Search Results (199)

Differential equations need boundary conditions (BCs) for their solution. It is widely acknowledged that differential equations and BCs are representative of independent physical processes, and no correlations between them are required. Two recent studies by Hilhorst, Chung et al. argue instead that, in the specific case of diffusion equations (DEs) in bounded systems, BCs are uniquely constrained by the form of transport coefficients. In this paper, we revisit how DEs emerge as fluid limits out of a picture of stochastic transport. We point out their limits of validity and argue that, in most physical systems, BCs and DEs are actually uncorrelated by virtue of the failure of diffusive approximation near the system’s boundaries. When, instead, the diffusive approximation holds everywhere, we show that the correct chain of reasoning goes in the direction opposite to that conjectured by Hilhorst and Chung: it is the choice of the BCs that determines the form of the DE in the surroundings of the boundary. Full article

We develop a symmetry-based variational theory that shows the coarse-grained balance of work inflow to heat outflow in a driven, dissipative system relaxed to the golden ratio. Two order-2 Möbius transformations—a self-dual flip and a self-similar shift—generate a discrete non-abelian subgroup of $\mathrm{PGL}(2,\mathbb{Q}\left(\sqrt{5}\right))$. Requiring any smooth, strictly convex Lyapunov functional to be invariant under both maps enforces a single non-equilibrium fixed point: the golden mean. We confirm this result by (i) a gradient-flow partial-differential equation, (ii) a birth–death Markov chain whose continuum limit is Fokker–Planck, (iii) a Martin–Siggia–Rose field theory, and (iv) exact Ward identities that protect the fixed point against noise. Microscopic kinetics merely set the approach rate; three parameter-free invariants emerge: a $62\%:38\%$ split between entropy production and useful power, an RG-invariant diffusion coefficient linking relaxation time and correlation length ${\mathcal{D}}_{\alpha }={\xi }^z/\tau$, and a $\vartheta ={45}^{\circ }$ eigen-angle that maps to the golden logarithmic spiral. The same dual symmetry underlies scaling laws in rotating turbulence, plant phyllotaxis, cortical avalanches, quantum critical metals, and even de-Sitter cosmology, providing a falsifiable, unifying principle for pattern formation far from equilibrium. Full article

A weak second-order numerical method for generating trajectories based on stochastic differential equations (SDE) is developed. The proposed approach bypasses direct noise realization by updating the system’s state using independent Gaussian random variables so as to reproduce the first three cumulants of the state variable at each time step to the second order in the time-step size. The update rule for the state variable is derived based on the system’s Fokker–Planck equation in an arbitrary number of dimensions. The high accuracy of the method as compared to the standard Milstein algorithm is demonstrated on the example of Büttiker’s ratchet. While the method is second-order accurate in the time step, it can be extended to systematically generate higher-order terms of the stochastic Taylor expansion approximating the solution of the SDE. 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

A class of dynamic equations containing a non-Maxwellian collision kernel is used to investigate the distribution of wealth. A trading rule, in which the trading propensity $\gamma$ of agents is a function of wealth w (namely, $\gamma =\gamma \left(w\right)$), is considered. Two different trading propensity functions are discussed. One is that $\gamma \left(w\right)$ increases with wealth. The other is that $\gamma \left(w\right)$ decreases with the increase in wealth. In a single transaction, when the transaction tendency increases with the increase in wealth, the rich invest more in transactions. The gap between the rich and the poor in society is reduced under suitable conditions. Through numerical simulation, we conclude that an escalation in market risk intensifies the inequality in wealth distribution. Full article

This paper investigates a multiple unmanned aerial vehicle (UAV) enabled network for supporting emergency communication services, where each drone acts as a base station (also called the drone small cell (DSC)). The novelty of this paper is that a mean field game (MFG)-based strategy is conceived for jointly controlling the three-dimensional (3D) locations of these drones to guarantee the distortion requirement of lossy communications, while considering the inter-cell interference and the flight energy consumption of drones. More explicitly, we derive the Hamilton–Jacobi–Bellman (HJB) and Fokker–Planck–Kolmogorov (FPK) equations, and propose an algorithm where both the Lax–Friedrichs scheme and the Lagrange relaxation are invoked for solving the HJB and FPK equations with 3D control vectors and state vectors. The numerical results show that the proposed algorithm can achieve a higher access rate with a similar flight energy consumption. Full article

We develop a unified analytical framework that systematically connects kinetic theory, optimal transport, and entropy dissipation through the novel integration of hypocoercivity methods with geometric structures. Building upon but distinctly extending classical hypocoercivity approaches, we demonstrate how geometric control, via commutators and curvature-like structures in probability spaces, resolves degeneracies inherent in kinetic operators. Centered around the Boltzmann and Fokker–Planck equations, we derive sharp exponential convergence estimates under minimal regularity assumptions, improving on prior methods by incorporating Wasserstein gradient flow techniques. Our framework is further applied to the study of hydrodynamic limits, collisional relaxation in magnetized plasmas, the Vlasov–Poisson system, and modern data-driven algorithms, highlighting the central role of entropy as both a physical and variational tool across disciplines. By bridging entropy dissipation, optimal transport, and geometric analysis, our work offers a new perspective on stability, convergence, and structure in high-dimensional kinetic models and applications. 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

