Fokker-Planck equation for interacting waves: Dispersion law renormalization and wave turbulence

The multiple timescale method for removing secularities is used to generate the Fokker-Planck (“FP”) equation for a system of interacting waves. This FP equation describes diffusion in the phase space of the angle, as well as the action, variables of all underlying modes. The first moment of the FP equation gives a kinetic (or Boltzmann-type) equation governing the averaged actions, and describing the diffusion of action in time. Angle diffusion leads to a renormalization of the dispersion law. Stationary solutions for the average action (or so-called spectral intensity) are derived for equilibrium and for the driven off-equilibrium state corresponding to a cascade of wave energy from low to high frequencies (wave turbulence). The reduced distribution function for these states is derived.The derivation of the FP equation from the Liouville equation, as well as the derivation of the kinetic equation from the FP equation, requires that the distribution of modes be sufficiently dense. In this limit, cumulants that are initially zero increase at a rate that is thermodynamically sm all. A Langevin equation, governing the evolution of a distinguished oscillator, that is applicable even in off-equilibrium conditions, is derived. The concept of winding numbers is extended to the general phase space motion of action-angle variables through the introduction of a multiple-valued probability density.     

分子动力学模拟Gd原子在Cu(110)表面的扩散过程

为了分析Gd吸附原子在Cu(110)表面的扩散机理，用分子动力学对该扩散过程进行模拟.模拟 结果表明在［1 1 0］方向Gd原子通过跳跃机理扩散，而且多步跳跃频率很高.而在［0 0 1］方向则通过交换机理扩散.吸附原子在［1 1 0］方向的扩散能力要比［0 0 1］ 方向强.通过对扩散频率的拟合，发现两种扩散机理都符合Arrhenius公式，从而确定了跳跃 机理的扩散势垒为0.097eV，交换机理的扩散势垒为0.33eV.另外还用能量弛豫的方法确定了 跳跃机理的扩散势垒. 关键词： 分子动力学 表面扩散 跳跃机理 交换机理 扩散势垒     

Dynamics of real financial markets: A reply to Frank’s comment

This reply addresses the assertion in the comment of T.D. Frank [T.D. Frank, Physica A 387 (2008) 773] on our paper [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343] that the approach to modeling financial markets that we propose is unrealistic. In our paper, we considered variable diffusion processes that have a diffusion coefficient that varies with both position (return in finance) and time, and used them to show that measuring a Hurst exponent H≠1/2 in a time series does not necessarily imply correlations between increments. We also proposed that such a variable diffusion process is the underlying stochastic process governing the dynamics of financial markets. Frank asserts that this is unrealistic because variable diffusion processes with H≠1/2 are driven with a “force” that varies in time as a power law. He claims, instead, that markets obey nonextensive thermostatistics. We discuss evidence from a recently published empirical study of the Euro-Dollar exchange rate [K.E. Bassler, J.L. McCauley, G.H. Gunaratne, PNAS 104 (2007) 17287] that shows that the market can be described with a variable diffusion process, but is inconsistent with nonextensive thermostatistics. This evidence demonstrates that our modeling approach is realistic and accurate.     

Nonlinear diffusion of indirect excitons in an ideal bilayer with an in-plane harmonic trap

The nonlinear diffusion of the spatially indirect excitons in an ideal bilayer with an in-plane harmonic trap is investigated based on the theories developed by Ivanov [A.L. Ivanov, Europhys. Lett. 59 (2002) 586; A.L. Ivanov, J. Phys.: Condens. Matter 16 (2004) S3629] and Rapaport et al. [R. Rapaport, G. Chen, S. Simon, O. Mitrofanov, L. Pfeiffer, P.M. Platzman, Phys. Rev. B 72 (2005) 075428]. A nonlinear equation for the diffusion of the indirect excitons in this structure is established. The two-dimensional density of the indirect excitons in this structure is calculated. The calculations show that the density adjacent to the trap center for different exciton temperatures can remain very high even long after the photo-excitation because of the confinement of the in-plane harmonic trap, and that the indirect excitons gather several tens of μm away from the trap center. The calculations are in good agreement qualitatively with the experimental results of Voros et al. [Z. Voros, D.W. Snoke, L. Pfeiffer, K. West, Phys. Rev. Lett. 97 (2006) 016803] and prove that an in-plane harmonic trap can indeed keep an exciton gas dense near its center.     

