Fokker-Planck Research Articles

Approximate and Exact Solutions of Some Nonlinear Differential Equations Using the Novel Coupling Approach in the Sense of Conformable Fractional Derivative

Several scientific fields utilize fractional nonlinear partial differential equations to model various phenomena. However, most of these equations lack exact solutions. Consequently, techniques for obtaining approximate solutions, which sometimes yield exact solutions, are essential. In this research, we develop a new approach by combining the homotopy perturbation method (HPM) and the conformable natural transform to solve the gas-dynamic equation (GDE), the Fokker-Planck equation (FPE), and the Swift-Hohenberg equation (SHE) in the context of conformable derivatives. The proposed approach is called the conformable natural homotopy perturbation method (CNHPM). This approach has the advantage of not requiring assumptions about significant or minor physical factors. Consequently, it eliminates some of the constraints associated with conventional perturbation methods and can solve both weak and highly nonlinear problems. We consider the absolute, relative, and residual errors numerically and graphically to assess the correctness of our approach. The results show that our approach serves as a suitable alternative to the approximate methods in the literature for solving fractional differential equations.

Spectral eigenfunction decomposition of a Fokker–Planck operator for relativistic heavy-ion collisions

A spectral solution method is proposed to solve a previously developed non-equilibrium statistical model describing partial thermalization of produced charged hadrons in relativistic heavy-ion collisions, thus improving the accuracy of the numerical solution. The particle’s phase-space trajectories are treated as a drift-diffusion stochastic process, leading to a Fokker–Planck equation (FPE) for the single-particle probability distribution function. The drift and diffusion coefficients are derived from the expected asymptotic states via appropriate fluctuation–dissipation relations, and the resulting FPE is then solved numerically using a spectral eigenfunction decomposition. The calculated time-dependent particle distributions are compared to Pb–Pb data from the ATLAS and ALICE collaborations at the Large Hadron Collider.

An optimization-based equilibrium measure describes fixed points of non-equilibrium dynamics: application to edge of chaos

Abstract Understanding neural dynamics is a central topic in machine learning, non-linear physics and neuroscience. However, the dynamics is non-linear, stochastic and particularly non-gradient, i.e., the driving force can not be written as gradient of a potential. These features make analytic studies very challenging. The common tool is the path integral approach or dynamical mean-field theory, but the drawback is that one has to solve the integro-differential or dynamical mean-field equations, which is computationally expensive and has no closed form solutions in general. From the aspect of associated Fokker-Planck equation, the steady state solution is generally unknown. Here, we treat searching for the steady states as an optimization problem, and construct an approximate potential related to the speed of the dynamics, and find that searching for the ground state of this potential is equivalent to running an approximate stochastic gradient dynamics or Langevin dynamics. Only in the zero temperature limit, the distribution of the original steady states can be achieved. The resultant stationary state of the dynamics follows exactly the canonical Boltzmann measure. Within this framework, the quenched disorder intrinsic in the neural networks can be averaged out by applying the replica method, which leads naturally to order parameters for the non-equilibrium steady states. Our theory reproduces the well-known result of edge-of-chaos, and further the order parameters characterizing the continuous transition are derived, and the order parameters are explained as fluctuations and responses of the steady states. Our method thus opens the door to analytically study the steady state landscape of the deterministic or stochastic high dimensional dynamics.

A multiscale stochastic particle method based on the Fokker-Planck model for nonequilibrium gas flows

The Fokker-Planck (FP) model characterizes the intermolecular collisions as continuous stochastic processes in velocity space. Compared to the Direct Simulation Monte Carlo (DSMC) method, which is grounded in the Boltzmann equation, the stochastic particle method based on the FP model, referred to as the SP-FP method, demonstrates a relative improvement in computational efficiency. However, the traditional SP-FP method, utilizing operator splitting scheme, encounters analogous limitations to DSMC, thereby limiting the application of large temporal-spatial discretization for simulating near-continuum and continuum flows. In this study, two critical advancements are introduced to improve the traditional SP-FP method. First, the exact integration scheme of the ellipsoidal-statistical FP (ESFP) model is derived, enabling the replication of the theoretical solutions for homogeneous relaxation with any time step. Furthermore, leveraging the integration scheme of the ESFP model, we propose a Multiscale Stochastic Particle (MSP) method. The MSP method quantifies the numerical errors stemming from the operator splitting scheme and corrects them in the relaxation step. This capability allows the MSP method to be employed with larger temporal-spatial discretization for inhomogeneous problems, achieving second-order accuracy while retaining implementation simplicity. Application of the MSP method to various benchmark problems, including Couette flow, Fourier flow, Poiseuille flow, Sod tube flow, Lid-driven cavity flow, and flow through a slit, demonstrates its promise for simulating multiscale nonequilibrium gas flows with high computational efficiency and accuracy.

