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*OSP 2015*

The 2015 Open Statistical Physics meeting took place on Wednesday, 25 March 2015. As in previous years, we had three parallel session of talks.

**Schedule**

Time | Session A | Session B | Session C |

10:00 | Coffee |
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10:30 | Welcome address |
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10:35 | Klages | Haug | Adamson |

11:10 | Mestel | Harris | Nold |

11:45 | Ford | Engel | Duncan |

12:20 | Lunch break |
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14:00 | Dettmann | Cavallaro | Durán Olivencia |

14:35 | Verschueren | Narros Gonzales | Reeks |

15:10 | Tea break |
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15:35 | Georgiou | GilGallegos | Tseluiko |

16:10 | Giles | Mieczkowski | Willis |

16:45 | Close |
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17:00 | Departure |

**Participants**

Luke Adamson (Portsmouth)

Massimo Cavallaro (Queen Mary)

Carl Dettmann (Bristol)

Andrew Duncan (Imperial)

Miguel A. Durán Olivencia (Imperial)

Megan Engel (Oxford)

Ian Ford (UCL)

Orestis Georgiou (Toshiba Research Europe Limited)

Alexander Giles (Bristol)

Sol GilGallegos (Queen Mary)

Uwe Grimm (Open)

Jim Hague (Open)

Rosemary Harris (Queen Mary)

Nils Haug (Queen Mary)

Rainer Klages (Queen Mary)

Ben Mestel (Open)

Piotr Mieczkowski (Open)

Arturo Narros Gonzalez (Queen Mary)

Andreas Nold (Imperial)

Marc Pradas (Open)

Thomas Prellberg (Queen Mary)

Mike Reeks (Newcastle)

Dmitri Tseluiko (Loughborough)

Paul Upton (Open)

Paul Verschueren (Open)

Gary Willis (Imperial)

**Talks and Abstracts**

**Luke Adamson**(Portsmouth)

*Analysis of autocorrelation functions arising from two different quasiperiodically forced systems*The talk will focus on results that provide a link between the autocorrelation function appearing in the study of symmetric barrier billiards, and that appearing in the study of a model which gives rise to strange non-chaotic attractors via the non-smooth pitchfork bifurcation. In particular, it is shown that after a simple parameter transformation, the correlation functions are identical.

**Massimo Cavallaro**(Queen Mary)

*Temporally correlated ZRP with open boundaries: steady state and fluctuations*We study an open boundary version of the on-off zero-range process introduced in [Hirschberg, Mukamel and Schutz, Phys. Rev. Lett.

**103**(2009) 090602].

In this model, the departure rates from a site depend not only on its occupation number, but also on previous arrivals. This condition promotes the instantaneous congestion of particles and excludes a factorized non-equilibrium stationary state.

We derive the exact stationary solution of the one-site system and a mean-field approximation for the one-dimensional lattice. Both solutions correspond to that of a Markovian ZRP with effective interaction. Then, we focus on the large deviations of current in this model.

Analytical and numerical calculations show that, although the particle distribution is well described by the effective Markovian solution, the probability of rare currents differs significantly from the Markovian case. In particular, we find evidence for a memory-induced dynamical phase transition.

**Carl Dettmann**(Bristol)

**Random geometric graphs (RGGs) are constructed from a Poisson point process by linking points with mutual distance below a fixed bound. At high density, the probability that the graph is connected is controlled by isolated points, which are more likely near boundaries of the domain. This probability can be estimated using a sum over boundary components, if the boundaries are smooth, and the connection probability approaches unity. In contrast, fractal boundaries lead to stretched exponential decay of the probability with density. Connectivity of RGGs is a useful model for many wireless networks.**

*Connectivity of confined geometric graphis in domains with smooth or fractal boundaries*

**Andrew Duncan**(Imperial)

*Brownian motion in a multiscale potential*In this talk, we will consider the problem of Brownian motion in a rough potential, modelled as a slowly-varying potential perturbed by periodic multiscale fluctuations. We show that the effective behaviour of this model can be described by an overdamped Langevin equation possessing multiplicative noise, for which the detailed balance condition will still hold. We will demonstrate how the small scale fluctuations in the potential can give rise to dynamical behaviour not present in the original, unperturbed model. Moreover, through numerical examples and analysis we will explore this behaviour in a number of regimes, particularly in the limit of increasingly many length scales.

