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.     

脉冲激励下超音速混合层涡结构的演化机理

采用大涡模拟方法对脉冲激励作用下的超音速混合层流场进行数值模拟,所得结果清晰展示了流场中涡结构的独特生长机理.基于涡核位置提取方法,对超音速混合层流场中涡结构的空间尺寸和瞬时对流速度等动态特性进行了定量计算.通过分析流场中涡结构的动态特性在不同频率脉冲激励下的变化,揭示出受脉冲激励超音速混合层流场中涡结构的演化机理:涡结构的生长不再是依靠相邻涡-涡结构之间的配对与融合,而是通过涡核外围的一串小涡旋结构被依次吸进涡核来实现,且受激励流场中各个涡结构的空间尺寸变化较小;流场中的涡结构数量与脉冲频率成正比例关系,而涡结构的空间尺寸与脉冲频率成反比例关系;涡结构的平均对流速度随脉冲频率的增大而减小.针对受脉冲激励超音速混合层,给出了能够表征涡结构特性与脉冲激励参数之间关系的方程式,即受激励流场中涡结构的平均对流速度与脉冲周期的乘积近似等于流场中涡结构的空间尺寸(涡结构平均直径).     

Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering

We report a method for generation of arbitrary shape and array of optical vortices by use of a superposition of coherent elementary vortices based on helical phase spatial filtering in spatial frequency domain. In this method, a helical phase spatial filter (HPSF) is placed in the spatial frequency plane of a 4-f imaging processing system. We demonstrated that the output field distribution represents the convolution between the input field and an elementary vortex field introduced by the HPSF, which results in a special shape or array of optical vortices determined by the “degenerate” properties of coherent elementary vortices and the distribution formats of the input field.     

Two-dimensional Brownian vortices

We introduce a stochastic model of 2D Brownian vortices associated with the canonical ensemble. The point vortices evolve through their usual mutual advection but they experience in addition a random velocity and a systematic drift generated by the system as a whole. The statistical equilibrium state of this stochastic model is the Gibbs canonical distribution. We consider a single species system and a system made of two types of vortices with positive and negative circulations. At positive temperatures, like-sign vortices repel each other (“plasma” case) and at negative temperatures, like-sign vortices attract each other (“gravity” case). We derive the stochastic equation satisfied by the exact vorticity field and the Fokker-Planck equation satisfied by the N-body distribution function. We present the BBGKY-like hierarchy of equations satisfied by the reduced distribution functions and close the hierarchy by considering an expansion of the solutions in powers of 1/N, where N is the number of vortices, in a proper thermodynamic limit. For spatially inhomogeneous systems, we derive the kinetic equations satisfied by the smooth vorticity field in a mean field approximation valid for N→+∞. For spatially homogeneous systems, we study the two-body correlation function, in a Debye-Hückel approximation valid at the order O(1/N). The results of this paper can also apply to other systems of random walkers with long-range interactions such as self-gravitating Brownian particles and bacterial populations experiencing chemotaxis. Furthermore, for positive temperatures, our study provides a kinetic derivation, from microscopic stochastic processes, of the Debye-Hückel model of electrolytes.     

Passive tracer dynamics in 4 point-vortex flow

The advection of passive tracers in a system of 4 identical point vortices is studied when the motion of the vortices is chaotic. The phenomenon of vortex-pairing has been observed and statistics of the pairing time is computed. The distribution exhibits a power-law tail with exponent ∼ 3.6 implying finite average pairing time. This exponents is in agreement with its computed analytical estimate of 3.5. Tracer motion is studied for a chosen initial condition of the vortex system. Accessible phase space is investigated. The size of the cores around the vortices is well approximated by the minimum inter-vortex distance and stickiness to these cores is observed. We investigate the origin of stickiness which we link to the phenomenon of vortex pairing and jumps of tracers between cores. Motion within the core is considered and fluctuations are shown to scale with tracer-vortex distance r as r ⁶. No outward or inward diffusion of tracers are observed. This investigation allows the separation of the accessible phase space in four distinct regions, each with its own specific properties: the region within the cores, the reunion of the periphery of all cores, the region where vortex motion is restricted and finally the far-field region. We speculate that the stickiness to the cores induced by vortex-pairings influences the long-time behavior of tracers and their anomalous diffusion. Received 28 September 2000 and Received in final form 9 February 2001     

Flux relaxation phenomena in the presence of weak AC magnetic fields

Relaxation of flux profile and magnetisation due to non-linear vortex diffusion in a superconducting slab settled in a parallel-to-the-surface DC and superimposed weak AC magnetic fields is studied for several kinds of the non-linear vortex diffusivity, corresponding to different possible shapes of the current–voltage characteristics of the superconductor. The evolution of the dynamic vortex response on applied weak AC field due to flux relaxation process is studied and relaxation characteristics of the AC magnetic susceptibility are calculated. The flux creep rate and magnetisation decay are shown to be enhanced significantly in the case of strongly non-linear regime of vortex diffusion if even rather weak AC magnetic field is applied. The possibility of ‘dynamical melting' of the vortex lattice occurring at rather high levels of the induced current density (j>j_c) is also demonstrated.     