Accounting for the energy dispersion of free carriers induced by powerful femtosecond laser radiation in dielectrics

Dynamics of free carriers in the conduction band of dielectrics induced by femtosecond near infrared laser radiation is analyzed. It is stressed the importance of accounting for the energy dispersion term in the Fokker-Planck equation describing such dynamics.

Collective oscillations in a three-dimensional spin model with non-reciprocal interactions

Abstract We study the onset of collective oscillations at low temperature in a three-dimensional spin model with non-reciprocal short-range interactions. Performing numerical simulations of the model, the presence of a continuous phase transition to global oscillations is confirmed by a finite-size scaling analysis, yielding values of the exponents β and ν compatible with both the three-dimensional XY and Ising equilibrium universality classes. By systematically varying the interaction range, we show that collective oscillations in this spin model actually result from two successive phase transitions: a mean-field phase transition over finite-size neighborhoods, which leads to the emergence of local noisy oscillators, and a synchronization transition of local noisy oscillators, which generates coherent macroscopic oscillations. Using a Fokker–Planck equation under a local mean-field approximation, we derive from the spin dynamics coupled Langevin equations for the complex amplitudes describing noisy oscillations on a mesoscopic scale. The phase diagram of these coupled equations is qualitatively obtained from a fully-connected (mean-field) approximation. This analytical approach allows us to clearly disentangle the onset of local and global oscillations, and to identify the two main control parameters, expressed as combinations of the microscopic parameters of the spin dynamics, that control the phase diagram of the model.

A Stochastic Approach in the Analysis of Compartmental Models in Epidemiology

The spread of infectious diseases is a significant public health issue. Mathematical modeling is being used to find appropriate answers to this question. In this paper we propose a stochastic approach to compartmental models. A new model allows to rely on the fokker-planck equation with a SEIR model, with constant rates, to take into account the random environment in the propagation of a disease, markov and kolmogorov properties have been used in the modeling of the process. We have also determined the basic reproduction rate R0 of our new stochastic model using the new generation matrix.

Beyond Time-Homogeneity for Continuous-Time Multistate Markov Models

–Multistate Markov models are a canonical parametric approach for data modeling of observed or latent stochastic processes supported on a finite state space. Continuous-time Markov processes describe data that are observed irregularly over time, as is often the case in longitudinal medical data, for example. Assuming that a continuous-time Markov process is time-homogeneous, a closed-form likelihood function can be derived from the Kolmogorov forward equations—a system of differential equations with a well-known matrix-exponential solution. Unfortunately, however, the forward equations do not admit an analytical solution for continuous-time, time- inhomogeneous Markov processes, and so researchers and practitioners often make the simplifying assumption that the process is piecewise time-homogeneous. In this article, we provide intuitions and illustrations of the potential biases for parameter estimation that may ensue in the more realistic scenario that the piecewise-homogeneous assumption is violated, and we advocate for a solution for likelihood computation in a truly time-inhomogeneous fashion. Particular focus is afforded to the context of multistate Markov models that allow for state label misclassifications, which applies more broadly to hidden Markov models (HMMs), and Bayesian computations bypass the necessity for computationally demanding numerical gradient approximations for obtaining maximum likelihood estimates (MLEs). Supplemental materials are available online.

Modeling local decoherence of a spin ensemble using a generalized Holstein–Primakoff mapping to a bosonic mode

We show how the decoherence that occurs in an entangling atomic spin–light interface can be simply modeled as the dynamics of a bosonic mode. Although one seeks to control the collective spin of the atomic system in the permutationally invariant (symmetric) subspace, diffuse scattering and optical pumping are local, making an exact description of the many-body state intractable. To overcome this issue we develop a generalized Holstein–Primakoff approximation for collective states which is valid when decoherence is uniform across a large atomic ensemble. In different applications the dynamics is conveniently treated as a Wigner function evolving according to a thermalizing diffusion equation, or by a Fokker–Planck equation for a bosonic mode decaying in a zero-temperature reservoir. We use our formalism to study the combined effect of Hamiltonian evolution, local and collective decoherence, and measurement backaction in preparing nonclassical spin states for application in quantum metrology.

Fluctuations and Power Low Distribution Function in Nonequilibrium Systems

The Fokker–Planck equation is formulated for the distribution functions of macroscopic open systems in the space of slowly changing physical variables (energy, adiabatic invariants, etc.). The stationary solution of such equations determines a quasi-equilibrium distribution function in the relevant space. The proposed approach involves the evolution of systems under the action of dissipation and diffusion in the space of the appropriate variables. It is shown that the well-known power law distribution can be obtained by considering internal and external fluctuations in statistical systems.

