33,567 research outputs found
Diffusion of energy in chains of oscillators with bulk noise
These notes are based on a mini-course given during the conference Particle
systems and PDE's - II which held at the Center of Mathematics of the
University of Minho in December 2013. We discuss the problem of normal and
anomalous diffusion of energy in systems of coupled oscillators perturbed by a
stochastic noise conserving energy
Asymptotic energy profile of a wavepacket in disordered chains
We investigate the long time behavior of a wavepacket initially localized at
a single site in translationally invariant harmonic and anharmonic chains
with random interactions. In the harmonic case, the energy profile averaged on time and disorder decays for large as a power
law where and 3/2 for
initial displacement and momentum excitations, respectively. The prefactor
depends on the probability distribution of the harmonic coupling constants and
diverges in the limit of weak disorder. As a consequence, the moments of the energy distribution averaged with respect to disorder
diverge in time as for , where
for . Molecular dynamics simulations yield good agreement with
these theoretical predictions. Therefore, in this system, the second moment of
the wavepacket diverges as a function of time despite the wavepacket is not
spreading. Thus, this only criteria often considered earlier as proving the
spreading of a wave packet, cannot be considered as sufficient in any model.
The anharmonic case is investigated numerically. It is found for intermediate
disorder, that the tail of the energy profile becomes very close to those of
the harmonic case. For weak and strong disorder, our results suggest that the
crossover to the harmonic behavior occurs at much larger and larger
time.Comment: To appear in Phys. Rev.
Lieb-Robinson Bounds in Quantum Many-Body Physics
We give an overview of recent results on Lieb-Robinson bounds and some of
their applications in the study of quantum many-body models in condensed matter
physics.Comment: Lecture Notes for the school "Entropy and the Quantum", 16-20 March
2009, Tucson, Arizona
Superdiffusion of energy in Hamiltonian systems perturbed by a conservative noise
We review some recent results on the anomalous diffusion of energy in systems
of 1D coupled oscillators and we revisit the role of momentum conservation.Comment: Proceedings of the conference PSPDE 2012
https://sites.google.com/site/meetingpspde
Role of conserved quantities in Fourier's law for diffusive mechanical systems
Energy transport can be influenced by the presence of other conserved
quantities. We consider here diffusive systems where energy and the other
conserved quantities evolve macroscopically on the same diffusive space-time
scale. In these situations the Fourier law depends also from the gradient of
the other conserved quantities. The rotor chain is a classical example of such
systems, where energy and angular momentum are conserved. We review here some
recent mathematical results about diffusive transport of energy and other
conserved quantities, in particular for systems where the bulk Hamiltonian
dynamics is perturbed by conservative stochastic terms. The presence of the
stochastic dynamics allows to define the transport coefficients (thermal
conductivity) and in some cases to prove the local equilibrium and the linear
response argument necessary to obtain the diffusive equations governing the
macroscopic evolution of the conserved quantities. Temperature profiles and
other conserved quantities profiles in the non-equilibrium stationary states
can be then understood from the non-stationary diffusive behaviour. We also
review some results and open problems on the two step approach (by weak
coupling or kinetic limits) to the heat equation, starting from mechanical
models with only energy conserved.Comment: Review Article for the CRAS-Physique, final versio
Kink plateau dynamics in finite-size lubricant chains
We extend the study of velocity quantization phenomena recently found in the
classical motion of an idealized 1D model solid lubricant -- consisting of a
harmonic chain interposed between two periodic sliding potentials [Phys. Rev.
Lett. 97, 056101 (2006)]. This quantization is due to one slider rigidly
dragging the commensurate lattice of kinks that the chain forms with the other
slider. In this follow-up work we consider finite-size chains rather than
infinite chains. The finite size (i) permits the development of robust velocity
plateaus as a function of the lubricant stiffness, and (ii) allows an overall
chain-length re-adjustment which spontaneously promotes single-particle
periodic oscillations. These periodic oscillations replace the quasi-periodic
motion produced by general incommensurate periods of the sliders and the
lubricant in the infinite-size model. Possible consequences of these results
for some real systems are discussed.Comment: 12 pages, 5 figures, ECOSS 200
Tagged particle diffusion in one-dimensional systems with Hamiltonian dynamics - II
We study various temporal correlation functions of a tagged particle in
one-dimensional systems of interacting point particles evolving with
Hamiltonian dynamics. Initial conditions of the particles are chosen from the
canonical thermal distribution. The correlation functions are studied in finite
systems, and their forms examined at short and long times. Various
one-dimensional systems are studied. Results of numerical simulations for the
Fermi-Pasta-Ulam chain are qualitatively similar to results for the harmonic
chain, and agree unexpectedly well with a simple description in terms of
linearized equations for damped fluctuating sound waves. Simulation results for
the alternate mass hard particle gas reveal that - in contradiction to our
earlier results [1] with smaller system sizes - the diffusion constant slowly
converges to a constant value, in a manner consistent with mode coupling
theories. Our simulations also show that the behaviour of the Lennard-Jones gas
depends on its density. At low densities, it behaves like a hard-particle gas,
and at high densities like an anharmonic chain. In all the systems studied, the
tagged particle was found to show normal diffusion asymptotically, with
convergence times depending on the system under study. Finite size effects show
up at time scales larger than sound traversal times, their nature being
system-specific.Comment: 15 pages, 12 figure
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