Abstract. Coralline algae are important habitat formers found on all rocky shores. While the impact of future ocean acidification on the physiological performance of the species has been well studied, little research has focused on potential changes in structural integrity in response to climate change. A previous study using 2-D Finite Element Analysis (FEA) suggested increased vulnerability to fracture (by wave action or boring) in algae grown under high CO 2 conditions. To assess how realistically 2-D simplified models represent structural performance, a series of increasingly biologically accurate 3-D FE models that represent different aspects of coralline algal growth were developed. Simplified geometric 3-D models of the genus Lithothamnion were compared to models created from computed tomography (CT) scan data of the same genus. The biologically accurate model and the simplified geometric model representing individual cells had similar average stresses and stress distributions, emphasising the importance of the cell walls in dissipating the stress throughout the structure. In contrast models without the accurate representation of the cell geometry resulted in larger stress and strain results. Our more complex 3-D model reiterated the potential of climate change to diminish the structural integrity of the organism. This suggests that under future environmental conditions the weakening of the coralline algal skeleton along with increased external pressures (wave and bioerosion) may negatively influence the ability for coralline algae to maintain a habitat able to sustain high levels of biodiversity.

We show that the time evolution of a quantum wavepacket in a periodic potential converges in a combined high-frequency/Boltzmann-Grad limit, up to second order in the coupling constant, to terms that are compatible with the linear Boltzmann equation. This complements results of Eng and Erdös for low-density random potentials, where convergence to the linear Boltzmann equation is proved in all orders. We conjecture, however, that the linear Boltzmann equation fails in the periodic setting for terms of order four and higher. Our proof uses Floquet-Bloch theory, multi-variable theta series and equidistribution theorems for homogeneous flows. Compared with other scaling limits traditionally considered in homogenisation theory, the Boltzmann-Grad limit requires control of the quantum dynamics for longer times, which are inversely proportional to the total scattering cross section of the single-site potential. R d R d ×R d a(x, y) b(x, y) dxdy, and the Hilbert-Schmidt inner product (1.8) A, B HS = Tr AB † .As is standard in semiclassics, we will measure momentum in units of h, and use the rescaling a(x, y) → h d/2 a(x, hy); the normalisation is chosen so that the L 2 -norm is preserved. In the classical picture of a point particle moving through an infinite field of scatterers, the Boltzmann-Grad scaling limit is one in which the radius of the scatterers is taken to zero, while space and time are simultaneously rescaled in order to ensure the mean free path length and mean free flight time remain finite. The classical mean free path length scales like r 1−d , and so we define the semiclassical Boltzmann-Grad scaling of a ∈ S(R d × R d ) by(1.9) D r,h a(x, y) = r d(d−1)/2 h d/2 a(r d−1 x, hy),

Bufetov, Bufetov-Forni and Bufetov-Solomyak have recently proved limit theorems for translation flows, horocycle flows and tiling flows, respectively. We present here analogous results for skew translations of a torus. §1. Bufetov [1] has recently established limit theorems for translation flows on flat surfaces and, in joint work with Forni [2], for horocycle flows on compact hyperbolic surfaces. Bufetov and Solomyak [3] have proved analogous theorems for tiling flows. The striking feature of these results is that the central limit theorem (a common feature of many "chaotic" dynamical systems) fails. The limit laws are instead characterised in terms of the corresponding renormalisation dynamics. The purpose of the present note is to point out that an analogous result holds in the simpler case of skew translations of the torus. Our approach uses the modular invariance of theta sums as in [14,15,16]. The limit theorems proved in these papers may be interpreted as limit theorems for random, rather than fixed, skew translations, cf. also [9, 10, 11] and [6]. The approach by Flaminio and Forni [8] developed for nilflows may yield an alternative route to our main result, but we have not explored this further. Another class of systems which exhibit non-normal limit laws are random translations of the torus; we refer the reader to Kesten's Cauchy limit theorem for circle rotations [12,13] and the recent higher-dimensional generalisations by Dolgopyat and Fayad [4,5]; see also Sinai and Ulcigrai [17] for limit theorems for circle rotations with non-integrable test functions. arXiv:1407.4320v1 [math.DS]

We study the macroscopic transport properties of the quantum Lorentz gas in a crystal with short-range potentials, and show that in the Boltzmann–Grad limit the quantum dynamics converges to a random flight process which is not compatible with the linear Boltzmann equation. Our derivation relies on a hypothesis concerning the statistical distribution of lattice points in thin domains, which is closely related to the Berry–Tabor conjecture in quantum chaos.

It is a fundamental problem in mathematical physics to derive macroscopic transport equations from microscopic models. In this paper, we derive the linear Boltzmann equation in the low-density limit of a damped quantum Lorentz gas for a large class of deterministic and random scatterer configurations. Previously this result was known only for the single-scatterer problem on the flat torus, and for uniformly random scatterer configurations where no damping is required. The damping is critical in establishing convergence—in the absence of damping the limiting behaviour depends on the exact configuration under consideration, and indeed, the linear Boltzmann equation is not expected to appear for periodic and other highly ordered configurations.

General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms Consider the 3-dimensional Laplacian with a potential described by point scatterers placed on the integer lattice. We prove that for Floquet-Bloch modes with fixed quasi-momentum satisfying a certain Diophantine condition, there is a subsequence of eigenvalues of positive density whose eigenfunctions exhibit equidistribution in position space and localisation in momentum space. This result complements the result of Ueberschaer and Kurlberg, J. Eur. Math. Soc. (JEMS) (to appear); [e-print arXiv:1409[e-print arXiv: .6878 (2014] who show momentum localisation for zero quasi-momentum in 2-dimensions and is the first result in this direction in 3-dimensions. Published by AIP Publishing. [http://dx

No abstract

No abstract