Classical Fokker-Planck Research Articles

Squeezed states generation in an array of Linear and Nonlinear Waveguides

We demonstrate the generation of squeezed states of light due to the second harmonic generation and Kerr effect in an array of nonlinear waveguides mediated through a linear one. We characterized the electromagnetic field by a quantum mechanical Hamiltonian and the density operator time evolution is obtained from the Von-Neumann equation of motion. Using the quasiprobability positive P of phase space representation, the classical Fokker-Planck equation is obtained from the master equation and translated to its classical matching set of nonlinear differential equations. We showed that because of the new possibilities of correlation between the linear and nonlinear channel waveguides, highly nonclassical light may be produced.

A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium

In this paper we are interested in the large time behavior of linear kinetic equations with heavy-tailed local equilibria. Our main contribution concerns the kinetic Levy-Fokker-Planck equation, for which we adapt hypocoercivity techniques in order to show that solutions converge exponentially fast to the global equilibrium. Compared to the classical kinetic Fokker-Planck equation, the issues here concern the lack of symmetry of the non-local Levy-Fokker-Planck operator and the understanding of its regularization properties. As a complementary related result, we also treat the case of the heavy-tailed BGK equation.

Effects of tunneling and multiphoton transitions on squeezed-state generation in bistable driven systems

Bistability of nonlinear resonantly driven oscillator in the presence of external noise is analyzed by means of classical Fokker-Planck equation in quasienergy space with account for tunneling effects and by quantum master equation in quasienergy states representation. Two time scales responsible for different stages of bistable system relaxation have been obtained. We found out that the slow relaxation rate caused by fluctuation--induced transitions between different stable states can be enhanced by several orders due to tunneling effects. It was also revealed that tunneling between nearly degenerate quasienergy states and resonant multiphoton transitions between the genuine eigenstates of the nonlinear oscillator are just the similar effects. It was demonstrated that the quasienergy states in the bistability region corresponding to higher amplitude are squeezed. The degree of squeezing is determined by the ratio between nonlinearity and detuning, so the uncertainty of one quadrature can be considerably smaller than the quantum limit. It was found out that tunneling effects can enhance the generation of output oscillator squeezed states. It is shown that 1D Fokker--Planck equation is a quasiclassical limit of a quantum master equation.

Particle Acceleration and Fractional Transport in Turbulent Reconnection

Abstract We consider a large-scale environment of turbulent reconnection that is fragmented into a number of randomly distributed unstable current sheets (UCSs), and we statistically analyze the acceleration of particles within this environment. We address two important cases of acceleration mechanisms when particles interact with the UCS: (a) electric field acceleration and (b) acceleration by reflection at contracting islands. Electrons and ions are accelerated very efficiently, attaining an energy distribution of power-law shape with an index 1–2, depending on the acceleration mechanism. The transport coefficients in energy space are estimated from test-particle simulation data, and we show that the classical Fokker–Planck (FP) equation fails to reproduce the simulation results when the transport coefficients are inserted into it and it is solved numerically. The cause for this failure is that the particles perform Levy flights in energy space, while the distributions of the energy increments exhibit power-law tails. We then use the fractional transport equation (FTE) derived by Isliker et al., whose parameters and the order of the fractional derivatives are inferred from the simulation data, and solving the FTE numerically, we show that the FTE successfully reproduces the kinetic energy distribution of the test particles. We discuss in detail the analysis of the simulation data and the criteria that allow one to judge the appropriateness of either an FTE or a classical FP equation as a transport model.

Fractional Transport in Strongly Turbulent Plasmas.

We analyze statistically the energization of particles in a large scale environment of strong turbulence that is fragmented into a large number of distributed current filaments. The turbulent environment is generated through strongly perturbed, 3D, resistive magnetohydrodynamics simulations, and it emerges naturally from the nonlinear evolution, without a specific reconnection geometry being set up. Based on test-particle simulations, we estimate the transport coefficients in energy space for use in the classical Fokker-Planck (FP) equation, and we show that the latter fails to reproduce the simulation results. The reason is that transport in energy space is highly anomalous (strange), the particles perform Levy flights, and the energy distributions show extended power-law tails. Newly then, we motivate the use and derive the specific form of a fractional transport equation (FTE), we determine its parameters and the order of the fractional derivatives from the simulation data, and we show that the FTE is able to reproduce the high energy part of the simulation data very well. The procedure for determining the FTE parameters also makes clear that it is the analysis of the simulation data that allows us to make the decision whether a classical FP equation or a FTE is appropriate.

