876 research outputs found
Classical dynamics on graphs
We consider the classical evolution of a particle on a graph by using a
time-continuous Frobenius-Perron operator which generalizes previous
propositions. In this way, the relaxation rates as well as the chaotic
properties can be defined for the time-continuous classical dynamics on graphs.
These properties are given as the zeros of some periodic-orbit zeta functions.
We consider in detail the case of infinite periodic graphs where the particle
undergoes a diffusion process. The infinite spatial extension is taken into
account by Fourier transforms which decompose the observables and probability
densities into sectors corresponding to different values of the wave number.
The hydrodynamic modes of diffusion are studied by an eigenvalue problem of a
Frobenius-Perron operator corresponding to a given sector. The diffusion
coefficient is obtained from the hydrodynamic modes of diffusion and has the
Green-Kubo form. Moreover, we study finite but large open graphs which converge
to the infinite periodic graph when their size goes to infinity. The lifetime
of the particle on the open graph is shown to correspond to the lifetime of a
system which undergoes a diffusion process before it escapes.Comment: 42 pages and 8 figure
Fractal Dimensions of the Hydrodynamic Modes of Diffusion
We consider the time-dependent statistical distributions of diffusive
processes in relaxation to a stationary state for simple, two dimensional
chaotic models based upon random walks on a line. We show that the cumulative
functions of the hydrodynamic modes of diffusion form fractal curves in the
complex plane, with a Hausdorff dimension larger than one. In the limit of
vanishing wavenumber, we derive a simple expression of the diffusion
coefficient in terms of this Hausdorff dimension and the positive Lyapunov
exponent of the chaotic model.Comment: 20 pages, 6 figures, submitted to Nonlinearit
The Fractality of the Hydrodynamic Modes of Diffusion
Transport by normal diffusion can be decomposed into the so-called
hydrodynamic modes which relax exponentially toward the equilibrium state. In
chaotic systems with two degrees of freedom, the fine scale structure of these
hydrodynamic modes is singular and fractal. We characterize them by their
Hausdorff dimension which is given in terms of Ruelle's topological pressure.
For long-wavelength modes, we derive a striking relation between the Hausdorff
dimension, the diffusion coefficient, and the positive Lyapunov exponent of the
system. This relation is tested numerically on two chaotic systems exhibiting
diffusion, both periodic Lorentz gases, one with hard repulsive forces, the
other with attractive, Yukawa forces. The agreement of the data with the theory
is excellent
Entropy Production in a Persistent Random Walk
We consider a one-dimensional persisent random walk viewed as a deterministic
process with a form of time reversal symmetry. Particle reservoirs placed at
both ends of the system induce a density current which drives the system out of
equilibrium. The phase space distribution is singular in the stationary state
and has a cumulative form expressed in terms of generalized Takagi functions.
The entropy production rate is computed using the coarse-graining formalism of
Gaspard, Gilbert and Dorfman. In the continuum limit, we show that the value of
the entropy production rate is independent of the coarse-graining and agrees
with the phenomenological entropy production rate of irreversible
thermodynamics.Comment: 21 pages, 8 figures, to appear in Physica
Kinetics and thermodynamics of first-order Markov chain copolymerization
We report a theoretical study of stochastic processes modeling the growth of
first-order Markov copolymers, as well as the reversed reaction of
depolymerization. These processes are ruled by kinetic equations describing
both the attachment and detachment of monomers. Exact solutions are obtained
for these kinetic equations in the steady regimes of multicomponent
copolymerization and depolymerization. Thermodynamic equilibrium is identified
as the state at which the growth velocity is vanishing on average and where
detailed balance is satisfied. Away from equilibrium, the analytical expression
of the thermodynamic entropy production is deduced in terms of the Shannon
disorder per monomer in the copolymer sequence. The Mayo-Lewis equation is
recovered in the fully irreversible growth regime. The theory also applies to
Bernoullian chains in the case where the attachment and detachment rates only
depend on the reacting monomer
Transport and dynamics on open quantum graphs
We study the classical limit of quantum mechanics on graphs by introducing a
Wigner function for graphs. The classical dynamics is compared to the quantum
dynamics obtained from the propagator. In particular we consider extended open
graphs whose classical dynamics generate a diffusion process. The transport
properties of the classical system are revealed in the scattering resonances
and in the time evolution of the quantum system.Comment: 42 pages, 13 figures, submitted to PR
Comparison of averages of flows and maps
It is shown that in transient chaos there is no direct relation between
averages in a continuos time dynamical system (flow) and averages using the
analogous discrete system defined by the corresponding Poincare map. In
contrast to permanent chaos, results obtained from the Poincare map can even be
qualitatively incorrect. The reason is that the return time between
intersections on the Poincare surface becomes relevant. However, after
introducing a true-time Poincare map, quantities known from the usual Poincare
map, such as conditionally invariant measure and natural measure, can be
generalized to this case. Escape rates and averages, e.g. Liapunov exponents
and drifts can be determined correctly using these novel measures. Significant
differences become evident when we compare with results obtained from the usual
Poincare map.Comment: 4 pages in Revtex with 2 included postscript figures, submitted to
Phys. Rev.
Emission from dielectric cavities in terms of invariant sets of the chaotic ray dynamics
In this paper, the chaotic ray dynamics inside dielectric cavities is
described by the properties of an invariant chaotic saddle. I show that the
localization of the far field emission in specific directions is related to the
filamentary pattern of the saddle's unstable manifold, along which the energy
inside the cavity is distributed. For cavities with mixed phase space, the
chaotic saddle is divided in hyperbolic and non-hyperbolic components, related,
respectively, to the intermediate exponential (t<t_c) and the asymptotic
power-law (t>t_c) decay of the energy inside the cavity. The alignment of the
manifolds of the two components of the saddle explains why even if the energy
concentration inside the cavity dramatically changes from tt_c, the
far field emission changes only slightly. Simulations in the annular billiard
confirm and illustrate the predictions.Comment: Corrected version, as published. 9 pages, 6 figure
Thermodynamic time asymmetry in nonequilibrium fluctuations
We here present the complete analysis of experiments on driven Brownian
motion and electric noise in a circuit, showing that thermodynamic entropy
production can be related to the breaking of time-reversal symmetry in the
statistical description of these nonequilibrium systems. The symmetry breaking
can be expressed in terms of dynamical entropies per unit time, one for the
forward process and the other for the time-reversed process. These entropies
per unit time characterize dynamical randomness, i.e., temporal disorder, in
time series of the nonequilibrium fluctuations. Their difference gives the
well-known thermodynamic entropy production, which thus finds its origin in the
time asymmetry of dynamical randomness, alias temporal disorder, in systems
driven out of equilibrium.Comment: to be published in : Journal of Statistical Mechanics: theory and
experimen
Unified Treatment of Quantum Fluctuation Theorem and Jarzynski Equality in Terms of microscopic reversibility
There are two related theorems which hold even in far from equilibrium,
namely fluctuation theorem and Jarzynski equality. Fluctuation theorem states
the existence of symmetry of fluctuation of entropy production, while Jarzynski
equality enables us to estimate the free energy change between two states by
using irreversible processes. On the other hand, relationship between these
theorems was investigated by Crooks for the classical stochastic systems. In
this letter, we derive quantum analogues of fluctuation theorem and Jarzynski
equality microscopic reversibility condition. In other words, the quantum
analogue of the work by Crooks is presented.Comment: 7pages, revised versio
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