28 research outputs found
Information dynamics: Temporal behavior of uncertainty measures
We carry out a systematic study of uncertainty measures that are generic to
dynamical processes of varied origins, provided they induce suitable continuous
probability distributions. The major technical tool are the information theory
methods and inequalities satisfied by Fisher and Shannon information measures.
We focus on a compatibility of these inequalities with the prescribed
(deterministic, random or quantum) temporal behavior of pertinent probability
densities.Comment: Incorporates cond-mat/0604538, title, abstract changed, text
modified, to appear in Cent. Eur. J. Phy
Big Entropy Fluctuations in Statistical Equilibrium: The Macroscopic Kinetics
Large entropy fluctuations in an equilibrium steady state of classical
mechanics were studied in extensive numerical experiments on a simple
2--freedom strongly chaotic Hamiltonian model described by the modified Arnold
cat map. The rise and fall of a large separated fluctuation was shown to be
described by the (regular and stable) "macroscopic" kinetics both fast
(ballistic) and slow (diffusive). We abandoned a vague problem of "appropriate"
initial conditions by observing (in a long run)spontaneous birth and death of
arbitrarily big fluctuations for any initial state of our dynamical model.
Statistics of the infinite chain of fluctuations, reminiscent to the Poincar\'e
recurrences, was shown to be Poissonian. A simple empirical relation for the
mean period between the fluctuations (Poincar\'e "cycle") has been found and
confirmed in numerical experiments. A new representation of the entropy via the
variance of only a few trajectories ("particles") is proposed which greatly
facilitates the computation, being at the same time fairly accurate for big
fluctuations. The relation of our results to a long standing debates over
statistical "irreversibility" and the "time arrow" is briefly discussed too.Comment: Latex 2.09, 26 pages, 6 figure
Big Entropy Fluctuations in Nonequilibrium Steady State: A Simple Model with Gauss Heat Bath
Large entropy fluctuations in a nonequilibrium steady state of classical
mechanics were studied in extensive numerical experiments on a simple 2-freedom
model with the so-called Gauss time-reversible thermostat. The local
fluctuations (on a set of fixed trajectory segments) from the average heat
entropy absorbed in thermostat were found to be non-Gaussian. Approximately,
the fluctuations can be discribed by a two-Gaussian distribution with a
crossover independent of the segment length and the number of trajectories
('particles'). The distribution itself does depend on both, approaching the
single standard Gaussian distribution as any of those parameters increases. The
global time-dependent fluctuations turned out to be qualitatively different in
that they have a strict upper bound much less than the average entropy
production. Thus, unlike the equilibrium steady state, the recovery of the
initial low entropy becomes impossible, after a sufficiently long time, even in
the largest fluctuations. However, preliminary numerical experiments and the
theoretical estimates in the special case of the critical dynamics with
superdiffusion suggest the existence of infinitely many Poincar\'e recurrences
to the initial state and beyond. This is a new interesting phenomenon to be
farther studied together with some other open questions. Relation of this
particular example of nonequilibrium steady state to a long-standing persistent
controversy over statistical 'irreversibility', or the notorious 'time arrow',
is also discussed. In conclusion, an unsolved problem of the origin of the
causality 'principle' is touched upon.Comment: 21 pages, 7 figure