In a recent paper by B. G. da Costa {\it et al.} [Phys. Rev. E 102, 062105(2020)], the phenomenological Langevin equation and the corresponding Fokker-Planck equation for an inhomogeneous medium with a position-dependent particle mass and position-dependent damping coefficient have been studied. The aim of this comment is to present a microscopic derivation of the Langevin equation for such a system. It is not equivalent to that in the commented paper.
We extend a recently introduced free-energy formalism for homogeneous Fokker-Planck equations to a wide, and physically appealing, class of inhomogeneous nonlinear Fokker-Planck equations. In our approach, the free-energy functional is expressed in terms of an entropic functional and an auxiliary potential, both derived from the coefficients of the equation. With reference to the introduced entropic functional, we discuss the entropy production in a relaxation process towards equilibrium. The properties of the stationary solutions of the considered Fokker-Planck equations are also discussed.
We discuss diffusion of particles in a spatially inhomogeneous medium. From the microscopic viewpoint we consider independent particles randomly evolving on a lattice. We show that the reversibility condition has a discrete geometric interpretation in terms of weights associated to un--oriented edges and vertices. We consider the hydrodynamic diffusive scaling that gives, as a macroscopic evolution equation, the Fokker--Planck equation corresponding to the evolution of the probability distribution of a reversible spatially inhomogeneous diffusion process. The geometric macroscopic counterpart …
The time and size distribution of the waves of topplings in the Abelian sandpile model are expressed as the first arrival at the origin distribution for a scale invariant, time inhomogeneous Fokker-Planck equation. Assuming a linear conjecture for the the time inhomogeneity exponent as function of loop erased random walk (LERW) critical exponent, suggested by numerical results, this approach allows to estimate the lower critical dimension of the model and the exact value of the critical exponent for LERW in three dimension. The avalanche size distribution in two dimensions is found to be the d…
We deduce a nonlinear and inhomogeneous Fokker-Planck equation within a generalized Stratonovich, or stochastic $α$-, prescription ($α=0$, $1/2$ and $1$ respectively correspond to the Itô, Stratonovich and anti-Itô prescriptions). We obtain its stationary state $p_{st}(x)$ for a class of constitutive relations between drift and diffusion and show that it has a $q$-exponential form, $p_{st}(x) = N_q[1 - (1-q)βV(x)]^{1/(1-q)}$, with an index $q$ which does not depend on $α$ in the presence of any nonvanishing nonlinearity. This is in contrast with the linear case, for which the index $q$ is $α$-…
Based on the well-known relation between Fokker-Planck equations and Schroedinger equations of quantum mechanics (QM), we propose new deformed Fokker-Planck (FP) equations associated with the Schroedinger equations of "discrete" QM. The latter is a natural discretization of QM and its Schroedinger equations are difference instead of differential equations. Exactly solvable FP equations are obtained corresponding to exactly solvable "discrete" QM, whose eigenfunctions include various deformations of the classical orthogonal polynomials.
In this paper new multi-variable deformed Fokker-Planck (FP) equations are presented. These deformed FP equations are associated with the Ruijsenaars-Schneider-van Diejen (RSvD) type systems in the same way that the usual one variable FP equation is associated with the one particle Schrödinger equation. As the RSvD systems are the "discrete" counterparts of the celebrated exactly solvable many-body Calogero-Sutherland-Moser systems, the deformed FP equations presented here can be considered as "discrete" deformations of the ordinary multi-variable FP equations.
We present the Fokker-Planck equation (FPE) for an inhomogeneous medium with a position-dependent mass particle by making use of the Langevin equation, in the context of a generalized deformed derivative for an arbitrary deformation space where the linear (nonlinear) character of the FPE is associated with the employed deformed linear (nonlinear) derivative. The FPE for an inhomogeneous medium with a position-dependent diffusion coefficient is equivalent to a deformed FPE within a deformed space, described by generalized derivatives, and constant diffusion coefficient. The deformed FPE is cons…
The superstatistics concept is a useful statistical method to describe inhomogeneous complex systems for which a system parameter $β$ fluctuates on a large spatio-temporal scale. In this paper we analyze a measured time series of wind speed fluctuations and extract the superstatistical distribution function $f(β)$ directly from the data. We construct suitable Langevin and Fokker-Planck models with a position dependent $β$-field and show that they reduce to standard type of superstatistics in the overdamped limit.
