Gradient Flow Formulation Research Articles

Variational Structures Beyond Gradient Flows: a Macroscopic Fluctuation-Theory Perspective

Macroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the large-deviation rate functional for the density of the particle system. While large deviations can be studied in considerable generality, these variational structures are often restricted to systems in detailed balance. Using insights from macroscopic fluctuation theory, in this work we aim to generalise this variational connection beyond dissipative systems by augmenting densities with fluxes, which encode non-dissipative effects. Our main contribution is an abstract theory, which for a given flux-density cost and a quasipotential, provides a decomposition into dissipative and non-dissipative components and a generalised orthogonality relation between them. We then apply this abstract theory to various stochastic particle systems—independent copies of jump processes, zero-range processes, chemical-reaction networks in complex balance and lattice-gas models—without assuming detailed balance. For macroscopic equations arising out of these particle systems, we derive new variational formulations that generalise the classical gradient-flow formulation.

An unconditionally energy stable linear scheme for Poisson–Nernst–Planck equations

This paper proposes a linear, unconditionally energy-stable scheme for the Poisson–Nernst–Planck (PNP) equations. Based on a gradient-flow formulation of the PNP equations, the energy factorization approach is applied to linearize the logarithm function at the previous time step, resulting in a linear semi-implicit scheme. Numerical analysis is conducted to illustrate that the proposed fully discrete scheme has desired properties at a discrete level, such as unconditional unique solvability, mass conservation, and energy dissipation. Numerical simulations verify that the proposed scheme, as expected, is first-order accurate in time and second-order accurate in space. Further numerical tests confirm that the proposed scheme can indeed preserve the desired properties. Applications of our numerical scheme to the simulations of electrolyte solutions demonstrate that, as a linear energy stable scheme of efficiency, it will be promising in simulating complicated transport phenomena of charged systems.

Generalized gradient structures for measure-valued population dynamics and their large-population limit

We consider the forward Kolmogorov equation corresponding to measure-valued processes stemming from a class of interacting particle systems in population dynamics, including variations of the Bolker–Pacala–Dieckmann-Law model. Under the assumption of detailed balance, we provide a rigorous generalized gradient structure, incorporating the fluxes arising from the birth and death of the particles. Moreover, in the large population limit, we show convergence of the forward Kolmogorov equation to a Liouville equation, which is a transport equation associated with the mean-field limit of the underlying process. In addition, we show convergence of the corresponding gradient structures in the sense of Energy-Dissipation Principles, from which we establish a propagation of chaos result for the particle system and derive a generalized gradient-flow formulation for the mean-field limit.

Gradient Flows for Coupling Order Parameters and Mechanics

We construct a formal gradient flow structure for phase-field evolution coupled to mechanics in Lagrangian coordinates, present common ways to couple the evolution, and provide an incremental minimization strategy. While the usual presentation of continuum mechanics is intentionally brief, we construct an extensible functional analytical framework and a discretization approach that preserves the underlying variational structure. We consider phase separation and swelling of gels and then study stationary states of multiphase systems with surface tension and contact lines and show the robustness of the general approach for large deformations. We highlight differences between compressible and incompressible models and discuss issues of the sharp-interface limit for different magnitudes of the Cahn–Hilliard mobility.

Optimal control problems of nonlocal interaction equations

In the present work we deal with the existence of solutions for optimal control problems associated to transport equations. The behaviour of a population of individuals will be influenced by the presence of a population of control agents whose role is to lead the dynamics of the individuals towards a specific goal. The dynamics of the population of individuals is described by a suitable nonlocal transport equation, while the role of the population of agents is designed by the optimal control problem. This model has been first studied in [12] for a class of continuous nonlocal potentials, while in the present project we consider the case of mildly singular potentials in a gradient flow formulation of the target transport equation.

Emergence of biological transportation networks as a self-regulated process

We study self-regulating processes modeling biological transportation networks. Firstly, we write the formal $L^2$-gradient flow for the symmetric tensor valued diffusivity $D$ of a broad class of entropy dissipations associated with a purely diffusive model. The introduction of a prescribed electric potential leads to the Fokker-Planck equation, for whose entropy dissipations we also investigate the formal $L^2$-gradient flow. We derive an integral formula for the second variation of the dissipation functional, proving convexity (in dependence of diffusivity tensor) for a quadratic entropy density modeling Joule heating. Finally, we couple in the Poisson equation for the electric potential obtaining the Poisson-Nernst-Planck system. The formal gradient flow of the associated entropy loss functional is derived, giving an evolution equation for $D$ coupled with two auxiliary elliptic PDEs.

