Explicit Asymptotic Velocity of the Boundary between Particles and Antiparticles

A. A. Zamyatin,V. A. Malyshev,A. D. Manita

doi:10.5402/2012/327298

A. A. Zamyatin, V. A. Malyshev + Show 1 more

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https://doi.org/10.5402/2012/327298

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Journal: ISRN Mathematical Physics	Publication Date: Sep 25, 2012
Citations: 4	License type: CC BY 3.0

Affiliation: Lomonosov Moscow State University

Abstract

On the real line initially there are infinite number of particles on the positive half line, each having one of -negative velocities . Similarly, there are infinite number of antiparticles on the negative half line, each having one of -positive velocities . Each particle moves with constant speed, initially prescribed to it. When particle and antiparticle collide, they both disappear. It is the only interaction in the system. We find explicitly the large time asymptotics of —the coordinate of the last collision before between particle and antiparticle.

Highlights

We consider one-dimensional dynamical model of the boundary between two phases particles and antiparticles, bears and bulls where the boundary moves due to reaction annihilation, transaction of pairs of particles of different phases.Assume that at time t 0 infinite number of -particles and − -particles are situated correspondingly on R and R− and have one-point correlation functions: KL f x, v ρi x δ v − vi, f− x, v ρj− x δ v − vj− . i1 j1for any i, j, vi < 0, vj− > 0, ISRN Mathematical Physics that is, two phases move towards each other
The only interaction in the system is the following, when two particles of different phases find themselves at the same point they immediately disappear annihilate
It follows that the phases stay separated, and one might call any point in-between them the phase boundary e.g., it could be the point of the last collision

Summary

Introduction

We consider one-dimensional dynamical model of the boundary between two phases particles and antiparticles, bears and bulls where the boundary moves due to reaction annihilation, transaction of pairs of particles of different phases. The main result of the paper is the explicit formula for the asymptotic velocity of the boundary as the function of 2 K L parameters—densities and initial velocities. It appears to be continuous but at some hypersurface some first derivatives in the parameters do not exist. We consider only the case of constant densities ρi , ρi− , that is, the period of very small volatility in the market This simplification allows us to get explicit formulae.

Initial Conditions

Dynamics

The following condition is assumed:

Another Scaling

Example of Phase Transition

Balance Equations—Physical Evidence

Associated Random Walk

Ergodic Case

Induced Process

Induced Vectors Introduce the plane:

Dynamical System

Simple Examples of Random Walks and Dynamical Systems