Global existence for a nonlocal and nonlinear Fokker–Planck equation

Dreyer, Wolfgang; Huth, Robert; Mielke, Alexander; Rehberg, Joachim; Winkler, Michael

doi:10.1007/s00033-014-0401-1

Cited by 12 publications

(15 citation statements)

References 12 publications

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“…In addition, this provides an alternative well-posedness result to [14,Lemma 1] that is based on a fixed point argument. Let us also note, that well-posedness in the case of compact state space is also obtained in [8].…”

Section: Introductionmentioning

confidence: 81%

“…In this section we investigate the evolution of the constrained Fokker-Planck equation (1.1) under the assumption that the external forcing becomes constant and under quadratic growth assumption at infinity of the potential H. The general idea is based on exploiting the entropy-dissipation identity (1.5). This strategy was partly also applied in [8,Chapter 5] and [10,Chapter 7] to derive the qualitative trend to equilibrium. We complemented this result with a quantitative rate of convergence to equilibrium based on the investigation of suitable relative entropies with respect to local equilibrium sates.…”

Section: Long Time Behaviourmentioning

confidence: 99%

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Gradient flow formulation and longtime behaviour of a constrained Fokker–Planck equation

2017

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Abstract. We consider a Fokker-Planck equation which is coupled to an externally given time-dependent constraint on its first moment. This constraint introduces a Lagrange-multiplier which renders the equation nonlocal and nonlinear.In this paper we exploit an interpretation of this equation as a Wasserstein gradient flow of a free energy F on a time-constrained manifold. First, we prove existence of solutions by passing to the limit in an explicit Euler scheme obtained by minimizing hF ( ) + W 2 2 ( 0 , ) among all satisfying the constraint for some 0 and time-step h > 0.Second, we provide quantitative estimates for the rate of convergence to equilibrium when the constraint converges to a constant. The proof is based on the investigation of a suitable relative entropy with respect to minimizers of the free energy chosen according to the constraint. The rate of convergence can be explicitly expressed in terms of constants in suitable logarithmic Sobolev inequalities.

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Section: Introductionmentioning

confidence: 81%

Section: Long Time Behaviourmentioning

confidence: 99%

Gradient flow formulation and longtime behaviour of a constrained Fokker–Planck equation

2017

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“…In particular, the inequality dE ≤ σ d holds along each trajectory and can be viewed as the Second Law of Thermodynamics, evaluated for the free energy of the many-particle ensemble in the presence of the dynamical control (FP 2 ); see [DGH11] for the physical interpretation of (4) and (5). Moreover, it has been shown in [DHM + 14] that the nonlocal equations (FP 1 )+(FP 2 ) can in fact be interpreted as a constraint gradient system with proper Lagrangian multiplier σ.…”

Section: Existence and Properties Of Solutionsmentioning

confidence: 99%

“…which turns (FP 1 ) into a nonlocal, nonlinear, and non-autonomous PDE. Nonlocal Fokker-Planck equations like (FP 1 )+(FP 2 ) have been introduced in [DGH11] in order to model the hysteretic behavior of many-particle storage systems such as modern Lithium-ion batteries (for the physical background, we also refer to [DJG + 10]). In this context, x ∈ R describes the thermodynamic state of a single particle (nano-particle made of iron-phosphate in the battery case), H is the free energy of each particle, and ν accounts for entropic effects.…”

Section: Introductionmentioning

confidence: 99%

Rate-Independent Dynamics and Kramers-Type Phase Transitions in Nonlocal Fokker–Planck Equations with Dynamical Control

Herrmann

Niethammer

Velázquez

2014

Arch Rational Mech Anal

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The hysteretic behavior of many-particle systems with non-convex free energy can be modeled by nonlocal Fokker-Planck equations that involve two small parameters and are driven by a timedependent constraint. In this paper we consider the fast reaction regime related to Kramers-type phase transitions and show that the dynamics in the small-parameter limit can be described by a rate-independent evolution equation with hysteresis. For the proof we first derive mass-dissipation estimates by means of Muckenhoupt constants, formulate conditional stability estimates, and characterize the mass flux between the different phases in terms of moment estimates that encode large deviation results. Afterwards we combine all these partial results and establish the dynamical stability of localized peaks as well as sufficiently strong compactness results for the basic macroscopic quantities.

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“…Our work is motivated by such nonlinear dynamics in the context of an ensemble of particles that exhibit bistable reaction kinetics, in particular electrochemical intercalation reactions. Dreyer et al [8][9][10][11][12] used Fokker-Planck equation to describe the probability distribution of the state of particles in many-particle storage systems such as lithium ion batteries and interconnected rubber balloons, in particular the phase transition due to regions of instability where the chemical potential decreases with lithium concentration and pressure decreases with volume. The model is capable of predicting voltage hysteresis between slow charge and discharge.…”

Section: Introductionmentioning

confidence: 99%

Population dynamics of driven autocatalytic reactive mixtures

Zhao

Bazant

2019

Phys. Rev. E

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Motivated by the theory of reaction kinetics based on nonequilibrium thermodynamics [1] and the linear stability of driven reaction-diffusion[2], we apply the Fokker-Planck equation to describe the population dynamics of an ensemble of reactive particles in contact with a chemical reservoir. We illustrate the effect of autocatalysis on the population dynamics by comparing systems with identical thermodynamics yet different reaction kinetics. The dynamic phase behavior of the system may be entirely different from what its thermodynamics may suggest. By defining phase separation for a particle ensemble to be when the probability distribution is bimodal, we find that thermodynamic phase separation may be suppressed by autoinhibitory reactions, while autocatalysis enhances phase separation and in some cases induce the ensemble that consists of thermodynamically single-phase systems to segregate into two distinct populations, which we term fictitious phase separation. Asymmetric reaction kinetics also results in qualitatively different population dynamics upon reversing the reaction direction. In the limit of negligible fluctuations, we use method of characteristics and linearization to study the evolution of the standard deviation of concentration as well as the condition for phase separation, in good agreement with the full numerical solution. Applications are discussed to Li-ion batteries and in situ x-ray diffraction.arXiv:1901.05575v1 [physics.chem-ph]

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Global existence for a nonlocal and nonlinear Fokker–Planck equation

Cited by 12 publications

References 12 publications

Gradient flow formulation and longtime behaviour of a constrained Fokker–Planck equation

Gradient flow formulation and longtime behaviour of a constrained Fokker–Planck equation

Rate-Independent Dynamics and Kramers-Type Phase Transitions in Nonlocal Fokker–Planck Equations with Dynamical Control

Population dynamics of driven autocatalytic reactive mixtures

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