## OSP 2011

The 2011 Open Statistical Physics meeting took place on Wednesday 2nd March 2011. It was well attended with over 40 participants, and 29 talks scheduled in three parallel sessions.

Schedule

 Time Session A Session B Session C 10:30 Welcome address 10:35 Prellberg Klages Touchette 11:10 Ali Sherkunov Nahum 11:35 Knight G Osmanovic Genway 12:00 Kopylov Mey Meneguz 12:25 Lunch break 13:30 Knight P Katzav Castelnovo 14:05 Vedmedenko O'Neill 14:40 Russell Concannon Bragg 15:05 Tea break 15:30 Vassilicos Giuggioli Dettman 16:05 Chleboun Black Tailor 16:30 Rahman Roberts Loscar 16:55 Close 18:00 Departure

Participants

Andrew Ali (Warwick)
Panayotis Benetatos (Cambridge)
Andy Bragg (Newcastle)
Andrew Black (Manchester)
Mick Bromilow (Open)
Claudio Castelnovo (Royal Holloway)
Paul Chleboun (Warwick)
Robert Concannon (Edinburgh)
Carl Dettman (Bristol)
Kym Eden-Jones (Edinburgh)
Ian Ford (UCL)
Sam Genway (Imperial College)
Uwe Grimm (Open)
Stefan Grosskinsky (Warwick)
Luca Giuggiou (Bristol)
Eytan Katzav (Kings College)
Rainer Klages (Queen Mary)
Georgie Knight (Queen Mary)
Philip Knight (Strathclyde)
Sergey Kopylov (Lancaster)
Bob Lambourne (Open)
Yasser Loscar (Nottingham)
Elena Meneguz (Newcastle)
Ben Mestel (Open)
Antonia Mey (Nottingham)
Kenneth O'Neill (Strathclyde)
Dino Osmanovic (UCL)
Thomas Prellberg (Queen Mary)
Mohammed Rizwanur Rahman (Bristol)
Ekaterina Roberts (Kings College)
Dom Russell (Edinburgh)
Rodrigo Sanchez (Queen Mary)
Yury Sherkunov (Lancaster)
Hemant Tailor (UCL)
Hugo Touchette (Queen Mary)
Andrey Umerski (Open)
Paul Upton (Open)
Christos Vassilicos (Imperial)
Elena Vedmedenko (Hamburg)
Michael Wilkinson (Open)

Talks and Abstracts

Segregation in growing populations
We analyse the growth of microbial populations of several different species, building on results in [Hallatschek et al., Proc. Natl. Acad. Sci., 104(50): 19926 - 19930 (2007)]. A well mixed population of fluorescently labelled microbes is grown in a circular geometry. As the population expands, a coarsening process driven by genetic drift gives rise to sectoring patterns with fractal boundaries, which show evidence of a non-trivial asymptotic distribution. We use concepts of local scale invariance in the KPZ universality class related to the scaling properties of the sector boundaries. We relate the observed population growth to stochastic surface growth models with roots in non-equilibrium statistical mechanics. Our analysis follows a computational and analytical route which we use to derive quantitative predictions and quantitative explanations.

Andrew Black (Manchester)
Stochastic amplification in an epidemic model with seasonal forcing
Stochastic models, subject to external forcing, can capture the regular oscillatory patterns of childhood epidemics, such as measles and whooping cough, but so far the mechanisms generating these patterns have not been well understood. We study the stochastic susceptible-infected-recovered (SIR) model with time-dependent forcing using analytic techniques which allow us to disentangle the interaction of stochasticity and external forcing. The model is formulated as a continuous time Markov process, which is decomposed into a deterministic dynamics together with stochastic corrections, by using an expansion in inverse system size. The forcing induces a limit cycle in the deterministic dynamics, and with the use of Floquet theory, a complete analysis of the fluctuations about this time-dependent solution is given. This analysis is applied when the limit cycle is annual, and after a period-doubling when it is biennial. The comprehensive nature of our approach allows us to give a coherent picture of the dynamics which unifies past work, but which also provides a systematic method for predicting the periods of oscillations seen in both whooping cough and measles epidemics.

