Search Results (55)

In this article, we establish the existence of global mild solutions to the Vlasov–Fokker–Planck equation with local alignment forces under specular reflection boundary conditions in the low-regularity function space $L_k^1{L}_T^{\infty }{L}_v^2$. A key difficulty is that the macroscopic averaged velocity u does not directly possess a dissipative structure in the equation. To overcome this, we rely on the dissipation $u-b$ from the linear part, combined with the dissipation of the macroscopic component b derived from the associated macroscopic equation. Moreover, since no direct energy functional is available for u, we fully exploit the dissipative mechanisms of both $u-b$ and b when handling the estimates for the nonlinear terms. Full article

In this article, the mechanism of relaxation polarization currents occurring at a constant temperature (isothermal process) in crystals with ionic-molecular chemical bonds (CIMBs) in an alternating electric field was investigated. Methods of the quasi-classical kinetic theory of dielectric relaxation, based on solutions of the nonlinear system of Fokker–Planck and Poisson equations (for the blocking electrode model) and perturbation theory (by expanding into an infinite series in powers of a dimensionless small parameter) were used. Generalized nonlinear mathematical expressions for calculating the complex amplitudes of relaxation modes of the volume-charge distribution of the main charge carriers (ions, protons, water molecules, etc.) were obtained. On this basis, formulas for the current density of relaxation polarization (for transient processes in a dielectric) in the k-th approximation of perturbation theory were constructed. The isothermal polarization currents are investigated in detail in the first four approximations (k = 1, 2, 3, 4) of perturbation theory. These expressions will be applied in the future to compare the results of theory and experiment, in analytical studies of the kinetics of isothermal ion-relaxation (in crystals with hydrogen bonds (HBC), proton-relaxation) polarization and in calculating the parameters of relaxers (molecular characteristics of charge carriers and crystal lattice parameters) in a wide range of field parameters (0.1–1000 MV/m) and temperatures (1–1550 K). Asymptotic (far from transient processes) recurrent formulas are constructed for complex amplitudes of relaxation modes and for the polarization current density in an arbitrary approximation k of perturbation theory with a multiplicity r by the polarizing field (a multiple of the fundamental frequency of the field). The high degree of reliability of the theoretical results obtained is justified by the complete agreement of the equations of the mathematical model for transient and stationary processes in the system with a harmonic external disturbance. This work is of a theoretical nature and is focused on the construction and analysis of nonlinear properties of a physical and mathematical model of isothermal ion-relaxation polarization in CIMB crystals under various parameters of electrical and temperature effects. The theoretical foundations for research (construction of equations and working formulas, algorithms, and computer programs for numerical calculations) of nonlinear kinetic phenomena during thermally stimulated relaxation polarization have been laid. This allows, with a higher degree of resolution of measuring instruments, to reveal the physical mechanisms of dielectric relaxation and conductivity and to calculate the parameters of a wide class of relaxators in dielectrics in a wide experimental temperature range (25–550 K). Full article

This paper presents a comprehensive dynamical analysis of a nonlinear oscillator subjected to both deterministic and stochastic excitations. Utilizing a diverse suite of analytical tools—including phase portraits, Poincaré sections, Lyapunov exponents, recurrence plots, Fokker–Planck equations, and sensitivity diagnostics—we investigate the transitions between quasi-periodicity, chaos, and stochastic disorder. The study reveals that quasi-periodic attractors exhibit robust topological structure under moderate noise but progressively disintegrate as stochastic intensity increases, leading to high-dimensional chaotic-like behavior. Recurrence quantification and Lyapunov spectra validate the transition from coherent dynamics to noise-dominated regimes. Poincaré maps and sensitivity analysis expose multistability and intricate basin geometries, while the Fokker–Planck formalism uncovers non-equilibrium steady states characterized by circulating probability currents. Together, these results provide a unified framework for understanding the geometry, statistics, and stability of noisy nonlinear systems. The findings have broad implications for systems ranging from mechanical oscillators to biological rhythms and offer a roadmap for future investigations into fractional dynamics, topological analysis, and data-driven modeling. Full article