Fick's law and Fokker-Planck equation in inhomogeneous environments

In inhomogeneous environments, the correct expression of the diffusive flux is not always given by the Fick's law Γ=−D∇n. The most general hydrodynamic equation modelling diffusion is indeed the Fokker-Planck equation (FPE). The microscopic dynamics of each specific system may affect the form of the FPE, either establishing connections between the diffusion and the convection term, as well as providing supplementary terms. In particular, the Fick's form for the diffusion equation may arise only in consequence of a specific kind of microscopic dynamics. It is also shown how, in the presence of sharp inhomogeneities, even the hydrodynamic FPE limit may becomes inaccurate and mask some features of the true solution, as computed from the Master equation.     

A class of asymptotic-preserving schemes for the Fokker–Planck–Landau equation

We present a class of asymptotic-preserving (AP) schemes for the nonhomogeneous Fokker–Planck–Landau (nFPL) equation. Filbet and Jin [16] designed a class of AP schemes for the classical Boltzmann equation, by penalization with the BGK operator, so they become efficient in the fluid dynamic regime. We generalize their idea to the nFPL equation, with a different penalization operator, the Fokker–Planck operator that can be inverted by the conjugate-gradient method. We compare the effects of different penalization operators, and conclude that the Fokker–Planck (FP) operator is a good choice. Such schemes overcome the stiffness of the collision operator in the fluid regime, and can capture the fluid dynamic limit without numerically resolving the small Knudsen number. Numerical experiments demonstrate that the schemes possess the AP property for general initial data, with numerical accuracy uniformly in the Knudsen number.     

Fabrication of a complex InAs ring-and-dot structure by droplet epitaxy

An InAs ring structure accompanying the formation of quantum dots (QDs) was fabricated on (1 0 0)GaAs using droplet epitaxy. The QDs were located in the vicinity of the ring, due to the diffusion of In atoms from the In droplets. In addition, the dots were found to have distributed elliptically and preferentially along the [0 1 1] direction, implying that In itself prefers to diffuse along the [0 1 1] direction, which is the opposite of the favorable diffusion orientation of group III atoms on (1 0 0)GaAs under a commonly used As-stabilized growth condition. This is the first observation of a ring structure accompanying the formation of quantum dots in droplet epitaxy.     

Electronic shot noise in fractal conductors

By solving a master equation in the Sierpiński lattice and in a planar random-resistor network, we determine the scaling with size L of the shot noise power P due to elastic scattering in a fractal conductor. We find a power-law scaling P proportional, variantL;{d_{f}-2-alpha}, with an exponent depending on the fractal dimension d_{f} and the anomalous diffusion exponent alpha. This is the same scaling as the time-averaged current I[over ], which implies that the Fano factor F=P/2eI[over ] is scale-independent. We obtain a value of F=1/3 for anomalous diffusion that is the same as for normal diffusion, even if there is no smallest length scale below which the normal diffusion equation holds. The fact that F remains fixed at 1/3 as one crosses the percolation threshold in a random-resistor network may explain recent measurements of a doping-independent Fano factor in a graphene flake.     

A decomposed equation for local entropy and entropy production in volume-preserving coarse-grained systems

In this study an equation for the local entropy is derived based on the formulation of a master equation and is applied to volume-preserving maps. The equation consists of the following terms: unsteady, convection, diffusion, probability-weighted phase space volume expansion rate, nonnegative entropy production, and residuals. The decomposition makes it possible to evaluate entropy production in terms of microscopic dynamics and is expected to be applicable to many coarse-grained systems on the phase space. When it is applied to two volume-preserving multibaker chain systems it is confirmed that the summation of the nonnegative entropy production on each site numerically coincides with the entropy production introduced by Gilbert et al. [T. Gilbert, J.R. Dorfman, P. Gaspard, Entropy production, fractals, and relaxation to equilibrium, Phys. Rev. Lett. 85 (2000) 1606-1609] and the phenomenological expression both in nonequilibrium steady and unsteady states. The coincidence is brought about by the fact that the residual terms vanish in the thermodynamic limit when they are integrated on each site. It follows that the entropy production is dominated by the nonnegative entropy production term and becomes positive in nonequilibrium states.     