Andy Bragg (Newcastle)
Stochastic amplification in an epidemic model with seasonal forcing

Claudio Castelnovo (Royal Holloway)
Hierarchy of critical behaviours in the interacting three colouring model on the honeycomb lattice
We investigate the behaviour of Baxter's three colouring model on the honeycomb lattice in presence of nearest neighbour interactions between the chirality spins. We unveil a hierarchy of critical behaviours whose long wavelength descriptions correspond to free majorana fermions merging pairwise into free bosons, and free bosons merging into the well-known SU(3)-symmetric critical behaviour that obtains in the non-interacting limit. Numerical transfer matrix calculations illustrate a paradoxically continuously varying central charge along a high-symmetry direction in parameter space, whose origin is yet to be explained.

Paul Chleboun (Warwick)
Large deviations and metastability in zero-range condensation
We discuss a general approach to understand the role of finite size effects and metastability in driven diffusive systems that exhibit a condensation transition. We present a detailed analysis of toy models, where the jump rates depend on the size of the system, including the zero-range process and the more recently introduced inclusion process. The systems are known to exhibit condensation as well as ergodicity breaking and metastability in the thermodynamic limit. Results on the equivalence of ensembles and metastability are characterised in terms of large deviation techniques. In light of these results we discuss the separation of time scales, switching between the condensed and fluid phase, and the dynamics of the condensate location.

The paper is still in preparation, an older related publication is:
P. Chleboun, S. Grosskinsky, Finite size effects and metastability in zero-range condensation, J. Stat. Phys. 140(5), 846-872 (2010) arXiv:1004.0408

Robert Concannon (Edinburgh)
A Non-Markovian Asymmetric Simple Exclusion Process
Abstract text to follow

Carl Dettman (Bristol)
New horizons in multidimensional diffusion: The Lorentz gas and the Riemann hypothesis
The Lorentz gas is a model of deterministic diffusion consisting of many fixed scatterers and a point particle making elastic collisions. In a two dimensional periodic array of spherical scatterers it is known that displacements are normally distributed, but with a nonstandard, superdiffusive scaling of the form sqrt(t ln t). I will describe attempts to extend this result to higher dimensions. In the case of
small scatterers, the superdiffusion coefficient is related to the Riemann hypothesis. There is also theoretical and numerical evidence for a qualitative change of behaviour in six dimensions and beyond.

Sam Genway (Imperial College)
Dynamics of thermalisation: a Gaussian regime
We study numerically the temporal evolution of the reduced density matrix for a two-site subsystem of a Hubbard model prepared far from equilibrium at a definite energy. Even for very small systems at relatively low energies, the subsystem can reach a steady state resembling equilibrium. This occurs for a non-perturbative coupling between the subsystem and the rest of the lattice where relaxation to equilibrium is Gaussian in time, in sharp contrast to perturbative results. We find similar results for random couplings, suggesting such behaviour is generic for small systems.

Luca Giuggioli (Bristol)
Animal movement and interaction and the collective emergence of territoriality
The formation and maintenance of animal territories can be cast as a collective movement process for which wandering animals (random walkers), that continuously deposit scent marks, avoid the locations recently marked by other conspecifics. It is possible to show that the dynamics of each territory i.e. the area delimited by the locations where the scent marks of neighbours remain active, becomes a 2D exclusion process, and that only two parameters, population density and the active scent time, control how territorial patterns emerge. A simplified version of the problem can be dealt within a Fokker-Planck formalism and provide answers to some of the transient characteristics observed in the stochastic simulation.

This work is based on two papers one to appear in PLoS Computational Biology, where the results of the stochastic simulations have been reported, and one on the analytic results recently uploaded on the arxiv <http://arxiv.org/abs/1102.0966>.

Eytan Katzav (Kings College)
Response--Correlation Inequality in Dynamical Systems
The flurry of activity in non equilibrium statistical phenomena covers many fields of theoretical and practical importance such as critical dynamics, growth models, front propagation, crack propagation and many more.

We will discuss an exact inequality relating the response function, measuring the steady state response of the physical field of interest to an external probe, and the correlation function. This inequality generalizes the Schwartz-Soffer inequality derived for quenched random systems. When turned then into an exponent inequality it bears important consequences for a wide set of dynamical problems, including critical dynamics and KPZ-like problems.

Rainer Klages (Queen Mary)
Anomalous dynamics of cell migration
Cell movement, for example, during embryogenesis or tumor metastasis, is a complex dynamical process resulting from an intricate interplay of multiple components of the cellular migration machinery. At first sight, the paths of migrating cells resemble those of thermally driven Brownian particles. However, cell migration is an active biological process putting a characterization in terms of normal Brownian motion into question. By analyzing the trajectories of two different types of kidney cells, we show experimentally that anomalous dynamics [1] characterizes cell migration. A superdiffusive increase of the mean squared displacement, non-Gaussian spatial probability distributions, and power-law decays of the velocity autocorrelations are the basis for this interpretation. Almost all results can be explained with a fractional Klein-Kramers equation allowing the quantitative classification of cell migration by a few parameters [2].

