154 research outputs found
Perturbative analysis of anharmonic chains of oscillators out of equilibrium
We compute the first-order correction to the correlation functions of the
stationary state of a stochastically forced harmonic chain out of equilibrium
when a small on-site anharmonic potential is added. This is achieved by
deriving a suitable formula for the covariance matrix of the invariant state.
We find that the first-order correction of the heat current does not depend on
the size of the system. Second, the temperature profile is linear when the
harmonic part of the on-site potential is zero. The sign of the gradient of the
profile, however, is opposite to the sign of the temperature difference of the
two heat baths.Comment: 26 pages, 2 figures, corrected typo
Probabilistic estimates for the Two Dimensional Stochastic Navier-Stokes Equations
We consider the Navier-Stokes equation on a two dimensional torus with a
random force, white noise in time and analytic in space, for arbitrary Reynolds
number . We prove probabilistic estimates for the long time behaviour of the
solutions that imply bounds for the dissipation scale and energy spectrum as
.Comment: 10 page
Normal Heat Conductivity in a strongly pinned chain of anharmonic oscillators
We consider a chain of coupled and strongly pinned anharmonic oscillators
subject to a non-equilibrium random forcing. Assuming that the stationary state
is approximately Gaussian, we first derive a stationary Boltzmann equation. By
localizing the involved resonances, we next invert the linearized collision
operator and compute the heat conductivity. In particular, we show that the
Gaussian approximation yields a finite conductivity
, for the anharmonic coupling
strength.Comment: Introduction and conclusion modifie
Macroscopic fluctuations theory of aerogel dynamics
We consider the thermodynamic potential describing the macroscopic
fluctuation of the current and local energy of a general class of Hamiltonian
models including aerogels. We argue that this potential is neither analytic nor
strictly convex, a property that should be expected in general but missing from
models studied in the literature. This opens the possibility of describing in
terms of a thermodynamic potential non-equilibrium phase transitions in a
concrete physical context. This special behaviour of the thermodynamic
potential is caused by the fact that the energy current is carried by particles
which may have arbitrary low speed with sufficiently large probability.Comment: final versio
Drastic fall-off of the thermal conductivity for disordered lattices in the limit of weak anharmonic interactions
We study the thermal conductivity, at fixed positive temperature, of a
disordered lattice of harmonic oscillators, weakly coupled to each other
through anharmonic potentials. The interaction is controlled by a small
parameter . We rigorously show, in two slightly different setups,
that the conductivity has a non-perturbative origin. This means that it decays
to zero faster than any polynomial in as . It
is then argued that this result extends to a disordered chain studied by Dhar
and Lebowitz, and to a classical spins chain recently investigated by
Oganesyan, Pal and Huse.Comment: 21 page
Nonequilibrium dynamics of a stochastic model of anomalous heat transport
We study the dynamics of covariances in a chain of harmonic oscillators with
conservative noise in contact with two stochastic Langevin heat baths. The
noise amounts to random collisions between nearest-neighbour oscillators that
exchange their momenta. In a recent paper, [S Lepri et al. J. Phys. A: Math.
Theor. 42 (2009) 025001], we have studied the stationary state of this system
with fixed boundary conditions, finding analytical exact expressions for the
temperature profile and the heat current in the thermodynamic (continuum)
limit. In this paper we extend the analysis to the evolution of the covariance
matrix and to generic boundary conditions. Our main purpose is to construct a
hydrodynamic description of the relaxation to the stationary state, starting
from the exact equations governing the evolution of the correlation matrix. We
identify and adiabatically eliminate the fast variables, arriving at a
continuity equation for the temperature profile T(y,t), complemented by an
ordinary equation that accounts for the evolution in the bulk. Altogether, we
find that the evolution of T(y,t) is the result of fractional diffusion.Comment: Submitted to Journal of Physics A, Mathematical and Theoretica
A stochastic model of anomalous heat transport: analytical solution of the steady state
We consider a one-dimensional harmonic crystal with conservative noise, in
contact with two stochastic Langevin heat baths at different temperatures. The
noise term consists of collisions between neighbouring oscillators that
exchange their momenta, with a rate . The stationary equations for the
covariance matrix are exactly solved in the thermodynamic limit ().
In particular, we derive an analytical expression for the temperature profile,
which turns out to be independent of . Moreover, we obtain an exact
expression for the leading term of the energy current, which scales as
. Our theoretical results are finally found to be consistent
with the numerical solutions of the covariance matrix for finite .Comment: Minor changes in the text. To appear in Journal of Physics
The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics
For low density gases the validity of the Boltzmann transport equation is
well established. The central object is the one-particle distribution function,
, which in the Boltzmann-Grad limit satisfies the Boltzmann equation. Grad
and, much refined, Cercignani argue for the existence of this limit on the
basis of the BBGKY hierarchy for hard spheres. At least for a short kinetic
time span, the argument can be made mathematically precise following the
seminal work of Lanford. In this article a corresponding programme is
undertaken for weakly nonlinear, both discrete and continuum, wave equations.
Our working example is the harmonic lattice with a weakly nonquadratic on-site
potential. We argue that the role of the Boltzmann -function is taken over
by the Wigner function, which is a very convenient device to filter the slow
degrees of freedom. The Wigner function, so to speak, labels locally the
covariances of dynamically almost stationary measures. One route to the phonon
Boltzmann equation is a Gaussian decoupling, which is based on the fact that
the purely harmonic dynamics has very good mixing properties. As a further
approach the expansion in terms of Feynman diagrams is outlined. Both methods
are extended to the quantized version of the weakly nonlinear wave equation.
The resulting phonon Boltzmann equation has been hardly studied on a rigorous
level. As one novel contribution we establish that the spatially homogeneous
stationary solutions are precisely the thermal Wigner functions. For three
phonon processes such a result requires extra conditions on the dispersion law.
We also outline the reasoning leading to Fourier's law for heat conduction.Comment: special issue on "Kinetic Theory", Journal of Statistical Physics,
improved versio
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