Uniform semigroup spectral analysis of the discrete, fractional and classical Fokker-Planck equations

In this paper, we investigate the spectral analysis (from the point of view of semi-groups) of discrete, fractional and classical Fokker-Planck equations. Discrete and fractional Fokker-Planck equations converge in some sense to the classical one. As a consequence, we first deal with discrete and classical Fokker-Planck equations in a same framework, proving uniform spectral estimates using a perturbation argument and an enlargement argument. Then, we do a similar analysis for fractional and classical Fokker-Planck equations using an argument of enlargement of the space in which the semigroup decays. We also handle another class of discrete Fokker-Planck equations which converge to the fractional Fokker-Planck one, we are also able to treat these equations in a same framework from the spectral analysis viewpoint, still with a semigroup approach and thanks to a perturbative argument combined with an enlargement one. Let us emphasize here that we improve the perturbative argument introduced in [7] and developed in [11], relaxing the hypothesis of the theorem, enlarging thus the class of operators which fulfills the assumptions required to apply it.

Convergence to equilibrium for linear Fokker-Planck equations

In this note, we investigate the spectral analysis and long time asymptotic convergence of semigroups associated to discrete, fractional and classical Fokker-Planck equations in some regime where the corresponding operators are close. We successively deal with the discrete and the classical Fokker-Planck model and the fractional and the classical Fokker-Planck model. In each case, we present results of uniform convergence to equilibrium based on perturbation and/or enlargement arguments and obtained in collaboration with S. Mischler in [7].

Efficient method to generate time evolution of the Wigner function for open quantum systems

The Wigner function is a useful tool for exploring the transition between quantum and classical dynamics, as well as the behavior of quantum chaotic systems. Evolving the Wigner function for open systems has proved challenging however; a variety of methods have been devised but suffer from being cumbersome and resource intensive. Here we present an efficient fast-Fourier method for evolving the Wigner function, that has a complexity of $O(N\log N)$ where $N$ is the size of the array storing the Wigner function. The efficiency, stability, and simplicity of this method allows us to simulate open system dynamics previously thought to be prohibitively expensive. As a demonstration we simulate the dynamics of both one-particle and two-particle systems under various environmental interactions. For a single particle we also compare the resulting evolution with that of the classical Fokker-Planck and Koopman-von Neumann equations, and show that the environmental interactions induce the quantum-to-classical transition as expected. In the case of two interacting particles we show that an environment interacting with one of the particles leads to the loss of coherence of the other.

Mapping between finite temperature classical and zero temperature quantum systems: Quantum critical jamming and quantum dynamical heterogeneities

Many electronic systems (e.g., the cuprate superconductors and heavy fermions) exhibit striking features in their dynamical response over a prominent range of experimental parameters. While there are some empirical suggestions of particular increasing length scales that accompany such transitions in some cases, this identification is not universal and in numerous instances no large correlation length is evident. To better understand, as a matter of principle, such behavior in quantum systems, we extend a known mapping (earlier studied in stochastic or supersymmetric quantum mechanics) between finite temperature classical Fokker-Planck systems and related quantum systems at zero temperature to include general nonequilibrium dynamics. Unlike Feynman mappings or stochastic quantization methods in field theories (as well as more recent holographic type dualities), the classical systems that we consider and their quantum duals reside in the same number of space-time dimensions. The upshot of our very broad and rigorous result is that a Wick rotation exactly relates (i) the dynamics in general finite temperature classical dissipative systems to (ii) zero temperature dynamics in the corresponding dual many-body quantum systems. Using this correspondence, we illustrate that, even in the absence of imposed disorder, many continuum quantum fluid systems (and possible lattice counterparts) may exhibit a zero-point ``quantum dynamical heterogeneity'' wherein the dynamics, at a given instant, is spatially nonuniform. While the static length scales accompanying this phenomenon do not seem to exhibit a clear divergence in standard correlation functions, the length scale of the dynamical heterogeneities can increase dramatically. We further study ``quantum jamming'' and illustrate how a hard-core bosonic system can undergo a zero temperature quantum critical metal-to-insulator-type transition with an extremely large effective dynamical exponent $z>4$ that is consistent with length scales that increase far more slowly than the relaxation time as a putative critical transition is approached. Similar results may hold for spin-liquid-type as well as interacting electronic systems. We suggest ways to analyze experimental data in order to adduce such phenomena. Our approach may be used to analyze other quenched quantum systems.