This study investigates the computation of fractional and higher integer-order moments for a stochastic process governed by a one-dimensional, non-homogeneous linear stochastic differential equation (SDE) driven by fractional Brownian motion (fBm). Unlike conventional approaches relying on moment-generating functions or Fokker–Planck equations, which often yield intractable expressions, we derive explicit closed-form formulas for these moments. Our methodology leverages the Wick–Itô calculus (fractional Itô formula) and the properties of Hermite polynomials to express moments efficiently. Additionally, we establish a recurrence relation for moment computation and propose an alternative approach based on generalized binomial expansions. To validate our findings, Monte Carlo simulations are performed, demonstrating a high degree of accuracy between theoretical and empirical results. The proposed framework provides novel insights into stochastic processes with long-memory properties, with potential applications in statistical inference, mathematical finance, and physical modeling of anomalous diffusion. Full article

We examine couplings, synchronization, pattern formation, and transformation in human dynamics. We consider intraindividual and interpersonal relations as coevolution dynamics of heterogeneous mixed couplings, synchronizations, and desynchronizations. They form the dynamic patterns of the embodied Self and organize intersubjective dynamics. We critically review various models with differing levels of complexity and degrees of freedom. The Fokker–Planck equation clarifies the balance between determinism and stochasticity. The HKB and Kuramoto models describe complex synchronization and pattern formation dynamics. Chimera states are ubiquitous in the mixed networks of human dynamics. Coupling, synchronization, and patterns form and transform, with gaps in between. We propose a formal model for these complex, mixed, and heterogeneous dynamics. Multidimensional theoretical models can represent the specific nature of human interactions and the dynamic structure of the embodied Self. The embodied Self emerges during a developmental process and retains its dynamical nature by continuously adapting to the ever-changing landscape of affordances of daily life. Full article

The Fokker–Planck (FP) equation represents the drift and diffusive processes in kinetic models. It can also be regarded as a model for the collision integral of the Boltzmann-type equation to represent thermo-hydrodynamic processes in fluids. The lattice Boltzmann method (LBM) is a drastically simplified discretization of the Boltzmann equation for simulating complex fluid motions and beyond. We construct new two FP-based LBMs, one for recovering the Navier–Stokes equations for fluid dynamics and the other for simulating the energy equation, where, in each case, the effect of collisions is represented as relaxations of different central moments to their respective attractors. Such attractors are obtained by matching the changes in various discrete central moments due to collision with the continuous central moments prescribed by the FP model. As such, the resulting central moment attractors depend on the lower-order moments and the diffusion tensor parameters, and significantly differ from those based on the Maxwell distribution. The diffusion tensor parameters for evolving higher moments in simulating fluid motions at relatively low viscosities are chosen based on a renormalization principle. Moreover, since the number of collision invariants of the FP-based LBMs for fluid motions and energy transport are different, the forms of the respective attractors are quite distinct. The use of such central moment formulations in modeling the collision step offers significant improvements in numerical stability, especially for simulations of thermal convective flows under a wide range of variations in the transport coefficients of the fluid. We develop new FP central moment LBMs for thermo-hydrodynamics in both two and three dimensions, and demonstrate the ability of our approach to simulate various cases involving thermal convective buoyancy-driven flows especially at high Rayleigh numbers with good quantitative accuracy. Moreover, we show significant improvements in the numerical stability of our FP central moment LBMs when compared to other existing central moment LBMs using the Maxwell distribution in achieving high Peclet numbers for mixed convection flows involving shear effects. 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

Here, we study the Navier–Stokes equation for the motion of a passive particle based on the Fokker–Planck equation in an incompressible conducting fluid induced by a magnetic field subject to an exponentially correlated Gaussian force in three-time domains. For the hydro-magnetic case of velocity and the time-dependent magnetic field, the mean squared velocity for the joint probability density of velocity and the magnetic field has a super-diffusive form that scales as $\sim t^3$ in $t>>\tau$, while the mean squared displacement for the joint probability density of velocity and the magnetic field reduces to time $\sim t^4$ in $t<<\tau$. The motion of a passive particle for $\tau =0$ and $t>>\tau$ behaves as a normal diffusion with the mean squared magnetic field being $<h^2(t)>\sim t$. In a short-time domain $t<<\tau$, the moment in the magnetic field of the incompressible conducting fluid undergoes super-diffusion with ${\mu }_{2,0,2}^h\sim t^6$, in agreement with our research outcome. Particularly, the combined entropy $H(v,h,t)$ ($H(h,v,t)$) for an active particle with the perturbative force has a minimum value of $\sim \ln t^2$ ($\sim \ln t^2$) in $t>>\tau$ ($\tau =0$), while the largest displacement entropy value is proportional to $\ln t^4$ in $t<<\tau$ and $\tau =0$. Full article

The self-assembly mechanisms of various complex biological structures, including viral capsids and carboxysomes, have been theoretically studied through numerous kinetic models. However, most of these models focus on the equilibrium aspects of a simplified kinetic description in terms of a single reaction coordinate, typically the number of proteins in a growing aggregate, which is often insufficient to describe the size and shape of the resulting structure. In this article, we use mesoscopic non-equilibrium thermodynamics (MNET) to derive the equations governing the non-equilibrium kinetics of viral capsid formation. The resulting kinetic equation is a Fokker–Planck equation, which considers viral capsid self-assembly as a diffusive process in the space of the relevant reaction coordinates. We discuss in detail the case of the self-assembly of a spherical (icosahedral) capsid with a fixed radius, which corresponds to a single degree of freedom, and indicate how to extend this approach to the self-assembly of spherical capsids that exhibit radial fluctuations, as well as to tubular structures and systems with higher degrees of freedom. Finally, we indicate how these equations can be solved in terms of the equivalent Langevin equations and be used to determine the rate of formation and size distribution of closed capsids, opening the door to the better understanding and control of the self- assembly process. 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