Nonlinear Particle Interaction Research Articles

Variable order fractional Fokker–Planck equations derived from Continuous Time Random Walks

Continuous Time Random Walk models (CTRW) of anomalous diffusion are studied, where the anomalous exponent β(x)∈(0,1) varies in space. This type of situation occurs e.g. in biophysics, where the density of the intracellular matrix varies throughout a cell. Scaling limits of CTRWs are known to have probability distributions which solve fractional Fokker–Planck type equations (FFPE). This correspondence between stochastic processes and FFPE solutions has many useful extensions e.g. to nonlinear particle interactions and reactions, but has not yet been sufficiently developed for FFPEs of the “variable order” type with non-constant β(x).In this article, variable order FFPEs (VOFFPE) are derived from scaling limits of CTRWs. The key mathematical tool is the 1-1 correspondence of a CTRW scaling limit to a bivariate Langevin process, which tracks the cumulative sum of jumps in one component and the cumulative sum of waiting times in the other. The spatially varying anomalous exponent is modelled by spatially varying β(x)-stable Lévy noise in the waiting time component. The VOFFPE displays a spatially heterogeneous temporal scaling behaviour, with generalized diffusivity and drift coefficients whose units are length2/timeβ(x) resp. length/timeβ(x). A global change of the time scale results in a spatially varying change in diffusivity and drift.A consequence of the mathematical derivation of a VOFFPE from CTRW limits in this article is that a solution of a VOFFPE can be approximated via Monte Carlo simulations. Based on such simulations, we are able to confirm that the VOFFPE is consistent under a change of the global time scale.

Effect of Viscosity on Nonlinear Tempering for Anomalous Diffusion of Viscous Particle: A Subdiffusive Case

In this study, we introduced another parameter on nonlinear particle interaction into sub- diffusive transport involving nonlinear effects such as adhesion, volume filling, etc. We also introduce an additional variable which is the effect of viscosity on the nonlinear escape rate of particle which affects the resulting integral escape rate. This paper focuses only on the in- vestigation of the effect of the added variable on the total escape rate. Lastly, we can see the importance of this study when dealing viscous macroparticles.

Nonlinear Tempering of Subdiffusion with Chemotaxis, Volume Filling, and Adhesion

The purpose of this paper is to implement nonlinear particle interactions into subdiffusive transport, involving chemotaxis and nonlinear effects such as volume filling and adhesion. We systematically derive nonlinear subdiffusive fractional equations with chemoattractant dependent forcing. We consider the diffusion limit of the master equation and analyse the role of nonlinear tempering in the stationary case. We study the interaction between attractive forces of anomalous aggregation and chemotaxis, with repulsive forces induced by nonlinear reactions. We show that this nonlinear interaction can prevent the phenomenon of anomalous aggregation when the local particle concentration grows too high. We also show that the effect of nonlinear tempering is to suppress the intermediate subdiffusive behaviour which results in a nonlinear advection diffusion equation in the long time limit.

Transport equations for subdiffusion with nonlinear particle interaction.

We show how the nonlinear interaction effects 'volume filling' and 'adhesion' can be incorporated into the fractional subdiffusive transport of cells and individual organisms. To this end, we use microscopic random walk models with anomalous trapping and systematically derive generic non-Markovian and nonlinear governing equations for the mean concentrations of the subdiffusive cells or organisms. We uncover an interesting interaction between the nonlinearities and the non-Markovian nature of the transport. In the subdiffusive case, this interaction manifests itself in a nontrivial combination of nonlinear terms with fractional derivatives. In the long time limit, however, these equations simplify to a form without fractional operators. This provides an easy method for the study of aggregation phenomena. In particular, this enables us to show that volume filling can prevent "anomalous aggregation," which occurs in subdiffusive systems with a spatially varying anomalous exponent.

Role of asymmetry in the geometric phase of a coupled atom-heteronuclear molecule system

We investigate the role of asymmetry in the geometric phase of an ultracold bosonic atom-heteronuclear molecule system where the asymmetry denotes the particle-number imbalance between two atomic species. Based on the mean-field approach, we derive the analytical expression of the geometric phase and confirm it by our numerical simulations. It is found that the geometric phase strongly depends both on the atom-number imbalance and the nonlinear particle interaction. We show that the geometric phase can be dramatically enhanced by increasing the imbalance or the repulsive interaction strength. For the symmetric case, a discontinuous behavior for the first-order derivative of the ground-state geometric phase at a critical point is seen which gives a distinct signal of a phase transition from mixed atom-molecule phase to pure molecule phase. The virtual magnetic field interpretation for the geometric phase is also given.

