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Considerable progress has recently been made with geometrical approaches to understanding and controlling small out-of-equilibrium systems, but a mathematically rigorous foundation for these methods has been lacking. Towards this end, we…
Optimal transport provides an inherently geometric and highly structured framework for studying spaces of probability measures, supplying a rich theoretical toolkit for contemporary statistics, machine learning, and generative modelling. In…
We introduce a new optimal transport distance between nonnegative finite Radon measures with possibly different masses. The construction is based on non-conservative continuity equations and a corresponding modified Benamou-Brenier formula.…
Advances in experimental techniques enable the precise manipulation of a large variety of active systems, which constantly dissipate energy to sustain nonequilibrium phenomena without any equilibrium equivalent. To design novel materials…
The three-body general problem is formulated as a problem of geodesic trajectories flows on the Riemannian manifold. It is proved that a curved space with local coordinate system allows to detect new hidden symmetries of the internal motion…
The optimal transport (OT) problem aims to find the most efficient mapping between two probability distributions under a given cost function, and has diverse applications in many fields such as machine learning, computer vision and computer…
We introduce Wasserstein-like dynamical transport distances between vector-valued densities on the real line. The mobility function from the scalar theory is replaced by a mobility matrix, that is subject to positivity and concavity…
The turnpike phenomenon stipulates that the solution of an optimal control problem in large time, remains essentially close to a steady-state of the dynamics, itself being the optimal solution of an associated static optimal control…
We introduce a numerical method for extracting minimal geodesics along the group of volume preserving maps, equipped with the L2 metric, which as observed by Arnold solve Euler's equations of inviscid incompressible fluids. The method…
We explore the idea that non-equilibrium steady states breaking detailed balance are obtained by deforming trajectories (lines in space-time) that have been sampled in a reference system with stochastic dynamics obeying detailed balance,…
Progress in miniaturized technology allows us to control physical systems at nanoscale with remarkable precision. Experimental advancements have sparked interest in control problems in stochastic thermodynamics, typically concerning a…
We prove that the transport of any differentiable scalar observable in $d$-dimensional non-equilibrium systems is bounded from above by the total entropy production scaled by the amount the observation "stretches" microscopic coordinates.…
We apply a finite-time geometric optimization framework to investigate thermal fluctuations and (non)equilibrium optimal processes in the $(2+1)$-dimensional BTZ black hole. Employing Hessian thermodynamic information metrics, we construct…
A relativistic self-gravitating equilibrium system with steady flow as well as spherical symmetry is discovered. The energy-momentum tensor contains the contribution of a current related to the flow and the metric tensor does an…
In this short note, we consider the problem of prescribing the Gauss curvature and image of the Gauss map for the graph of a function over a domain in Euclidean space. The prescription of the image of the Gauss map turns this into a second…
We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong…
The divergence of the heat conductivity in the thermodynamic limit is investigated in 2d-lattice models of anharmonic solids with nearest-neighbour interaction from single-well potentials. Two different numerical approaches based on…
Direct thermal-to-electric energy converters typically operate in the linear regime, where the ratio of actual maximum power relative to the ideal maximum power, the so-called fill factor (FF), is 0.25. By increasing the FF one can…
This paper contains a fully geometric formulation of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC). Although GENERIC, which is the sum of Hamiltonian mechanics and gradient dynamics, is a framework…
In this work we consider thermodynamic geometries defined as Hessians of different potentials and derive some useful formulae that show their complementary role in the description of thermodynamic systems with two degrees of freedom that…