Abstract:
The origin of deterministic diffusion is a matter of discussion. We study the asymptotic distributions of the sums $y_n(x)=\sum_{k=0}^{n-1}\psi (x+k\alpha)$, where $\psi$ is a periodic function of bounded variation and $\alpha$ an irrational number. It is known that no diffusion process will be observed. Nevertheless, we find a picewise constant function $\psi$ and an increasing sequence of integer $(n_j)_j$ such that the limit distribution of the sequence $(y_{n_j}/\sqrt j)_j$ is Gaussian (with stricly positive variance). If $\alpha$ is of constant type, we show that the sequence $(n_j)_j$ may be taken to grow exponentially (this is close to optimal in some sense, and one has $||y_{n_j}||_{\mathrm L^2}\sim \max_{0\le k\le n_j}||y_k||_{\mathrm L^2}$ as $j\to\infty$). We give an heuristic link with the theory of expanding maps of the interval.

Abstract:
We study the energy current in a model of heat conduction, first considered in detail by Casher and Lebowitz. The model consists of a one-dimensional disordered harmonic chain of n i.i.d. random masses, connected to their nearest neighbors via identical springs, and coupled at the boundaries to Langevin heat baths, with respective temperatures T_1 and T_n. Let EJ_n be the steady-state energy current across the chain, averaged over the masses. We prove that EJ_n \sim (T_1 - T_n)n^{-3/2} in the limit n \to \infty, as has been conjectured by various authors over the time. The proof relies on a new explicit representation for the elements of the product of associated transfer matrices.

Abstract:
We study a one-dimensional hamiltonian chain of masses perturbed by an energy conserving noise. The dynamics is such that, according to its hamiltonian part, particles move freely in cells and interact with their neighbors through collisions, made possible by a small overlap of size $\epsilon > 0$ between near cells. The noise only randomly flips the velocity of the particles. If $\epsilon \rightarrow 0$, and if time is rescaled by a factor $1/{\epsilon}$, we show that energy evolves autonomously according to a stochastic equation, which hydrodynamic limit is known in some cases. In particular, if only two different energies are present, the limiting process coincides with the simple symmetric exclusion process.

Abstract:
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 $\epsilon > 0$. 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 $\epsilon$ as $\epsilon\rightarrow 0$. 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.

Abstract:
We introduce a class of stochastic weakly coupled map lattices, as models for studying heat conduction in solids. Each particle on the lattice evolves according to an internal dynamics that depends on its energy, and exchanges energy with its neighboors at a rate that depends on its internal state. We study energy fluctuations at equilibrium in a diffusive scaling. In some cases, we derive the hydordynamic limit of the fluctuation field.

Abstract:
We investigate the possibility of Many-Body Localization in translation invariant Hamiltonian systems, which was recently brought up by several authors. A key feature of Many-Body Localized disordered systems is recovered, namely the fact that resonant spots are rare and far-between. However, we point out that resonant spots are mobile, unlike in models with strong quenched disorder, and that these mobile spots constitute a possible mechanism for delocalization, albeit possibly only on very long timescales. In some models, this argument for delocalization can be made very explicit in first order of perturbation theory in the hopping. For models where this does not work, we present instead a non-perturbative argument that relies solely on ergodicity inside the resonant spots.

Abstract:
The maternal-embryonic nutritional relationship in chondrichthyans has been poorly explored. Consequently, accurately discerning between their different reproductive modes is difficult; especially lecithotrophy and incipient histotrophy. This present study is the first to assess changes in mass throughout embryonic development of an oviparous chondrichthyan other than Scyliorhinus canicula. Heterodontus portusjacksoni egg cases were collected and used to quantify the gain or loss of wet mass, dry mass, water content, inorganic and organic matter from freshly deposited eggs (without macroscopically visible embryos) to near full-term embryos. A loss in organic mass of ~40% found from this study is approximately double the values previously obtained for S. canicula. This raises concerns for the validity of the current threshold value used to discern between lecithotrophic and matrotrophic species. Accordingly, 26 studies published in the primary literature between 1932 and 2012 addressing the maternal-embryonic nutritional relationship in sharks were reviewed. Values for changes in mass reported for over 20 different shark species were synthesised and recalculated, revealing multiple typographical, transcribing, calculation and rounding errors across many papers. These results suggest that the current threshold value of ？20% established by previous studies is invalid and should be avoided to ascertain the reproductive mode of aplacental viviparous species.

Abstract:
In this paper we study a Hamiltonian system with a spatially asymmetric potential. We are interested in the effects on the dynamics when the potential becomes symmetric slowly in time. We focus on a highly simplified non-trivial model problem (a metaphor) to be able to pursue explicit calculations as far as possible. Using the techniques of averaging and adiabatic invariants, we are able to study all bounded solutions, which reveals significant asymmetric dynamics even when the asymmetric contributions to the potential have become negligibly small.

Abstract:
We study the thermal properties of a pinned disordered harmonic chain weakly perturbed by a noise and an anharmonic potential. The noise is controlled by a parameter $\lambda \rightarrow 0$, and the anharmonicity by a parameter $\lambda' \le \lambda$. Let $\kappa$ be the conductivity of the chain, defined through the Green-Kubo formula. Under suitable hypotheses, we show that $\kappa = \mathcal O (\lambda)$ and, in the absence of anharmonic potential, that $\kappa \sim \lambda$. This is in sharp contrast with the ordered chain for which $\kappa \sim 1/\lambda$, and so shows the persitence of localization effects for a non-integrable dynamics.