Applied Mathematics Research Centre

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

**Towards wall functions for the prediction of solute segregation in plane front directional solidification**

*Thursday 07 June, 2018, 14:00 h, DH Seminar Room*

The presented work focuses on solute segregation occurring in directional solidification processes with sharp solid/liquid interface as for the production of large photovoltaic silicon ingots. In these applications, various stirring techniques aim at enhancing segregation, leading to fully turbulent flows in the melt with very thin solute boundary layer ahead the solid/liquid front. A major difficulty for simulating such processes is their inherently multi-scale nature: the impurity segregation problem is controlled at the solute boundary layer scale (tens of micrometers) while the thermal problem is ruled at the crucible scale (meters in the targeted photovoltaic silicon processes). The solute boundary layer is convection-controlled and requires a specific refinement in numerical models mesh.

In order to lighten numerical simulations, analytical wall functions describing solute boundary layers are derived from a scaling analysis. These wall functions provide segregation profiles from purely thermo-hydrodynamic simulations, which do not require solute boundary layer refinement at the solid/liquid interface. The proposed scaling analysis accounts for turbulent solute transport. A procedure to predict concentration fields in the solid phase from a hydrodynamic simulation of the solidification process is proposed. This procedure only uses wall shear-stress profiles at the solidification front as input data.

Validation is made with respect to 2D configurations by comparison to transient segregation simulations. Convective regimes ranging from laminar to fully turbulent and several growth rates and molecular diffusivities are considered. Both boundary layer thicknesses during the growth and solid phase concentration fields are fairly predicted in these simplified solidification cases.

**Hydrodynamic Approaches to Non-Equilibrium Steady States***Wednesday 16 May, 2018, 14:00 h, DH Seminar Room*

We discuss advances in characterising non-equilibrium steady states using insights from condensed matter physics, gauge-gravity duality and hydrodynamics.

[1] Bhaseen, Doyon, Lucas and Schalm, "Energy flow in quantum critical systems far from equilibrium", Nature Physics 11, 509 (2015).

[2] Lucas, Schalm, Doyon, and Bhaseen, "Shock waves, rarefaction waves, and nonequilibrium steady states in quantum critical systems", Phys. Rev. D 94, 025004 (2016).

**Transition to turbulence, then collapse of turbulence, in pipe flow**

*Wednesday 28 March, 2018, 14:00 h, DH Seminar Room*

In 1883, Osborne Reynolds observed that the transition to turbulence in a pipeoccurs in the absence of a linear instability -- disturbances to the laminar flow of finite amplitude are required to trigger turbulence. But in recent experiments, disturbances have been observed to have the opposite effect, causing turbulence to collapse. These effects are a result of the complicated nature of the laminar-turbulent boundary.

Over the last decade the laminar-turbulent boundary has been observed to be mediated by travelling-wave solutions. In this talk I will briefly describe how travelling wave solutions were discovered, and how we now understand how to find such nonlinear solutions much more rapidly. I will then discuss how modifications to the large-scale flow can alter the laminar-turbulent boundary to bring about the collapse of turbulence

**Disorder-Free Localization**

I will present an exactly solvable quantum model of fermions interacting with spins-1/2 showing dynamical localization without any quenched disorder. I will also discuss a number of extensions of this model which are related to the physics of many-body localization (MBL), and to quantum quenches in two-dimensional Z2 lattice gauge theories.

**Homeostatic plasticity and external input shape neural network dynamics**

*Wednesday 21 March, 2018, 14:00 h, DH Seminar Room*

Dynamics of spiking neural networks exhibit clear differences, depending whether originating from living organisms or artificially grown cultures. In living organisms, the neural activity shows continuous, fluctuating dynamics, whereas cultured networks develop strong bursts separated by periods of silence. We propose that this is a result of an interplay between (1) network input, which is much weaker in isolated cultures than in the intact brain, and (2) homeostatic plasticity, a slow negative feedback mechanism adapting the neural spike rate. Based on our theoretical work, we predict that homeostasis can be harnessed to tune the dynamic state of a network by altering its input strength. Most importantly, this could allow to abolish the bursts in cultured neurons and render the dynamics brain-like instead - a key prerequisite to study neurological and psychiatric disorders on the network level under laboratory conditions.

**Large scale vortices and zonal flows in spherical rotating convection**

*Wednesday 14 March, 2018, 14:00 h, DH Seminar Room*

Large-scale coherent structures have been observed recently in numerical simulations of rapidly rotating convection in Cartesian boxes. However, similar structures have not yet been reported in the spherical geometry, which is more relevant to geophysical applications. In this talk, we will report numerical evidence of such large-scale coherent structures in a rapidly rotating whole sphere driven by internal heating. When the Rayleigh number is well above the critical value and a stress-free boundary condition is used for the velocity field, small-scale convective flows merge into large-scale structures, leading to a long-lived, axial-invariant cyclone around the rotation axis. We will discuss the formation of such large-scale cyclones and the heat transport associated with these large-scale structures.

**Coherent Structures in Stably Stratified Couette Flow**

*Wednesday 28 February, 2018, 14:00 h, DH Seminar Room*

The study of the fully nonlinear exact solutions known as Exact Coherent Structures (ECS), has helped to elucidate how turbulence arises. The relevance of ECS has been backed up by a large body of work, which agree with the view that the existence of these solutions is a necessary (but not sufficient) condition for turbulent dynamics.

The plane Couette flow problem - a fluid sheared between two parallel, differentially moving plates - is a classic canonical flow that has continuously provided significant insights in the nature of turbulent flow. This flow is linearly stable for all Reynolds numbers, Re, yet turbulence can occur at a finite Re. In this work we add extra physics to plane Couette flow in the form of stable stratification (gravity perpendicular to the plates) to explore transition in a stably stratified shear flow and investigate the influence of ECS.

**Shear-thinning: A stabilising effect? Yes, no, maybe?**

*Wednesday 21 February, 2018, 14:00 h, DH Seminar Room*

In this talk we will investigate how viscosity effects the stability of a fluid flow. By assuming a shear-thinning viscosity relationship, where an increase in shear-rate results in a decrease in fluid viscosity, we show that flows can be interpreted as being either stabilised or destabilised, depending on the definition of the Reynolds number. Using a two-dimensional boundary-layer flow as our ‘toy model’ we are able to show equivalence between different shear-thinning models. The effect shear-thinning has on important parameters such as the critical Reynolds number and the maximum frequency of the disturbances will be discussed and interpreted in the wider context. Furthermore, to gain an insight into the underlying physical mechanisms affecting the destabilisation of the disturbances, an integral energy equation is derived and energy calculations are presented.

**Continuous transition to turbulence in a planar shear flow**

*Wednesday 14 February, 2018, 14:00 h, DH Seminar Room*

Classifying the transition to turbulence in planar shear flows is a long-standing question without a definitive answer. More specifically, the question is one of continuous or discontinuous transition, whether an arbitrarily small turbulence fraction can be maintained in the long-time limit. To attack this problem, either in simulations or experiments, requires domains that are large relative to the building blocks of transition, turbulent spots and bands. The combination of large domains and long time integrations result in a computational burden too large for 3D DNS. To overcome this obstacle we study a model flow, Waleffe flow, with the same hallmarks of laminar-turbulent intermittency. Using this flow we can address the question of whether transition is discontinuous or continuous.