Thermodynamic quantum Fokker–Planck equations and their application to thermostatic Stirling engine

We developed a computer code for the thermodynamic quantum Fokker–Planck equations (T-QFPE), derived from a thermodynamic system–bath model. This model consists of an anharmonic subsystem coupled to multiple Ohmic baths at different temperatures, which are connected to or disconnected from the subsystem as a function of time. The code numerically integrates the T-QFPE and their classical expression to simulate isothermal, isentropic, thermostatic, and entropic processes in both quantum and classical cases. The accuracy of the results was verified by comparing the analytical solutions of the Brownian oscillator. In addition, we illustrated a breakdown of the Markovian Lindblad-master equation in the pure quantum regime. As a demonstration, we simulated a thermostatic Stirling engine employed to develop non-equilibrium thermodynamics [S. Koyanagi and Y. Tanimura, J. Chem. Phys. 161, 114113 (2024)] under quasi-static conditions. The quasi-static thermodynamic potentials, described as intensive and extensive variables, were depicted as work diagrams. In the classical case, the work done by the external field is independent of the system–bath coupling strength. In contrast, in the quantum case, the work decreases as the coupling strength increases due to quantum entanglement between the subsystem and bath. The codes were developed for multicore processors using Open Multi-Processing (OpenMP) and for graphics processing units using the Compute Unified Device Architecture. These codes are provided in the supplementary material.

Classical and quantum thermodynamics in a non-equilibrium regime: Application to thermostatic Stirling engine

We have developed a thermodynamic theory in the non-equilibrium regime, which we describe as a thermodynamic system–bath model [Koyanagi and Tanimura, J. Chem. Phys. 160, 234112 (2024)]. Based on the dimensionless (DL) minimum work principle, non-equilibrium thermodynamic potentials are expressed in terms of non-equilibrium extensive and intensive variables in time derivative form. This is made possible by incorporating the entropy production rate into the definition of non-equilibrium thermodynamic potentials. These potentials can be evaluated from the DL non-equilibrium-to-equilibrium minimum work principle, which is derived from the principle of DL minimum work and is equivalent to the second law of thermodynamics. We thus obtain the non-equilibrium Massieu–Planck potentials as entropic potentials and the non-equilibrium Helmholtz–Gibbs potentials as free energies. Unlike the fluctuation theorem and stochastic thermodynamics theory, this theory does not require the assumption of a factorized initial condition and is valid in the full quantum regime, where the system and bath are quantum mechanically entangled. Our results are numerically verified by simulating a thermostatic Stirling engine consisting of two isothermal processes and two thermostatic processes using the quantum hierarchical Fokker–Planck equations and the classical Kramers equation derived from the thermodynamic system–bath model. We then show that, from weak to strong system–bath interactions, the thermodynamic process can be analyzed using a non-equilibrium work diagram analogous to the equilibrium one for given time-dependent intensive variables. The results can be used to develop efficient heat machines in non-equilibrium regimes.

Long-time behavior and quasi-density function for a stochastic epidemic model with relapse and reinfection

A stochastic susceptible–infected–recovered–infected (SIRI) epidemic model with relapse and reinfection is established in this paper. First, we prove that the solution to the epidemic model is unique and globally positive. Next, we determine some sufficient conditions for the extinction of the disease when [Formula: see text] and for the persistence in mean in the case of [Formula: see text]. Furthermore, we prove the existence of at least one ergodic stationary distribution of the stochastic model if [Formula: see text]. Additionally, by solving the corresponding three-dimensional Fokker–Planck equation, it is theoretically shown that the epidemic model has a log-normal probability density function when [Formula: see text], then we obtain the exact expression of density function of the stationary distribution. Finally, we give some numerical simulations to support our theoretical results.

Quantifying random collisions between particles inside and outside a circle

Random collisions of particles occur in various biophysical and physical systems. Inspired by the binding of receptor and ligand on the cell membrane, we devised a method based on stochastic dynamical modeling to quantify the likelihood of two random particles colliding on a circle. We consider the dynamics of a receptor binding to a ligand on the cell membrane, where the receptor and ligand perform different motions and are thus modeled by stochastic differential equations with non-Gaussian noise. We use neural networks based on the Onsager–Machlup function to compute the probability P1 of an unbounded receptor diffusing to the cell membrane. Meanwhile, we compute the probability P2 of the extracellular ligand arriving at the cell membrane by solving the associated nonlocal Fokker–Planck equation. We can then calculate the most probable binding probability by combining P1 and P2. In this way, we conclude with some indication of how the receptors could distribute on the membrane, as well as where the ligand will most probably encounter the receptor, contributing to a better understanding of the cell’s response to external stimuli and communication with other cells.

Random Transitions of a Binary Star in the Canonical Ensemble.

