Extended Nernst–Planck Equation Incorporating Partial Dehydration Effect

Dense fluids and liquids molecules are in constant interaction; hence, they do not fit into the Boltzmann’s picture of a clearcut separation between free-streaming and collisional interactions. Since the interactions are soft and do not involve large scattering angles, an effective way of describing dense fluids is to formulate stochastic models of particle motion, as pioneered by Einstein’s theory of Brownian motion and later extended by Paul Langevin. Besides its practical value for the study of the kinetic theory of dense fluids, Brownian motion bears a central place in the historical development of kinetic theory. Among others, it provided conclusive evidence in favor of the atomistic theory of matter. This chapter introduces the basic notions of stochastic dynamics and its connection with other important kinetic equations, primarily the Fokker–Planck equation, which bear a complementary role to the Boltzmann equation in the kinetic theory of dense fluids.

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Stochastic resonance in periodically driven bistable systems subjected to anomalous diffusion

SN Applied Sciences ◽

10.1007/s42452-021-04418-6 ◽

2021 ◽

Vol 3 (4) ◽

Author(s):

F. Naha Nzoupe ◽

Alain M. Dikandé

Keyword(s):

Stochastic Resonance ◽

Signal To Noise Ratio ◽

Planck Equation ◽

Relevant Information ◽

Periodic Signal ◽

Signal To Noise ◽

Periodically Driven ◽

Loop Area ◽

Bistable Systems ◽

Noise Ratio

AbstractThe occurrence of stochastic resonance in bistable systems undergoing anomalous diffusions, which arise from density-dependent fluctuations, is investigated with an emphasis on the analytical formulation of the problem as well as a possible analytical derivation of key quantifiers of stochastic resonance. The nonlinear Fokker–Planck equation describing the system dynamics, together with the corresponding Ito–Langevin equation, is formulated. In the linear response regime, analytical expressions of the spectral amplification, of the signal-to-noise ratio and of the hysteresis loop area are derived as quantifiers of stochastic resonance. These quantifiers are found to be strongly dependent on the parameters controlling the type of diffusion; in particular, the peak characterizing the signal-to-noise ratio occurs only in close ranges of parameters. Results introduce the relevant information that, taking into consideration the interactions of anomalous diffusive systems with a periodic signal, can provide a better understanding of the physics of stochastic resonance in bistable systems driven by periodic forces.

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An Incompressible Polymer Fluid Interacting with a Koiter Shell

Journal of Nonlinear Science ◽

10.1007/s00332-021-09678-5 ◽

2021 ◽

Vol 31 (1) ◽

Author(s):

Dominic Breit ◽

Prince Romeo Mensah

Keyword(s):

Moving Boundary ◽

Classical Model ◽

Elastic Shell ◽

Planck Equation ◽

Navier Stokes ◽

Flexible Shell ◽

Navier Stokes Equation ◽

Shell System ◽

Forward Equation ◽

Polymer Fluid

AbstractWe study a mutually coupled mesoscopic-macroscopic-shell system of equations modeling a dilute incompressible polymer fluid which is evolving and interacting with a flexible shell of Koiter type. The polymer constitutes a solvent-solute mixture where the solvent is modelled on the macroscopic scale by the incompressible Navier–Stokes equation and the solute is modelled on the mesoscopic scale by a Fokker–Planck equation (Kolmogorov forward equation) for the probability density function of the bead-spring polymer chain configuration. This mixture interacts with a nonlinear elastic shell which serves as a moving boundary of the physical spatial domain of the polymer fluid. We use the classical model by Koiter to describe the shell movement which yields a fully nonlinear fourth order hyperbolic equation. Our main result is the existence of a weak solution to the underlying system which exists until the Koiter energy degenerates or the flexible shell approaches a self-intersection.

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Electrochemical deposition induced continuum damage-healing framework for the cementitious composite

International Journal of Damage Mechanics ◽

10.1177/1056789521991871 ◽

2021 ◽

pp. 105678952199187

Author(s):

Hehua Zhu ◽

Qing Chen ◽

J Woody Ju ◽

Zhiguo Yan ◽

Zhengwu Jiang

Keyword(s):

Electrochemical Deposition ◽

Planck Equation ◽

Healing Time ◽

Aqueous Environment ◽

Continuum Damage ◽

Deposition Method ◽

Cementitious Composite ◽

Current Conservation ◽

Electrochemical Deposition Method ◽

Damage Healing

The electrochemical deposition method is a promising approach to repair the deteriorated concrete in the aqueous environment. In this paper, a continuum damage-healing framework is presented for the electrochemical deposition method based on the multi-field coupling growth process of the electrochemical deposition products. The ion transportation and the electrode reactions are characterized by employing the Nernst-Planck equation and the current conservation equation. The level set method is adopted to capture the growth of the deposition products. Based on the deposition process, a new empirical healing law is presented, with which a new continuum damage-healing framework is presented for electrochemical deposition method. Numerical examples are conducted by applying the presented framework to the damaged cementitious composite under the tensile loadings. The presented framework is compared with the classic continuum damage-healing theory and the experimental data. The results show that the presented models can describe the electrochemical deposition method induced damage-healing for the cementitious composite. Furthermore, the effects of the healing time, the solution concentration and the external voltage on the damage-healing behaviors are investigated based on our proposed framework.

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Numerical Solution of the Fokker-Planck Equation by Spectral Collocation and Finite-Element Methods for Stochastic Magnetization Dynamics

IEEE Transactions on Magnetics ◽

10.1109/tmag.2021.3084335 ◽

2021 ◽

pp. 1-1

Author(s):

Valentino Scalera ◽

Patrizio Ansalone ◽

Salvatore Perna ◽

Claudio Serpico ◽

Massimiliano d'Aquino

Keyword(s):

Finite Element ◽

Numerical Solution ◽

Finite Element Methods ◽

Planck Equation ◽

Magnetization Dynamics ◽

Spectral Collocation ◽

Fokker Planck Equation ◽

Fokker Planck

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Time-changed fractional Ornstein-Uhlenbeck process

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0022 ◽

2020 ◽

Vol 23 (2) ◽

pp. 450-483 ◽

Cited By ~ 2

Author(s):

Giacomo Ascione ◽

Yuliya Mishura ◽

Enrica Pirozzi

Keyword(s):

Planck Equation ◽

Uhlenbeck Process ◽

Fokker Planck Equation ◽

Ornstein Uhlenbeck Process ◽

Fokker Planck

AbstractWe define a time-changed fractional Ornstein-Uhlenbeck process by composing a fractional Ornstein-Uhlenbeck process with the inverse of a subordinator. Properties of the moments of such process are investigated and the existence of the density is shown. We also provide a generalized Fokker-Planck equation for the density of the process.

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