**Energy spectra and fluxes of buoyancy-driven flows**

*Wednesday 31 January, 2018, 14:00 h, DH Seminar Room*

Buoyancy-driven flows are often encountered in geophysics, astrophysics, atmospheric and solar physics, and engineering. In general, these flows come in two categories: stably stratified flows and Rayleigh-Bénard convection (RBC). Turbulent aspects of these flows are an active field of research. An important unsolved problem in this field is how to quantify the small-scale quantities, e.g., spectra and fluxes of kinetic energy (KE) and potential energy (PE) of these flows. Using direct numerical simulations performed at high resolution, we demonstrate that the stably stratified turbulence at moderate stratification exhibits Bolgiano-Obukhov scaling, due to the conversion of kinetic energy to potential energy via buoyancy. We show that the KE flux decreases with the wavenumber (k) which yield k^{-11/5} and k^{-7/5} scaling for KE and PE spectra respectively. For RBC, we performed simulation at grid resolution 4096^3 on a cubical box and have shown a delicate balance of dissipation and energy supply rate by buoyancy. This balance leads to a constant KE flux and rules out the Bolgiano-Obukhov scaling, and we observe Kolmogorov’s spectrum.

**The Route to Turbulence**

*Wednesday 15 November, 2017, 16:00 h, DH Seminar Room*

Explaining the route to turbulence in wall-bounded shear flows has been a long and tortuous journey. After years of missteps, controversies, and uncertainties, we are at last converging on a unified and fascinating picture of transition in flows such as pipes, channels, and ducts. Classically, subcritical transition (such as in a pipe), was thought to imply a discontinuous route to turbulence. We now know that this is not the case -- subcritical shear flows may, and often do, exhibit continuous transition. I will discuss recent developments in experiments, simulations, and theory that have established a deep connection between transition in subcritical shear flows and a class of non-equilibrium statistical phase transitions known as directed percolation. From this we understand how to define precise critical points for systems without linear instabilities and how to characterize the onset of turbulence in terms of non-trivial, but universal power laws. I will discuss the physics responsible for the complex turbulent structures ubiquitously observed near transition and end with thoughts on outstanding open questions.

**Chromosome conformations and topologically stabilized polymer states**

It has been known for some time that apart from traditional equilibrium states of polymer systems, such as swollen polymer coils and melts of linear polymer chains, which have been studied by classical polymer physics, there exist a class of states whose properties are mostly controlled by the topological interactions of the chains. The archetypical system of this type is a melt of non-concatenated polymer rings. In recent years, it became more and more clear that such states can be observed, at least as metastable ones, in various other contexts including, for example, rapid collapse of a linear chain under external force (in this context they are often called crumpled or fractal globules), and conformation-dependent polymerization. Most importantly, such states seem to be a good candidate for the description of chromosome packing in living cells.

In my talk I will start with a brief review of the classical polymer theory, and then talk about recent advances in theoretical and numerical understanding of the structure and dynamics of these topologically stabilized states, paying special attention to the biological relevance of the results.

**Structure of Plasma Heating in Gyrokinetic Alfvenic Turbulence**

*Wednesday 8 November, 2017, 14:00 h, DH Seminar Room*

We analyse plasma heating in weakly collisional kinetic Alfvén wave turbulence using high resolution gyrokinetic simulations spanning the range of scales between the ion and the electron gyroradii. Real space structures that have a higher than average heating rate are shown not to be confined to current sheets. Furthermore, we show that electrons are dominated by parallel heating while the ions prefer the perpendicular heating route. We comment on the implications of the results.

**MHD Couette flow: a support for surface rheology of liquid metals in the Gibbs approach**** **

*Thursday 2 November, 2017, 16:00 h, DH Seminar Room*

The investigation of the two-way coupling between bulk magnetohydrodynamics (MHD) and surface rheology is particularly motivated by metallurgy and microelectronics applications where flows of liquid / melted metals, often topped with oxide layers, are usually encountered. MHD and surface rheology are briefly introduced with special emphasis on the following concepts: shear and dilatational surface viscosities, Joule extinction, magnetic diffusion of angular momentum. The development of a new annular MHD surface viscometer is presented and justified. Based on analytical and numerical calculations, the surface shear viscosity is demonstrated to be responsible for the electrical activation of a Hartmann layer, leading to a variety of MHD flow patterns. MHD is finally presented as a means to identify properly the surface shear viscosity alone, with no need for considering the surface dilatational viscosity. From quite recent experiments, an (original) estimate of the surface shear viscosity of Galinstan (liquid alloy at room temperature) is given.

**A new approach to the self-excited dynamo problem (54 years late)***Wednesday November 1, 2017, 14:00 h, DH Seminar Room*

Over the last twenty years there have been numerous computations of self-excited dynamos driven by thermal convection in a spherical geometry. This has put planetary dynamos on a firm footing, and demonstrated the basic mechanisms at play. But the computations suffer from being too viscous by many orders of magnitude, and thus it is unclear that the correct force balance is being achieved. I will discuss an alternative approach to the problem due to Taylor (1963), in which viscosity and inertia are neglected at the outset. This has the distinct advantage of ensuring a balance between Coriolis, Lorentz and buoyancy forces, together with pressure, at the outset. This approximation has been termed Magnetostrophy. I will show how control theory can be used as a robust method to discover new solutions to this problem, and the implications.

**Nonlinear dynamics of viscous flows ***Wednesday October 18, 2017, 14:00 h, DH Seminar Room*

In this talk I will introduce some of my research interests in theoretical fluid dynamics. In particular, I will focus on the following two topics: (a) Creep closure of cavities in viscous/non-Newtonian fluids, which has application in geophysics and possibly industry, and (b) Self-similar singular behaviour (rupture, blow-up, and more complicated things) of fourth order partial differential equations that are used to describe thin film flow, which are of great interest both in applications and theory. I will also take the chance to describe problems I am working on or would like to explore in future.

**Ensemble inequivalence in the mean-field Blume-Emery-Griffiths model***Wednesday August 2, 2017, 16:00 h, DH Seminar Room*

We study inequivalence of the canonical and microcanonical ensembles in the mean-field Blume-Emery-Griffiths model. The thermodynamic phase diagrams of the model strongly depend on the value of the biquadratic exchange interaction. At small values there is a transition between a ferromagnetic and a paramagnetic state, and the location of the tricritical point, separating second order and first order transitions, differs between the ensembles. At higher values, in addition, a transition between two different paramagnetic states appears, characterized by a critical point. Again, the position of the critical point is not the same in both ensembles. Furthermore, we study ergodicity breaking: gaps in the accessible magnetization are present at low energies.

**Critical percolation in 2d dynamics***Wednesday July 5, 2017, 16:00 h, DH Seminar Room*

I will present recent results on the study of early time dynamics of the bidimensional ferromagnetic Ising model instantaneously quenched from the disordered to the ordered, low temperature, phase. In particular, I will explain how we can identify a rapid approach to random critical percolation in a time-scale much shorter than the equilibration time. I will also discuss how this depends (or not) on the choice of dynamics (conserved versus non-conserved), the type of lattice or the addition of disorder.

**Lorentz force velocimetry for weakly conducting fluids***Wednesday June 28, 2017, 14:00 h, DH Seminar Room*

We report about the contactless flow measurement in low-conducting fluids using Lorentz force velocimetry LFV. We show that with highly sensitive force sensors we can record flow rates in tap water. For LFV there is nearly no influence of the flow profile to the signal. In two-phase flow it is possible to measure size and velocity of large gas bubbles with the LFV.

**Random Field Ising Model with Conserved Kinetics***Wednesday May 24, 2017, 16:00 h, DH Seminar Room*

We perform comprehensive Monte Carlo (MC) simulations to study ordering dynamics in the random field Ising model with conserved order parameter (C-RFIM) in d = 2, 3. The observations from this study are: (a) For a fixed value of the disorder, the correlation function C(r, t) exhibits dynamical scaling. (b) The scaling function is not robust with respect to the disorder strength i.e., super-universality (SU) is violated. (c) At early times, the domains follow algebraic growth with a disorder-dependent exponent. At late times, there is a cross-over to logarithmic growth. (d) The small-r behavior of the correlation function exhibits a cusp singularity. (e) The corresponding structure factor exhibits a non-Porod tail, and obeys a generalized Tomita sum rule.

