Non-integrable fermionic chains near criticality

We compute the Drude weight and the critical exponents as functions of the density in non-integrable generalizations of XXZ or Hubbard chains, in the critical zero temperature regime where Luttinger liquid description breaks down and Bethe ansatz cannot be used. Even in the regions where irrelevant terms dominate, no difference between integrable and non integrable models appear in exponents and conductivity. Our results are based on a fully rigorous two-regime multiscale analysis and a recently introduced partially solvable model

Perturbative methods in coupled lattice maps and applications

Issued as final reportNational Science Foundation (U.S.

Fourier's Law for a Harmonic Crystal with Self-consistent Stochastic Reservoirs

We consider a d-dimensional harmonic crystal in contact with a stochastic Langevin type heat bath at each site. The temperatures of the "exterior" left and right heat baths are at specified values T_L and T_R, respectively, while the temperatures of the "interior" baths are chosen self-consistently so that there is no average flux of energy between them and the system in the steady state. We prove that this requirement uniquely fixes the temperatures and the self consistent system has a unique steady state. For the infinite system this state is one of local thermal equilibrium. The corresponding heat current satisfies Fourier's law with a finite positive thermal conductivity which can also be computed using the Green-Kubo formula. For the harmonic chain (d=1) the conductivity agrees with the expression obtained by Bolsterli, Rich and Visscher in 1970 who first studied this model. In the other limit, d>>1, the stationary infinite volume heat conductivity behaves as 1/(l_d*d) where l_d is the coupling to the intermediate reservoirs. We also analyze the effect of having a non-uniform distribution of the heat bath couplings. These results are proven rigorously by controlling the behavior of the correlations in the thermodynamic limit.Comment: 33 page

Quasi linear flows on tori: regularity of their linearization

Under suitable conditions a flow on a torus $C^{(p)}$--close, with $p$ large enough, to a quasi periodic diophantine rotation is shown to be conjugated to the quasi periodic rotation by a map that is analytic in the perturbation size. This result is parallel to Moser's theorem stating conjugability in class $C^{(p')}$ for some $p'<p$. The extra conditions restrict the class of perturbations that are allowed.Comment: 27 page

Entropy decay for the Kac evolution

We consider solutions to the Kac master equation for initial conditions where $N$ particles are in a thermal equilibrium and $M\le N$ particles are out of equilibrium. We show that such solutions have exponential decay in entropy relative to the thermal state. More precisely, the decay is exponential in time with an explicit rate that is essentially independent on the particle number. This is in marked contrast to previous results which show that the entropy production for arbitrary initial conditions is inversely proportional to the particle number. The proof relies on Nelson's hypercontractive estimate and the geometric form of the Brascamp-Lieb inequalities due to Franck Barthe. Similar results hold for the Kac-Boltzmann equation with uniform scattering cross sections.Comment: 26 page