On the basis of the quantum q-oscillator algebra in the framework of quantum groups and non-commutative q-differential calculus, we investigate a possible q-deformation of the classical Poisson bracket in order to extend a generalized q-deformed dynamics in the classical regime. In this framework, classical q-deformed kinetic equations, Kramers and Fokker-Planck equations, are also studied. Pacs: 05.20.Dd, 45.20.-d, 02.20.Uw Keywords: Kinetic theory, q-deformed classical mechanics, quantum groups, quantum algebras
Diffusion in inhomogeneous materials can be described by both the Fick and Fokker--Planck diffusion equations. Here, we study a mixed Fick and Fokker-Planck diffusion problem with coefficients rapidly oscillating both in space and time. We obtain macroscopic models performing the homogenization limit by means of the unfolding technique.
General self-consistent expressions for the coefficients of diffusion and dynamical friction in a stable, bound, multicomponent self-gravitating and inhomogeneous system are derived. They account for the detailed dynamics of the colliding particles and their self-consistent dressing by collective gravitational interactions. The associated Fokker-Planck equation is shown to be fully consistent with the corresponding inhomogeneous Balescu-Lenard equation and, in the weak self-gravitating limit, to the inhomogeneous Landau equation. Hence it provides an alternative derivation to both and demonstr…
It is pointed out that, for the fractional Fokker-Planck equation for subdiffusion proposed by Metzler, Barkai, and Klafter [Phys. Rev. Lett. 82 (1999) 3563], there are four types of infinitely many exact solutions associated with the newly discovered exceptional orthogonal polynomials. They represent fractionally deformed versions of the Rayleigh process and the Jacobi process.
We revisit the problem of the overdamped (large friction) limit of the Brownian dynamics in an inhomogeneous medium characterized by a position-dependent friction coefficient and a multiplicative noise (local temperature) in one space dimension. Starting from the Kramers equation and analyzing it through the expansion in terms of eigenfunctions of a quantum harmonic oscillator, we derive analytically the corresponding Fokker-Planck equation in the overdamped limit. The result is fully consistent with the previous finding by Sancho, San Miguel, and Dürr \cite{Sanc82}. Our method allows us to ge…
The focus of this work is to create benchmark simulations of decay rates to statistical equilibrium in transport plasma models for Coulomb particle interactions given by a coupled Vlasov-Poisson Fokker-Planck-Landau equation, as well as with Maxwell type and hard sphere interactions. The qualitative decay to the equilibrium Maxwell-Boltzmann distribution through relative entropy is studied in detail for all three types of particle interactions by means of a conservative hybrid spectral and discontinuous Galerkin scheme adapted from previous work. More precisely, the Coulomb case shows that the…
On the basis of multivariate Langevin processes we present a realization of Levy flights as a continuous process. For the simple case of a particle moving under the influence of friction and a velocity dependent stochastic force we explicitly derive the generalized Langevin equation and the corresponding generalized Fokker-Planck equation describing Levy flights. Our procedure is similar to the treatment of the Kramers-Fokker Planck equation in the Smoluchowski limit. The proposed approach forms a feasible way of tackling Levy flights in inhomogeneous media or systems with boundaries what is u…
We study generic aspects of bubble dynamics in DNA under time dependent perturbations, for example temperature change, by mapping the associated Fokker-Planck equation to a quantum time-dependent Schroedinger equation with imaginary time. In the static case we show that the eigenequation is exactly the same as that of the $β$-deformed nuclear liquid drop model, without the issue of non-integer angular momentum. A universal breathing dynamics is demonstrated by using an approximate method in quantum mechanics. The calculated bubble autocorrelation function qualitatively agrees with experimental…
We have recently presented an extension of the standard variational calculus to include the presence of deformed derivatives in the Lagrangian of a system of particles and in the Lagrangian density of field-theoretic models. Classical Euler-Lagrange equations and the Hamiltonian formalism have been re-assessed in this approach. Whenever applied to a number of physical systems, the resulting dynamical equations come out to be the correct ones found in the literature, specially with mass-dependent and with non-linear equations for classical and quantum-mechanical systems. In the present contribu…
