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

Diagnosing the Trouble With Quantum Mechanics

We discuss an article by Steven Weinberg expressing his discontent with the usual ways to understand quantum mechanics. We examine the two solutions that he considers and criticizes and propose another one, which he does not discuss, the pilot wave theory or Bohmian mechanics, for which his criticisms do not apply.Comment: 23 pages, 4 figure

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]

Schr\"odinger's paradox and proofs of nonlocality using only perfect correlations

We discuss proofs of nonlocality based on a generalization by Erwin Schr\"odinger of the argument of Einstein, Podolsky and Rosen. These proofs do not appeal in any way to Bell's inequalities. Indeed, one striking feature of the proofs is that they can be used to establish nonlocality solely on the basis of suitably robust perfect correlations. First we explain that Schr\"odinger's argument shows that locality and the perfect correlations between measurements of observables on spatially separated systems implies the existence of a non-contextual value-map for quantum observables; non-contextual means that the observable has a particular value before its measurement, for any given quantum system, and that any experiment "measuring this observable" will reveal that value. Then, we establish the impossibility of a non-contextual value-map for quantum observables {\it without invoking any further quantum predictions}. Combining this with Schr\"odinger's argument implies nonlocality. Finally, we illustrate how Bohmian mechanics is compatible with the impossibility of a non-contextual value-map.Comment: 30 pages, 2 figure

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