Planck System Research Articles

Propagation of chaos for the Vlasov–Poisson–Fokker–Planck equation with a polynomial cut-off

We consider a [Formula: see text]-particle system interacting through the Newtonian potential with a polynomial cut-off in the presence of noise in velocity. We rigorously prove the propagation of chaos for this interacting stochastic particle system. Taking the cut-off like [Formula: see text] with [Formula: see text] in the force, we provide a quantitative error estimate between the empirical measure associated to that [Formula: see text]-particle system and the solutions of the [Formula: see text]-dimensional Vlasov–Poisson–Fokker–Planck (VPFP) system. We also study the propagation of chaos for the Vlasov–Fokker–Planck equation with less singular interaction forces than the Newtonian one.

Global existence of solutions for the Poisson–Nernst–Planck system with steric effects

In this work, we study the global existence of a modified Poisson–Nernst–Planck system with steric effects. This model involves cross-diffusion and non-local terms. The main idea of the proof is to approximate the system by truncating the diffusion matrix and apply the Schauder’s fixed-point theorem. Moreover, we obtain L2 uniform in time estimates from the energy inequality (3.5). The global existence then follows.

Existence of large-data global-in-time finite-energy weak solutions to a compressible FENE-P model

A compressible FENE-P-type model with stress diffusion is derived from an approximate macroscopic closure of a compressible Navier–Stokes–Fokker–Planck system arising in the kinetic theory of dilute polymeric fluids, where polymer chains immersed in a barotropic, compressible, isothermal, viscous Newtonian solvent are idealised as pairs of massless beads connected with finitely extensible nonlinear elastic (FENE) springs. We develop a priori bounds for the model, including logarithmic bounds, which guarantee the non-negativity of the conformation tensor and a bound on its trace, and we prove the existence of large-data global-in-time finite-energy weak solutions in two and three space dimensions.

Instability in a Vlasov–Fokker–Planck binary mixture

This paper is concerned with a kinetic model of a Vlasov–Fokker–Planck system used to describe the evolution of two species of particles interacting through a potential and a thermal reservoir at given temperature. We prove that at low temperature, the homogeneous equilibrium is dynamically unstable under certain perturbations. Our work is motivated by a problem arising in [4].

Asymptotic models for transport in large aspect ratio nanopores

Ion flow in charged nanopores is strongly influenced by the ratio of the Debye length to the pore radius. We investigate the asymptotic behaviour of solutions to the Poisson–Nernst–Planck (PNP) system in narrow pore like geometries and study the influence of the pore geometry and surface charge on ion transport. The physical properties of real pores motivate the investigation of distinguished asymptotic limits, in which either the Debye length and pore radius are comparable or the pore length is very much greater than its radius This results in a quasi-one-dimensional (1D) PNP model, which can be further simplified, in the physically relevant limit of strong pore wall surface charge, to a fully 1D model. Favourable comparison is made to the two-dimensional (2D) PNP equations in typical pore geometries. It is also shown that, for physically realistic parameters, the standard 1D area averaged PNP model for ion flow through a pore is a very poor approximation to the (real) 2D solution to the PNP equations. This leads us to propose that the quasi-1D PNP model derived here, whose computational cost is significantly less than 2D solution of the PNP equations, should replace the use of the 1D area averaged PNP equations as a tool to investigate ion and current flows in ion pores.

The same class of stationary solutions to some multidimensional kinetic systems with extensive background density

In this study, we investigate the existence and uniqueness of the same class of stationary solutions for some kinetic systems comprising the Vlasov–Poisson–Fokker–Planck system, Vlasov–Poisson–Boltzmann system, and Vlasov–Maxwell–Boltzmann system in arbitrary space dimensions N(N≥3). Essentially, the problem can be reduced to solving the problem of a second order elliptic equation with exponential nonlinearity. This result had been proved in three spatial dimensions but the extension to a higher-dimensional setting makes the existence proof nontrivial. In particular, it necessary to highlight that the requirement regarding the background density function is relaxed compared with previous studies in the case of three spatial dimensions.