**Miguel A. Durán Olivencia**(Imperial)

*A DDFT equation for arbitrary-shape colloidal fluids including inertia and hydrodynamic interactions*Dynamic density functional theory (DDFT), a time-dependent extension of density functional theory, has become a remarkably powerful tool in the study of colloidal fluids. In recent years there has been extensive research to improve this framework finally yielding a general DDFT equation which takes account of the effects of inertia and hydrodynamic interactions (HI) for multi-species spherical colloidal systems, e.g. Goddard et al. [1]. Our aim here is to generalise these previous studies to systems of anisotropic colloidal particles. We discuss the derivation of a new DDFT for system of particles of arbitrary shape starting from the Kramers equation for the time evolution of the phase-space distribution function. The resultant equations show an inevitable translational-rotational coupling, which could be related to important effects in anisotropic systems. In the overdamped (high friction) limit the theory is notably simplified leading to the DDFT equation previously derived by Wittkowski and Löwen [2].

[1] B. D. Goddard, A. Nold, N. Savva, P. Yatsyshin, and S. Kalliadasis,

*Unification of dynamic density functional theory for colloidal fluids to include inertia and hydrodynamic interactions: derivation and numerical experiments,*Journal of Physics: Condensed Matter

**25**(2013) 035101.

[2] R. Wittkowski and H. Löwen,

*Dynamical density functional theory for colloidal particles with arbitrary shape*, Molecular Physics

**109**(2011) 2935–2943.

**Megan Engel**(Oxford)

**The energy landscapes that drive structure formation in biopolymers are difficult to measure. Here we validate experimentally a novel method to reconstruct landscape profiles from single-molecule pulling curves using an inverse Weierstrass transform (IWT) of the Jarzysnki free-energy integral. The method was applied to unfolding measurements of a DNA hairpin, replicating the results found by the more-established weighted histogram (WHAM) and inverse Boltzmann methods. Applying both WHAM and IWT methods to reconstruct the folding landscape for a RNA pseudoknot having a stiff energy barrier, we found that landscape features with sharper curvature than the force probe stiffness could not be recovered with the IWT method. The IWT method is thus best for analyzing data from stiff force probes such as atomic force microscopes.**

*Reconstructing folding energy landscape profiles from nonequilibrium pulling curves with an inverse Weierstrass integral transform*

(

__http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.113.238104__)

**Ian Ford**(UCL)

**It is generally observed that if a dynamical system is sufficiently complex, then as time progresses it will share out energy and other properties amongst its component parts, eliminating any initial imbalances. Given knowledge of the dynamics and the statistics of the initial condition, attempts have been made to quantify the irreversibility of such behaviour. The concepts of stochastic entropy production and the dissipation function have been introduced as measures of irreversibility under certain conditions, principally for stochastic and deterministic dynamics, respectively. Here, we reinterpret the dissipation function as a broader measure of irreversibility in the context of deterministic dynamics where the initial probability density function (pdf) of the system is not necessarily symmetric in velocity, as is usually assumed. It tests for a failure of what we call the obversibility of the system, to be contrasted with the more usual reversibility. We argue that imposing symmetry on the pdf of velocity coordinates within a framework of deterministic dynamics might be an acceptable ansatz for some applications, but introduces difficulties when discussing irreversible behaviour.**

*Contrasting measures of irreversibility in stochastic and deterministic dynamics*

**Orestis Georgiou**(Toshiba Research Europe Limited)

*Maximum likelihood based multihop localization in wireless sensor networks*For data sets retrieved from wireless sensors to be insightful, it is often of paramount importance that the data be accurate and also location stamped. We describe a maximum-likelihood based multihop localization algorithm called kHopLoc for use in wireless sensor networks that is strong in both isotropic and anisotropic network deployment regions. During an initial training phase, a Monte Carlo simulation is utilized to produce multihop connection density functions. Then, sensor node locations are estimated by maximizing local likelihood functions of the hop counts to anchor nodes.

**Alexander Giles**(Bristol)

*Betweenness centrality in dense random geometric networks*Random geometric networks are mathematical structures consisting of a set of nodes placed randomly within a bounded set V mutually coupled with a probability dependent on their Euclidean separation, and are the classic model used within the expanding field of ad-hoc wireless net- works. In order to rank the importance of network nodes, we consider the well established ‘betweenness’ centrality measure (quantifying how often a node is on a shortest path of links between any pair of nodes), providing an analytic treatment of betweenness within a random graph model using a continuum approach by deriving a closed form expression for the expected betweenness of a node placed within a dense random geometric network formed inside a disk of radius R. We confirm this with numerical simulations, and discuss the importance of the formula for mitigating the ‘boundary effect’ connectivity phenomenon, for cluster head node election protocol design and for detecting the location of a network’s ‘vulnerability backbone'.