Inelastic cross-sections and natural lifetimes for the 62D3/2, 5/2 and 82S1/2 states of Rb

The results of an experimental study of population dynamics following excitation of [0pt] and [0pt] states of rubidium are reported. Excitation transfer and quenching cross-sections in collisions with ground state rubidium atoms, and natural lifetimes have been measured. The experiment was performed in a vapour cell, using pulsed two-photon excitation and photon counting detection. The analysis of time dependent signals was based on a rate equation model in which transitions induced by thermal radiation have been accounted for. The measurements yielded following results: (1) [0pt] state J-mixing cross-section: [0pt] ; (2) cross-sections for [0pt] excitation transfer process: [0pt] ; (3) quenching cross-sections: [0pt] , [0pt] , [0pt] ; [0pt](4) radiative lifetimes: [0pt] ns, [0pt] ns, [0pt] ns. Received 1st December 1998 and Received in final form 17 May 1999     

Effect of Solvent Polarity on Fluorescence Quenching of New Indole Derivatives by CCl4

ABSTRACT

The fluorescence quenching of solutes 3-[5′-methyl-3′-phenylindol-2′-yl]-s-triazolo [3,4-b] [1,3,4] thiadiazol-6(5H)-thione (MPITTT) and 3-phenyl-2,5-bis-[thiosemicarbazido] indole (PbisTI) by carbon tetrachloride (CCl₄) in dioxane and acetonitrile mixtures has been studied at room temperature by steady-state fluorescence measurements. The positive deviation from linearity has been observed in the Stern–Volmer (S-V) plots for both fluorophores in different composition of mixed solvents even at moderate CCl₄ concentration (0.10 mol dm^?3). Various quenching parameters of the quenching processes have been determined using the extended S-V equation and have been found to be dependent on the solvent polarity. Further, with the use of the finite sink approximation model, it is concluded that the bimolecular quenching reactions are diffusion limited, and the distance parameter R′ and mutual diffusion coefficient D are estimated independently.     

Effects of Buoyancy on Langmuir Circulation

Based on the Navier-Stokes equation, an equation describing the Langmuir circulation is derived by a perturbation method when the influences of Coriolis force and buoyancy force are both considered. The approach used in the analysis is similar to the works carried out by Craik and Leibovieh [3. Fluid Mech. 73 （1976）401], Leibovieh [3. Fluid Mech. 79 （1977） 715] and Huang [3. Fluid Mech. 91 （1979） 191]. Potential applications of the equation proposed are discussed in the area of Antarctic circumpolar current.     

Link between the laws of geometrical optics and the radiative transfer equation in media with a spatially varying refractive index

We proposed in a previous paper [J.-M. Tualle, E. Tinet, Opt. Commun. 228 (2003) 33] a modified radiative transfer equation to describe radiative transfer in a medium with a spatially varying refractive index. The present paper is devoted to the demonstration that this equation perfectly works in the non-absorbing/non-scattering limit, what was contested by Martí-López and coworkers [L. Martí-López, J. Bouza-Domínguez, R.A. Martínez-Celorio, J.C. Hebden, Opt. Commun. 266 (2006) 44]. The assertion that this equation would imply a zero divergence of the rays is also commented.     

Local equilibrium in the inertial layer of wall bounded turbulence

Recently Brouwers [Dissipation equals production in the log layer of wall-induced turbulence. Phys Fluids. 2007;19:101702] carried out an asymptotic analysis using the RANS based turbulence energy transport equation and showed that the energy dissipation equals its production in the inertial layer of wall-induced turbulence. Assuming log-law profile to the mean velocity, pressure, viscous and energy diffusion terms were estimated and shown to be negligibly small compared to the production and dissipation terms thereby proving local equilibrium. However, based on scale relations Tennekes and Lumley [A first course in turbulence. Cambridge (MA): MIT Press; 1994] have already established that the pressure and energy diffusion terms appearing in the energy transport equation are of the same order of magnitude especially in the inertial layer thus leading to a contradiction. Hence we have attempted here to re-estimate the turbulence energy budgets in a different way by invoking the Kolomogrov’s similarity hypotheses and (4/5)th law. Magnitudes of pressure and energy diffusion terms are determined explicitly and found to match well with the DNS data. The striking point of the present analysis is that no prior assumption is enforced on the mean velocity profile. Further, two main advantages of the present study are, (i) reasonable estimates for both the diffusion terms are obtained explicitly that were unavailable before and (ii) these estimates help us to tweak the production/dissipation terms to reflect the influence of turbulent diffusion mechanisms without the necessity to model them as in the case of elliptic relaxation and Reynold stress RANS models.     