Nonlinear particle interaction effects on Ramsey interferometry in an atom–heteronuclear molecule conversion system

We investigate the Ramsey interferometry with a nonlinear atom–heteronuclear molecule conversion boson system, where the nonlinearity arises from the interaction between coherent atoms. In this coupled system, the nonlinear interaction between particles plays a significant role in a dynamic process. The nice Ramsey-type interference fringes in the time domain have been obtained when a sequence of two identical magnetic field pulses near the Feshbach resonance is applied. We show that a blue shift (or a red shift) of frequency of Ramsey-type fringes emerges due to the repulsive particle interaction (or the attractive particle interaction). In particular, we find that the visibility of Ramsey-type patterns can be strongly modified by the nonlinear particle interaction and a much higher visibility of Ramsey-type fringes can be accessed by tuning the Ramsey pulse duration appropriately.

Control of frictional dynamics of a one-dimensional particle array

Control of frictional forces is required in many applications of tribology. While the problem is approached by chemical means traditionally, a recent approach was proposed to control the system mechanically to tune frictional responses. We design feedback control laws for a one-dimensional particle array sliding on a surface subject to friction. The Frenkel–Kontorova model describing the dynamics is a nonlinear interconnected system and the accessible control elements are average quantities only. We prove local stability of equilibrium points of the un-controlled system in the presence of linear and nonlinear particle interactions, respectively. We then formulate a tracking control problem, whose control objective is for the average system to reach a designated targeted velocity using accessible elements. Sufficient stabilization conditions are explicitly derived for the closed-loop error systems using the Lyapunov theory based methods. Simulation results show satisfactory performances. The results can be applied to other physical systems whose dynamics is described by the Frenkel–Kontorova model.

On the integration of equations of motion for particle-in-cell codes

An area-preserving implementation of the 2nd order Runge–Kutta integration method for equations of motion is presented. For forces independent of velocity the scheme possesses the same numerical simplicity and stability as the leapfrog method, and is not implicit for forces which do depend on velocity. It can be therefore easily applied where the leapfrog method in general cannot. We discuss the stability of the new scheme and test its performance in calculations of particle motion in three cases of interest. First, in the ubiquitous and numerically demanding example of nonlinear interaction of particles with a propagating plane wave, second, in the case of particle motion in a static magnetic field and, third, in a nonlinear dissipative case leading to a limit cycle. We compare computed orbits with exact orbits and with results from the leapfrog and other low-order integration schemes. Of special interest is the role of intrinsic stochasticity introduced by time differencing, which can destroy orbits of an otherwise exactly integrable system and therefore constitutes a restriction on the applicability of an integration scheme in such a context [A. Friedman, S.P. Auerbach, J. Comput. Phys. 93 (1991) 171]. In particular, we show that for a plane wave the new scheme proposed herein can be reduced to a symmetric standard map. This leads to the nonlinear stability condition Δ t ω B ⩽ 1, where Δ t is the time step and ω B the particle bounce frequency.

Improving regular acceleration in the nonlinear interaction of particles and waves

In this work one studies the effects arising from the inclusion of a stationary extraordinary mode in the resonant interaction of magnetized particles and perpendicularly propagating electrostatic waves. It is found that for a stationary mode frequency of the order of the Larmor frequency and with a suitably chosen amplitude, one is able to suppress the resonance which drives the weakly relativistic dynamics into chaos. Improved regular acceleration of initially low energy particles is thus attained. Analytical estimates of the optimal stationary mode amplitude is presented. A detailed study of the topological effects due to resonance suppression based on bifurcation analysis is performed. The main results are verified with the help of single particle numerical simulations.

Theory of nonlinear interaction of particles and waves in an inverse plasma maser. Part 1. Collision integral

An expression is obtained for the collision integral describing the simultaneous interaction of plasma particles with resonant and non-resonant waves. It is shown that this collision integral is determined by two processes: a ‘direct’ nonlinear interaction of particles and waves, and the influence of the non-stationarity of the system. The expression for the nonlinear collision integral is found to be quite different from the expression for a quasi-linear collision integral; in particular, the nonlinear integral contains higher-order derivatives of the distribution function with respect to momentum than the quasi-linear one.

