A convergent string method: Existence and approximation for the Hamiltonian boundary-value problem

This article studies the existence of long-time solutions to the Hamiltonian boundary value problem, and their consistent numerical approximation. Such a boundary value problem is, for example, common in Molecular Dynamics, where one aims at finding a dynamic trajectory that joins a given initial state with a final one, with the evolution being governed by classical (Hamiltonian) dynamics. The setting considered here is sufficiently general so that long time transition trajectories connecting two configurations can be included, provided the total energy $E$ is chosen suitably. In particular, the formulation presented here can be used to detect transition paths between two stable basins and thus to prove the existence of long-time trajectories. The starting point is the formulation of the equation of motion of classical mechanics in the framework of Jacobi's principle; a curve shortening procedure inspired by Birkhoff's method is then developed to find geodesic solutions. This approach can be viewed as a string method

Entropy production and the geometry of dissipative evolution equations

Purely dissipative evolution equations are often cast as gradient flow structures, $\dot{\mathbf{z}}=K(\mathbf{z})DS(\mathbf{z})$, where the variable $\mathbf{z}$ of interest evolves towards the maximum of a functional $S$ according to a metric defined by an operator $K$. While the functional often follows immediately from physical considerations (e.g., the thermodynamic entropy), the operator $K$ and the associated geometry does not necessarily so (e.g., Wasserstein geometry for diffusion). In this paper, we present a variational statement in the sense of maximum entropy production that directly delivers a relationship between the operator $K$ and the constraints of the system. In particular, the Wasserstein metric naturally arises here from the conservation of mass or energy, and depends on the Onsager resistivity tensor, which, itself, may be understood as another metric, as in the Steepest Entropy Ascent formalism. This new variational principle is exemplified here for the simultaneous evolution of conserved and non-conserved quantities in open systems. It thus extends the classical Onsager flux-force relationships and the associated variational statement to variables that do not have a flux associated to them. We further show that the metric structure $K$ is intimately linked to the celebrated Freidlin-Wentzell theory of stochastically perturbed gradient flows, and that the proposed variational principle encloses an infinite-dimensional fluctuation-dissipation statement

Travelling wavefronts in nonlocal diffusion equations with nonlocal delay effects

This paper deals with the existence, monotonicity, uniqueness and asymptotic behaviour of travelling wavefronts for a class of temporally delayed, spatially nonlocal diffusion equations

Geometrical interpretation of fluctuating hydrodynamics in diffusive systems

We discuss geometric formulations of hydrodynamic limits in diffusive systems. Specifically, we describe a geometrical construction in the space of density profiles --- the Wasserstein geometry --- which allows the deterministic hydrodynamic evolution of the systems to be related to steepest descent of the free energy, and show how this formulation can be related to most probable paths of mesoscopic dissipative systems. The geometric viewpoint is also linked to fluctuating hydrodynamics of these systems via a saddle point argument.Comment: 19 page

Well-posedness for a regularised inertial Dean-Kawasaki model for slender particles in several space dimensions

A stochastic PDE, describing mesoscopic fluctuations in systems of weakly interacting inertial particles of finite volume, is proposed and analysed in any finite dimension $d\in\mathbb{N}$. It is a regularised and inertial version of the Dean-Kawasaki model. A high-probability well-posedness theory for this model is developed. This theory improves significantly on the spatial scaling restrictions imposed in an earlier work of the same authors, which applied only to significantly larger particles in one dimension. The well-posedness theory now applies in $d$-dimensions when the particle-width $\epsilon$ is proportional to $N^{-1/\theta}$ for $\theta>2d$ and $N$ is the number of particles. This scaling is optimal in a certain Sobolev norm. Key tools of the analysis are fractional Sobolev spaces, sharp bounds on Bessel functions, separability of the regularisation in the $d$-spatial dimensions, and use of the Fa\`a di Bruno's formula.Comment: 28 pages, no figure

On the Γ\Gamma-limit for a non-uniformly bounded sequence of two phase metric functionals

In this study we consider the $\Gamma$-limit of a highly oscillatory Riemannian metric length functional as its period tends to 0. The metric coefficient takes values in either $\{1,\infty\}$ or $\{1,\beta \varepsilon^{-p}\}$ where $\beta,\varepsilon > 0$ and $p \in (0,\infty)$. We find that for a large class of metrics, in particular those metrics whose surface of discontinuity forms a differentiable manifold, the $\Gamma$-limit exists, as in the uniformly bounded case. However, when one attempts to determine the $\Gamma$-limit for the corresponding boundary value problem, the existence of the $\Gamma$-limit depends on the value of $p$. Specifically, we show that the power $p=1$ is critical in that the $\Gamma$-limit exists for $p < 1$, whereas it ceases to exist for $p \geq 1$. The results here have applications in both nonlinear optics and the effective description of a Hamiltonian particle in a discontinuous potential.Comment: 31 pages, 1 figure. Submitte