation algorithm)对五维对流非线性动力系统的定常解和周期解进行了数值计算。在参数平面Ri-Re上计算出实分岔点曲线、极限点曲线、Hopf分岔点曲线,绘出了分岔图。在分岔图上的不同区域,存在性质不同的稳定解如定常吸引子、周期吸引子等。分析了定常解、周期解的分岔过程。计算结果很好地说明大气中由基本态到对流态再到波动态最后到湍流态的物理转换过程。 连续算法对研究非线性动力系统的分岔以及耗散结构是很有效的计算方法。     

Avalanches in a nonlinear oscillator chain in a periodic potential

We consider the dynamics of a chain of coupled units evolving in a periodic substrate potential. The chain is initially in a flat state and situated in a potential well. A bias force, acting as a weak driving mechanism, is applied at a single unit of the chain. We study the instigation of directed transport in two types of system: (i) a microcanonical situation associated with deterministic and conservative dynamics and (ii) the Langevin dynamics when the system is in contact with a heat bath. Interestingly, for the deterministic and conservative dynamics the directed transport is drastically enhanced compared with its Langevin counterpart. In particular, in the deterministic and conservative regime a self-organised redistribution of energy triggers huge-sized avalanches yielding ultimately accelerated transport of the chain. In contrast, in the thermally-assisted process between avalanches the chain settles always into a pinned metastable state impeding continual accelerated chain motion.     

Random turn walk on a half line with creation of particles at the origin

We consider a version of random motion of hard core particles on the semi-lattice 1,2,3,… , where in each time instant one of three possible events occurs, viz., (a) a randomly chosen particle hops to a free neighboring site, (b) a particle is created at the origin (namely, at site 1) provided that site 1 is free and (c) a particle is eliminated at the origin (provided that the site 1 is occupied). Relations to the BKP equation are explained. Namely, the tau functions of two different BKP hierarchies provide generating functions respectively (I) for transition weights between different particle configurations and (II) for an important object: a normalization function which plays the role of the statistical sum for our non-equilibrium system. For time t→∞ we obtain the asymptotic configuration of particles obtained from the initial empty state (the state without particles).     

A gas of Brownian particles in statistical dynamics

We consider a large number of particles on a one-dimensional latticel Z in interaction with a heat particle; the latter is located on the bond linking the position of the particle to the point to which it jumps. The energy of a single particle is given by a potentialV(x), x∈Z. In the continuum limit, the classical version leads to Brownian motion with drift. A quantum version leads to a local drift velocity which is independent of the applied force. Both these models obey Einstein's relation between drift, diffusion, and applied force. The system obeys the first and second laws of thermodynamics, with the time evolution given by a pair of coupled non linear heat equations, one for the density of the Brownian particles and one for the heat occupation number; the equation for a tagged Brownian particle can be written as a stochastic differential equation.