After reviewing the peculiar thermodynamics and statistical mechanics of self-gravitating systems, we consider the case of a "binary star" consisting of two particles of size a in gravitational interaction in a box of radius R. The caloric curve of this system displays a region of negative specific heat in the microcanonical ensemble, which is replaced by a first-order phase transition in the canonical ensemble. The free energy viewed as a thermodynamic potential exhibits two local minima that correspond to two metastable states separated by an unstable maximum forming a barrier of potential. By introducing a Langevin equation to model the interaction of the particles with the thermal bath, we study the random transitions of the system between a "dilute" state, where the particles are well separated, and a "condensed" state, where the particles are bound together. We show that the evolution of the system is given by a Fokker-Planck equation in energy space and that the lifetime of a metastable state is given by the Kramers formula involving the barrier of free energy. This is a particular case of the theory developed in a previous paper (Chavanis, 2005) for N Brownian particles in gravitational interaction associated with the canonical ensemble. In the case of a binary star (N=2), all the quantities can be calculated exactly analytically. We compare these results with those obtained in the mean field limit N→+∞.

Neoclassical transport computations in non-isothermal tokamak plasmas

Neoclassical transport in a non-isothermal plasma in which each plasma species has different equilibrium temperature is investigated by solving the drift kinetic equation with a Fokker–Planck (FP) collision operator in a circular tokamak model. Since it is known that a linearized FP operator does not have a self-adjoint property in a non-isothermal plasma, approximate models are developed for comparison to intend to have the self-adjoint property in the non-isothermal case. To achieve this, we set a common temperature that the system should reach after a long time, and the individual temperature of each particle species is expressed by a parameter to measure a shift of the individual temperature from the common one. Then, both the Vlasov part and the collision term of the kinetic equation are expanded around the common temperature, taking the temperature shift parameter up to the first order. It is found that the lowest order collision term of expansion preserves the self-adjointness while the first-order, nonlinear FP term does not. A large difference of the ion heat neoclassical transport is found in comparison between the developed models with and without the self-adjointness and the original FP collision term in the non-isothermal plasma, especially in a strong temperature equilibration regime, showing that a contribution of the collision term without the self-adjointness seems to be significant. Furthermore, when an impurity species is included, the result is complicated where the usual enhancement in the main ion particle transport coefficient, due to the impurity effect, is rather suppressed with the increase in the ion heat transport coefficients by the non-isothermal effects.

Field theory of active Brownian particles with dry friction

Abstract We present a field theoretic approach to capture the motion of a particle with dry friction for one- and two-dimensional (2D) diffusive particles, and further expand the framework for 2D active Brownian particles. Starting with the Fokker–Planck equation and introducing the Hermite polynomials as the corresponding eigen-functions, we obtain the actions and propagators. Using a perturbation expansion, we calculate the effective diffusion coefficient in the presence of both wet and dry frictions in a perturbative way via the Green–Kubo relation. We further compare the analytical result with the numerical simulation. Our result can be used to estimate the values of dry friction coefficient in experiments.

Sufficient condition for reliable logic operations in an over damped bistable system driven by Gaussian white noise.

In this paper, we investigate an overdamped bistable system subject to Gaussian white noise and logical inputs. By solving and estimating the steady distribution of the corresponding Fokker-Planck equation and considering the two essential features of the reliable logic operations (RLOs)-initial value independence and sign invariance-we establish the sufficient condition for RLO occurrence and identify parametric resonance phenomena in the system. Our numerical simulations confirm the reliability and accuracy of the theoretical results. This work offers insights into enhancing the accuracy of stochastic systems, particularly in the realm of logical stochastic resonance, thereby contributing to advancements in understanding and controlling stochastic dynamical systems.

Transient dispersion of settling gyrotactic microorganisms in an open channel flow

Dispersion of microorganisms is a key issue in bio-physics and has many applications in the fields of algae cultivation, biomass energy, and wetland ecology. However, there has been limited exploration of the effects of settling behavior and initial release conditions on the transient dispersion of gyrotactic microorganisms. This paper explores the transient dispersion of settling gyrotactic microorganisms in an open channel flow. The moment equations derived from the Smoluchowski equation are solved by the biorthogonal expansion method, and the results are compared with random walk simulations, showing good agreement. The time variations of concentration distribution, drift velocity, and dispersivity of settling gyrotactic microorganism suspension are explored in detail under typical initial release conditions. As illustrated and characterized, settlement weakens the gravitactic focusing of microorganisms near the free surface, leads to accumulation at the bottom, and increases the dispersivity; from a line source release, the relaxation time is shortest, and the microorganisms scatter fastest in the longitudinal direction, while the point source at the water surface leads to the most concentrated longitudinal distribution and the highest drift velocity; furthermore, the initial release condition assumes an important role in shaping the concentration distribution and drift velocity.

A machine learning framework for efficiently solving Fokker–Planck equations

A machine learning framework for efficiently solving Fokker–Planck equations