Soliton-like solutions of certain types of nonlinear diffusion-reaction equations with variable coefficient

With a view to exploring new soliton-like solutions of certain types of nonlinear diffusion-reaction (DR) equations with a variable coefficient, we demonstrate the viability of a method which is the combination of both the symbolic computation technique of Gao and Tian [Y.T. Gao, B. Tian, Comput. Phys. Commun. 133 (2001) 158] and auxiliary equation method of Sirendaoreji [Sirendaoreji, Phys. Lett. A 356 (2006) 124] and used recently for the KdV equation. In particular, the DR equations with quadratic and cubic nonlinearities with a time-dependent velocity in the convective flux term are studied and the existence of soliton-like solutions is shown.     

Brownian motion on a manifold

The question of the existence and correct form of equations describing Brownian motion on a manifold cannot be answered by mathematics alone, but requires a study of the underlying physics. As in classical mechanics, manifolds enter through the transformation of variables needed to account for the presence of constraints. The constraints are either due to a physical agency that forces the motion to remain on a manifold, or they represent conserved quantities of the equation of motion themselves. Also the Brownian motion is described either by a Smoluchowski diffusion equation or by a Kramers equation. The four cases lead to the following conclusions, (i) Smoluchowski diffusion with a conserved quantity reduces to a diffusion equation on the manifold; (ii) The same is true for diffusion with a physical constraint in three dimensions, but in more dimensions it may happen thatno autonomous equation on the manifold results; (iii) A Kramers equation with a conserved quantity reduces to an equation on the manifold, but in general not of the form of a Kramers equation; (iv) The Kramers equation with a physical constraint reduces to an autonomous Kramers equation on the manifold only for a special shape of that constraint. Throughout, only a certain type of physical constraints has been envisaged, and global questions are ignored. Finally, the customary heuristic construction of a Fokker-Planck equation for a mechanical system on a manifold is demonstrated for the case of Brownian rotation of a rigid body, and its shortcomings are emphasized.     

Results for a fractional diffusion equation with a nonlocal term in spherical symmetry

We obtain time dependent solutions for a fractional diffusion equation containing a nonlocal term by considering the spherical symmetry and using the Green function approach. The nonlocal term incorporated in the diffusion equation may also be related to the spatial and time fractional derivative and introduces different regimes of spreading of the solution with the time evolution. In addition, a rich class of anomalous diffusion processes may be described from the results obtained here.     

Methodology for the solutions of some reduced Fokker‐Planck equations in high dimensions

In this paper, a new methodology is formulated for solving the reduced Fokker‐Planck (FP) equations in high dimensions based on the idea that the state space of large‐scale nonlinear stochastic dynamic system is split into two subspaces. The FP equation relevant to the nonlinear stochastic dynamic system is then integrated over one of the subspaces. The FP equation for the joint probability density function of the state variables in another subspace is formulated with some techniques. Therefore, the FP equation in high‐dimensional state space is reduced to some FP equations in low‐dimensional state spaces, which are solvable with exponential polynomial closure method. Numerical results are presented and compared with the results from Monte Carlo simulation and those from equivalent linearization to show the effectiveness of the presented solution procedure. It attempts to provide an analytical tool for the probabilistic solutions of the nonlinear stochastic dynamics systems arising from statistical mechanics and other areas of science and engineering.     

Anharmonic oscillators energies via artificial perturbation method

A new pseudoperturbative (artificial in nature) methodical proposal [#!ref15!#] is used to solve for Schr?dinger equation with a class of phenomenologically useful and methodically challenging anharmonic oscillator potentials . The effect of the [#!ref4!#,#!ref5!#] Padé approximant on the leading eigenenergy term is studied. Comparison with results from numerical (exact) and several eligible (approximation) methods is made. Received 2 July 1999 and Received in final form 18 November 1999