Hypocoercivity and Uniform Regularity for the Vlasov--Poisson--Fokker--Planck System with Uncertainty and Multiple Scales

We study the Vlasov--Poisson--Fokker--Planck system with uncertainty and multiple scales. Here the uncertainty, modeled by random variables, enters the solution through initial data, while the multiple scales lead the system to its high-field or parabolic regimes. With the help of proper Lyapunov-type inequalities, under some mild conditions on the initial data, the regularity of the solution in the random space, as well as exponential decay of the solution to the global Maxwellian, are established under Sobolev norms, which are uniform in terms of the scaling parameters. These are the first hypocoercivity results for a nonlinear kinetic system with random input, which are important for the understanding of the sensitivity of the system under random perturbations, and for the establishment of spectral convergence of popular numerical methods for uncertainty quantification based on (spectrally accurate) polynomial chaos expansions.

Regularity Criteria for the 3D Dissipative System Modeling Electro-Hydrodynamics

In this paper, we study the three-dimensional dissipative fluid-dynamical model, which is a strongly coupled nonlinear nonlocal system characterized by the Navier–Stokes/Poisson–Nernst–Planck system. It is proved that the local smooth solution can be continued beyond the time T provided that the vorticity $$\omega $$ satisfies $$\begin{aligned} \int _{0}^{T}\frac{\Vert \omega (\cdot ,t)\Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}^{\frac{2}{2-\alpha }}}{1+\ln \left( e+\Vert \omega (\cdot ,t)\Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}\right) }\mathrm{d}t<\infty \quad \text {for} \,\, 0<\alpha <2. \end{aligned}$$ Moreover, two regularity criteria for the marginal cases $$\alpha =0$$ and $$\alpha =2$$ are also established, respectively.

Mathematical modeling of a discontinuous solution of the generalized Poisson--Nernst--Planck problem in a two-phase medium

In this paper a mathematical model generalizing Poisson--Nernst--Planck system is considered. The generalized model presents electrokinetics of species in a two-phase medium consisted of solid particles and a pore space. The governing relations describe cross-diffusion of the charged species together with the overall electrostatic potential. At the interface between the pore and the solid phases nonlinear electro-chemical reactions are taken into account provided by jumps of field variables. The main advantage of the generalized model is that the total mass balance is kept within our setting. As the result of the variational approach, well-posedness properties of a discontinuous solution of the problem are demonstrated and supported by the energy and entropy estimates.

A local meshless method for solving multi-dimensional Vlasov–Poisson and Vlasov–Poisson–Fokker–Planck systems arising in plasma physics

In this paper, we use a linear combination of the shape functions of reproducing kernel particle method (RKPM) and RBFs for achieving the unknown weights into each stencil. We obtain an error bound for the new shape function. Also, in this paper, we investigate a numerical procedure based on the presented technique for solving the Vlasov---Poisson and Vlasov---Poisson---Fokker---Planck systems. The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma. The Vlasov---Poisson equations are used to describe various phenomena in plasma, in particular Landau damping and the distributions in a double layer plasma. We use the RKPM/RBF-FD technique for discretization of space direction and employ the method of lines to achieve a high-order accuracy in temporal direction. Numerical examples are reported which demonstrate the theoretical results and the efficiency of proposed scheme.