**Pattern Formation in the Kinetics of Phase Transitions***Wednesday May 17, 2017, 16:00 h, DH Seminar Room*

Consider a system which is rendered thermodynamically unstable by a sudden change of parameters, e.g., temperature, pressure, etc. The system evolves towards its new equilibrium state via the emergence and growth of domains enriched in the preferred phase. Problems in this area of "kinetics of phase transitions" have received much research attention. In this talk, we review our understanding of this area. We conclude by discussing the problem of surface-directed spinodal decomposition, i.e., the interplay of wetting and phase separation at surfaces.

**Critical speeding-up***Wednesday April 5, 2017, 16:00 h, DH Seminar Room*

Critical slowing-down is a dynamical manifestation of phase transitions, deteriorating the efficiency of many Markov chain Monte Carlo algorithms. In a nutshell, it describes the super-linear scaling of dynamical timescales, such as relaxation, mixing or autocorrelation times, with the volume of the underlying physical system. This effect often occurs in form of a power law, characterised by a positive dynamical critical exponent. Quite a number of (cluster) algorithms have been devised in order to reduce or eliminate the effect of critical slowing-down. The standard lore is that large-scale altering cluster algorithms are superior to locally perturbing Markov chains in reducing critical slowing-down.

Surprisingly, a number of recent numerical studies have not only shown that locally perturbing Markov chains for geometric loop and Fortuin-Kasteleyn representations of Ising and Potts models can have smaller dynamical critical exponents than some cluster algorithms, they also revealed that these Markov chains exhibit critical speeding-up. In contrast to the “negative” slowing-down mechanism, critical speeding-up is a “positive” manifestation of collective fluctuations at criticality, yielding a sub-linear scaling of dynamical timescales with negative dynamical critical exponents. Moreover critical speeding-up was recently observed in quantum-mechanical systems and in spin ice, providing further evidence for its ubiquitous relevance.

In this talk we present a spectral characterisation of the critical speeding-up phenomenon, which we then utilize to prove critical speeding-up for a number of geometric observables, including the susceptibility, in two- and high-dimensional percolation.

**T****oroidal vortices in protoplanetary discs, a solution to the dust migration problem?**

Present observational techniques have allowed us to witness the process of planet formation up to an unprecedented level of detail. However, we are still far from having a universally accepted theoretical model for planet formation. Planets form around young stars in protoplanetary discs. Such discs are initially composed of a mixture of gas and small dust grains, with sizes ranging from a few microns up to a few millimeters. In order to form planets, such grains must gather together and stick, building bigger and bigger aggregates until the first planetary embryos (the so-called planetesimals) are formed. However, the exact mechanism capable of gathering the grains, letting them stick and grow is still unknown. In this talk, I will present a recently discovered hydrodynamical instability, driven by the presence of dust in protoplanetary discs, that may be able to offer a simple solution to this problem.

**Magnetization Curves and Thermal Entanglement of Low-Dimensional AFM Materials**

The magnetization plateaus have played a great role in understanding of a large family of nontrivial quantum phenomena of spin systems. We have studied some magnetic properties and thermal entanglement of the Ising–Heisenberg model using transfer-matrix and dynamical method.The pyrochlore edge-shared tetrahedron ladder with spin-1/2 is an excellent candidate to realize the antiferromagnetic properties. We have applied the separation of the Heisenberg intra-rung and Ising inter- rung interactions on a pyrochlore ladder with antiferromagnetic spin-1/2 couplings and obtained the magnetization plateau, thermal concurrence and magnetic susceptibility. The second and third parts of the seminar will be devoted to the spin-1 Ising-Heisenberg models of Ni-containing compounds [Ni3(fum)(m3-OH)2(H2O4)]n on a diamond chain and [Ni (NN’-dmen)(µ-N3)2]n polymer, which have both antiferromagnetic and ferromagnetic couplings. The magnetization plateaus and negativity (thermal entanglement) are obtained. We study in the last part of seminar the connection between Lyapunov exponents and magnetization plateaus using the dynamical technique.

**How physics is helping bring new perspectives into important economic problems?**

TBC

**Acoustic streaming in liquids & a study of gas entrainment**

Acoustic streaming denotes the flow induced by a propagating acoustic wave in liquid. On the one hand, acoustic streaming offers a contactless way of stirring either corrosive or particularly delicate fluids such as liquid silicon in the semiconductors manufacturing process. On the other hand, recent works show that it can unwillingly spoil velocities measurements obtained with ultrasounds, a crucial technique when dealing with liquid metals and opaque fluids for instance. It is investigated with both experimental and numerical approaches. A 2MHz circular plane transducer, used as an acoustic source, is introduced inside a water tank. The measurements concern the acoustic pressure field (hydrophone) and the velocity field (PIV). Numerical simulations are also performed with the software STARCCM+TM. They solve the incompressible Navier Stokes equations with an acoustic force source term. A good agreement is obtained between the experimental and numerical results through several configurations.

In Sodium cooled fast nuclear reactors, the presence of eddies at the free surface combined with the downward flow created by the suction of hot liquid sodium (by the intermediate heat exchanger) can provoke entrainment of gas (argon). Such entrainment induces safety issue since gas could be transported to the bottom of the vessel and may lead to the accumulation of gas pockets close to the core of the reactor. The study focuses on gas entrainment from surface swirls through an experimental apparatus in water. A shear flow is generated between a horizontal flow and a stagnant flow, and a vertical pumping is added, at the bottom of the test section, to produce gas entrainment. The objectives are: first, to identify the experimental condition of gas entrainment occurrence, second, to describe and quantify the occurrence of gas entrainment (shadowgraphy, image processing) and third, to characterize the physical mechanisms involved (PIV). These experiments support the validation of TRIO_CFD, a CEA internal code.

**Finite-size corrections for universal boundary entropy in bond percolation**

We compute the boundary entropy for bond percolation on the square lattice in the presence of a boundary loop weight, and prove explicit and exact expressions on a strip and on a cylinder of size L. For the cylinder we provide a rigorous asymptotic analysis which allows for the computation of finite-size corrections to arbitrary order. For the strip we provide exact expressions that have been verified using high-precision numerical analysis. Our rigorous and exact results corroborate an argument based on conformal field theory, in particular concerning universal logarithmic corrections for the case of the strip due to the presence of corners in the geometry. We furthermore observe a crossover at a special value of the boundary loop weight.

**Non-equililibrium studies of some complex systems with disorder and long-range interactions***Wednesday March 1, 2017, 15:00 h, DH Seminar Room*

Many physical systems are inherently disordered or have complex interactions between their constituents. The nature of the disorder and the interactions play a major role in determining their static and dynamical properties. They are characterized by complex free energy landscapes with deep minima and often show interesting non-equililibrium behaviour such as ageing, slow relaxation, anomalous transport properties, etc. We present our recent results on some complex systems which are of interest to us.

The first part of the talk will be on ground-state properties of the random-field Ising model (RFIM) on isometric lattices [1]. The RFIM is an archetypal model of a system with quenched disorder. Using a computationally efficient Graph-Cut algorithm we obtain the exact ground states of the RFIM on the simple cubic, body centered cubic and face centered cubic lattices. The small-r behaviour of the correlation function exhibits a cusp regime signifying fractal interfaces. The difference in the lattice structures gives rise to slightly different fractal exponents in the ferromagnetic phase. As a result, we find that the energy cost of forming an interface in the three lattices are significantly different for the different lattices. This has important implications on their relaxation behaviour.

