1,091 research outputs found
Microscopic derivation of an adiabatic thermodynamic transformation
We obtain macroscopic adiabatic thermodynamic transformations by space-time
scalings of a microscopic Hamiltonian dynamics subject to random collisions
with the environment. The microscopic dynamics is given by a chain of
oscillators subject to a varying tension (external force) and to collisions
with external independent particles of "infinite mass". The effect of each
collision is to change the sign of the velocity without changing the modulus.
This way the energy is conserved by the resulting dynamics. After a diffusive
space-time scaling and cross-graining, the profiles of volume and energy
converge to the solution of a deterministic diffusive system of equations with
boundary conditions given by the applied tension. This defines an irreversible
thermodynamic transformation from an initial equilibrium to a new equilibrium
given by the final tension applied. Quasi-static reversible adiabatic
transformations are then obtained by a further time scaling. Then we prove that
the relations between the limit work, internal energy and thermodynamic entropy
agree with the first and second principle of thermodynamics.Comment: New version accepted for the publication in Brazilian Journal of
Probability and Statistic
Equilibrium fluctuations for the disordered harmonic chain perturbed by an energy conserving noise
We investigate the macroscopic behavior of the disordered harmonic chain of
oscillators, through energy diffusion. The Hamiltonian dynamics of the system
is perturbed by a degenerate conservative noise. After rescaling space and time
diffusively, we prove that energy fluctuations in equilibrium evolve according
to a linear heat equation. The diffusion coefficient is obtained from the
non-gradient Varadhan's approach, and is equivalently defined through the
Green-Kubo formula. Since the perturbation is very degenerate and the symmetric
part of the generator does not have a spectral gap, the standard non-gradient
method is reviewed under new perspectives.Comment: One major correction has been done. An author has been adde
Crossover to the stochastic Burgers equation for the WASEP with a slow bond
We consider the weakly asymmetric simple exclusion process in the presence of
a slow bond and starting from the invariant state, namely the Bernoulli product
measure of parameter . The rate of passage of particles to the
right (resp. left) is
(resp.
) except at
the bond of vertices where the rate to the right (resp. left) is
given by (resp.
). Above,
, , . For , we show that
the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on
the Schwartz space if , while for it is an
energy solution of the stochastic Burgers equation. For , it
is an Ornstein-Uhlenbeck process associated to the heat equation with Robin's
boundary conditions. For , the limit density fluctuation
field is an Ornstein-Uhlenbeck process associated to the heat equation with
Neumann's boundary conditions
Second order Boltzmann-Gibbs principle for polynomial functions and applications
In this paper we give a new proof of the second order Boltzmann-Gibbs
principle. The proof does not impose the knowledge on the spectral gap
inequality for the underlying model and it relies on a proper decomposition of
the antisymmetric part of the current of the system in terms of polynomial
functions. In addition, we fully derive the convergence of the equilibrium
fluctuations towards 1) a trivial process in case of supper-diffusive systems,
2) an Ornstein-Uhlenbeck process or the unique energy solution of the
stochastic Burgers equation, in case of weakly asymmetric diffusive systems.
Examples and applications are presented for weakly and partial asymmetric
exclusion processes, weakly asymmetric speed change exclusion processes and
hamiltonian systems with exponential interactions
Nonlinear Perturbation of a Noisy Hamiltonian Lattice Field Model: Universality Persistence
In [2] it has been proved that a linear Hamiltonian lattice field perturbed
by a conservative stochastic noise belongs to the 3/2-L\'evy/Diffusive
universality class in the nonlinear fluctuating theory terminology [15], i.e.
energy superdiffuses like an asymmetric stable 3/2-L\'evy process and volume
like a Brownian motion. According to this theory this should remain valid at
zero tension if the harmonic potential is replaced by an even potential. In
this work we consider a quartic anharmonicity and show that the result obtained
in the harmonic case persists up to some small critical value of the
anharmonicity
Interpolation process between standard diffusion and fractional diffusion
We consider a Hamiltonian lattice field model with two conserved quantities,
energy and volume, perturbed by stochastic noise preserving the two previous
quantities. It is known that this model displays anomalous diffusion of energy
of fractional type due to the conservation of the volume [5, 3]. We superpose
to this system a second stochastic noise conserving energy but not volume. If
the intensity of this noise is of order one, normal diffusion of energy is
restored while it is without effect if intensity is sufficiently small. In this
paper we investigate the nature of the energy fluctuations for a critical value
of the intensity. We show that the latter are described by an
Ornstein-Uhlenbeck process driven by a L\'evy process which interpolates
between Brownian motion and the maximally asymmetric 3/2-stable L\'evy process.
This result extends and solves a problem left open in [4].Comment: to appear in AIHP
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Hydrodynamic limit for a chain with thermal and mechanical boundary forces
We prove the hydrodynamic limit for a one dimensional harmonic chain with a
random flip of the momentum sign. The system is open and subject to two
thermostats at the boundaries and to an external tension at one of the
endpoints. Under a diffusive scaling of space-time, we prove that the empirical
profiles of the two locally conserved quantities, the volume stretch and the
energy, converge to the solution of a non-linear diffusive system of
conservative partial differential equations
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