10,804 research outputs found
Noise induced dissipation in Lebesgue-measure preserving maps on dimensional torus
We consider dissipative systems resulting from the Gaussian and
-stable noise perturbations of measure-preserving maps on the
dimensional torus. We study the dissipation time scale and its physical
implications as the noise level \vep vanishes.
We show that nonergodic maps give rise to an O(1/\vep) dissipation time
whereas ergodic toral automorphisms, including cat maps and their
-dimensional generalizations, have an O(\ln{(1/\vep)}) dissipation time
with a constant related to the minimal, {\em dimensionally averaged entropy}
among the automorphism's irreducible blocks. Our approach reduces the
calculation of the dissipation time to a nonlinear, arithmetic optimization
problem which is solved asymptotically by means of some fundamental theorems in
theories of convexity, Diophantine approximation and arithmetic progression. We
show that the same asymptotic can be reproduced by degenerate noises as well as
mere coarse-graining. We also discuss the implication of the dissipation time
in kinematic dynamo.Comment: The research is supported in part by the grant from U.S. National
Science Foundation, DMS-9971322 and Lech Wolowsk
Compactifications of moduli spaces inspired by mirror symmetry
We study moduli spaces of nonlinear sigma-models on Calabi-Yau manifolds,
using the one-loop semiclassical approximation. The data being parameterized
includes a choice of complex structure on the manifold, as well as some ``extra
structure'' described by means of classes in H^2. The expectation that this
moduli space is well-behaved in these ``extra structure'' directions leads us
to formulate a simple and compelling conjecture about the action of the
automorphism group on the K\"ahler cone. If true, it allows one to apply
Looijenga's ``semi-toric'' technique to construct a partial compactification of
the moduli space. We explore the implications which this construction has
concerning the properties of the moduli space of complex structures on a
``mirror partner'' of the original Calabi-Yau manifold. We also discuss how a
similarity which might have been noticed between certain work of Mumford and of
Mori from the 1970's produces (with hindsight) evidence for mirror symmetry
which was available in 1979. [The author is willing to mail hardcopy preprints
upon request.]Comment: 25 pp., LaTeX 2.09 with AmS-Font
The hyperbolic, the arithmetic and the quantum phase
We develop a new approach of the quantum phase in an Hilbert space of finite
dimension which is based on the relation between the physical concept of phase
locking and mathematical concepts such as cyclotomy and the Ramanujan sums. As
a result, phase variability looks quite similar to its classical counterpart,
having peaks at dimensions equal to a power of a prime number. Squeezing of the
phase noise is allowed for specific quantum states. The concept of phase
entanglement for Kloosterman pairs of phase-locked states is introduced.Comment: accepted for publication for the special issue of J. Opt. B, in
relation to ICSSUR, Puebla (Mexico): Foundations of Quantum Optics, to be
published in June 200
Statistical Mechanics of 2+1 Gravity From Riemann Zeta Function and Alexander Polynomial:Exact Results
In the recent publication (Journal of Geometry and Physics,33(2000)23-102) we
demonstrated that dynamics of 2+1 gravity can be described in terms of train
tracks. Train tracks were introduced by Thurston in connection with description
of dynamics of surface automorphisms. In this work we provide an example of
utilization of general formalism developed earlier. The complete exact solution
of the model problem describing equilibrium dynamics of train tracks on the
punctured torus is obtained. Being guided by similarities between the dynamics
of 2d liquid crystals and 2+1 gravity the partition function for gravity is
mapped into that for the Farey spin chain. The Farey spin chain partition
function, fortunately, is known exactly and has been thoroughly investigated
recently. Accordingly, the transition between the pseudo-Anosov and the
periodic dynamic regime (in Thurston's terminology) in the case of gravity is
being reinterpreted in terms of phase transitions in the Farey spin chain whose
partition function is just a ratio of two Riemann zeta functions. The mapping
into the spin chain is facilitated by recognition of a special role of the
Alexander polynomial for knots/links in study of dynamics of self
homeomorphisms of surfaces. At the end of paper, using some facts from the
theory of arithmetic hyperbolic 3-manifolds (initiated by Bianchi in 1892), we
develop systematic extension of the obtained results to noncompact Riemannian
surfaces of higher genus. Some of the obtained results are also useful for 3+1
gravity. In particular, using the theorem of Margulis, we provide new reasons
for the black hole existence in the Universe: black holes make our Universe
arithmetic. That is the discrete Lie groups of motion are arithmetic.Comment: 69 pages,11 figures. Journal of Geometry and Physics (in press
Statistical Mechanics of 2+1 Gravity From Riemann Zeta Function and Alexander Polynomial:Exact Results
In the recent publication (Journal of Geometry and Physics,33(2000)23-102) we
demonstrated that dynamics of 2+1 gravity can be described in terms of train
tracks. Train tracks were introduced by Thurston in connection with description
of dynamics of surface automorphisms. In this work we provide an example of
utilization of general formalism developed earlier. The complete exact solution
of the model problem describing equilibrium dynamics of train tracks on the
punctured torus is obtained. Being guided by similarities between the dynamics
of 2d liquid crystals and 2+1 gravity the partition function for gravity is
mapped into that for the Farey spin chain. The Farey spin chain partition
function, fortunately, is known exactly and has been thoroughly investigated
recently. Accordingly, the transition between the pseudo-Anosov and the
periodic dynamic regime (in Thurston's terminology) in the case of gravity is
being reinterpreted in terms of phase transitions in the Farey spin chain whose
partition function is just a ratio of two Riemann zeta functions. The mapping
into the spin chain is facilitated by recognition of a special role of the
Alexander polynomial for knots/links in study of dynamics of self
homeomorphisms of surfaces. At the end of paper, using some facts from the
theory of arithmetic hyperbolic 3-manifolds (initiated by Bianchi in 1892), we
develop systematic extension of the obtained results to noncompact Riemannian
surfaces of higher genus. Some of the obtained results are also useful for 3+1
gravity. In particular, using the theorem of Margulis, we provide new reasons
for the black hole existence in the Universe: black holes make our Universe
arithmetic. That is the discrete Lie groups of motion are arithmetic.Comment: 69 pages,11 figures. Journal of Geometry and Physics (in press
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