Modelling compression sensing in ionic polymer metal composites

Ionic polymer metal composites (IPMCs) consist of an ionomeric membrane, including mobile counterions, sandwiched between two thin noble metal electrodes. IPMCs find application as sensors and actuators, where an imposed mechanical loading generates a voltage across the electrodes, and, vice versa, an imposed electric field causes deformation. Here, we present a predictive modelling approach to elucidate the dynamic sensing response of IPMCs subject to a time-varying through-the-thickness compression (‘compression sensing’). The model relies on the continuum theory recently developed by Porfiri and co-workers, which couples finite deformations to the modified Poisson–Nernst–Planck (PNP) system governing the IPMC electrochemistry. For the ‘compression sensing’ problem we establish a perturbative closed-form solution along with a finite element (FE) solution. The systematic comparison between these two solutions is a central contribution of this study, offering insight on accuracy and mathematical complexity. The method of matched asymptotic expansions is employed to find the analytical solution. To this end, we uncouple the force balance from the modified PNP system and separately linearise the PNP equations in the ionomer bulk and in the boundary layers at the ionomer–electrode interfaces. Comparison with FE results for the fully coupled nonlinear system demonstrates the accuracy of the analytical solution to describe IPMC sensing for moderate deformation levels. We finally demonstrate the potential of the modelling scheme to accurately reproduce experimental results from the literature. The proposed model is expected to aid in the design of IPMC sensors, contribute to an improved understanding of IPMC electrochemomechanical response, and offer insight into the role of nonlinear phenomena across mechanics and electrochemistry.

Non-localness of Excess Potentials and Boundary Value Problems of Poisson–Nernst–Planck Systems for Ionic Flow: A Case Study

Poisson–Nernst–Planck (PNP) type systems are basic primitive models for ionic flow through ion channels. Important properties of ion channels, such as current-voltage relations, permeation and selectivity, can be extracted from solutions of boundary value problems (BVP) of PNP type models. Many issues of BVP of PNP type systems with local excess potentials (including particularly classical PNP systems that treat ions as point-charges) are extensively examined analytically and numerically. On the other hand, for PNP type systems with nonlocal excess potentials, even the issue of well-posedness of BVP is poorly understood. In fact, the formulation of correct boundary conditions seems to be overlooked, even though complications of ionic behavior near the boundaries (locations of applied electrodes) have been long experienced in experiments and simulations. PNP type systems with nonlocal excess potentials can be viewed as functional differential systems and, for many approximation models of nonlocal excess potentials, as differential equations with both delays and advances. Thus PNP type systems with nonlocal excess potentials have infinite degree of freedoms and BVP with the traditional “two-point-boundary-conditions” would be severely under determined. The mathematical theory for PNP with nonlocal excess potential would be significantly different from that for PNP with local excess potentials. Taking into considerations of experimental designs of ionic flow through ion channels and in a relatively simple setting, we present a form of natural “boundary conditions” so that the corresponding BVP of PNP type systems with nonlocal excess potentials are generally well-posed. This work, at an early stage toward a better understanding of related issues, provides some insights on interpretations of experimental designs of imposing boundary conditions and for correct formulations of numerical simulations, and hopefully, will stimulate further mathematical analysis on this important issue.

Simulation of diffuse-charge capacitance in electric double layer capacitors

We use a Lattice Boltzmann Model (LBM) in order to simulate diffuse-charge dynamics in Electric Double Layer Capacitors (EDLCs). Simulations are carried out for both the charge and the discharge processes on 2D systems of complex random electrode geometries (pure random, random spheres and random fibers). The steric effect of concentrated solutions is considered by using a Modified Poisson–Nernst–Planck (MPNP) equations and compared with regular Poisson–Nernst–Planck (PNP) systems. The effects of electrode microstructures (electrode density, electrode filler morphology, filler size, etc.) on the net charge distribution and charge/discharge time are studied in detail. The influence of applied potential during discharging process is also discussed. Our studies show how electrode morphology can be used to tailor the properties of supercapacitors.

