17 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
Quantization of the Riemann Zeta-Function and Cosmology
Quantization of the Riemann zeta-function is proposed. We treat the Riemann
zeta-function as a symbol of a pseudodifferential operator and study the
corresponding classical and quantum field theories. This approach is motivated
by the theory of p-adic strings and by recent works on stringy cosmological
models. We show that the Lagrangian for the zeta-function field is equivalent
to the sum of the Klein-Gordon Lagrangians with masses defined by the zeros of
the Riemann zeta-function. Quantization of the mathematics of Fermat-Wiles and
the Langlands program is indicated. The Beilinson conjectures on the values of
L-functions of motives are interpreted as dealing with the cosmological
constant problem. Possible cosmological applications of the zeta-function field
theory are discussed.Comment: 14 pages, corrected typos, references and comments adde
Derivation of the Boltzmann equation and entropy production in functional mechanics
A derivation of the Boltzmann equation from the Liouville equation by the use
of the Grad limiting procedure in a finite volume is proposed. We introduce two
scales of space-time: macro- and microscale and use the BBGKY hierarchy and the
functional formulation of classical mechanics. According to the functional
approach to mechanics, a state of a system of particles is formed from the
measurements, which are rational numbers. Hence, one can speak about the
accuracy of the initial probability density function in the Liouville equation.
We assume that the initial data for the microscopic density functions are
assigned by the macroscopic one (so, one can say about a kind of hierarchy and
subordination of the microscale to the macroscale) and derive the Boltzmann
equation, which leads to the entropy production.Comment: 14 page
Randomness in Classical Mechanics and Quantum Mechanics
The Copenhagen interpretation of quantum mechanics assumes the existence of
the classical deterministic Newtonian world. We argue that in fact the Newton
determinism in classical world does not hold and in classical mechanics there
is fundamental and irreducible randomness. The classical Newtonian trajectory
does not have a direct physical meaning since arbitrary real numbers are not
observable. There are classical uncertainty relations, i.e. the uncertainty
(errors of observation) in the determination of coordinate and momentum is
always positive (non zero).
A "functional" formulation of classical mechanics was suggested. The
fundamental equation of the microscopic dynamics in the functional approach is
not the Newton equation but the Liouville equation for the distribution
function of the single particle. Solutions of the Liouville equation have the
property of delocalization which accounts for irreversibility. The Newton
equation in this approach appears as an approximate equation describing the
dynamics of the average values of the position and momenta for not too long
time intervals. Corrections to the Newton trajectories are computed. An
interpretation of quantum mechanics is attempted in which both classical and
quantum mechanics contain fundamental randomness. Instead of an ensemble of
events one introduces an ensemble of observers.Comment: 12 pages, Late