The Fokker-Planck equation provides complete statistical description of a particle undergoing random motion in a solvent. In the presence of Lorentz force due to an external magnetic field, the Fokker-Planck equation picks up a tensorial coefficient, which reflects the anisotropy of the particle's motion. This tensor, however, can not be interpreted as a diffusion tensor; there are antisymmetric terms which give rise to fluxes perpendicular to the density gradients. Here, we show that for an inhomogeneous magnetic field these nondiffusive fluxes have finite divergence and therefore affect the …
We straight-forwardly derive the Onsager-Machlup Lagrangian from the Fokker-Planck equation and show that friction and dissipation are a natural property of the equation of motion. We develop a method to calculate the local variance $σ_{2}\,b(q)^{2}$ and identify this function as a Helmholtz-factor. In both meanings the function $b(q)$ describes properties of the environment. For application, we examine the free fall through a barometric medium and model a blow of wind by a solitonic pulse running through the medium. We treat harmonic oscillators immersed in a thermal bath, finding intuitive a…
We present a kinetic theory for inhomogeneous systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory, we obtain a closed kinetic equation describing the relaxation of the distribution function of the system as a whole due to resonances between different orbits. This equation is written in angle-action variables. It conserves mass and energy and increases the Boltzmann entropy (H-theorem). Using a thermal bath approximation, we derive a Fokker-Planck equation that describes the relaxation of a test particle towards the Boltzmann distrib…
We develop the kinetic theory of Hamiltonian systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory, we obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. This equation is valid at order 1/N in a proper thermodynamic limit and it coincides with the kinetic equation obtained from the BBGKY hierarchy. For N tending to infinity, it reduces to the Vlasov equation describing collisionless systems. We describe the process of phase mixing and violent relaxation …
In this work we develop the path integral optimization in a class of inhomogeneous 2d CFTs constructed by putting an ordinary CFT on a space with a position dependent metric. After setting up and solving the general optimization problem, we study specific examples, including the Möbius, SSD and Rainbow deformed CFTs, and analyze path integral geometries and complexity for universal classes of states in these models. We find that metrics for optimal path integrals coincide with particular slices of $AdS_3$ geometries, on which Einstein's equations are equivalent to the condition for minimal pat…
We show that the general two-variable Langevin equations with inhomogeneous noise and friction can generate many different forms of power-law distributions. By solving the corresponding stationary Fokker-Planck equation, we can obtain a condition under which these power-law distributions are accurately created in a system away from equilibrium. This condition is an energy-dependent relation between the diffusion coefficient and the friction coefficient and thus it provides a fluctuation-dissipation relation for nonequilibrium systems with power-law distributions. Further, we study the specific…
This is the final version of the author's diploma thesis written at the Humboldt University of Berlin in 1995. The topic is the flow of granular material in narrow vertical pipes, driven by the gravity, that is described by Langevin equations. Neglecting the interactions, we can solve the resulting Fokker-Planck equation for the homogeneous case. The consideration of inelastic collisions leads to a Boltzmann equation. Assuming local equilibrium, the hydrodynamic equations lead to the extension of the Langevin equation formalism for the inhomogeneous case. For certain parameter ranges, our form…
The problem of diffusion in a time-dependent (and generally inhomogeneous) external field is considered on the basis of a generalized master equation with two times, introduced in [1,2]. We consider the case of the quasi Fokker-Planck approximation, when the probability transition function for diffusion (PTD-function) does not possess a long tail in coordinate space and can be expanded as a function of instantaneous displacements. The more complicated case of long tails in the PTD will be discussed separately. We also discuss diffusion on the basis of hydrodynamic and kinetic equations and sho…