Second-order decoupled energy-stable schemes for Cahn-Hilliard-Navier-Stokes equations

The Cahn-Hilliard-Navier-Stokes (CHNS) equations represent the fundamental building blocks of hydrodynamic phase-field models for multiphase fluid flow dynamics. Due to the coupling between the Navier-Stokes equation and the Cahn-Hilliard equation, the CHNS system is non-trivial to be solved numerically. Traditionally, a numerical extrapolation for the coupling terms is used. However, such brute-force extrapolation usually destroys the intrinsic thermodynamic structures of this CHNS system. This paper proposes a new strategy to reformulate the CHNS system into a constraint gradient flow formulation, where the reversible and irreversible structures are clearly revealed. This guides us to propose operator splitting schemes that have several advantageous properties. First of all, the proposed schemes lead to several decoupled systems in smaller sizes to be solved at each time marching step. This significantly reduces computational costs. Secondly, the proposed schemes still guarantee the thermodynamic laws of the CHNS system at the discrete level. In addition, unlike the recently populated IEQ or SAV approaches using auxiliary variables, our resulting energy laws are formulated in the original variables. This is a significant improvement, as the modified energy laws with auxiliary variables sometimes deviate from the original energy law. Our proposed framework lays a foundation for designing decoupled and energy stable numerical algorithms for hydrodynamic phase-field models. Furthermore, various numerical algorithms can be obtained given different splitting steps, making this framework rather general. The proposed numerical algorithms are implemented. Their second-order temporal and spatial accuracy are verified numerically. Some numerical examples and benchmark problems are calculated to verify the effectiveness of the proposed schemes.

Large Deviations for Kac-Like Walks

We introduce a Kac’s type walk whose rate of binary collisions preserves the total momentum but not the kinetic energy. In the limit of large number of particles we describe the dynamics in terms of empirical measure and flow, proving the corresponding large deviation principle. The associated rate function has an explicit expression. As a byproduct of this analysis, we provide a gradient flow formulation of the Boltzmann-Kac equation.

Gradient flow formulation and second order numerical method for motion by mean curvature and contact line dynamics on rough surface

We study the dynamics of a droplet moving on an inclined rough surface in the absence of inertial and viscous stress effects. In this case, the dynamics of the droplet is a purely geometric motion in terms of the wetting domain and the capillary surface. Using a single graph representation, we interpret this geometric motion as a gradient flow on a manifold. We propose unconditionally stable first/second order numerical schemes to simulate this geometric motion of the droplet, which is described using motion by mean curvature coupled with moving contact lines. The schemes are based on (i) explicit moving boundaries, which decouple the dynamic updates of the contact lines and the capillary surface, (ii) an arbitrary Lagrangian–Eulerian method on moving grids and (iii) a predictor-corrector method with a nonlinear elliptic solver up to second order accuracy. For the case of quasi-static dynamics with continuous spatial variable in the numerical schemes, we prove the stability and convergence of the first/second order numerical schemes. To demonstrate the accuracy and long-time validation of the proposed schemes, several challenging computational examples – including breathing droplets, droplets on inhomogeneous rough surfaces and quasi-static Kelvin pendant droplets – are constructed and compared with exact solutions to quasi-static dynamics obtained by desingularized differential-algebraic system of equations (DAEs).

On the Completely Separable State for the Lohe Tensor Model

We study completely separable states of the Lohe tensor model and their asymptotic collective dynamics. Here, the completely separable state means that it is a tensor product of rank-1 tensors. For the generalized Lohe matrix model corresponding to the Lohe tensor model for rank-2 tensors with the same size, we observe that the component rank-1 tensors of the completely separable states satisfy the swarm double sphere model introduced in [Lohe in Physica D 412, 2020]. We also show that the swarm double sphere model can be represented as a gradient system with an analytic potential. Using this gradient flow formulation, we provide the swarm multisphere model on the product of multiple unit spheres with possibly different dimensions, and then we construct a completely separable state of the swarm multisphere model as a tensor product of rank-1 tensors which is a solution of the proposed swarm multisphere model. This concept of separability has been introduced in the theory of quantum information. Finally, we also provide a sufficient framework leading to the complete aggregation of completely separable states.

Existence of weak solutions to a cross-diffusion Cahn-Hilliard type system

The aim of this article is to study a Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects, degenerate mobility and where only one of the species does separate from the others. We define a notion of weak solution adapted to possible degeneracies and our main result is (global in time) existence. In order to overcome the lack of a-priori estimates, our proof uses the formal gradient flow structure of the system and an extension of the boundedness by entropy method which involves a careful analysis of an auxiliary variational problem. This allows to obtain solutions to an approximate, time-discrete system. Letting the time step size go to zero, we recover the desired weak solution where, due to their low regularity, the Cahn-Hilliard terms require a special treatment.