**Sol GilGallegos**(Queen Mary)

*Deterministic diffusion in smooth periodic potentials*A dynamical system that has been widely studied to understand the origin of deterministic diffusion is the periodic Lorentz gas. For this Hamiltonian particle billiard it has been proved that the diffusion coefficient is well defined for a certain range of control parameters, and its parameter dependence and random walk properties have been explored. On the other side it is known that when perturbing (smoothing) such a billiard, tangent periodic orbits produce islands in the phase space of the perturbed system (Rom-Kedar & Turaev 1998). It is also known that elliptic islands and stickiness can cause anomalies/irregularities in transport properties (Zaslavsky 2002). Here we study the behaviour of the diffusion coefficient when the Lorentz gas is perturbed by softening the potential walls. Numerical results show that the diffusion coefficient in the smooth system is a highly irregular function of a control parameter which measures the spacing between two scatterers. In particular we observe significant peaks at certain parameter values. We argue that these peaks are generated by specific periodic orbits and corresponding small islands in the phase space of our system which bifurcate under parameter variation...

**Rosemary Harris**(Queen Mary)

*Random walkers with extreme value memory*Motivated by the psychological literature on the 'peak-end rule' for remembered experience, we perform an analysis within a random walk framework of a discrete choice model where agentsʼ future choices depend on the peak memory of their past experiences. In particular, we use this approach to investigate whether increased noise/disruption always leads to more switching between decisions. Extreme value theory illuminates different classes of behaviour in the system indicating that the outcome is dependent on the scale used for reflection; this could have implications, for example, in questionnaire design.

**Nils Haug**(Queen Mary)

*Scaling functions for vesicle models*Geometric cluster models are a class of physical toy models used to mirror the physical behaviour of systems such as polymer chains and cell membranes. Examples include Dyck paths, staircase polygons and partially directed self-avoiding walks.

In many cases, the grand canonical partition functions (or bivariate generating functions) of geometric cluster models can be expressed in terms of basic hypergeometric series. In order to understand the behaviour of the models around special points of the phase diagram, one reaches for uniform asymptotic expansions of their partition functions.

At the example of Dyck paths, I will show how uniform asymptotic expansions of the partition functions of several geometric cluster models can be obtained by expressing the involved basic hypergeometric series as contour integrals and applying the saddle point method, generalized to the case of several coincident saddles.

**Rainer Klages**(Queen Mary)

*Diffusion in randomly perturbed dissipative dynamics*Dynamical systems having many coexisting attractors present interesting properties from both fundamental theoretical and modelling points of view. When such dynamics is under bounded random perturbations, the basins of attraction are no longer invariant and there is the possibility of transport among them. Here we introduce a basic theoretical setting which enables us to study this hopping process from the perspective of anomalous transport using the concept of a random dynamical system with holes. We apply it to a simple model by investigating the role of hyperbolicity for the transport among basins. We show numerically that our system exhibits non-Gaussian position distributions, power-law escape times, and subdiffusion. Our simulation results are reproduced consistently from stochastic Continuous Time Random Walk theory.

[1] C.S. Rodrigues, A.V. Chechkin, A.P.S. de Moura, C. Grebogi and R. Klages, Europhys. Lett.

**108**(2014) 40002.

**Ben Mestel**(Open)

*The barrier billiard renormalization strange set: golden mean trajectory*The analysis of correlations in symmetric barrier billiards for the golden mean trajectory is characterized by the action of a renormalization operator on the space of piecewise constant functions taking the values +/-1 We show that orbits of the renormalization operator explore a strange attractor in function space dubbed the barrier billiard renormalization strange set, which is given by an embedding of a sub-shift of finite type.

