Applied Mathematics Research Centre

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

**Exploring different regimes in finite-size scaling of the droplet ****condensation/evaporation transition**

We present a finite-size scaling analysis of the droplet condensation/evaporation transition of a lattice gas (2D, 3D) and a Lennard-Jones gas (3D) at fixed density. Parallel multicanonical simulations allow to sample the required system sizes with precise equilibrium estimates. In the limit of large systems, we verify the theoretical leading-order scaling prediction for both the transition temperature and the finite-size rounding. In addition, we present an emerging intermediate scaling regime, consistent in all considered cases. This implies that care has to be taken when considering scaling ranges, possibly leading to completely wrong prediction of the thermodynamic limit. Moreover, we show that the rounding of the transition seems to be a suitable measure of the scaling regimes for the phase transition between a homogenous and a mixed phase.

**Nonequilibrium Statistical Physics of Interacting Many Body Systems**

The time evolution for closed quantum many body systems is usually given as a one-parameter group of unitary operators on a Hilbert space representing a group of automorphisms on an underlying C*-algebra of observables. Dissipative processes, irreversible phenomena, decay of unstable particles, approach to thermodynamic equilibrium or quantum measurement processes are difficult to accommodate within this framework. For infinite systems the characterization of completeness or incompleteness of this dynamical prescription is poorly understood and constitutes a nontrivial first step in the analysis of the dynamics of such systems. The time evolution of macroscopic states for classical and quantum many body systems in statistical physics need not correspond to a translation group or semigroup. Instead convolution semigroups appears generically. The presentation will discuss the implications of this result for the foundations of nonequilibrium statistical physics as well as possible applications to experiment.

**Study of Ising antiferromagnet on the stacked triangular lattice**

The effective-field theory with correlations (EFT) is applied in order to study geometrically frustrated Ising spin systems on the stacked triangular lattice. Spins in adjacent layers form linear chains, which are either ferromagnetically or antiferromagnetically coupled. The frustration is induced by the antiferromagnetic interaction between individual chains. In fact, the stacked model with ferromagnetic chains is suitable for description of quasi-one-dimensional frustrated magnetic materials such as Ca_3Co_2O_6 and CsCoX_3, where X = Cl a Br. We focused on the study of the critical behavior, thermodynamic properties, but mainly on the nature of the magnetocaloric effect, that determines the potential of using these geometrically frustrated materials in the process of magnetic cooling. We also study the impact of spin-size changes on this phenomenon, including mixed spins, which can lead to fairly nontrivial behavior of the system. We try to verify the most interesting results obtained by EFT by more sophisticated numerical methods such as parallel tempering Monte Carlo simulations or Wang-Landau algorithm.

**Correlated Extreme Values in Branching Brownian Motion***Thursday September 18, 2014, 14:00 h, DH Seminar Room*

We investigate one dimensional branching Brownian motion in which at each time step particles either diffuse (with diffusion constant D), die (with rate d), or split into two particles (with rate b). When the birth rate exceeds the death rate (b > d), there is an exponential proliferation of particles and the process is explosive. When b < d, the process eventually dies. At the critical point (b = d) this system is characterized by a fluctuating number of particles with a constant average. Quite remarkably, although the individual positions of these particles have a non-trivial finite time behaviour, the average distances between successive particles (the gaps) become stationary at large times, implying strong correlations between them. We compute the probability distribution functions (PDFs) of these gaps, by conditioning the system to have a fixed number of particles at a given time t. At large times we show that these PDFs are characterized by a power law tail ~1/g^3 (for large gaps g) at the critical point and ~exp(- g/c) otherwise, with a correlation length c~\sqrt(D/|b - d|). We discuss the emergence of these two length scales, the dominant overall length scale of the individual positions, and the sub-dominant gap length scale in this system. We also extend our study to the spatial extent of this process (the distance between the rightmost and leftmost visted sites). We derive exact results for the PDF of this spatial extent for the cases b <= d where the two extreme points are strongly correlated. Once again we find an emergent power law at the critical point with a correlation length ~\sqrt(D/|b - d|) away from criticality. Direct Monte Carlo simulations confirm our predictions.

**Self-Assembly and Phase Behaviour of Indented Colloids***Wednesday September 17, 2014, 14:00 h, DH Seminar Room*

Spherical colloids with an indentation can chain via the process of lock-key binding whereby the indented side of one particle (the lock) fits neatly into the convex side of another (the key). Assembly occurs spontaneously when the colloids are immersed in a sea of much smaller polymers leading to depletion interactions which promote bond formation. We report simulation and theoretical studies of this process in two models. Firstly we have considered an explicit mixture of indented colloids in a bath of much smaller hard spheres. The morphology of the chains that are formed depends sensitively on the size of the colloidal indentation, with smaller values additionally permitting chain branching. In contrast to the case of spheres with attractive patches, Wertheim's thermodynamic perturbation theory fails to provide a fully quantitative description of the polymerization transition. We trace this failure to a neglect of packing effects and we introduce a modified theory that accounts better for the shape of the colloids, yielding improved agreement with simulation[1]. Secondly we have mapped simulations of the full mixture to a greatly simplified model allowing us to study the phase behaviour of large systems. As the ratio between directional and non- directional binding increases the liquid phases become very stringy with the critical points moving to very low density. The resulting liquids have a rich, open structure with interesting percolation properties.

**Planetary Dynamos Driven by Inertial Waves***Wednesday July 2, 2014, 14:00 h, DH Seminar Room*

Great progress has been made in the numerical simulation of planetary dynamos, though these numerical experiments still operate in a regime very far from real planets. For example, it seems unlikely that viscous forces are at all significant in planetary interiors, yet the more weakly forced numerical simulations display a significant dependence on viscosity, and indeed in some simulations the dynamo mechanism itself is viscously driven, taking the form of helical Ekman pumping within Busse-like convection rolls. Given the similarity of the external fields observed in the terrestrial planets and gas giants, and the tiny value of the Ekman number in all such cases, it seems natural to suppose that the underlying dynamo mechanism is simple, robust, independent of viscosity, and insensitive to mechanical boundary conditions. In this talk one such mechanism is proposed which relies on the spontaneous emission of inertial waves within the planetary core.

**How superspreading works? Understanding the mechanism with molecular ****dynamics simulations**

The superspreading mechanism by which aqueous droplets laden with trisiloxane surfactants wet hydrophobic substrates is poorly understood. Superspreading has been attributed to various factors over the last decades, e.g., the peculiar T-shape geometry of the surfactant, Marangoni flow, etc. However, evidence of the exact superspreading mechanism has not been provided. Here, we use molecular dynamics simulations of a coarse-grained model to capture the surfactant microscopic behaviour, which is a key to understand the superspreading mechanism. For the very first time, we are able to propose a plausible superspreading mechanism by means of molecular simulations, and we anticipate that these results will enable a design of surfactants with enhanced spreading ability.

**Lattice Boltzmann method for simulating fluid flow**

Computational fluid flow focusses on discretisations of Navier-Stokes equations, and numerical analysts try to answer the question of how close to the model equation is the discretisation. There is another point of view. The continuum model is not good when we approach turbulence, and we should have a different model in that regime. The Lattice Boltzmann model is a different model of fluid flow, which converges to the Navier-Stokes equations when life is good. When life is bad it is different. We look at stabilisation methods for the LBM which allow simulation at very high Reynolds number.