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EP/L002302/1 - Advances in Kinetic Theory: Cercignani's Many Body Conjecture, Chaoticity and Entropic Chaos.

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Dr A Einav EP/L002302/1 - Advances in Kinetic Theory: Cercignani's Many Body Conjecture, Chaoticity and Entropic Chaos.

Fellow - Pure Maths and Mathematical Statistics, University of Cambridge

Scheme

EPSRC Fellowship

Research Areas

Mathematical Analysis Mathematical Analysis

Non-linear Systems Non-linear Systems

Start Date

10/2013

End Date

09/2016

Value

£234,019

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Grant Description

Summary and Description of the grant

The subject of Kinetic Theory investigates systems consisting of a large amount of objects and how they evolve. Some examples for such systems are galaxies, plasma and dilute gases.
One of the most important equations in Kinetic theory, one that describes the evolution of the distribution function of a dilute gas, is the Boltzmann equation. Discovered originally by Maxwell (in some form), the equation is named after Boltzmann due to his many contributions to its investigation.
In his work, Boltzmann introduced a new quantity called the entropy, measuring the disorder in the problem, and showed his famous H-theorem: The entropy must decrease along solutions to the Boltzmann equation. Combining this with the observation that the Gaussian function is a time-independent equilibrium to the equation, Boltzmann managed to deduce that solutions to the Boltzmann equation must converge to an appropriate Gaussian function as the time goes to infinity.
Unfortunately the Boltzmann equation has an intrinsic problem - it is not reversible in time. If we assume that the laws governing the evolution of the particles are reversible in time, we would expect to find that after long enough time the system had returned to its initial state, contradicting Boltzmann's convergence result. The key to see why there isn't any contradiction was given by Boltzmann himself: the idea is that the equation may still be valid in a time scale that is much shorter than that it will take the system to return to its original state. As such, the question of how quickly the solution to the Boltzmann equation converges to the equilibrium state is of paramount importance in Kinetic Theory. The main goal of many studies, this programme included, is to show that the rate of convergence is exponential.
There have been many attempts to try and show how reversible Newtonian systems can lead to an irreversible equation. The most successful ones so far involve many particle systems, which converge in some sense to an irreversible equation as the number of particles goes to infinity. A prime example to this is Kac's and McKean's Models. In both models we consider a system of many particles undergoing binary collisions with randomness in the scattering process. Focusing on only one particle and letting the number of particles go to infinity yields the spatially homogeneous Boltzmann equation (or a simplified version of it in one dimension) as long as we assume chaoticity, that is, that any fixed number of particles become more and more independent as the total number of particles increases.
Since the many particle models are linear, while the Boltzmann equation is very non-linear, we may try to find the rate of convergence to equilibrium in such models, and 'transfer' it to the limit. This programme has started at the late 1950s, and had several successes, yet to this day a satisfactory convergence rate from the linear models to the Boltzmann equation hasn't been established.
The goal of the proposed research is to prove a conjecture by Cercignani's on the rate of convergence to equilibrium using the entropy concept in the many particle system. We propose to do it by further investigation of the affect of moments of the one particle marginal on the expression attained from formally differentiating the entropy under the linear evolution, as well as investigating connection between it and the important concept of entropic chaoticity, which measures how far our distribution is from a completely 'independent' case. Another avenue of research this project will undertake is to try and see if one can find a better quantity to measure the distance between the evolved distribution and the equilibrium state. One prominent distance, that has proved useful in recent studies, is the Wasserstein distance.
Lastly, time permitting, we will investigate other possible convergence results, as well as the concept of chaoticity and entropic chaoticity in other many particle models.

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Grant Event Details:
Name: Advances in Kinetic Theory: Cercignani's Many Body Conjecture, Chaoticity and Entropic Chaos. - EP/L002302/1
Start Date: 2013-10-01T00:00:00+00:00
End Date: 2016-09-30T00:00:00+00:00

Organization: University of Cambridge

Description: The subject of Kinetic Theory investigates systems consisting of a large amount of objects and how they evolve. Some examples for such systems are galaxies, plasma and dilute gases. One of the most important equations in Kinetic theory, one that describes ...