The second part of the talk will be on kinetics of phase ordering in dipolar Ising systems [2]. The dipole-dipole interactions are long-ranged, alternating in sign and anisotropic in nature. Equilbrium studies have revealed novel consequences of these complitated interactions but their effect on non-equilbrium aspects is unexplored. Using Monte Carlo simulations, we study the Glauber dynamics of this model quenched into the ferromagnetic phase. Our main observations are the emergence of elongated domains and anisotropic domain growth laws with distinct dynamical exponents along the spin axis and perpendicular to it.

The third and last part of the talk will be a quick review of our work on anomalous transport in rough thin films [3]. We use the non-Porod decay of the structure factor to accurately determine the fractal properties of some prototypical nanoparticle films. Using scaling arguments, we find that the resistance of rough films scales as a non-integer power law of its lateral size, in contrast to integer scaling laws for compact films. Our results are valuable for understanding reports of anisotropic electrical properties in rough thin films.

[1] A. Bupathy, V. Banerjee, and S. Puri, "Random-field Ising model on isometric lattices: Ground states and non-Porod scattering," Physical Review E 93, 012104 (2016).

[2] A. Bupathy, V. Banerjee, and S. Puri, "Columnar domains and anisotropic growth laws in dipolar systems," Submitted to Physical Review Letters.

[3] A. Bupathy, R. Verma, V. Banerjee and S. Puri, "Non-Porod scattering and non-integer scaling of resistance in rough films," Journal of Physics and Chemistry of Solids 103, 33 (2017).

**Waves in spherical dynamo simulations and implications for the Earth’s core**

MHD waves excited in rapidly rotating Earth's core can produce secular time-variations of the planetary magnetic field. In the magnetostrophic state, in which Lorentz force and Coriolis force balance, unique wave motions can occur in both axisymmetric and nonaxisymmetric modes, including torsional Alfven waves and slow magnetic Rossby waves. To explore the relevance of these waves and the dynamics, we use DNS of convection-driven dynamos in rotating spherical shells. Waves in data could provide us with information about the physical properties, such as the field strength, within the deep interior of the planet.

**Studying polymers using coarse-grained models and generalized-ensemble computer simulations**

Understanding single-chain behavior of polymers has become a widespread interdisciplinary interest. Conformational changes based on specific internal properties, like bonded and non-bonded interactions can be studied qualitatively to great extend via computer simulations. Coarse-graining of specific polymers can help mapping qualitative findings back to realistic quantities. Utilizing the generalized-ensemble simulations and sophisticated statistical analysis methods we investigate polymers undergoing conformational changes when interacting with substrates of different geometries.

**The Porpoise of Power: The hydrodynamic advantage of dolphins**

Most creatures do not wish to give up their secrets easily and the dolphin is no exception to the rule. For centuries they have been observed as incredibly good at swimming (porpoising) and generally frolicking about at high speed entertaining the observer and providing Engineers, Scientists and Mathematicians many questions in the fields of hydrodynamics, material science and animal propulsion. In 1936 Sir James Gray concluded that a dolphin’s muscles would have to generate seven times the amount of power than a human in order for it to travel at approximately 8m/s. Sir James believed that it was not possible for the dolphin’s muscles to generate this and that there must be other mechanisms at play. This talk outlines some of the work conducted by the author and others in the field and looks at the various fluid flow mechanisms that could exist and how they behave. We also bring the field up to date with current findings and answer the question: Has the dolphin really given up all its secrets or are they still doing it on porpoise?

*Apologies, this awful pun has not been included on porpoise.

**Rotating Rayleigh-Taylor Instability**

We examine theoretically and experimentally the effect of rotation upon the two-layer Rayleigh-Taylor instability in three dimensions. Following the classical, variational approach of Miles [1], we find the dispersion relation that characterizes the instability at the interface between the two co-rotating fluids. Despite Chandrasekhar’s [2] statement that rotation cannot stabilize an otherwise unstable stratification, we derive a critical rotation rate for stabilizing the most unstable mode in a non-rotating arrangement — a new and apparently contradictory result.

Using the magnetic field of a superconducting solenoid magnet we trigger the Rayleigh-Taylor instability in an otherwise gravitationally stable cylindrical tank containing two miscible liquid layers: a light paramagnetic liquid above a heavier diamagnetic liquid below. We find that rotation acts to retard the instability’s growth rate and stabilize long wavelength modes; the scale of the observed structures decreases with increasing rotation rate, asymptoting to a minimum wavelength controlled by viscosity. Our experimental evidence is consistent with our earlier theoretical prediction.

[1] Miles, J.W. 1964 Free-surface oscillations in a slowly rotating liquid. J. Fluid Mech. 18(2), 187–194.

[2] Chandrasekhar, S. 1964 Hydrodynamic and Hydromagnetic Stability. New York: Dover.

**Large-scale vortices and dynamos in rotating convection**

Recent computational studies have described the formation of large-scale vortices in rotating turbulent convection in planar geometry. These vortices, which are long-lived and depth-invariant, form by the merger of convective thermal plumes. In the absence of magnetic fields, these vortices grow to the horizontal size of the computational domain. In the first part of the talk, I will describe how these vortices form and how they affect the transport of heat through the system. In the second part of the talk, I will discuss their interaction with self-sustained magnetic fields.

**Experimental evidence of symmetry-breaking supercritical transition in pipe flow of shear-thinning fluids**

Unique experimental results reveal that the asymmetric flow of shear-thinning fluid through a cylindrical pipe, which was previously associated with the laminar-turbulent transition process, actually has the characteristics of a non-hysteretic and reversible, supercritical instability of the laminar base state. Contrary to what was previously believed, classical transition is actually responsible for returning symmetry to the flow. These novel and unexpected discoveries offer new insights into the stability of the flow of shear-thinning fluids and the fundamental nature of the laminar-turbulent transition process.

**Optimisation of Dynamos**

In some planets and stars, e.g., the Earth, a magnetic field is constantly being generated and sustained by the complex motion of a conducting fluid in the interior. This phenomenon can be understood in the framework of dynamo theory, which mathematically describes the interaction between the flow and the magnetic field. In this theory, the growth of a seed magnetic field is determined by the competition between magnetic induction and magnetic diffusion. The ratio of these two effects is given by a dimensionless parameter called the magnetic Reynolds number (Rm). The value at which the magnetic field becomes self-sustaining , i.e., the system acts like a magnetic dynamo, is called the critical magnetic Reynolds number. An outstanding question is which kind of flow can amplify a seed magnetic field most efficiently so that the critical Rm is the lowest. We adapt a variational method [1] to search for the minimal threshold of dynamos. This method allows us to maximize the growth of the magnetic field over a time window T by optimizing the steady flow field and a seed magnetic field. In this talk, I will present the general principle and numerical methods used to solve this optimization problem in a sphere with no-slip and insulating boundary conditions and the results we obtained. In addition, I will show the preliminary results from extended models in which we study how the boundary conditions and symmetry separations change the minimal threshold in a sphere.

[1] A.P. Willis. Optimization of the magnetic dynamo, Phys. Rev. Lett. 109 (25), 251101. (2012).

**Modeling Multiphase Flow - Old and new in my research**

This presentation gives a general overview on different aspects of modelling and simulation of multiphase flow problems.

The sedimentation of polydisperse suspensions has various applications as wastewater treatment and the tailing condensation and water recovery in mining industry. A description by system of conservation laws allows to discuss modelling aspects, elaborate a mathematical analysis as the solution of Riemann problems and motivates the development of numerical simulation tools. One application of the presented sedimentation model is the description of rock separation in a magma chamber. One special feature of suspensions, where the solid materials have varying density, are spatial segregation effects described as “fingering”. The related instabilities can be interpreted by the elliptic degeneracy of the first-order hyperbolic equations.