Nanomechanical resonator coupled linearly via its momentum to a quantum point contact

We use a Born-Markov approximated master equation approach to study the symmetrized-in-frequency current noise spectrum and the oscillator steady state of a nanoelectromechanical system where a nanoscale resonator is coupled linearly via its momentum to a quantum point contact (QPC). Our current noise spectra exhibit clear signatures of the quantum correlations between the QPC current and the back-action force on the oscillator at a value of the relative tunneling phase (\eta = -\pi/2) where such correlations are expected to be maximized. We also show that the steady state of the oscillator obeys a classical Fokker-Planck equation, but can experience thermomechanical noise squeezing in the presence of a momentum-coupled detector bath and a position-coupled environmental bath. Besides, the full master equation clearly shows that half of the detector back-action is correlated with electron tunneling, indicating a departure from the model of the detector as an effective bath and suggesting that a future calculation valid at lower bias voltage, stronger tunneling and/or stronger coupling might reveal interesting quantum effects in the oscillator dynamics.

Master Equation in Phase Space for a Spin in an Arbitrarily Directed Uniform External Field

The time evolution equation for the probability density function of spin orientations in the phase space representation of the polar and azimuthal angles is derived for the nonaxially symmetric problem of a quantum paramagnet subjected to a uniform magnetic field of arbitrary direction. This is accomplished by first rotating the coordinate system into one in which the polar axis is collinear with the field vector, then writing the reduced density matrix equation in the new coordinate system as an explicit inverse Wigner-Stratonovich transformation so that the phase space master equation may be derived just as in the axially symmetric case [Yu.P. Kalmykov et al., J. Stat. Phys. 131:969, 2008]. The properties of this equation, resembling the corresponding Fokker-Planck equation, are investigated. In particular, in the large spin limit, S→∞, the master equation becomes the classical Fokker-Planck equation describing the magnetization dynamics of a classical paramagnet in an arbitrarily directed uniform external field.

Spreading of Collimated Particle Beams Within a Generalized Fokker-Planck Diffusion Description

Recently, an expansion of the Boltzmann scattering operator describing the angular spreading of particle beams was given that included the effects of large angle scattering processes, thus generalizing the classical Fokker-Planck equation, valid in the limit of small angle scattering. The present work aims at making an analytical comparison between predictions based on the classical Fokker-Planck equation and those based on a generalized one, which includes a first-order correction term in the expansion of the Boltzmann scattering operator. The analysis is carried out for thin slabs where backscattering effects can be neglected and makes use of a moment approach, which leads to an infinite system of recursively coupled ordinary differential equations. The system is truncated in a consistent manner, and the effects of large angle scattering on the evolution of the moments are determined in explicit analytical form. An approximate similarity solution of the generalized Fokker-Planck equation is also found, and the results of both approaches provide a clear picture of the increased diffusive beam spreading due to large angle scattering. A comparison with previously published Monte Carlo simulation results shows good agreement.

Theory of the superglass phase

A superglass is a phase of matter which is characterized at the same time by superfluidity and a frozen amorphous structure. We introduce a model of interacting bosons in three dimensions that displays this phase unambiguously and that can be analyzed exactly or using controlled approximations. Employing a mapping between quantum Hamiltonians and classical Fokker-Planck operators, we show that the ground-state wave function of the quantum model is proportional to the Boltzmann measure of classical hard spheres. This connection allows us to obtain quantitative results on static and dynamic quantum correlation functions. In particular, by translating known results on the glassy dynamics of Brownian hard spheres we work out the properties of the superglass phase and of the quantum phase transition between the superfluid and the superglass phase.

Generalized diffusion: A microscopic approach

The Fokker-Planck equation for the probability f(r,t) to find a random walker at position r at time t is derived for the case that the probability to make jumps depends nonlinearly on f(r,t) . The result is a generalized form of the classical Fokker-Planck equation where the effects of drift, due to a violation of detailed balance, and of external fields are also considered. It is shown that in the absence of drift and external fields a scaling solution, describing anomalous diffusion, is possible only if the nonlinearity in the jump probability is of the power law type [ approximately f;{eta}(r,t)] , in which case the generalized Fokker-Planck equation reduces to the porous media equation. Monte Carlo simulations are shown to confirm the theoretical results.