Aiming at the state estimation problem of nonlinear systems (NLSs), the traditional typical nonlinear filtering methods (e.g., Particle Filter, PF) have large errors in system state, resulting in low accuracy and high computational speed. To perfect the imperfections, a new Bayesian estimation method based on particle flow velocity (PFV-BEM) is proposed in this paper. Firstly, a symmetrical projection space based on the state information is selected, the basis function is determined by a set of Fourier series with symmetric properties, the state update is carried out according to the projection principle to calculate the prior information of the state, and select its particle points. Secondly, the particle flow velocity is defined, which describes the evolution process of random samples from the prior distribution to the posterior distribution. The posterior information of the state is calculated by solving the parameters related to the particle flow velocity. Finally, the estimated mean and standard deviation of the state are solved. Simulation experiments are carried out based on two instances of one-dimensional general nonlinear examples and multi-target motion tracking, The newly proposed algorithm is compared with the Particle Filter (PF), and the simulation results clearly indicate the feasibility of this novel Bayesian estimation algorithm. Full article

Stem cells therapies offer a promising approach in translational oncology, as well as in regenerative medicine due to the tropism of these cells to the damage site. To track the distribution of stem cells, the latter could be labeled by MRI-sensitive superparamagnetic (SPM) iron oxide nanoparticles. In the current study, magnetic properties of the magnetic nanoparticles (MNPs) incorporated into the bone marrow-derived fetal mesenchymal stem cells (FetMSCs) were evaluated employing nonlinear magnetic response measurements. Synthesized dextran-coated iron oxide nanoparticles were additionally characterized by X-ray diffraction, transmission electron microscopy, and dynamic light scattering. The MNP uptake by the FetMSCs 24 h following coincubation was studied by longitudinal nonlinear response to weak alternating magnetic field with registration of the second harmonic of magnetization. Subsequent data processing using a formalism based on the numerical solution of the Fokker–Planck kinetic equation allowed us to determine magnetic and dynamic parameters and the state of MNPs in the cells, as well as in the culture medium. It was found that MNPs formed aggregates in the culture medium; they were absorbed by the cells during coincubation. The aggregates exhibited SPM regime in the medium, and the parameters of the MNP aggregates remained virtually unchanged in the cells, indicating the preservation of the aggregation state of MNPs inside the cells. This implies also the preservation of the organic shell of the nanoparticles inside FetMSCs. The accumulation of MNPs by mesenchymal stem cells gradually increased with the concentration of MNPs. Thus, the study confirmed that the labeling of MSCs with MNPs is an effective method for subsequent cell tracking as incorporated nanoparticles retain their magnetic properties. Full article

The authors present an in-depth theoretical study of two nonlinear circuits capable of transient thermal energy harvesting at one temperature. The first circuit has a storage capacitor and diode connected in series. The second circuit has three storage capacitors, and two diodes arranged for full wave rectification. The authors solve both Ito–Langevin and Fokker–Planck equations for both circuits using a large parameter space including capacitance values and diode quality. Surprisingly, using diodes one can harvest thermal energy at a single temperature by charging capacitors. However, this is a transient phenomenon. In equilibrium, the capacitor charge is zero, and this solution alone satisfies the second law of thermodynamics. The authors found that higher quality diodes provide more stored charge and longer lifetimes. Harvesting thermal energy from the ambient environment using diode nonlinearity requires capacitors to be charged but then disconnected from the circuit before they have time to discharge. Full article

Optimal control theory aims to find an optimal protocol to steer a system between assigned boundary conditions while minimizing a given cost functional in finite time. Equations arising from these types of problems are often non-linear and difficult to solve numerically. In this article, we describe numerical methods of integration for two partial differential equations that commonly arise in optimal control theory: the Fokker–Planck equation driven by a mechanical potential for which we use the Girsanov theorem; and the Hamilton–Jacobi–Bellman, or dynamic programming, equation for which we find the gradient of its solution using the Bismut–Elworthy–Li formula. The computation of the gradient is necessary to specify the optimal protocol. Finally, we give an example application of the numerical techniques to solving an optimal control problem without spacial discretization using machine learning. Full article