Similar modelling and simulation strategies apply in the general framework of convection-reaction-diffusion equations, for example in the modelling of pedestrian crowds by convection-diffusion equations and the modelling of epidemics by diffusion-reaction equation. An equation class complementary to conservation laws are Hamilton-Jacobi equations. The concept of of Level Set Method allows to track implicitly defined fronts, and can be adapted to simulate the foam front propagation in oil reservoirs or of fire fronts when modelling bushfire.

**Turbulence in planetary cores**

An overview of the work of Henri-Claude and collaborators.

**Ising model on plane: numerical solution**

Using the bond-propagation algorithm [1], we study the Ising model on a plane. We obtain the free energy, internal energy, and specific heat numerically on square lattices with a square shape and various combinations of the boundary conditions [2]. The numerical data are analyzed with finite-size scaling. The exact results are conjectured for the corner logarithmic terms in the free energy and it’s agreed with the conformal field theory very well. The exact results are conjectured also for the corner logarithmic term and the edge logarithmic term in the internal energy and in the specific heat.

[1] Y.L. Loh and E.W. Carlson, Phys. Rev. Letts. 97, 227205 (2006).

[2] X. Wu and N. Izmailian, Phys. Rev. E 91, 012102 (2015).

**A Brief Review on the Properties of the Spin-1 Ising-Heisenberg ****Diamond Chain Models**

We study the properties of the exactly solvable spin-1 Ising-Heisenberg diamond chain models using the transfer matrix method [1-4]. In particular, exact results for the ground state, magnetization process, specific heat, partition function zeros and thermal entanglement are presented. The contributions of the single-ion anisotropy, bilinear XXZ Heisenberg, biquadratic XXZ- and Ising-Ising interaction parameters into the properties of the models are discussed. As we show, the magnetization curves may include either one, two or three intermediate magnetization plateaus at zero, one-third and two-thirds of the saturation magnetization.

[1] N.S. Ananikian, J. Stre?ka, V. Hovhannisyan, Solid State Commun.194 (2014) 48.

[2] V.S. Abgaryan, N.S. Ananikian, L.N. Ananikyan, V.V. Hovhannisyan, Solid State Commun.224 (2015) 15.

[3] V.V. Hovhannisyan, N.S. Ananikian, R. Kenna, Physica A 453 (2016) 116.

[4] V.V. Hovhannisyan, J. Stre?ka, N.S. Ananikian, J. Phys.: Condens. Matter 28 (2016) 085401.

**Diffusion Limited Aggregation fractals: current state**

Diffusion Limited Aggregation (DLA) is a model introduced in 1983 by Witten and Sander as idealization of the process to form random fractals. These fractals are looks somehow similar to the real objects known in nature. We present current state of the knowledge on DLA fractals, which seems to be governed by just one non-integer exponent. We emphasize that theoretical progress in two-dimensional case may be obtained in the vicinity of the special point corresponding to the fractal objects with five-fold symmetry. We shortly review different approaches to the problem that is the different techniques for solving Laplace equation.

**Morphologies of Bottle-Brush Block Copolymers***Wednesday March 9, 2016, 15:00 h, DH Seminar Room*

We investigate the self-assembly of bottle-brush block copolymers into well-defined periodic morphologies by using molecular dynamics simulations of a bead-spring model. The microphase separation is driven by the chemical incompatibility between the different blocks of side chains leading to the formation of two- and three-domain lamellae and hexagonally packed cylinders. The molecular asymmetry required for the formation of cylindrical domains is not introduced by the difference in volume fractions, but by the asymmetry of the side chain lengths and the stiffness of the backbone. Such behaviour deviates significantly from what is typically known for linear block copolymers. The obtained morphology maps provide a genuine way of understanding the role of molecular architecture in achieving experimentally desired structures for nanoporous and photonic materials based on the self-assembly of bottle-brush block copolymers.

**Fermionic quantum criticality in honeycomb and pi-flux Hubbard models**

We numerically investigate the critical behavior of the Hubbard model on the honeycomb and the pi-flux lattice, which exhibits a direct transition from a Dirac semimetal to an antiferromagnetically ordered Mott insulator. We use projective auxiliary-field quantum Monte Carlo simulations and a careful finite-size scaling analysis that exploits approximately improved renormalization-group-invariant observables. This approach, which is successfully verified for the three-dimensional XY transition of the Kane-Mele-Hubbard model, allows us to extract estimates for the critical couplings and the critical exponents. The results confirm that the critical behavior for the semimetal to Mott insulator transition in the Hubbard model belongs to the Gross-Neveu-Heisenberg universality class on both lattices.

Ref: F. Parisen Toldin, M. Hohenadler, F. F. Assaad, I. F. Herbut, Phys. Rev. B 91, 165108 (2015), arXiv:1411.2502

**Exactly solvable model of needle crystal growth**

I am going to speak about a simple dynamical model which is supposed to catch the main properties of a set of needle crystals (like, e.g., diamond needles) growing from a flat substrate.

We consider needles which start growing at time zero in random directions from a set of randomly positioned seeds. Any collision of such two growing needles implies a tip of one needle hitting the body of another one. Our assumption is that on such a collision the needle whose tip hits another one stops growing (‘dies’), while another one continues to grow unperturbed.

I will discuss the properties of this model both for an unlimited uniform initial distribution of seeds (i.e., when one has an unlimited line or plane where seeds are distributed with fixed density), and a seed distribution with limited support (i.e., all seeds are located within a given interval or half-line).

In the unlimited case we were able to find

- scaling behavior of the density of growing needles and of their angle distribution as a function of time in both (1+1)D and (2+1)D;
- the exact needle angle/density distribution as a function of time in (1+1)D;
- asymptotic solution of a Boltzmann equation in (1+1)D, which substantially differs from the exact solution.

For the case with final support in (1+1)D we have

- given that there was a finite initial number of seeds, we can calculate the full probability distribution of the number of needles surviving at infinite time;
- given that one half-line is filled with a fixed density of seeds, and another half-line is empty, we can estimate the average number of needles infiltrating the originally empty half-plane up to a given time.

However, in the second case there are several tantalizingly simple observations which we are still unable to prove. I plan to outline them as well.

**Using strong gradient magnetic field to study the equilibrium shapes and stability of liquids in weightlessness and in different gravity**

The shape of the spinning Earth (or any object bound by its own gravity) is determined by competing gravitational and rotational forces. Many famous mathematicians have worked on this problem beginning with Newton who discovered an elegant proof showing that the equilibrium shape is an oblate spheroid for small angular velocities. Later illustrious mathematicians, including the astrophysicist Chandrasekhar, discovered other theoretical shapes which include rugby ball-shaped prolate spheroids and two-lobed peanut shapes. The shapes of a spinning liquid droplet are directly related; here, surface tension acts similarly to gravity in holding the droplet together.

In this talk I will present recent work in which we use the magnetic field of a superconducting magnet to levitate liquid droplets in order to study the shapes of spinning and electrically charged droplets.

The same magnetic body force that levitates the liquid drops may also be used to apply differential body forces to a stratified liquid. I will introduce the use of this technique to study the Rayleigh Taylor instability under rotation, and present results showing that rotation slows the growth of the instability.

**Impulsively excited disturbances in non-uniform boundary layers**

Results from numerical simulations will be presented for the linearized disturbance impulse responses of non-uniform boundary layers. Two distinct forms of boundary layer non-uniformity have been studied.

Firstly, we consider temporally steady rotating disc boundary layers, where there is a spatial inhomogeneity which stems from the radially increasing circumferential velocity. This turns out to have a very significant impact on the radial propagation of disturbances and their long-term growth behaviour. For example, the introduction of flow control measures such as surface suction, which are chosen to be locally stabilizing, can in fact lead to a global destabilisation. Moreover, this destabilisation may be associated with a novel kind of faster then exponential disturbance growth.

