Class Of Fokker-Planck Equations Research Articles

QSLE-v1.0: A New Software Package for the Calculation of Coupled Quantum-Classical Dynamics in Condensed Phases Based on a Stochastic Approach.

Recently, a stochastic version of the quantum-classical Liouville equation has been proposed [Campeggio, J.; Cortivo, R.; Zerbetto, M. J. Chem. Phys. 2023, 158, 244104], to compute the coupled quantum-classical dynamics of molecules in condensed phases. The approach, called quantum-stochastic Liouville equation (QSLE), is based on coupling the time evolution of electronic states to a stochastic description of the relevant (classical) nuclear coordinates. Natural internal coordinates are used, i.e., bond lengths, bond angles, and dihedral angles. The approach is tailored to directly propagate the populations of the electronic states over time, based on a classical Fokker-Planck equation for the nuclear degrees of freedom coupled to a master equation for the jumps among the electronic states. The QSLE is a multiscale approach requiring many ingredients to be assembled in order to carry out the numerical solution. To make the approach accessible, a software package that handles the main (and most cumbersome) details of the numerical workflow has been implemented into the software package QSLE-v1.0, which is introduced in the present paper. Here, it is considered the case of one torsional nuclear coordinate and two nonadiabatic electronic potential energy surfaces. This is a very basic model for interpreting photoisomerization or charge transfer phenomena, but despite its simplicity, it can be applied even in complex systems where the relevant quantum/classical parts affecting the phenomena under study are highly localized. A sand-box model system for describing photoisomerization processes is reported to demonstrate the usage of the software package. QSLE-v1.0 is open source and distributed under the GPLv2.0 license.

Kinetic equation for stochastic vector bundles

The kinetic equation is crucial for understanding the statistical properties of stochastic processes, yet current equations, such as the classical Fokker–Planck, are limited to local analysis. This paper derives a new kinetic equation for stochastic systems on vector bundles, addressing global scale randomness. The kinetic equation was derived by cumulant expansion of the ensemble-averaged local probability density function, which is a functional of state transition trajectories. The kinetic equation is the geodesic equation for the probability space. It captures global and historical influences, accounts for non-Markovianity, and can be reduced to the classical Fokker–Planck equation for Markovian processes. This paper also discusses relative issues concerning the kinetic equation, including non-Markovianity, Markov approximation, macroscopic conservation equations, gauge transformation, and truncation of the infinite-order kinetic equation, as well as limitations that require further attention.

Birth–death dynamics for sampling: global convergence, approximations and their asymptotics

Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth–death dynamics. We improve results in previous works (Liu et al 2023 Appl. Math. Optim. 87 48; Lu et al 2019 arXiv:1905.09863) and provide weaker hypotheses under which the probability density of the birth–death governed by Kullback–Leibler divergence or by χ 2 divergence converge exponentially fast to the Gibbs equilibrium measure, with a universal rate that is independent of the potential barrier. To build a practical numerical sampler based on the pure birth–death dynamics, we consider an interacting particle system, which is inspired by the gradient flow structure and the classical Fokker–Planck equation and relies on kernel-based approximations of the measure. Using the technique of Γ-convergence of gradient flows, we show that on the torus, smooth and bounded positive solutions of the kernelised dynamics converge on finite time intervals, to the pure birth–death dynamics as the kernel bandwidth shrinks to zero. Moreover we provide quantitative estimates on the bias of minimisers of the energy corresponding to the kernelised dynamics. Finally we prove the long-time asymptotic results on the convergence of the asymptotic states of the kernelised dynamics towards the Gibbs measure.

The Onsager–Machlup theory of fluctuations and time-dependent generalized normal distribution

Abstract Generalization of the variational formulation of the Onsager–Machlup thermodynamic theory of fluctuation is considered. Within the framework of variational theory, we introduce the time-dependent generalized normal distribution and Hamilton–Jacobi equation. The family of higher-order partial differential equations, which generalize classical Fokker–Planck equation, is considered. It is shown that proposed theory can be used for describing anomalous diffusion.

Generalized second Einstein relation in heterogeneous media

Generalized second Einstein relation is obtained from a class of Fokker-Planck equations with generic time-space-dependent coefficients. The relation is also obtained from an integro-differential approach.

A second-order accurate numerical scheme for a time-fractional Fokker–Planck equation

Abstract A second-order accurate time-stepping scheme for solving a time-fractional Fokker–Planck equation of order $\alpha \in (0, 1)$, with a general driving force, is investigated. A stability bound for the semidiscrete solution is obtained for $\alpha \in (1/2,1)$ via a novel and concise approach. Our stability estimate is $\alpha $-robust in the sense that it remains valid in the limiting case where $\alpha $ approaches $1$ (when the model reduces to the classical Fokker–Planck equation), a limit that presents practical importance. Concerning the error analysis, we obtain an optimal second-order accurate estimate for $\alpha \in (1/2,1)$. A time-graded mesh is used to compensate for the singular behavior of the continuous solution near the origin. The time-stepping scheme scheme is associated with a standard spatial Galerkin finite element discretization to numerically support our theoretical contributions. We employ the resulting fully discrete computable numerical scheme to perform some numerical tests. These tests suggest that the imposed time-graded meshes assumption could be further relaxed, and we observe second-order accuracy even for the case $\alpha \in (0,1/2]$, that is, outside the range covered by the theory.

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.