A generalized scientific review with elements of additions and clarifications has been carried out on the methods of theoretical research on the electrophysical properties of crystals with ionic–molecular chemical bonds (CIMBs). The main theoretical tools adopted are the methods of quasi-classical kinetic theory as applied to ionic subsystems relaxing in layered dielectrics (natural silicates, crystal hydrates, various types of ceramics, and perovskites) in an electric field. A universal (applicable for any CIMBs class crystals) nonlinear quasi-classical kinetic equation of theoretical and practical importance has been constructed. This equation describes, in complex with the Poisson equation, the mechanism of ion-relaxation polarization and conductivity in a wide range of polarizing field parameters (0.1–1000 MV/m) and temperatures (1–1550 K). The physical model is based on a system of non-interacting ions (due to the low concentration in the crystal) moving in a one-dimensional, spatially periodic crystalline potential field, perturbed by an external electric field. The energy spectrum of ions is assumed to be continuous. Elements of quantum mechanical theory in a quasi-classical model are used to mathematically describe the influence of tunnel transitions of hydrogen ions (protons) during the interaction of proton and anion subsystems in hydrogen-bonded crystals (HBC) on the polarization of the dielectric in the region of nitrogen (50–100 K) and helium (1–10 K) temperatures. The mathematical model is based on the solution of a system of nonlinear Fokker-Planck and Poisson equations, solved by perturbation theory methods (via expanding solutions into infinite power series in a small dimensionless parameter). Theoretical frequency and temperature spectra of the dielectric loss tangent were constructed and analyzed, the molecular parameters of relaxers were calculated, and the physical nature of the maxima of the experimental temperature spectra of dielectric losses for a number of HBC crystals was discovered. The low-temperature maximum, which is caused by the quantum tunneling of protons and is absent in the experimental spectra, was theoretically calculated and investigated. The most effective areas of scientific and technical application of the theoretical results obtained were identified. The application of the equations and recurrent formulas of the constructed model to the study of nonlinear optical effects in elements of laser technologies and nonlinear radio wave effects in elements of microwave signal control systems is of the greatest interest. Full article

We present a new formulation of the central moment lattice Boltzmann (LB) method based on a minimal continuous Fokker-Planck (FP) kinetic model, originally proposed for stochastic diffusive-drift processes (e.g., Brownian dynamics), by adapting it as a collision model for the continuous Boltzmann equation (CBE) for fluid dynamics. The FP collision model has several desirable properties, including its ability to preserve the quadratic nonlinearity of the CBE, unlike that based on the common Bhatnagar-Gross-Krook model. Rather than using an equivalent Langevin equation as a proxy, we construct our approach by directly matching the changes in different discrete central moments independently supported by the lattice under collision to those given by the CBE under the FP-guided collision model. This can be interpreted as a new path for the collision process in terms of the relaxation of the various central moments to “equilibria”, which we term as the Markovian central moment attractors that depend on the products of the adjacent lower order moments and a diffusion coefficient tensor, thereby involving of a chain of attractors; effectively, the latter are nonlinear functions of not only the hydrodynamic variables, but also the non-conserved moments; the relaxation rates are based on scaling the drift coefficient by the order of the moment involved. The construction of the method in terms of the relevant central moments rather than via the drift and diffusion of the distribution functions directly in the velocity space facilitates its numerical implementation and analysis. We show its consistency to the Navier-Stokes equations via a Chapman-Enskog analysis and elucidate the choice of the diffusion coefficient based on the second order moments in accurately representing flows at relatively low viscosities or high Reynolds numbers. We will demonstrate the accuracy and robustness of our new central moment FP-LB formulation, termed as the FPC-LBM, using the D3Q27 lattice for simulations of a variety of flows, including wall-bounded turbulent flows. We show that the FPC-LBM is more stable than other existing LB schemes based on central moments, while avoiding numerical hyperviscosity effects in flow simulations at relatively very low physical fluid viscosities through a refinement to a model founded on kinetic theory. Full article

The objective of this study is to model astrophysical systems using the nonlinear Fokker–Planck equation, with the Adomian method chosen for its iterative and precise solutions in this context, applying boundary conditions relevant to data from the Rossi X-ray Timing Explorer (RXTE). The results include analysis of 156 X-ray intensity distributions from X-ray binaries (XRBs), exhibiting long-tail profiles consistent with Tsallis q-Gaussian distributions. The corresponding q-values align with the principles of Tsallis thermostatistics. Various diffusion hypotheses—classical, linear, nonlinear, and anomalous—are examined, with q-values further supporting Tsallis thermostatistics. Adjustments in the parameter α (related to the order of fractional temporal derivation) reveal the extent of the memory effect, strongly correlating with fractal properties in the diffusive process. Extending this research to other XRBs is both possible and recommended to generalize the characteristics of X-ray scattering and electromagnetic waves at different frequencies originating from similar astronomical objects. Full article