The second type of boundary layer non-uniformity that we will consider allows us to address the effects of base-flow unsteadiness on the global development of disturbances. We conducted simulations for the oscillatory Stokes layer that is driven by the time-periodic in-plane motion of a bounding flat plate. The unsteadiness was found to give rise to multiple wavepackets for the impulse response, which displayed an intricate tree-like spatial-temporal structure.

For both types of flow configuration, we will illustrate various intriguing features of the new patterns of behaviour that were discovered. These had not been anticipated by previous studies for simpler configurations, where a steady in time boundary layer could be treated as being approximately spatially homogeneous along a dominant disturbance propagation direction. The aim will be to provide an overview and to highlight some of the novelties.

**The Driven Random Field Ising Model: Interfaces, textures and Scaling Laws**

We use a computationally efficient graph-cut method (GCM) to obtain the ground-state (GS) morphologies at zero temperature of the random-field Ising model in presence of a uniformly varying external field H. We have defined pinned sites as those sites at which spins are pointing against the direction of negative applied field, and pinned clusters as those which are nucleated by the pinned sites when the field is incremented. We study the growth of pinning clusters, size distribution, domain textures, and interfacial properties.

**Some recent results about fractal polymer globules as a model for genome spatial organization**

Recent experimental results on the spatial structure of eukaryotic chromatin, in particular those obtained by genome-wide chromosome conformation capture (Hi-C) method [1], led to a renewed interest in the so-called crumpled globule or fractal globule state of polymers first described almost 30 years ago in [2]. Indeed, the analysis carried out in [3] shows that the spatial conformation of chromatin (at least on the level of characteristics averaged along the genome) is more compatible with the fractal globule model of genome packing than with any other known simple physical model.

In this presentation we will review three of our recent contributions to the development of a general theory of the fractal globule state of polymers.

First, we show that combination of (i) hierarchical chain folding implied by the fractal globule model, (ii) block-coplymer structure of chromatin, and (iii) averaging over conformations, can qualitatively reproduce all the features seen on the experimental Hi-C maps [4]. Second, we address a computational problem of how to generate fractal-globule-like polymer conformations in computer experiment. In particular, we propose and characterize a method of generating such structures based self-reinforced random walks with excluded volume interaction. Third, we study, be means of scaling theory and DPD computer simulations, the thermal motion of chain links in a fractal globule, and show it be not compatible with either Rouse or reputation model of polymer dynamics. We show[5], that monomer motion in this state is subdiffusive with a scaling exponent of 0.4 (instead of 0.5 for the Rouse model), which seems to be in good agreement with the available data from biological experiments.

[1] J. Dekker, et al. Science, 295, 1306 (2002).

[2] A.Yu. Grosberg, S.K. Nechaev, E.I. Shakhnovich J. Phys. France 49, 2095 (1988).

[3] E. Lieberman-Aiden, et al. Science, 326, 289 (2009).

[4] L. Nazarov, M. Tamm, V. Avetisov, S. Nechaev, Soft Matter, 11, 1019 (2015).

[5] M. Tamm, L. Nazarov, A. Gavrilov, A. Chertovich, Phys. Rev. Lett., 114, 178102 (2015).

**Efficient Large Scale Simulation of Stochastic Lattice Models on GPUs**

With growing importance of nano-patterned surfaces and nano-composite materials in many applications from energy technologies to nano-electronics, a thorough understanding of the self-organized evolution of nano-structures needs to be established. Modelling and simulations of such processes can help in this endeavor and provide predictions for the turnout of manufacturing processes.

In this talk GPGPU-enabled implementations of two stochastic lattice models will be discussed, shedding light on the complications which arise when simulations of stochastic processes are to make efficient use of massively parallel GPU architectures. A single-GPU implementation of the (2+1)-dimensional Roof-Top-model allows very precise large-scale studies of surface growth processes in the Kardar-Parisi-Zhang universality class.[1] Furthermore a multi-GPU enabled version of the 3d kinetic Metropolis lattice Monte-Carlo method[2] provides the capability to study the evolution of nano-structures both towards and out-of-equilibrium at spatio-temporal scales of experiments using only small to medium-sized GPU clusters.

[1] J. Kelling, G. Ódor Extremely large-scale simulation of a Kardar-Parisi-Zhang model using graphics cards, Physical Review E 84, 061150 (2011)

[2] J. Kelling, G. Ódor, F. Nagy, H. Schulz, K. Heinig Comparison of different parallel implementations of the 2+1-dimensional KPZ model and the 3-dimensional KMC model, The European Physical Journal - Special Topics 210, 175-187 (2012)

**On mathematical and computational problems of genomics: from sequencing to population structure**

Genomics is a young research area in genetics and bioinformatics. It has appeared recently due to a significant decrease in the value of DNA sequencing. This drop in sequencing prices led to a rapid growth of the amount of genomic data. Mathematical and computational analysis is of a great importance at all stages of the genomic study: from the identification of variable sites (such as mutations, insertions and deletions) to the study of the population structure. We will discuss what kind of major mathematical and computational problems arise in genomics and will introduce their most basic models and concepts.

**Network Geometry**

Networks are mathematical structures that are universally used to describe a large variety of complex systems such as the brain or the Internet. Characterizing the geometrical properties of these networks has become increasingly relevant for routing problems, inference and data mining. In real growing networks, topological, structural and geometrical properties emerge spontaneously from their dynamical rules. Here we show that a single two parameter model of emergent network geometry,constructed by gluing triangles, can generate complex network geometries with non-trivial distribution of curvatures, combining exponential growth and small-world properties with finite spectral dimensionality. In one limit, the non-equilibrium dynamical rules of these networks can generate scale-free networks with clustering and communities, in another limit 2 dimensional manifolds with non-trivial modularity. When manifolds of arbitrary dimension are constructed, and energies are assigned to their nodes these networks can be mapped to quantum network states and they follows quantum statistics despite they do not obey equilibrium statistical mechanics.

**The Discrete Multi-Hybrid System for the simulation of solid-liquid flows**

The flow of solid-liquid suspensions is a generic problem which poses many challenges to scientists and industrialists across many different areas. Applications range widely from processing of food and pharmaceuticals, through oil and mining industries, to blood and biological applications. Such flows involve a large array of complex phenomena, which often includes particle deformation, breakage, degradation, melting, swelling, erosion, aggregation etc. Traditionally, specific modelling techniques have been developed by focusing on certain specific aspects of the flow and simplifying the others. Computational Fluid Dynamics, for instance, accurately describes the fluid dynamics, but the solids phase is simplified by the point-particle assumption. Granular mechanics provides a good account of the inter-particle contact forces, but it cannot handle phenomena such as solid-liquid mass transfer or melting. Solid mechanics can describe the elastic and plastic deformations in the solid, but not the fluid dynamics of the liquid phase.

In order to achieve a more sophisticated description of these systems, a new modelling approach, called the Discrete Multi-Hybrid System, is presented. This technique combines Smoothed Particle Hydrodynamics for the fluid phase, Coarse Grained Molecular Dynamics for the internal solid structure, and the Discrete Element Method to account for the interparticle collisions. In this way, the model can deal with a large variety of particles (e.g. non-spherical, elastic, breakable, melting, solidifying, swelling), and flow conditions (e.g. confined, free-surface, microscopic). Various examples are presented and discussed.

**From particle condensation to polymer aggregation**

TBA

**Impulsively excited disturbances in non-uniform boundary layers**

Results from numerical simulations will be presented for the linearized disturbance impulse responses of non-uniform boundary layers. Two distinct forms of boundary layer non-uniformity have been studied.