Simultaneous measurement of D and T2 using the distant dipolar field

The presence of long-range dipolar fields in liquids is known to introduce a non-linear term in the Bloch-Torrey equations which is responsible for many interesting effects in nuclear magnetic resonance as well as in magnetic resonance imaging. We show here, for the first time, that the diffusion coefficient D and the spin-spin relaxation time T2 can be obtained simultaneously from the time evolution profile of the long-range dipolar field refocused signal. In a COSY Revamped by Z-asymmetric Echo Detection sequence, the analytical first-order approximation solution of the Bloch-Torrey equations modified to include the effect of the distant dipolar field is used to demonstrate the technique in an experiment using doped water.     

Singular Limit of a Reaction-Diffusion Equation with a Spatially Inhomogeneous Reaction Term

We study reaction-diffusion equations with a spatially inhomogeneous reaction term. If the coefficient of these reaction term is much larger than the diffusion coefficient, a sharp interface appears between two different phases. We show that the equation of motion of such an interface involves a drift term despite the absence of drift in the original diffusion equations. In particular, we show that the same rich spatial patterns observed for a chemotaxis-growth model can be realized by a system without a drift term.     

Approach to a Stationary State in an External Field

We study relaxation towards a stationary out-of-equilibrium state by analyzing a one-dimensional stochastic process followed by a particle accelerated by an external field and propagating through a thermal bath. The effect of collisions is described within one-dimensional formulation of Boltzmann’s kinetic theory. We present analytical solutions for the Maxwell gas and for the very hard particle model. The exponentially fast relaxation of the velocity distribution towards the stationary form is demonstrated. In the reference frame moving with constant drift velocity the hydrodynamic diffusive mode is shown to govern the distribution in the position space. We show that the exact value of the diffusion coefficient for any value of the field is correctly predicted by the Green-Kubo autocorrelation formula generalized to the stationary state.     

Astigmatic telescopic transformation of a high-order optical vortex

Properties of an optical vortex light beam formed after the astigmatic telescopic transformation of a circular Laguerre-Gaussian mode are considered both theoretically and experimentally. The beam evolution is found to be in conformity with the general notions on the high-order optical vortex symmetry breakdown. Upon propagation, the asymmetric beam shows a sort of rotation of its transverse profile in accord with the energy circulation in the original circular mode; this process is described on the base of the beam intensity moments and the vortex and asymmetry components of its orbital angular momentum. An l-charged optical vortex converts into |l| secondary first-order vortices positioned on a straight line crossing the beam axis. Orientation of this straight line in the beam cross section and spatial separation of the secondary vortex cores depend on the propagation distance. Morphology (orientation and anisotropy) of all the secondary vortices is the same and depends on the propagation distance; the anisotropy can be characterized by the vortex component of the beam angular momentum. At certain distance, relative separation of secondary vortices with respect to the beam transverse size reaches its maximum that corresponds to the minimum anisotropy of the vortices. The results can be useful in the context of current research of the optical vortex arrays.     

Topological Vortices in Superfluid Films

We study the topological structure of the vortex system in a superfluid film. Explicit expressions for the vortex density and velocity field as functions of the superfluid order parameter are derived. The evolution of vortices is also studied from the topological properties of the superfluid order parameter field.     

Topological Vortices in Superfluid Films

We study the topological structure of the vortex system in a superfiuid film. Explicit expressions for the vortex density and velocity field as functions of the superfluid order parameter are derived. The evolution of vortices is also studied from the topological properties of the superfluid order parameter field.     

