363 research outputs found

### Approach to equilibrium for the phonon Boltzmann equation

We study the asymptotics of solutions of the Boltzmann equation describing
the kinetic limit of a lattice of classical interacting anharmonic oscillators.
We prove that, if the initial condition is a small perturbation of an
equilibrium state, and vanishes at infinity, the dynamics tends diffusively to
equilibrium. The solution is the sum of a local equilibrium state, associated
to conserved quantities that diffuse to zero, and fast variables that are
slaved to the slow ones. This slaving implies the Fourier law, which relates
the induced currents to the gradients of the conserved quantities.Comment: 23 page

### Infinite dimensional SRB measures

We review the basic steps leading to the construction of a Sinai-Ruelle-Bowen
(SRB) measure for an infinite lattice of weakly coupled expanding circle maps,
and we show that this measure has exponential decay of space-time correlations.
First, using the Perron-Frobenius operator, one connects the dynamical system
of coupled maps on a $d$-dimensional lattice to an equilibrium statistical
mechanical model on a lattice of dimension $d+1$. This lattice model is, for
weakly coupled maps, in a high-temperature phase, and we use a general, but
very elementary, method to prove exponential decay of correlations at high
temperatures.Comment: 19 page

### Global Large Time Self-similarity of a Thermal-Diffusive Combustion System with Critical Nonlinearity

We study the initial value problem of the thermal-diffusive combustion
system: $u_{1,t} = u_{1,x,x} - u_1 u^2_2, u_{2,t} = d u_{2,xx} + u_1 u^2_2, x
\in R^1$, for non-negative spatially decaying initial data of arbitrary size
and for any positive constant $d$. We show that if the initial data decays to
zero sufficiently fast at infinity, then the solution $(u_1,u_2)$ converges to
a self-similar solution of the reduced system: $u_{1,t} = u_{1,xx} - u_1 u^2_2,
u_{2,t} = d u_{2,xx}$, in the large time limit. In particular, $u_1$ decays to
zero like ${\cal O}(t^{-\frac{1}{2}-\delta})$, where $\delta > 0$ is an
anomalous exponent depending on the initial data, and $u_2$ decays to zero with
normal rate ${\cal O}(t^{-\frac{1}{2}})$. The idea of the proof is to combine
the a priori estimates for the decay of global solutions with the
renormalization group (RG) method for establishing the self-similarity of the
solutions in the large time limit.Comment: 22pages, Latex, [email protected],[email protected],
[email protected]

### 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 $R$. We prove probabilistic estimates for the long time behaviour of the
solutions that imply bounds for the dissipation scale and energy spectrum as
$R\to\infty$.Comment: 10 page

### Some new results on an old controversy: is perturbation theory the correct asymptotic expansion in nonabelian models?

Several years ago it was found that perturbation theory for two-dimensional
O(N) models depends on boundary conditions even after the infinite volume limit
has been taken termwise, provided $N>2$. There ensued a discussion whether the
boundary conditions introduced to show this phenomenon were somehow anomalous
and there was a class of `reasonable' boundary conditions not suffering from
this ambiguity. Here we present the results of some computations that may be
interpreted as giving some support for the correctness of perturbation theory
with conventional boundary conditions; however the fundamental underlying
question of the correctness of perturbation theory in these models and in
particular the perturbative $\beta$ function remain challenging problems of
mathematical physics.Comment: 4 pages, 3 figure

### Renormalization Group and Asymptotics of Solutions of Nonlinear Parabolic Equations

We present a general method for studying long time asymptotics of nonlinear
parabolic partial differential equations. The method does not rely on a priori
estimates such as the maximum principle. It applies to systems of coupled
equations, to boundary conditions at infinity creating a front, and to higher
(possibly fractional) differential linear terms. We present in detail the
analysis for nonlinear diffusion-type equations with initial data falling off
at infinity and also for data interpolating between two different stationary
solutions at infinity.Comment: 29 page

### KAM Theorem and Quantum Field Theory

We give a new proof of the KAM theorem for analytic Hamiltonians. The proof
is inspired by a quantum field theory formulation of the problem and is based
on a renormalization group argument treating the small denominators inductively
scale by scale. The crucial cancellations of resonances are shown to follow
from the Ward identities expressing the translation invariance of the
corresponding field theory.Comment: 32 page

### Relaxation to quantum equilibrium for Dirac fermions in the de Broglie-Bohm pilot-wave theory

Numerical simulations indicate that the Born rule does not need to be
postulated in the de Broglie-Bohm pilot-wave theory, but arises dynamically
(relaxation to quantum equilibrium). These simulations were done for a particle
in a two-dimensional box whose wave-function obeys the non-relativistic
Schroedinger equation and is therefore scalar. The chaotic nature of the de
Broglie-Bohm trajectories, thanks to the nodes of the wave-function which act
as vortices, is crucial for a fast relaxation to quantum equilibrium. For
spinors, we typically do not expect any node. However, in the case of the Dirac
equation, the de Broglie-Bohm velocity field has vorticity even in the absence
of nodes. This observation raises the question of the origin of relaxation to
quantum equilibrium for fermions. In this article, we provide numerical
evidence to show that Dirac particles also undergo relaxation, by simulating
the evolution of various non-equilibrium distributions for two-dimensional
systems (the 2D Dirac oscillator and the Dirac particle in a spherical 2D box).Comment: 11 pages, 9 figure

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