Convergence to Equilibrium in Wasserstein distance for damped Euler equations with interaction forces

We develop tools to construct Lyapunov functionals on the space of probability measures in order to investigate the convergence to global equilibrium of a damped Euler system under the influence of external and interaction potential forces with respect to the 2-Wasserstein distance. We also discuss the overdamped limit to a nonlocal equation used in the modelling of granular media with respect to the 2-Wasserstein distance, and provide rigorous proofs for particular examples in one spatial dimension

Structure preserving schemes for the continuum Kuramoto model: phase transitions

The construction of numerical schemes for the Kuramoto model is challenging due to the structural properties of the system which are essential in order to capture the correct physical behavior, like the description of stationary states and phase transitions. Additional difficulties are represented by the high dimensionality of the problem in presence of multiple frequencies. In this paper, we develop numerical methods which are capable to preserve these structural properties of the Kuramoto equation in the presence of diffusion and to solve efficiently the multiple frequencies case. The novel schemes are then used to numerically investigate the phase transitions in the case of identical and non identical oscillators

Non-Gaussianity and gravitational wave background in curvaton with a double well potential

We study the density perturbation by a curvaton with a double well potential and estimate the nonlinear parameters for non-Gaussianity and the amplitude of gravitational wave background generated during inflation. The predicted nonlinear parameters strongly depend on the size of a curvaton self-coupling constant as well as the reheating temperature after inflation for a given initial amplitude of the curvaton. The difference from usual massive self-interacting curvaton is also emphasized.Comment: 23 pages, 6 figure

Modulated reheating by curvaton

There might be a light scalar field during inflation which is not responsible for the accelerating inflationary expansion. Then, its quantum fluctuation is stretched during inflation. This scalar field could be a curvaton, if it decays at a late time. In addition, if the inflaton decay rate depends on the light scalar field expectation value by interactions between them, density perturbations could be generated by the quantum fluctuation of the light field when the inflaton decays. This is modulated reheating mechanism. We study curvature perturbation in models where a light scalar field does not only play a role of curvaton but also induce modulated reheating at the inflaton decay. We calculate the non-linearity parameters as well as the scalar spectral index and the tensor-to-scalar ratio. We find that there is a parameter region where non-linearity parameters are also significantly enhanced by the cancellation between the modulated effect and the curvaton contribution. For the simple quadratic potential model of both inflaton and curvaton, both tensor-to-scalar ratio and nonlinearity parameters could be simultaneously large.Comment: 26 pages, 22 figure

An analytical framework for a consensus-based global optimization method

In this paper we provide an analytical framework for investigating the efficiency of a consensus-based model for tackling global optimization problems. This work justifies the optimization algorithm in the mean-field sense showing the convergence to the global minimizer for a large class of functions. Theoretical results on consensus estimates are then illustrated by numerical simulations where variants of the method including nonlinear diffusion are introduced

Mean-field limit for collective behavior models with sharp sensitivity regions

We rigorously show the mean-field limit for a large class of swarming individual based models with local sharp sensitivity regions. For instance, these models include nonlocal repulsive-attractive forces locally averaged over sharp vision cones and Cucker-Smale interactions with discontinuous communication weights. We construct global-in-time defined notion of solutions through a differential inclusion system corresponding to the particle descriptions. We estimate the error between the solutions to the differential inclusion system and weak solutions to the expected limiting kinetic equation by employing tools from optimal transport theory. Quantitative bounds on the expansion of the 1-Wasserstein distance along flows based on a weak-strong stability estimate are obtained. We also provide different examples of realistic sensitivity sets satisfying the assumptions of our main results