Related papers: Matrix displacement convexity along density flows
We present a new formulation of some basic differential geometric notions on a smooth manifold M, in the setting of nonstandard analysis. In place of classical vector fields, for which one needs to construct the tangent bundle of M, we…
The matrix version of the entropy-power inequality for real or complex coefficients and variables is proved using a transportation argument that easily settles the equality case. An application to blind source extraction is given.
Convexity properties of the entropy along displacement interpolations are crucial in the Lott-Sturm-Villani theory of lower bounded curvature of geodesic measure spaces. As discrete spaces fail to be geodesic, an alternate analogous theory…
The paper considers the Euler system of PDE on a smooth compact Riemannian manifold of positive curvature without boundary, and the sphere ${\mathbb{S}}^2$ in particular. The paper interprets the Euler equations as a transport problem for…
Two dimensional flows on fixed smooth surfaces have been studied in the point of view of vorticity dynamics. Firstly, the related deformation theory including kinematics and kinetics is developed. Secondly, some primary relations in…
Locally broken symmetries are used across fields to transport matter, particles and information in preferential directions. Beyond local mechanisms, spatially distributed nonlinearities in crystalline media have enabled non-reciprocal…
We consider non-linear evolution equations arising from mean-field limits of particle systems on discrete spaces. We investigate a notion of curvature bounds for these dynamics based on convexity of the free energy along interpolations in a…
In an incompressible flow, fluid density remains invariant along fluid element trajectories. This implies that the spatial distribution of non-interacting noninertial particles in such flows cannot develop density inhomogeneities beyond…
The article demonstrates that the internal circulation velocity and patterns in sessile droplets on superhydrophobic surfaces is governed by the surface curvature. Particle Image Velocimetry reveals that increasing convexity deteriorates…
We show that transport in the presence of entropic barriers exhibits peculiar characteristics which makes it distinctly different from that occurring through energy barriers. The constrained dynamics yields a scaling regime for the particle…
The theory of disordered elastic systems is one of the most powerful frameworks to assess the physics of multiple systems that span from ferromagnets to migrating biological cells. In this formalism, one assumes that the system can be…
We study the quantitative convergence of drift-diffusion PDEs that arise as Wasserstein gradient flows of linearly convex functions over the space of probability measures on ${\mathbb R}^d$. In this setting, the objective is in general not…
Conserved integrals and invariants (advected scalars) are studied for the equations of radial compressible fluid/gas flow in $n>1$ dimensions. Apart from entropy, which is a well-know invariant, three additional invariants are found from an…
The larger the distance to instability from a matrix is, the more robustly stable the associated autonomous dynamical system is in the presence of uncertainties and typically the less severe transient behavior its solution exhibits.…
Our recent work identifies material surfaces in incompressible flows that extremize the transport of an arbitrary, weakly diffusive scalar field relative to neighboring surfaces. Such barriers and enhancers of transport can be located…
We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in…
We initiate a systematic study of intrinsic dimensional versions of classical functional inequalities which capture refined properties of the underlying objects. We focus on model spaces: Euclidean space, Hamming cube, and manifolds of…
A basic idea in optimal transport is that optimizers can be characterized through a geometric property of their support sets called cyclical monotonicity. In recent years, similar "monotonicity principles" have found applications in other…
The study of passive scalar transport in a turbulent velocity field leads naturally to the notion of generalized flows which are families of probability distributions on the space of solutions to the associated ODEs, which no longer satisfy…
Normalizing flows are deep generative models that enable efficient likelihood estimation and sampling through invertible transformations. A key challenge is to design linear layers that enhance expressiveness while maintaining efficient…