Master Equation in Phase Space for a Uniaxial Spin System

A master equation, for the time evolution of the quasi-probability density function of spin orientations in the phase space representation of the polar and azimuthal angles is derived for a uniaxial spin system subject to a magnetic field parallel to the axis of symmetry. This equation is obtained from the reduced density matrix evolution equation (assuming that the spin-bath coupling is weak and that the correlation time of the bath is so short that the stochastic process resulting from it is Markovian) by expressing it in terms of the inverse Wigner-Stratonovich transformation and evaluating the various commutators via the properties of polarization operators and spherical harmonics. The properties of this phase space master equation, resembling the Fokker-Planck equation, are investigated, leading to a finite series (in terms of the spherical harmonics) for its stationary solution, which is the equilibrium quasi-probability density function of spin “orientations” corresponding to the canonical density matrix and which may be expressed in closed form for a given spin number. Moreover, in the large spin limit, the master equation transforms to the classical Fokker-Planck equation describing the magnetization dynamics of a uniaxial paramagnet.

Thermal symmetry of the Markovian master equation

The quantum Markovian master equation of the reduced dynamics of a harmonic oscillator coupled to a thermal reservoir is shown to possess thermal symmetry. This symmetry is revealed by a Bogoliubov transformation that can be represented by a hyperbolic rotation acting on the Liouville space of the reduced dynamics. The Liouville space is obtained as an extension of the Hilbert space through the introduction of tilde variables used in the thermofield dynamics formalism. The angle of rotation depends on the temperature of the reservoir, as well as the value of Planck's constant. This symmetry relates the thermal states of the system at any two temperatures. This includes absolute zero, at which purely quantum effects are revealed. The Caldeira-Leggett equation and the classical Fokker-Planck equation also possess thermal symmetry. We compare the thermal symmetry obtained from the Bogoliubov transformation in related fields and discuss the effects of the symmetry on the shape of a Gaussian wave packet.

Solution of the master equation for Wigner’s quasiprobability distribution in phase space for the Brownian motion of a particle in a double well potential

Quantum effects in the Brownian motion of a particle in the symmetric double well potential V(x)=ax(2)2+bx(4)4 are treated using the semiclassical master equation for the time evolution of the Wigner distribution function W(x,p,t) in phase space (x,p). The equilibrium position autocorrelation function, dynamic susceptibility, and escape rate are evaluated via matrix continued fractions in the manner customarily used for the classical Fokker-Planck equation. The escape rate so yielded has a quantum correction depending strongly on the barrier height and is compared with that given analytically by the quantum mechanical reaction rate solution of the Kramers turnover problem. The matrix continued fraction solution substantially agrees with the analytic solution. Moreover, the low-frequency part of the spectrum associated with noise assisted Kramers transitions across the potential barrier may be accurately described by a single Lorentzian with characteristic frequency given by the quantum mechanical reaction rate.

Semiclassical master equation in Wigners phase space applied to Brownian motion in a periodic potential

The quantum Brownian motion of a particle in a cosine periodic potential V(x)= -V{0}cos(x/x{0}) is treated using the master equation for the time evolution of the Wigner distribution function W(x,p,t) in phase space (x,p) . The dynamic structure factor, escape rate, and jump-length probabilities are evaluated via matrix continued fractions in the manner customarily used for the classical Fokker-Planck equation. The escape rate so yielded is compared with that given analytically by the quantum-mechanical reaction rate solution of the Kramers turnover problem. The matrix continued fraction solution substantially agrees with the analytic solution.

Wigner function approach to the quantum Brownian motion of a particle in a potential

Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.

Classical limit of master equation for a harmonic oscillator coupled to an oscillator bath with separable initial conditions

The Wigner transform of the master equation describing the reduced dynamics of the system, of a harmonic oscillator coupled to an oscillator bath, was obtained by Karrlein and Grabert [Phys. Rev. E 55, 153 (1997)]. It was shown that for some special correlated initial conditions the master equation reduces, in the classical limit, to the corresponding classical Fokker-Planck equation obtained by Adelman [J. Chem Phys. 64, 124 (1976)]. However, for separable initial conditions the Adelman equations were not recovered. We resolve this problem by showing that, for separable initial conditions, the classical Langevin equations are somewhat different from the one considered by Adelman. We obtain the corresponding Fokker-Planck equation and show that they exactly match the classical limit of the evolution of the Wigner function obtained from the master equation for separable initial conditions. We also discuss why thermal initial conditions correspond to Adelman's solution.