Transport of charged particles: Entropy production and Maximum Dissipation Principle

Hsieh, Chung-Bao; Hyon, Yun Kyong; Lee, Hijin; Lin, Tai-Chia; Li, Chun

doi:10.1016/j.jmaa.2014.07.078

Cited by 24 publications

(28 citation statements)

References 21 publications

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“…Since we cannot expect that ∇u i ∈ L q (Ω), q > d, for d ≥ 3, we employ the technique of Gajewski [9] which avoids this regularity. The method seems to work only for linear mobilities u i , which is the reason why we cannot apply it to (13). The idea is to introduce the semimetric…”

Section: Theorem 3 (Uniqueness Of Weak Solutions) Let Assumptionsmentioning

confidence: 99%

Analysis of a degenerate parabolic cross-diffusion system for ion transport

Gerstenmayer

Jüngel

2018

Journal of Mathematical Analysis and Applications

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A cross-diffusion system describing ion transport through biological membranes or nanopores in a bounded domain with mixed Dirichlet-Neumann boundary conditions is analyzed. The ion concentrations solve strongly coupled diffusion equations with a drift term involving the electric potential which is coupled to the concentrations through a Poisson equation. The global-in-time existence of bounded weak solutions and the uniqueness of weak solutions under moderate regularity assumptions are shown. The main difficulties of the analysis are the cross-diffusion terms and the degeneracy of the diffusion matrix, preventing the use of standard tools. The proofs are based on the boundedness-by-entropy method, extended to nonhomogeneous boundary conditions, and the uniqueness technique of Gajewski. A finite-volume discretization in one space dimension illustrates the large-time behavior of the numerical solutions and shows that the equilibration rates may be very small.

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Section: Theorem 3 (Uniqueness Of Weak Solutions) Let Assumptionsmentioning

confidence: 99%

Analysis of a degenerate parabolic cross-diffusion system for ion transport

Gerstenmayer

Jüngel

2018

Journal of Mathematical Analysis and Applications

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“…The motion of these quasi-particles are described by mean field models, evaluated both by simulations [109,210] and theory [109,152,156,193,210]. For example, the Poisson drift diffusion equations [113], often called PNP (for Poisson Nernst Planck) in biophysics and nanotechnology [12,16,21,24,41,51,53,99,214]. PNP is of particular importance because it is used widely, nearly universally, to design and understand the devices of our semiconductor technology, from transistors to computer chips.…”

Section: Arxiv:150207251 [Q-bioot] 11 August 2015mentioning

confidence: 99%

“…PNP is not enough, however, when ionic solutions are involved. A great deal of effort has been spent applying PNP equations to electrochemical systems [12,16,21,24,41,51,53,99,214] hoping they might serve as adequate robust constitutive equations, but that is not the case. The nonideality of ionic solutions, arising in large measure from saturation effects produced by the finite size of ions, demands more powerful mathematics than the partial differential equations of PNP used in computational electronics.…”

Section: Arxiv:150207251 [Q-bioot] 11 August 2015mentioning

confidence: 99%

“…Energetic variational methods are particularly useful because they allow multiscale derivation of partial differential equations (and far field boundary conditions) from physical principles when multiple fields are involved, like convection, diffusion, steric exclusion, and migration in an electric field. Energetic variational principles have recently become available for systems involving friction [62,106,215], that is to say, for systems involving ionic solutions [50, 99,161,213]. Computations must be done in three dimensions because spheres do not exist in one and two dimensions.…”

Section: ˆˆˆˆmentioning

confidence: 99%

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Mass Action and Conservation of Current

Eisenberg¹

2016

Hungarian Journal of Industry and Chemistry

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The law of mass action does not force a series of chemical reactions to have the same current flow everywhere. Interruption of far-away current does not stop current everywhere in a series of chemical reactions (analyzed according to the law of mass action), and so does not obey Maxwell's equations. An additional constraint and equation is needed to enforce global continuity of current. The additional constraint is introduced in this paper in the special case that the chemical reaction describes spatial movement through narrow channels. In that case, a fully consistent treatment is possible using different models of charge movement. The general case must be dealt with by variational methods that enforce consistency of all the physical laws involved. Violations of current continuity arise away from equilibrium, when current flows, and the law of mass action is applied to a nonequilibrium situation, different from the systems considered when the law was originally derived. Device design in the chemical world is difficult because simple laws are not obeyed in that way. Rate constants of the law of mass action are found experimentally to change from one set of conditions to another. The law of mass action is not robust in most cases and cannot serve the same role that circuit models do in our electrical technology. Robust models and device designs in the chemical world will not be possible until continuity of current is embedded in a generalization of the law of mass action using a consistent variational model of energy and dissipation.

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“…There have been many results on wellposedness and long-time behavior of the solutions for the model [2,5,6,13,14,31]. Also, various versions of the PNP system are concerned for various applications [15,17,20,21,26,27,38,39]. In this paper, we consider the PNP system with variable dielectric coefficients, which takes the form…”

mentioning

confidence: 99%

Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains

Hsieh¹

2019

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, we consider radial solutions of the Poisson-Nernst-Planck (PNP) system with variable dielectric coefficients εg(x) in N-dimensional annular domains, N ≥ 2. When the parameter ε tends to zero, the PNP system admits a boundary layer solution as a steady state, which satisfies the charge conserving Poisson-Boltzmann (CCPB) equation. For the stability of the radial boundary layer solutions to the time-dependent radial PNP system, we estimate the radial solution of the perturbed problem with global electroneutrality. We generalize the argument of the one spatial dimension case (cf. [18]) and find a new way to transform the perturbed problem. By choosing a suitable weighted norm, we then derive the associated energy law which can be used to prove that the H −1 x norm of the solution of the perturbed problem decays exponentially in time with the exponent independent of ε if the coefficient of the Robin boundary condition of electrostatic potential has a suitable positive lower bound.

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Transport of charged particles: Entropy production and Maximum Dissipation Principle

Cited by 24 publications

References 21 publications

Analysis of a degenerate parabolic cross-diffusion system for ion transport

Analysis of a degenerate parabolic cross-diffusion system for ion transport

Mass Action and Conservation of Current

Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains

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