Firstly, we consider temporally steady rotating disc boundary layers, where there is a spatial inhomogeneity which stems from the radially increasing circumferential velocity. This turns out to have a very significant impact on the radial propagation of disturbances and their long-term growth behaviour. For example, the introduction of flow control measures such as surface suction, which are chosen to be locally stabilizing, can in fact lead to a global destabilisation. Moreover, this destabilisation may be associated with a novel kind of faster then exponential disturbance growth.

The second type of boundary layer non-uniformity that we will consider allows us to address the effects of base-flow unsteadiness on the global development of disturbances. We conducted simulations for the oscillatory Stokes layer that is driven by the time-periodic in-plane motion of a bounding flat plate. The unsteadiness was found to give rise to multiple wavepackets for the impulse response, which displayed an intricate tree-like spatial-temporal structure.

For both types of flow configuration, we will illustrate various intriguing features of the new patterns of behaviour that were discovered. These had not been anticipated by previous studies for simpler configurations, where a steady in time boundary layer could be treated as being approximately spatially homogeneous along a dominant disturbance propagation direction. The aim will be to provide an overview and to highlight some of the novelties.

**Quantum spin operator and decoration transformation***Friday October 19, 2015, 16:00 h, DH Seminar Room*

The classical version of decoration transformation has been used to map some spin lattice models into another equivalent spin lattice models. This transformation is a very useful tool for identifying a class of spin lattice models. Because, in some particular cases, it is possible to show a class of spin lattice models that can be map into another class of exactly solvable models. Unfortunately, the quantum spin decoration transformation cannot be mapped exactly onto another quantum spin lattice model. However, it is possible to perform a mapping in the "classical" limit, establishing the equivalence between two quantum spin lattice models. Using the cumulant expansion, one can demonstrate that this transformation is valid up to third order in the inverse of temperature, such as for Heisenberg-like models. Here, we present some typical examples for Heisenberg models illustrating this mapping into another Heisenberg model. This transformation could be useful to study the equivalence between to quantum spin lattice.

**Influence of carbon nanotube surface modification on structural properties of polyimide based nanocomposites***Wednesday September 23, 2015, 14:30 h, DH Seminar Room*

Recently, composites based on conjugated polymers filled with carbon nanomaterials have been applied extensively in the field of organic electronics. Regioregular poly-3-hexylthiophene (P3HT) is a polyconjugated heterocyclic polymer considered as one of the best choices for the electron donor material in such composites. The efficiency of organic photovoltaic cells made of P3HT-based composites highly depends on the structure of polymer matrix within the interface region between components. P3HT is semicrystalline polymer and its structure is usually described as combination of crystalline and amorphous domains. The most common P3HT crystalline domain structure is represented by layers of parallel elongated all syn-conformation P3HT chains. However, the potential existence of helical conformation of P3HT chains have been suggested in several works. In the present study helical chain of P3HT was simulated using atomistic molecular dynamics approach in order to investigate the potential impact of temperature regime and amorphous polymer surrounding on structure evolution.

**Molecular dynamics simulationss P3HT helical structure***Wednesday September 23, 2015, 14:00 h, DH Seminar Room*

Recently, composites based on conjugated polymers filled with carbon nanomaterials have been applied extensively in the field of organic electronics. Regioregular poly-3-hexylthiophene (P3HT) is a polyconjugated heterocyclic polymer considered as one of the best choices for the electron donor material in such composites. The efficiency of organic photovoltaic cells made of P3HT-based composites highly depends on the structure of polymer matrix within the interface region between components. P3HT is semicrystalline polymer and its structure is usually described as combination of crystalline and amorphous domains. The most common P3HT crystalline domain structure is represented by layers of parallel elongated all syn-conformation P3HT chains. However, the potential existence of helical conformation of P3HT chains have been suggested in several works. In the present study helical chain of P3HT was simulated using atomistic molecular dynamics approach in order to investigate the potential impact of temperature regime and amorphous polymer surrounding on structure evolution.

**Non-additive measures in social choice and equilibrium theory**

Spaces of probability measures play an important role in the game theory and related areas. In particular, some properties of these spaces are used in the proof of existence of equilibria in mixed strategies. However, numerous paradoxes in decision theory require using non-additive measures, i.e., capacities in describing decision-maker's behaviour under uncertainty. We will apply some category theory arguments to formalise games in non-additive strategies.

We will also discuss a possibility of a non-additive version of the Gibbard-Satterthwaite theorem on manipulability of voting systems.

**Beyond Moore's Law? Seeking Quantum Speedup Through Spin Glasses**

Can quantum computers indeed meet the promise of doing complex calculations faster than classical computers based on transistor technologies? While the holy grail of a programmable universal quantum computer will probably still take decades to reach, one can already begin to answer this question by testing programmable quantum annealing machines that are currently being built. These machines, such as D-Wave Two, use a non-mainstream method known as adiabatic quantum annealing to perform optimization tasks. After a brief introduction to optimization methods, I summarize recent progress in the design and construction of quantum annealing machines. Unfortunately, to date, a conclusive detection of quantum speedup remains elusive. Based on insights from the study of spin glasses, combined with large-scale Monte Carlo simulations and data mining techniques, in this talk I present ideas on how to construct tunable hard benchmark problems that work around the intrinsic noise and technical constraints of current quantum optimization machines. Our results show that a careful design of the hardware architecture and benchmark problems is key when building quantum annealers.

Work done in collaboration with

F. Hamze (D-Wave Systems Inc),

H. Munoz-Bauza (Texas A&M University),

A. Ochoa (Texas A&M University),

S. Schnabel (Leipzig University),

Z. Zhu (Texas A&M University)

**Unzipping and melting of DNA**

We discuss a thermodynamic approach for the force induced unzipping of DNA, based on a set of hypotheses about its response functions. The thermal melting, in contrast, is dominated by the fluctuating bubbles of broken hydrogen bonds. A few consequences of this fluctuation are to be discussed, e.g., an Efimov like effect in a three stranded DNA near melting. We discuss a scaling approach for this problem and the connection with a renormalization group limit cycle.

**Controlled Frustration and Chaos, Critical Phases and Lower-Critical Dimension in Spin Glasses**

Various spin systems with frozen disorder, in which ferromagnetic and antiferromagnetic interactions do not match and local degrees of freedom inject entropy, have been studied by renormalization-group theory in various integer and fractional dimensions. We show that with local rearrangements of ferromagnetic and antiferromagnetic interactions, the interaction non-matching (aka frustration) can be tuned without changing the material components. Thus, the spin-glass phase never before seen in two dimensions has been obtained and the spin-glass phase always seen in three dimensions has been removed.[2] In the spin-glass phase in even-q-state clock spin models, we have shown the equivalence of the chaotic distribution of interactions at a given position appearing under scale changes and the chaotic distribution of interactions in all positions of the system at a given scale. A universality has been found across the different q values.[1] In odd-q-state clock spin models, differently, asymmetric phase diagrams as in quantum systems and many phases in which every point is a critical point have been found.[3] 23 sequenced hierarchical models have been solved and the lower-critical dimension where the spin-glass phase disappears has been precisely determined as 2.5210.[4]

[1] “High q-State Clock Spin Glasses in Three Dimensions and the Lyapunov Exponents of Chaotic Phases and Chaotic Phase Boundaries”, E. Ilker and A.N. Berker, Phys. Rev. E 87, 032124 (2013).

[2] “Overfrustrated and Underfrustrated Spin Glasses in d=3 and 2: Evolution of Phase Diagrams and Chaos Including Spin-Glass Order in d=2”, E. Ilker and A.N. Berker, Phys. Rev. E 89, 042139 (2014).

