2 research outputs found
Rolling in the Higgs Model and Elliptic Functions
Asymptotic methods in nonlinear dynamics are used to improve perturbation
theory results in the oscillations regime. However, for some problems of
nonlinear dynamics, particularly in the case of Higgs (Duffing) equation and
the Friedmann cosmological equations, not only small oscillations regime is of
interest but also the regime of rolling (climbing), more precisely the rolling
from a top (climbing to a top). In the Friedman cosmology, where the slow
rolling regime is often used, the rolling from a top (not necessary slow) is of
interest too.
In the present work a method for approximate solution to the Higgs equation
in the rolling regime is presented. It is shown that in order to improve
perturbation theory in the rolling regime turns out to be effective not to use
an expansion in trigonometric functions as it is done in case of small
oscillations but use expansions in hyperbolic functions instead. This regime is
investigated using the representation of the solution in terms of elliptic
functions. An accuracy of the corresponding approximation is estimated.Comment: Latex, 36 Pages, 8 figures, typos correcte
Derivation of the particle dynamics from kinetic equations
We consider the microscopic solutions of the Boltzmann-Enskog equation
discovered by Bogolyubov. The fact that the time-irreversible kinetic equation
has time-reversible microscopic solutions is rather surprising. We analyze this
paradox and show that the reversibility or irreversibility property of the
Boltzmann-Enskog equation depends on the considered class of solutions. If the
considered solutions have the form of sums of delta-functions, then the
equation is reversible. If the considered solutions belong to the class of
continuously differentiable functions, then the equation is irreversible. Also,
we construct the so called approximate microscopic solutions. These solutions
are continuously differentiable and they are reversible on bounded time
intervals. This analysis suggests a way to reconcile the time-irreversible
kinetic equations with the time-reversible particle dynamics. Usually one tries
to derive the kinetic equations from the particle dynamics. On the contrary, we
postulate the Boltzmann-Enskog equation or another kinetic equation and treat
their microscopic solutions as the particle dynamics. So, instead of the
derivation of the kinetic equations from the microdynamics we suggest a kind of
derivation of the microdynamics from the kinetic equations.Comment: 18 pages; some misprints have been corrected, some references have
been adde