Theory of nonlinear interaction of particles and waves in an inverse plasma maser. Part 2. Stationary solution and evolution of initial distributions

The evolution of the distribution function due to the simultaneous nonlinear interaction of plasma particles with resonant and non-resonant waves is studied. A stationary particle distribution resulting from a balance of the quasi-linear interaction and the nonlinear one is found. The temporal evolution of an initial δ-function-shaped distribution (like a ‘beam’) is examined in the one-dimensional case. General formulae are obtained for stochastic particle acceleration (taking account of the nonlinear interaction studied here).

Binary correlation and phase space granulation in a turbulent plasma

The time evolution of the average plasma distribution function is essentially determined by the coUective binary correlation ( 91 (x 1, v v t) 3f (x2, v2, t) ), where 91 is the fluctuating part of the distribution function. In the quasilinear theory, in the presence of a broad spectrum, the free stream part of the binary correlation is shown to phase mix to zero within a short interval of time (Davidson 1972) while in the resonance velocity domain the coherent part shows a strong positive correlation ,-,,O(~-2k), where ~k is the growth rate, but is devoid of any fine grained spatial structure. On the other hand a sufficiently narrow spectrum of a BGK wave shows a strong correlation as well as phase space structure. Dupree (1972) has pointed out an interesting departure from the above results when nonlinear interaction of particles with a wide spectrum of waves is taken into consideration. For a wide spectrum turbulence, the concept of trapping of a particle by the entire wave field is indeed vague. The particle 1 (x 1, v 0 is trapped only by the portion (k + 3k, ~o~+,k) ir'Ir 11 = oJJk and [ 91 I < I kl; with respect to the remaining portion of the spectrum it is untrapped. So the renormalized quasilinear (RQL) approximation (Misguich and Balescu 1975a) is not expected to yield an appreciably different result from that of the quasilinear theory so lar as the normal modes are eoneerned. But the situation changes when we attempt to calculate ( 91 t) 91 t)). The particle 2(x2, v2) is trapped by the free streaming mode (k, k'v 0 whenever v 2 .-. v 1 for all k belonging to the spectrum. The contribution to the correlation arising from the resonant interaction with the free streaming mode may therefore become important and on applying the RQL approximation theory, we find that i t i s indeed so. Two particle streams, each resonant with a wave of the same speed, modulate each other and give rise to phase space structure, although on a scale much smaller than those generated e.g., by trapped particles in a BGK wave. The purpose of the present paper is to obtain an expression for the collective binary correlation and determine the so caUed ' c lump' part in the RQL approximation theory. The generalized Dupree damping, in which is incorporated the effect of the turbulence mediated change in the slope of the distribution function, is found to

Stabilization of Flute Instability by High Frequency Field

The effects of external inhomogeneous high-frequency field on the flute instability is studied theoretically from a point of view of non-linear particle interaction instead of an approach using a time averaged quasi-static potential or the Mathieu stability diagram. In this paper, it is assumed that the electric field component of the high-frequency field is perpendicular to the steady state confining magnetic field. It turns out that the h f field can compensate the perturbation of the flute instability with the aid of the ion drift induced by the field. The results appear to show that this field has the stabilizing effect in case that ω 0 >ω c i and has the exciting effect in case that ω 0 <ω c i , provided that the h f intensity increases with distance outwardly from the plasma, where ω 0 is the h f field frequency and ω c i is the ion cyclotron frequency.

STABILIZATION OF FLUTE INSTABILITY BY MEANS OF HIGH FREQUENCY FIELD

Flute instability is studied theoretically from the point of view of non-linear particle interaction under inhomogeneous high frequency field whose electric field component is parallel to confining magnetic. field. At first, kinetic equations with nonlinear term are solved by neglecting the non-linear term, i.e. by linear approximation. Using the linear solution, a first order non-linear solution is obtained and similarly a second order non-linear equation is solved. These solutions of the kinetic equations are utilized for induction of the dispersion relation of Flute instability. The interaction of the high frequency field with the instability are supposed to scour as a result of the additional induced particle motions paraller to the wave number Vector of the instability. Effects of the field on the instability are considered from the two points of view; one is attributed to the electron drifts and the other is owing to the periodic displacement of ions or electrons oscillating with the various inherent frequencies. The instability can be stabilized by the electron drifts whose velocities are larger than those of the ions drifted by the driving force of the instability and moreover may be easily stabilized by the additional periodic displacements of ions or electrons, superimposed on the above electron drifts. These displacement are owing to the high frequency field whose frequency is more resonant to half the ion or electron cyclotron frequency.