[3] “Odd q-State Clock Spin-Glass Models in Three Dimensions, Asymmetric Phase Diagrams, and Multiple Algebraically Ordered Phases”, E. Ilker and A.N. Berker, Phys. Rev. E 90, 062112 (2014).

[4] “Lower-Critical Spin-Glass Dimension from 23 Sequenced Hierarchical Models”, M. Demirtas, A. Tuncer, and A.N. Berker, arXiv:1502.06443 (2015).

**An introduction to the computational complexity of spin systems***Wednesday May 13, 2015, 14:00 h, DH Seminar Room*

A spin system is defined by a graph, a number q of "spins", and an matrix of interaction strengths between pairs of spins. Each configuration, i.e., assignment of spins to vertices, has a certain weight, obtained by taking a product over edges of the interaction strength between the spins at the endpoints of the edge. The problem is to approximate the partition function of the system, i.e., the sum of weights over all configurations.

We explore the computational complexity of this problem. The complexity of computing exact solutions is by now well understood. However, given the preponderance of negative results, it is natural to consider the complexity of computing approximate solutions with guaranteed error bounds. The situation here is less well understood. I'll discuss how complexity theory needs to be adapted to prove intractability results, and what algorithmic techniques exist to establish tractability results. A striking feature of recent work is the strong connection between computational intractability and phase transitions.

The work on which this talk is based is due to too many authors to list here, so credits will be left to the talk itself.

**Inequivalence of Ensembles in a Blume-Emery-Griffiths model**

We study inequivalence of the canonical and microcanonical ensembles in a mean-field Blume-Emery-Griffiths model with biquadratic exchange interaction. Inequivalence of the phase diagrams of the two ensembles below the tricritical point of the canonical ensemble is shown. We observe negative specific heat and temperature jumps at low temperatures in the microcanonical ensemble. The gaps in the accessible magnetization interval appear at low energies, which show the absence of the microscopic configuration in this region.

**Statistical modelling of financial time series and discussion around some algo trading malicious practices and future challenges***Wednesday April 1, 2015, 14:00 h, DH Seminar Room*

The revolution of computing combined with application of sophisticated mathematical theories, profoundly impacted modern world, and especially finance. In recent years, algorithmic trading raised from the synergy of multiple disciplines such as mathematics, informatics, finance and data sciences. Nowadays, financial markets become increasingly complex: we can imagine a (social) network where individual, human and/or computer investors, interact in constantly changing circumstances. In this presentation, I will talk about some already known facts and specifics of algorithmic trading. Then, I will describe some malicious practices and introduce methods to detect them and protect financial markets. I will also present some challenges that will be object of further research of the scientific community.

**Phase transitions in antiferromagnetic ****Potts models**

I survey some recent results concerning the existence and absence of phase transitions for antiferromagnetic $q$-state Potts models on two-dimensional lattices. These models provide some striking examples of entropy-driven phase transitions.

**Neoclassical tearing modes in tokamak plasmas**

The neoclassical tearing mode (NTM) instability is expected to pose a problem for ITER. The current theory on NTM growth is not fully qualitative so any control scheme is not fully informed. Quantifying the effect of the magnetic island introduced by the tearing mode will require large scale numerical work. My project aims to investigate the validity of certain assumptions to simplify the ionic distribution function, and obtain density profiles in the collisionless regime.

**Second order Moller-Plesset calculations by means of GPU-computing***Thursday March 26, 2015, 14:00 h, DH Seminar Room*

Being able to carry out accurate calculations for electron correlations in molecular systems is crucial when simulating chemical processes with application to biology or medicine. An approach to such correlations is 2nd Order Moller-Plesset perturbation theory (MP2), which yields a value for the corresponding correlation energy. However, MP2's main drawback is its tremendous computational cost, preventing from applying the method to large biological molecular systems. Acceleration is performed by means of GPU computing : a recent, affordable, powerful and complementary tool to parallel computing on CPU clusters. Consequently, simple and rather straightforward implementation considerations lead to a significant speed up compared to updated version of serial legacy computational chemistry codes.

**Stabilization and destabilization mechanisms in shear layers**

Although numerical solutions to linear stability equations are readily available, we still need to understand instability mechanisms if we are to anticipate how modifications to basic velocity profiles will affect the stability properties of the flow. For example, as is well known, inviscid instability is suppressed if inflexion points can be removed. However, the role of viscosity in producing instability in the absence of inflexion points is less easy to understand. While there are concepts to help us to interpret viscous instability, like `negative energy waves', and the role of viscous wall layers in generating Reynolds stress, it isn't very clear how they can help us to control viscous instability.

We suggest here an alternative explanation of the origin of viscous instability, and show how it indicates strategies for the control of viscous instability. We argue that the primary effect of weak viscosity is to generate a discrete spectrum of damped viscous modes. If there is a lightly damped inviscid mode then this can resonate (form a branch point) with one of the viscous modes to create instability. Existence of a lightly damped inviscid mode is therefore a necessary condition for viscous instability. An increase in critical Reynolds number, or even complete stabilization, can be achieved if the branch point can be stabilized. The approach is illustrated using plane Poiseuille-Couette flow, and a boundary layer flow, with very substantial stabilization achieved in the latter case despite using a profile modification that creates inflexion points.

**Application of Shannon’s Sampling Theorem in quantum ****mechanics**

Cellular Automata (CA) are studied in various fields of science, ranging from physics to the theory of computation, from mathematics to theoretical biology. They present dynamical systems and are used to study, in particular, the evolution of discrete complex systems. They are defined by cells, for example sites on a lattice, which evolve in a synchronized way which is determined in detail by an updating rule. R.P. Feynman proposed the scheme of a quantum computer (extending the idea of CA) and showed that such a computer is capable of simulating quantum phenomena using quantum mechanics for its operations.

A CA is naturally defined on a lattice which introduces a natural unit of length (or time). The state of a CA can then be viewed as a string of data that contains all relevant information about the corresponding related quantum state. In information theory, the Shannon Theorem is one of the most important results, which states that the information contained in the denumerable set of all samples of a function on a lattice is equivalent to that contained in a band-limited continous function; it relates the band-limit to the frequency of sampling. Its use is fundamental for the transmission and evaluation of information, since it allows to convert arrays of data (bits) to analogue signals (such as sound or video).

Introducing both an action for a class of CA variational principle and a variational principle we obtain the equations of motion for the CA. Those equations can be coupled in order to obtain a discrete version of the Schrödinger equation. A general solution is constructed and we show how it is possible to find discrete conservation laws. Using the Shannon Theorem we can map the discrete equation into a continuous Schrödinger equation showing differences and analogies. On a lattice based theory is generally possible to have various definition of the derivative. We analyse two of those definitions showing that they leads to different uncertainty relations, studying under which conditions we can recover the usual relation of indeterminacy.

We also present some ideas how to extend this discussion also to relativistic equation.

**Singular Capillary Microflows: Modelling, Computation and Scaling***Wednesday March 11, 2015, 15:00 h, DH Seminar Room*

Understanding the interaction of liquids with solids (wetting) and other liquid bodies (coalescence) holds the key to optimizing a whole host of technological processes, including a number of emerging microfluidic devices such as 3D-printers. Accurate experimental observation of these phenomena is complex due to the small spatio-temporal scales or interest and, consequently, mathematical modelling and computational simulation become key tools with which to probe such flows.

Dynamic wetting and coalescence are both so-called `singular' capillary flows, in which classical modelling approaches lead to infinite values of flow variables and computation becomes increasingly complex. In this talk, I will describe the mathematical models proposed for this class of flows and the techniques which have been used to obtain both approximate and exact computational solutions. Simulations will reveal (a) the dominant physical mechanisms in these flows, (b) the accuracy of scaling laws proposed for them and (c) a number of previous mis-conceptions in the published literature.