966 research outputs found
The Poisson equations in the nonholonomic Suslov problem: Integrability, meromorphic and hypergeometric solutions
We consider the problem of integrability of the Poisson equations describing
spatial motion of a rigid body in the classical nonholonomic Suslov problem. We
obtain necessary conditions for their solutions to be meromorphic and show that
under some further restrictions these conditions are also sufficient. The
latter lead to a family of explicit meromorphic solutions, which correspond to
rather special motions of the body in space. We also give explicit extra
polynomial integrals in this case.
In the more general case (but under one restriction), the Poisson equations
are transformed into a generalized third order hypergeometric equation. A study
of its monodromy group allows us also to calculate the "scattering" angle: the
angle between the axes of limit permanent rotations of the body in space
Solutions for the General, Confluent and Biconfluent Heun equations and their connection with Abel equations
In a recent paper, the canonical forms of a new multi-parameter class of Abel
differential equations, so-called AIR, all of whose members can be mapped into
Riccati equations, were shown to be related to the differential equations for
the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection
between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and
Biconfluent (BHE) equations is presented. This connection fixes the value of
one of the Heun parameters, expresses another one in terms of those remaining,
and provides closed form solutions in terms of pFq functions for the resulting
GHE, CHE and BHE, respectively depending on four, three and two irreducible
parameters. This connection also turns evident what is the relation between the
Heun parameters such that the solutions admit Liouvillian form, and suggests a
mechanism for relating linear equations with N and N-1 singularities through
the canonical forms of a non-linear equation of one order less.Comment: Original version submitted to Journal of Physics A: 16 pages, related
to math.GM/0002059 and math-ph/0402040. Revised version according to
referee's comments: 23 pages. Sign corrected (June/17) in formula (79).
Second revised version (July/25): 25 pages. See also
http://lie.uwaterloo.ca/odetools.ht
Preservation of Positivity by Dynamical Coarse-Graining
We compare different quantum Master equations for the time evolution of the
reduced density matrix. The widely applied secular approximation (rotating wave
approximation) applied in combination with the Born-Markov approximation
generates a Lindblad type master equation ensuring for completely positive and
stable evolution and is typically well applicable for optical baths. For phonon
baths however, the secular approximation is expected to be invalid. The usual
Markovian master equation does not generally preserve positivity of the density
matrix. As a solution we propose a coarse-graining approach with a dynamically
adapted coarse graining time scale. For some simple examples we demonstrate
that this preserves the accuracy of the integro-differential Born equation. For
large times we analytically show that the secular approximation master equation
is recovered. The method can in principle be extended to systems with a
dynamically changing system Hamiltonian, which is of special interest for
adiabatic quantum computation. We give some numerical examples for the
spin-boson model of cases where a spin system thermalizes rapidly, and other
examples where thermalization is not reached.Comment: 18 pages, 7 figures, reviewers suggestions included and tightened
presentation; accepted for publication in PR
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