9 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
Noise Induced Dissipation in Discrete-Time Classical and Quantum Dynamical Systems
We introduce a new characteristics of chaoticity of classical and quantum
dynamical systems by defining the notion of the dissipation time which enables
us to test how the system responds to the noise and in particular to measure
the speed at which an initially closed, conservative system converges to the
equilibrium when subjected to noisy (stochastic) perturbations. We prove fast
dissipation result for classical Anosov systems and general exponentially
mixing maps. Slow dissipation result is proved for regular systems including
non-weakly mixing maps. In quantum setting we study simultaneous semiclassical
and small noise asymptotics of the dissipation time of quantized toral
symplectomorphisms (generalized cat maps) and derive sharp bounds for
semiclassical regime in which quantum-classical correspondence of dissipation
times holds.Comment: PhD Dissertation, University of California at Davis, June 2004,
LaTex, 195 page
Pade approximants of random Stieltjes series
We consider the random continued fraction S(t) := 1/(s_1 + t/(s_2 + t/(s_3 +
>...))) where the s_n are independent random variables with the same gamma
distribution. For every realisation of the sequence, S(t) defines a Stieltjes
function. We study the convergence of the finite truncations of the continued
fraction or, equivalently, of the diagonal Pade approximants of the function
S(t). By using the Dyson--Schmidt method for an equivalent one-dimensional
disordered system, and the results of Marklof et al. (2005), we obtain explicit
formulae (in terms of modified Bessel functions) for the almost-sure rate of
convergence of these approximants, and for the almost-sure distribution of
their poles.Comment: To appear in Proc Roy So
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Noise Induced Dissipation in Discrete-Time Classical and Quantum Dynamical Systems
We introduce a new characteristics of chaoticity of classical and quantum dynamical
systems by defining the notion of the dissipation time which enables us to test how the
system responds to the noise and in particular to measure the speed at which an initially
closed, conservative system converges to the equilibrium when subjected to noisy
(stochastic) perturbations. We prove fast dissipation result for classical Anosov systems
and general exponentially mixing maps. Slow dissipation result is proved for regular
systems including non-weakly mixing maps. In quantum setting we study simultaneous
semiclassical and small noise asymptotics of the dissipation time of quantized toral
symplectomorphisms (generalized cat maps) and derive sharp bounds for semiclassical regime
in which quantum-classical correspondence of dissipation times holds