Applied Mathematics Research Centre

The AMRC hosts a series of seminars on subjects in statistical physics and fluid dynamics.

**Coupled convective phenomena**

The research activities of the Coupled Convective Phenomena team of the LMFA (Laboratory of Fluid Mechanics and Acoustics, in Lyon, France) are developed along two axes. The first one is connected to coating processes in which a thin layer of a material is deposited in liquid state over a substrate and also to natural rapid gravity driven mass movements like avalanches and debris/mud flows. The typical configuration in our studies is that of a liquid film (possibly non-newtonian) flowing down an inclined plane under the effect of gravity. In particular we investigate the stability of the uniform flow both theoretically and experimentally. The second axis is that of directional solidification of metallic alloys and semi-conducting crystals growth from a melt. We have a fundamental standpoint on these applications since we consider the stability of natural convection flows in differentially heated cavities and their control by external fields -- magnetic or acoustic. We are also interested in improving stirring solutions for these processes. Focus will be made on the case of acoustic fields, which can create quasi-steady flows. Our experimental and theoretical investigation of this phenomenon, often referred to as ‘acoustic streaming’, will be presented.

The URL for the team page is : lmfa.ec-lyon.fr/index.php

**Recreating the Big Bang with the World's Largest Machine: The LHC at CERN**

The 27km Large Hadron Collider (LHC), situated 150 metres under the Swiss-French boarder at CERN near Geneva, is the World's most powerful particle accelerator.

Protons (hydrogen nuclei) are smashed together at 0.999999991 times the speed of light recreating, for a tiny instant, the violent particle collisions which would have existed less than a billionth of a second after the Big Bang. At the end of each year, lead nuclei are accelerated and collided in the LHC producing the highest temperatures and densities ever made in an experiment and recreating the exotic primordial soup which existed at the birth of our Universe.

**Polymers in anisotropic environment: conformational properties***Wednesday December 11, 2013, 16:00 h, DH Seminar Room*

The conformational properties of flexible polymers in d dimensions in environments with extended defects are analyzed both analytically and numerically. We consider the case, when structural defects are correlated in \varepsilon_d dimensions and randomly distributed in the remaining d-\varepsilon_d. Within the lattice model of self-avoiding random walks (SAW), we apply the pruned enriched Rosenbluth method (PERM) and find the estimates for scaling exponents and universal shape parameters of polymers in environment with parallel rod-like defects (\varepsilon_d=1). An analytical description of the model is developed within the des Cloizeaux direct polymer renormalization scheme.

**Transmuted Finite-Size Scaling at First-Order Phase Transitions with Exponential Degeneracy of Ordered States**

We note that the standard inverse system volume scaling for finite-size corrections at a first-order phase transition (i.e. $1/L3$ for an $L \times L \times L$ lattice in $3D$) is transmuted to $1/L2$ scaling if there is an exponential low-temperature phase degeneracy. The gonihedric Ising model which has a four-spin interaction, plaquette Hamiltonian provides an exemplar of just such a system. We use multicanonical simulations of this model to generate high-precision data which provides strong confirmation of the transmuted finite-size scaling law. The dual to the gonihedric model, which is an anisotropically coupled Ashkin-Teller model, has a similar degeneracy and also displays the transmuted scaling.

**Effects of Bending Stiffness to a Polymer inside a Spherical Cage**

We study the change of the pseudo phase transition of a simple homopolymer inside a spherical confinement. Of our particular interest is the shift of the collapse and freezing transitions with shrinking radius of the sphere and how it changes with varying stiffness of the polymer.

Therefore we use a modified bead-stick model, with an additional cosine potential to introduce a bending stiffness. We used the multicanonical Monte Carlo method to get an overview of the complete the phase space of this model. To characterize the pseudo phase transition we analyse fluctuations of energetic and conformational observables. We observed a clear dependence of the shift of the collapse transition with the stiffness of the polymer. For very flexible polymers the transition temperature goes to lower temperatures with shrinking sphere radius. This effect turns around with increasing stiffness of the polymer.

**High q-State Clock Spin Glasses in Three Dimensions and the Lyapunov Exponents of Chaotic Phases and Chaotic Phase Boundaries**

Spin-glass phases and phase transitions for q-state clock models and their q -> oo limit the XY model, in spatial dimension d = 3, are studied by a detailed renormalization-group study that is exact for the d=3 hierarchical lattice and approximate for the cubic lattice.[1] In addition to the now well-established chaotic rescaling behavior of the spin-glass phase, each of the two types of spin-glass phase boundaries displays, under renormalization-group trajectories, their own distinctive chaotic behavior. These chaotic renormalization-group trajectories subdivide into two categories, namely as strong-coupling chaos (in the spin-glass phase and, distinctly, on the spinglass-ferromagnetic phase boundary) and as intermediate-coupling chaos (on the spinglass-paramagnetic phase boundary). We thus characterize each different phase and phase boundary exhibiting chaos by its distinct Lyapunov exponent, which we calculate. We show that, under renormalization-group, chaotic trajectories and fixed distributions are mechanistically and quantitatively equivalent. The phase diagrams of arbitrary even q-state clock spin-glass models in d=3 are calculated. These models, for all non-infinite q, have a finite-temperature spin-glass phase. Furthermore, the spin-glass phases exhibit a universal ordering behavior, independent of q. The spin-glass phases and the spinglass-paramagnetic phase boundaries exhibit universal fixed distributions, chaotic trajectories and Lyapunov exponents. In the XY model limit, our calculations indicate a zero-temperature spin-glass phase.

**Interior Point Methods for Large Scale Optimization**

Interior Point Methods (IPMs) for linear, quadratic and nonlinear optimization have been around for more than 25 years and have completely changed the field of optimization. In the first part of my talk I will focus on the major features responsible for their spectacular success:

(a) nice properties (self-concordance) of logarithmic barriers which are responsible for the polynomial complexity of IPMs,

(b) a unified view of IPMs for linear, quadratic, convex nonlinear, second-order cone, and semidefinite optimization,

(c) IPMs' ability to solve very large problems.

In the second part of my talk I will address the theoretical issues of applying the *inexact* Newton method in an IPM and a redesign of the method bearing two objectives in mind:

(a) to avoid an explicit access to the problem data and to allow only matrix-vector products to be executed with the Hessian and Jacobian and its transpose; and

(b) to allow the method work in a limited-memory regime.

If time permits, I will present the numerical results which demonstrate the practical performance of the new approach applied to four classes of challenging optimization problems:

- relaxations of quadratic assignment problems,

- quantum information problems,

- sparse approximation problems arising in compressed sensing,

- PageRank (google) problems.We prove entropic long range