**Piotr Mieczkowski**(Open)

*Modelling cell orientation in artificial tissues***Arturo Narros Gonzalez**(Queen Mary)

*Multi-blob coarse graining for ring polymer solutions*We present a multi-scale molecular modelling of concentrated solutions of unknotted and non-concatenated ring polymers in good solvent conditions. The approach is based on a multi-blob representation of each ring polymer, which is capable of over- coming the shortcomings of single-blob approaches that lose their validity at concentrations exceeding the overlap density of the solution [A. Narros, A. J. Moreno and C. N. Likos, Soft Matter

**6**(2010) 2435]. By means of a first principles coarse-graining strategy based on analytically determined effective pair potentials between the blobs, computed at zero density, we quantitatively reproduce single molecule and solution properties of a system with well-defined topological constraints. Detailed comparisons with the underlying, monomer-resolved model demonstrate the validity of our approach, which employs fully transferable pair potentials between connected and unconnected blobs. We demonstrate that the pair structure between the centers of mass of the rings is accurately reproduced by the multi-blob approach, opening thus the way for simulation of arbitrarily long polymers. Finally, we show the importance of the topological constraint of non-concatenation in the structure of the concentrated solution, and in particular on the size of the correlation hole and the shrinkage of the rings as melt concentrations are approached.

**Andreas Nold**(Imperial)

*Insights into nanoscale effects at the moving contact line using density functional theory*A self-contained description of the moving contact line has been a challenge in fluid-mechanics for almost half a century. The main reason for this is that at the vicinity of this triple juncture of interfaces, continuum formulations break down and nanoscale ingredients have to be considered. Here, we discuss a statistical mechanics approach to this problem, based on density functional theory (DFT). First, we discuss the main ideas of equilibrium DFT and present numerical results obtained for the equilibrium contact line. Second, DFT is extended to dynamical systems, hence providing a link between particle-based and continuum approaches for (a) colloidal systems and (b) atomic simple fluid systems. A careful presentation of the assumptions needed to go from the Langevin via the Kramers equation to the Navier-Stokes-like equations provides insights into the nanoscale effects which have to be included for a self-contained and numerically tractable description of the moving contact line.

**Mike Reeks**(Newcastle)

*The concept of particle pressure in a suspension of particles in a turbulent flow*The Clausius Virial theorem of Classical Kinetic Theory is used to evaluate the pressure of a suspension of small particles at equilibrium in an isotropic homogeneous and stationary turbulent flow. It then follows a similar approach to the way Einstein [1] evaluated the diffusion coefficient of Brownian particles (leading to the Stokes-Einstein relation) to similarly evaluate the long term diffusion coefficient of the suspended particles. In contrast to Brownian motion, the analogue of temperature in the equation of state which relates pressure to particle density is not the kinetic energy per unit particle mass except when the particle equation of motion approximates to a Langevin Equation.

**Dmitri Tseluiko**(Loughborough)

*Coherent structures in non-local dispersive active-dissipative systems*We analyse coherent structures in non-local dispersive active-dissipative nonlinear systems, using as a prototype the Kuramoto-Sivashinsky equation with an additional non-local term that contains stabilising/destabilising and dispersive parts. Such equations arise, for example, in the modelling of a liquid film flow in the presence of various external effects. We analyse pulse solutions and formation of bound states in such equations. Since the Shilnikov-type approach is not applicable for analysing bound states in non-local equations, we develop a weak-interaction theory. The non-locality results in several features that are not present in local equations. For example, the standard first-neighbour approximation is not applicable anymore. We, in addition, compare the theoretical predictions with numerical results for reduced model equations and Stokes flow.

**Paul Verschueren**(Open)

**Quasiperiodic Birkhoff sums and products arise in the study of critical phenomena in diverse fields such as the emergence of Strange Non-Chaotic Attractors, Critical KAM Theory, and String Theory (q-series growth at the unit boundary). They have also been studied in Complex Analysis, and Partition Theory (Number Theory).**

*Quasiperiodic Birkhoff sums and product*

In this talk we will introduce these functions and their interesting fractal properties, and then review results on the most heavily studied example, Sudler’s product of sines. We present new results which provide a rigorous theoretical foundation for experimental results reported recently by Knill & Tangerman.

**Gary Willis**(Imperial)

**Explicit Poor Convergence of the Real-Space RG for 2- and 3-Dimensional Percolation Problems**

**Making recourse to a well-known argument often encountered in undergraduate textbooks which provides grounds for optimism as regards the effectiveness of the real-space RG to tackle percolation problems on small latices, the argument is extended to slightly larger systems as well as higher dimensions. A robust but extremely poor convergence as the lattice size is increased is observed as well as a more surprising result. Finally, a connection is made between this and an implementation of the RG for the Ising Model which was presented as OSP 2014.**