A gradient flow approach to linear Boltzmann equations

We introduce a gradient flow formulation of linear Boltzmann equations. Under a diffusive scaling we derive a diffusion equation by using the machinery of gradient flows.

A gradient flow formulation for the stochastic Amari neural field model.

We study stochastic Amari-type neural field equations, which are mean-field models for neural activity in the cortex. We prove that under certain assumptions on the coupling kernel, the neural field model can be viewed as a gradient flow in a nonlocal Hilbert space. This makes all gradient flow methods available for the analysis, which could previously not be used, as it was not known, whether a rigorous gradient flow formulation exists. We show that the equation is well-posed in the nonlocal Hilbert space in the sense that solutions starting in this space also remain in it for all times and space-time regularity results hold for the case of spatially correlated noise. Uniqueness of invariant measures, ergodic properties for the associated Feller semigroups, and several examples of kernels are also discussed.

An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems

As an extension of our previous work in Sun et.al (2018) [41], we develop a discontinuous Galerkin method for solving cross-diffusion systems with a formal gradient flow structure. These systems are associated with non-increasing entropy functionals. For a class of problems, the positivity (non-negativity) of solutions is also expected, which is implied by the physical model and is crucial to the entropy structure. The semi-discrete numerical scheme we propose is entropy stable. Furthermore, the scheme is also compatible with the positivity-preserving procedure in Zhang (2017) [42] in many scenarios. Hence the resulting fully discrete scheme is able to produce non-negative solutions. The method can be applied to both one-dimensional problems and two-dimensional problems on Cartesian meshes. Numerical examples are given to examine the performance of the method.

The two-loop energy\u2013momentum tensor within the gradient-flow formalism

The gradient-flow formulation of the energy–momentum tensor of QCD is extended to NNLO perturbation theory. This means that the Wilson coefficients which multiply the flowed operators in the corresponding expression for the regular energy–momentum tensor are calculated to this order. The result has been obtained by applying modern tools of regular perturbation theory, reducing the occurring two-loop integrals, which also include flow-time integrations, to a small set of master integrals which can be calculated analytically.

Uniformly Compressing Mean Curvature Flow

Michor and Mumford showed that the mean curvature flow is a gradient flow on a Riemannian structure with a degenerate geodesic distance. It is also known to destroy the uniform density of gridpoints on the evolving surfaces. We introduce a related geometric flow which is free of these drawbacks. Our flow can be viewed as a formal gradient flow on a certain submanifold of the Wasserstein space of probability measures endowed with Otto's Riemannian structure. We obtain a number of analytic results concerning well-posedness and long-time stability which are however restricted to the 1D case of evolution of loops.

Displacement convexity for the entropy in semi-discrete non-linear Fokker–Planck equations

The displacement λ-convexity of a non-standard entropy with respect to a non-local transportation metric in finite state spaces is shown using a gradient flow approach. The constant λ is computed explicitly in terms ofa prioriestimates of the solution to a finite-difference approximation of a non-linear Fokker–Planck equation. The key idea is to employ a new mean function, which defines the Onsager operator in the gradient flow formulation.

Nonlinear degenerate cross-diffusion systems with nonlocal interaction

We investigate a class of systems of partial differential equations with nonlinear cross-diffusion and nonlocal interactions, which are of interest in several contexts in social sciences, finance, biology, and real world applications. Assuming a uniform “coerciveness” assumption on the diffusion part, which allows to consider a large class of systems with degenerate cross-diffusion (i.e. of porous medium type) and relaxes sets of assumptions previously considered in the literature, we prove global-in-time existence of weak solutions by means of a semi-implicit version of the Jordan–Kinderlehrer–Otto scheme. Our approach allows to consider nonlocal interaction terms not necessarily yielding a formal gradient flow structure.

Well-posedness of evolution equations with time-dependent nonlinear mobility: A modified minimizing movement scheme

AbstractWe prove the existence of nonnegative weak solutions to a class of second- and fourth-order nonautonomous nonlinear evolution equations with an explicitly time-dependent mobility function posed on the whole space {{{\mathbb{R}}^{d}}}, for arbitrary {d\geq 1}. Exploiting a very formal gradient flow structure, the cornerstone of our proof is a modified version of the classical minimizing movement scheme for gradient flows. The mobility function is required to satisfy – at each time point separately – the conditions by which one can define a modified Wasserstein distance on the space of probability densities with finite second moment. The explicit dependency on the time variable is assumed to be at least of Lipschitz regularity. We also sketch possible extensions of our result to the case of bounded spatial domains and more general mobility functions.

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

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 implicit Euler scheme obtained by minimizing hF(ϱ)+W22(ϱ0,ϱ) among all ϱ satisfying the constraint for some ϱ0 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.