The Vlasov--Poisson--Fokker--Planck System with Uncertainty and a One-dimensional Asymptotic Preserving Method

We develop a stochastic asymptotic preserving (s-AP) scheme for the Vlasov--Poisson--Fokker--Planck system in the high field regime with uncertainty based on the generalized polynomial chaos stochastic Galerkin framework (gPC-SG). We first prove that, for a given electric field with uncertainty, the regularity of initial data in the random space is preserved by the analytical solution at a later time, which allows us to establish the spectral convergence of the gPC-SG method. We follow the framework developed in [S. Jin and L. Wang, Acta Math. Sci., 31 (2011), pp. 2219--2232] to numerically solve the resulting system in one space dimension and show formally that the fully discretized scheme is s-AP in the high field regime. Numerical examples are given to validate the accuracy and s-AP properties of the proposed method.

Erratum to “Global classical solutions of the ‘one and one-half’ dimensional Vlasov–Maxwell–Fokker–Planck system”

Erratum to “Global classical solutions of the ‘one and one-half’ dimensional Vlasov–Maxwell–Fokker–Planck system”

A free energy satisfying discontinuous Galerkin method for one-dimensional Poisson–Nernst–Planck systems

We design an arbitrary-order free energy satisfying discontinuous Galerkin (DG) method for solving time-dependent Poisson–Nernst–Planck systems. Both the semi-discrete and fully discrete DG methods are shown to satisfy the corresponding discrete free energy dissipation law for positive numerical solutions. Positivity of numerical solutions is enforced by an accuracy-preserving limiter in reference to positive cell averages. Numerical examples are presented to demonstrate the high resolution of the numerical algorithm and to illustrate the proven properties of mass conservation, free energy dissipation, as well as the preservation of steady states.

A Wasserstein gradient flow approach to Poisson−Nernst−Planck equations

The Poisson−Nernst−Planck system of equations used to model ionic transport is interpreted as a gradient flow for the Wasserstein distance and a free energy in the space of probability measures with finite second moment. A variational scheme is then set up and is the starting point of the construction of global weak solutions in a unified framework for the cases of both linear and nonlinear diffusion. The proof of the main results relies on the derivation of additional estimates based on the flow interchange technique developed by Matthes et al. in [D. Matthes, R.J. McCann and G. Savare, Commun. Partial Differ. Equ. 34 (2009) 1352–1397].

Hydrodynamic singular regimes in 1 + 1 kinetic models and spectral numerical methods

Classical results from spectral theory of stationary linear kinetic equations are applied to efficiently approximate two physically relevant weakly nonlinear kinetic models: a model of chemotaxis involving a biased velocity-redistribution integral term, and a Vlasov–Fokker–Planck (VFP) system. Both are coupled to an attractive elliptic equation producing corresponding mean-field potentials. Spectral decompositions of stationary kinetic distributions are recalled, based on a variation of Case's elementary solutions (for the first model) and on a Sturm–Liouville eigenvalue problem (for the second one). Well-balanced Godunov schemes with strong stability properties are deduced. Moreover, in the stiff hydrodynamical scaling, an hybridized algorithm is set up, for which asymptotic-preserving properties can be established under mild restrictions on the computational grid. Several numerical validations are displayed, including the consistency of the VFP model with Burgers–Hopf dynamics on the velocity field after blowup of the macroscopic density into Dirac masses.

Simulating transient dynamics of the time-dependent time fractional Fokker–Planck systems

For a physically realistic type of time-dependent time fractional Fokker–Planck (FP) equation, derived as the continuous limit of the continuous time random walk with time-modulated Boltzmann jumping weight, a semi-analytic iteration scheme based on the truncated (generalized) Fourier series is presented to simulate the resultant transient dynamics when the external time modulation is a piece-wise constant signal. At first, the iteration scheme is demonstrated with a simple time-dependent time fractional FP equation on finite interval with two absorbing boundaries, and then it is generalized to the more general time-dependent Smoluchowski-type time fractional Fokker–Planck equation. The numerical examples verify the efficiency and accuracy of the iteration method, and some novel dynamical phenomena including polarized motion orientations and periodic response death are discussed.

A Numerical Strategy for Nernst-Planck Systems with Solvation Effect

A Numerical Strategy for Nernst-Planck Systems with Solvation Effect