Kinetic equation for particle transport in turbulent flows

With a new approach based on the ensemble average over particle state transition paths in phase space, a kinetic equation for particles transported in turbulent flows is derived. The probability density function (PDF) for particles is defined as an ensemble average of a special fine-grained PDF, referred to as the local path density operator. The kinetic equation is derived from a Taylor series expansion of the PDF in terms of the cumulants with respect to particle paths in phase space and leads to a closed expression for its diffusion terms. It shows that the random forcing of eddy fluctuations, non-stationarity of turbulence, and inertia of particles are explicitly presented in the diffusion coefficient, which could help us to understand how particles are diffused by these underlying mechanisms. The kinetic equation is applicable to non-Markovian, non-Gaussian, and non-stationary stochastic processes, while for Markovian processes, it recovers the classical Fokker–Planck equation. The macroscopic equations for particle phase are derived based on the kinetic equation and compared with the direct numerical simulation of particles transported in turbulent flows.

Weak invariants in dissipative systems: action principle and Noether charge for kinetic theory.

In non-equilibrium classical thermostatistics, the state of a system may be described by not only dynamical/thermodynamical variables but also a kinetic distribution function. This 'double structure' bears some analogy with that in quantum thermodynamics, where both dynamical variables and the Hilbert space are involved. Recently, the concept of weak invariants has repeatedly been discussed in the context of quantum thermodynamics. A weak invariant is defined in such a way that its value changes in time but its expectation value is conserved under time evolution prescribed by a kinetic equation. Here, a new aspect of a weak invariant is revealed for the classical Fokker-Planck equation as an example of classical kinetic equations. The auxiliary field formalism is applied to the construction of the action for the kinetic equation. Then, it is shown that the auxiliary field is a weak invariant and is the Noether charge. The action is invariant under the transformation generated by the weak invariant. The result may shed light on possible roles of the symmetry principle in the kinetic descriptions of non-equilibrium systems. This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.

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].

Generalized local and nonlocal master equations for some stochastic processes

In this paper, we present a study on generalized local and nonlocal equations for some stochastic processes. By considering the net flux change in a region determined by the transition probability, we derive the master equation to describe the evolution of the probability density function. Some examples, such as classical Fokker–Planck equations, models for Lévy process, and stochastic coagulation equations, are provided as illustrations. A particular application is a consistent derivation of coupled dynamical systems for spatially inhomogeneous stochastic coagulation processes.

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.

Channeling experiments with sub-GeV electrons in flat silicon single crystals

Various planar channeling experiments, performed at the Mainz Microtron MAMI with electrons at silicon single crystals, have been reanalyzed. Two types of signals have been employed. The low energy loss signal originates from emission of channeling radiation in the energy domain between 0.4 and 9MeV while the high energy loss signal from electrons which have lost about 50% of their primary energy by emission of bremsstrahlung photons. The (110) planar channeling data, taken at a beam energy of 855MeV with the former signal, can well be described on the basis of the solution of the classical Fokker–Planck equation. The measurements with the latter signal at beam energies between 195 and 855MeV indicate quantum state phenomena. For (111) planar channeling calculations with the Fokker–Planck equation have also been performed at a beam energy of 6.3GeV. The results indicate that data taken with a crystal of 60μm thickness [U. Wienands et al., Phys. Rev. Lett. 114, 074801 (2015)] are probably not suited to determine the predicted “asymptotic” dechanneling length of 265μm which applies for about 48% of all electrons.

On the solution of the Fokker‐Planck equation on infinite domains using problem‐specific orthonormal basis functions in a Galerkin‐type method

AbstractAbstract: The dynamical behavior of a mechanical system under random excitation can be described by its probability density function (PDF). The PDF can be determined by random sampling or directly computed as the solution of the corresponding FOKKER‐PLANCK equation (FPE). Since the class of FPE with known exact solution is very small, the solution is mostly approximated by numerical methods including finite elements, finite differences or other types of local discretization methods that are best suited for finite domains. Whilst this restriction can be desired for some models, the domain needs to be unbounded in the general case. The authors present a method of calculating a semi‐analytical approximate solution for the stationary FPE over an unbounded domain using a GALERKIN‐type method. The utilization of a problem‐specific orthonormal basis for the ansatz‐space ensures that only a small number of improper integrals needs to be computed. In fact, these computations are inexpensive and can be performed prior to the more costly algorithm that constructs the system of linear equations. The computation of the specific basis and its properties are discussed in detail. In order to demonstrate the method and its application on technical systems, the PDF for a nonlinear bistable magneto‐piezo‐electric energy harvester is calculated. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An Analog of the 2-Wasserstein Metric in Non-Commutative Probability Under Which the Fermionic Fokker–Planck Equation is Gradient Flow for the Entropy

Let $${\mathfrak{C}}$$ denote the Clifford algebra over $${\mathbb{R}^n}$$ , which is the von Neumann algebra generated by n self-adjoint operators Q j , j = 1,…,n satisfying the canonical anticommutation relations, Q i Q j + Q j Q i = 2δ ij I, and let τ denote the normalized trace on $${\mathfrak{C}}$$ . This algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom. Let $${\mathfrak{P}}$$ denote the set of all positive operators $${\rho\in\mathfrak{C}}$$ such that τ(ρ) = 1; these are the non-commutative analogs of probability densities in the non-commutative probability space $${(\mathfrak{C},\tau)}$$ . The fermionic Fokker–Planck equation is a quantum-mechanical analog of the classical Fokker–Planck equation with which it has much in common, such as the same optimal hypercontractivity properties. In this paper we construct a Riemannian metric on $${\mathfrak{P}}$$ that we show to be a natural analog of the classical 2-Wasserstein metric, and we show that, in analogy with the classical case, the fermionic Fokker–Planck equation is gradient flow in this metric for the relative entropy with respect to the ground state. We derive a number of consequences of this, such as a sharp Talagrand inequality for this metric, and we prove a number of results pertaining to this metric. Several open problems are raised.