Hyper-ballistic diffusion is shown to arise from a simple model of microswimmers moving through a porous media while competing for resources. By using a mean-field model where swimmers interact through the local concentration, we show that a non-linear Fokker–Planck equation arises. The solution exhibits hyper-ballistic superdiffusive motion, with a diffusion exponent of four. A microscopic simulation strategy is proposed, which shows excellent agreement with theoretical analysis. Full article

We study the entropy production in a fractal system composed of two subsystems, each of which is subjected to an external force. This is achieved by using the H-theorem on the nonlinear Fokker–Planck equations (NFEs) characterizing the diffusing dynamics of each subsystem. In particular, we write a general NFE in terms of Hausdorff derivatives to take into account the metric of each system. We have also investigated some solutions from the analytical and numerical point of view. We demonstrate that each subsystem affects the total entropy and how the diffusive process is anomalous when the fractal nature of the system is considered. Full article

The sustainable exploitation of groundwater resources is a multifaceted and complex problem, which is controlled, among many other factors and processes, by water flow in porous soils and sediments. Modeling water flow in unsaturated, non-deformable porous media is commonly based on a partial differential equation, which translates the mass conservation principle into mathematical terms. Such an equation assumes that the variation of the volumetric water content ($\theta$) in the medium is balanced by the net flux of water flow, i.e., the divergence of specific discharge, if source/sink terms are negligible. Specific discharge is in turn related to the matric potential (h), through the non-linear Darcy–Buckingham law. The resulting equation can be rewritten in different ways, in order to express it as a partial differential equation where a single physical quantity is considered to be a dependent variable. Namely, the most common instances are the Fokker–Planck Equation (for $\theta$), and the Richards Equation (for h). The other two forms can be given for generalized matric flux potential ($\Phi$) and for hydraulic conductivity (K). The latter two cases are shown to limit the non-linearity to multiplicative terms for an exponential K-to-h relationship. Different types of boundary conditions are examined for the four different formalisms. Moreover, remarks given on the physico-mathematical properties of the relationships between K, h, and $\theta$ could be useful for further theoretical and practical studies. Full article

We investigate the dynamics of a system composed of two different subsystems when subjected to different nonlinear Fokker–Planck equations by considering the H–theorem. We use the H–theorem to obtain the conditions required to establish a suitable dependence for the system’s interaction that agrees with the thermodynamics law when the nonlinearity in these equations is the same. In this framework, we also consider different dynamical aspects of each subsystem and investigate a possible expression for the entropy of the composite system. Full article

Increasing interest has been shown in the subject of non-additive entropic forms during recent years, which has essentially been due to their potential applications in the area of complex systems. Based on the fact that a given entropic form should depend only on a set of probabilities, its time evolution is directly related to the evolution of these probabilities. In the present work, we discuss some basic aspects related to non-additive entropies considering their time evolution in the cases of continuous and discrete probabilities, for which nonlinear forms of Fokker–Planck and master equations are considered, respectively. For continuous probabilities, we discuss an H-theorem, which is proven by connecting functionals that appear in a nonlinear Fokker–Planck equation with a general entropic form. This theorem ensures that the stationary-state solution of the Fokker–Planck equation coincides with the equilibrium solution that emerges from the extremization of the entropic form. At equilibrium, we show that a Carnot cycle holds for a general entropic form under standard thermodynamic requirements. In the case of discrete probabilities, we also prove an H-theorem considering the time evolution of probabilities described by a master equation. The stationary-state solution that comes from the master equation is shown to coincide with the equilibrium solution that emerges from the extremization of the entropic form. For this case, we also discuss how the third law of thermodynamics applies to equilibrium non-additive entropic forms in general. The physical consequences related to the fact that the equilibrium-state distributions, which are obtained from the corresponding evolution equations (for both continuous and discrete probabilities), coincide with those obtained from the extremization of the entropic form, the restrictions for the validity of a Carnot cycle, and an appropriate formulation of the third law of thermodynamics for general entropic forms are discussed. Full article