6,258 research outputs found

### The fractality of the relaxation modes in deterministic reaction-diffusion systems

In chaotic reaction-diffusion systems with two degrees of freedom, the modes
governing the exponential relaxation to the thermodynamic equilibrium present a
fractal structure which can be characterized by a Hausdorff dimension. For long
wavelength modes, this dimension is related to the Lyapunov exponent and to a
reactive diffusion coefficient. This relationship is tested numerically on a
reactive multibaker model and on a two-dimensional periodic reactive Lorentz
gas. The agreement with the theory is excellent

### 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

### Self-Organization at the Nanoscale Scale in Far-From-Equilibrium Surface Reactions and Copolymerizations

An overview is given of theoretical progress on self-organization at the
nanoscale in reactive systems of heterogeneous catalysis observed by field
emission microscopy techniques and at the molecular scale in copolymerization
processes. The results are presented in the perspective of recent advances in
nonequilibrium thermodynamics and statistical mechanics, allowing us to
understand how nanosystems driven away from equilibrium can manifest
directionality and dynamical order.Comment: A. S. Mikhailov and G. Ertl, Editors, Proceedings of the
International Conference "Engineering of Chemical Complexity", Berlin Center
for Studies of Complex Chemical Systems, 4-8 July 201

### Heat transport in stochastic energy exchange models of locally confined hard spheres

We study heat transport in a class of stochastic energy exchange systems that
characterize the interactions of networks of locally trapped hard spheres under
the assumption that neighbouring particles undergo rare binary collisions. Our
results provide an extension to three-dimensional dynamics of previous ones
applying to the dynamics of confined two-dimensional hard disks [Gaspard P &
Gilbert T On the derivation of Fourier's law in stochastic energy exchange
systems J Stat Mech (2008) P11021]. It is remarkable that the heat conductivity
is here again given by the frequency of energy exchanges. Moreover the
expression of the stochastic kernel which specifies the energy exchange
dynamics is simpler in this case and therefore allows for faster and more
extensive numerical computations.Comment: 21 pages, 5 figure

### Connection formulas between Coulomb wave functions

The mathematical relations between the regular Coulomb function
$F_{\eta\ell}(\rho)$ and the irregular Coulomb functions
$H^\pm_{\eta\ell}(\rho)$ and $G_{\eta\ell}(\rho)$ are obtained in the complex
plane of the variables $\eta$ and $\rho$ for integer or half-integer values of
$\ell$. These relations, referred to as "connection formulas", form the basis
of the theory of Coulomb wave functions, and play an important role in many
fields of physics, especially in the quantum theory of charged particle
scattering. As a first step, the symmetry properties of the regular function
$F_{\eta\ell}(\rho)$ are studied, in particular under the transformation
$\ell\mapsto-\ell-1$, by means of the modified Coulomb function
$\Phi_{\eta\ell}(\rho)$, which is entire in the dimensionless energy
$\eta^{-2}$ and the angular momentum $\ell$. Then, it is shown that, for
integer or half-integer $\ell$, the irregular functions
$H^\pm_{\eta\ell}(\rho)$ and $G_{\eta\ell}(\rho)$ can be expressed in terms of
the derivatives of $\Phi_{\eta,\ell}(\rho)$ and $\Phi_{\eta,-\ell-1}(\rho)$
with respect to $\ell$. As a consequence, the connection formulas directly lead
to the description of the singular structures of $H^\pm_{\eta\ell}(\rho)$ and
$G_{\eta\ell}(\rho)$ at complex energies in their whole Riemann surface. The
analysis of the functions is supplemented by novel graphical representations in
the complex plane of $\eta^{-1}$.Comment: 24 pages, 4 figures, 39 reference

### Growth and dissolution of macromolecular Markov chains

The kinetics and thermodynamics of free living copolymerization are studied
for processes with rates depending on k monomeric units of the macromolecular
chain behind the unit that is attached or detached. In this case, the sequence
of monomeric units in the growing copolymer is a kth-order Markov chain. In the
regime of steady growth, the statistical properties of the sequence are
determined analytically in terms of the attachment and detachment rates. In
this way, the mean growth velocity as well as the thermodynamic entropy
production and the sequence disorder can be calculated systematically. These
different properties are also investigated in the regime of depolymerization
where the macromolecular chain is dissolved by the surrounding solution. In
this regime, the entropy production is shown to satisfy Landauer's principle

### Hamiltonian dynamics, nanosystems, and nonequilibrium statistical mechanics

An overview is given of recent advances in nonequilibrium statistical
mechanics on the basis of the theory of Hamiltonian dynamical systems and in
the perspective provided by the nanosciences. It is shown how the properties of
relaxation toward a state of equilibrium can be derived from Liouville's
equation for Hamiltonian dynamical systems. The relaxation rates can be
conceived in terms of the so-called Pollicott-Ruelle resonances. In spatially
extended systems, the transport coefficients can also be obtained from the
Pollicott-Ruelle resonances. The Liouvillian eigenstates associated with these
resonances are in general singular and present fractal properties. The singular
character of the nonequilibrium states is shown to be at the origin of the
positive entropy production of nonequilibrium thermodynamics. Furthermore,
large-deviation dynamical relationships are obtained which relate the transport
properties to the characteristic quantities of the microscopic dynamics such as
the Lyapunov exponents, the Kolmogorov-Sinai entropy per unit time, and the
fractal dimensions. We show that these large-deviation dynamical relationships
belong to the same family of formulas as the fluctuation theorem, as well as a
new formula relating the entropy production to the difference between an
entropy per unit time of Kolmogorov-Sinai type and a time-reversed entropy per
unit time. The connections to the nonequilibrium work theorem and the transient
fluctuation theorem are also discussed. Applications to nanosystems are
described.Comment: Lecture notes for the International Summer School Fundamental
Problems in Statistical Physics XI (Leuven, Belgium, September 4-17, 2005

### Fluctuation relations for equilibrium states with broken discrete symmetries

Relationships are obtained expressing the breaking of spin-reversal symmetry
by an external magnetic field in Gibbsian canonical equilibrium states of spin
systems under specific assumptions. These relationships include an exact
fluctuation relation for the probability distribution of the magnetization, as
well as a relation between the standard thermodynamic entropy, an associated
spin-reversed entropy or coentropy, and the product of the average
magnetization with the external field, as a non-negative Kullback-Leibler
divergence. These symmetry relations are applied to the model of noninteracting
spins, the 1D and 2D Ising models, and the Curie-Weiss model, all in an
external magnetic field. The results are drawn by analogy with similar
relations obtained in the context of nonequilibrium physics

### Signatures of classical bifurcations in the quantum scattering resonances of dissociating molecules

A study is reported of the quantum scattering resonances of dissociating
molecules using a semiclassical approach based on periodic-orbit theory. The
dynamics takes place on a potential energy surface with an energy barrier
separating two channels of dissociation. Above the barrier, the unstable
symmetric-stretch periodic orbit may undergo a supercritical pitchfork
bifurcation, leading to a classically chaotic regime. Signatures of the
bifurcation appear in the spectrum of resonances, which have a shorter lifetime
than classically expected. A method is proposed to evaluate semiclassically the
energy and lifetime of the quantum resonances in this intermediate regime

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