Large-eddy simulation of impinging jets with embedded azimuthal vortices

We have carried out large-eddy simulations of an impinging jet with embedded azimuthal vortices, a model of the wake of a helicopter hovering in ground effect. The azimuthal vortices are generated by sinusoidal forcing of the velocity at the jet exit. They strengthen while they are advected towards the ground; when they are close to the solid surface, a layer of opposite-sign vorticity is formed at the wall, and lifted up to form a secondary vortex that interacts with the primary one. Regions of reversed flow are caused by the strong, localised, adverse pressure gradient. After this interaction, the primary vortices begin to decay, mostly due to the Reynolds shear stresses, which contribute to the turbulent diffusion of vorticity term in the budget of the phase-averaged azimuthal vorticity. This mechanism is extremely robust, and plays the most important role in the vortex decay even if no turbulence is initially present in the jet, or if the no-slip condition is removed. A three-dimensional instability also plays a role: removing it leads to slower decay. Our results also point out some challenges for turbulence models for the unsteady Reynolds-averaged Navier–Stokes equations.     

Formation and relaxation of two-dimensional vortex crystals in a magnetized pure-electron plasma

Systematic examinations are carried out experimentally about the contribution of background vorticity distributions (BGVD's) to the spontaneous formation and decay of ordered arrays (vortex crystals) composed of strong vortices (clumps) by using a pure-electron plasma. It is found that the BGVD level needs to be higher for an increasing number of clumps to form vortex crystals and that the number of the clumps constituting the crystal decreases in time as proportional to gamma lnt in contrast to proportional to t (-xi) with xi approximately 1 as accepted well in turbulence models. The decay rate gamma increases with the BGVD level. The observed configurations of the clumps cover the theoretically predicted catalogue of vortex arrays in superfluid helium, suggesting a possible relaxation path of the crystal states.     

Chirality dependent component of vortex advection in excitable media

An advective field induces drift of a vortex in excitable media. The component of the drift velocity C( perpendicular ) perpendicular to the field is known to change its sign with the chirality of the vortex. In an experiment with vortices in an electric field in a chemical excitable medium, we have found unexpectedly that C( perpendicular ) changes its sign also independently of chirality with changing composition of the medium. We did not succeed to explain this phenomenon by using existing mathematical models of chemical excitable media. The experiment described calls for more realistic models.(c) 1999 American Institute of Physics.     

Topological Vortices in the Modified Two-Dimensional Gross—Pitaevskii Theory

It is shown that there exists an elementary vortex topological current in the modified two-dimensional Gross—Pitaevskii theory. An explicit expression for the vortex velocity field as a function of the order parameter is derived. The evolution of vortices is studied from the topological properties of the order parameter field. It is found that vortices generate or annihilate at the limit points and encounter, split, or merge at the bifurcation points of the order parameter field .     

Stationary solutions of Liouville equations for non-Hamiltonian systems

We consider the class of non-Hamiltonian and dissipative statistical systems with distributions that are determined by the Hamiltonian. The distributions are derived analytically as stationary solutions of the Liouville equation for non-Hamiltonian systems. The class of non-Hamiltonian systems can be described by a non-holonomic (non-integrable) constraint: the velocity of the elementary phase volume change is directly proportional to the power of non-potential forces. The coefficient of this proportionality is determined by Hamiltonian. The constant temperature systems, canonical-dissipative systems, and Fermi-Bose classical systems are the special cases of this class of non-Hamiltonian systems.     

Quantized vortices in the ideal bose gas: a physical realization of random polynomials

We propose a physical system allowing one to experimentally observe the distribution of the complex zeros of a random polynomial. We consider a degenerate, rotating, quasi-ideal atomic Bose gas prepared in the lowest Landau level. Thermal fluctuations provide the randomness of the bosonic field and of the locations of the vortex cores. These vortices can be mapped to zeros of random polynomials, and observed in the density profile of the gas.     

Critical dynamics of gauge systems: spontaneous vortex formation in 2D superconductors

We examine the formation of vortices during the nonequilibrium relaxation of a high-temperature initial state of an Abelian-Higgs system. We equilibrate the scalar and gauge fields using gauge-invariant Langevin equations and relax the system by instantaneously removing thermal fluctuations. For couplings near critical, kappa(c) = square root[lambda]/e = 1, we observe the formation of large clusters of like-sign magnetic vortices. Their appearance has implications for the dynamics of the phase transition, for the distribution of topological defects, and for late-time phase ordering kinetics. We offer explanations for both the observed vortex densities and vortex configurations.     

Lateral phase drift of the topological charge density in stochastic optical fields

The statistical distributions of optical vortices or topological charge in stochastic optical fields can be inhomogeneous in both transverse directions. Such two-dimensional inhomogeneous vortex or topological charge distributions evolve in a complex way during free-space propagation. While the evolution of one-dimensional topological charge densities can be described by a linear diffusion process, the evolution of two-dimensional topological charge densities exhibits some additional nonlinear dynamics. Here we propose a phase drift mechanism as a partial explanation for this additional nonlinear dynamics. Numerical results are presented in support of this proposal.