References
[1] R. Klages, G.Radons, I.M.Sokolov (Eds.), Anomalous transport (Wiley-VCH, Weinheim,
2008).
[2] P.Dieterich, R.Klages, R.Preuss, A.Schwab, PNAS 105, 459 (2008).

Georgie Knight (Queen Mary)
Understanding fractal diffusion coefficients: Approximation methods
We consider various methods of analytically approximating parameter dependent diffusion coefficients, in one-dimensional dynamical systems. We compare the utility of different methods in helping us to understand the complicated, often fractal structures that we find.

Philip Knight (Strathclyde)
Bak-Sneppen type models and rank-driven processes
Michael Grinfeld, Philip A. Knight, Andrew R. Wade

The Bak-Sneppen model is a simple stochastic model of evolution that exhibits self-organized criticality and for which few analytical results have been established. In the original Bak--Sneppen model and many subsequent variants, interactions among the evolving species are tied to a specified topology. We report a surprising connection between Bak--Sneppen type models and more tractable Markov processes that evolve without any reference to an underlying topology. Specifically, we show that in the case of a large number of species, the long time behaviour of the fitness profile in the anisotropic Bak-Sneppen model can be replicated by a model with a purely rank-based update rule whose asymptotics can be studied rigorously.

Sergey Kopylov (Lancaster)
Transport anomaly at the ordering transition for adatoms on graphene
Impurities in graphene experience a long-range 1/r^3 RKKY interaction due to polarization of the electron Fermi sea (Friedel oscillations). For surface adsorbents such an interaction may result in their structural ordering, repeating the pattern of the Friedel oscillations of electron density. We analyze a manifestation of the partial ordering transition of adatoms on graphene in resistivity measurements. We find that Kekule mosaic ordering of adatoms increases sheet resistance of graphene, due to a gap opening in its spectrum, and that critical fluctuations of the order parameter lead to a non-monotonic temperature dependence of resistivity, with a cusp-like minimum at T=T_c.

Yasser Loscar (Nottingham)
Thermodynamics of trajectories of the 1D-Ising model
We present a numerical study of the one-dimensional Ising model with Glauber dynamics where we apply the large deviations formalism [1] to study the properties of ensembles of trajectories. We confirm the dynamical ferro-magnetic transition which has been theoretically predicted recently at zero magnetic field [2]. The transition can be understood with finite size scaling theory taking the observational time as the size of the system. In this way we have estimated the exponents of the transition which are compared with those belonging to the 2D-Ising universality class. Also, we have extended the phase diagram of [2] by considering the case of a non-zero external magnetic field. We discuss general implications of our results for the relation between thermodynamic and dynamic phase structure.

References
[1] J. P. Garrahan, R. L. Jack, V. Lecomte, E. Pitard, K. van Duijvendijk, and F. van Wijland, Phys. Rev. Lett. 98, 195702 (2007).
[2] Robert L Jack and Peter Sollich, Prog. Theor. Phys. Supp. 184, 304 (2010).

Elena Meneguz (Newcastle)
Statistical properties of particle segregation in homogeneous isotropic turbulence
The Full Lagrangian Method (FLM) is used in a DNS of incompressible homogeneous isotropic turbulent flow to measure the statistical properties of the segregation of small inertial particles advected with Stokes drag by the turbulence. Qualitative good agreement is observed with previous simulations in synthetic turbulent flow fields, IJzermans et al. (2010): in particular the existence of singularities in the particle concentration field and a threshold value for the particle Stokes number above which the net compressibility of the particle concentration changes sign (from compression to dilation). We extend the previous analysis of segregation in KS random flow fields by examining the distribution in time of the compression of an elemental volume of particles and show that it is close to log normal as far as the 3rd and 4th moments but becomes highly non Gaussian for higher order moments when the contribution of singularities increasingly dominates the statistics. Measurements of the rate of occurrence of singularities show that it reaches a maximum at a Stokes number ~ 1, with the distribution of times between singularities following a Poisson process. Following the approach used by Fevrier et al. (2005), we also measured the random uncorrelated motion (RUM) and mesoscopic components of the compression and show that their ratio follows the same dependence on Stokes number as that for the particle turbulent kinetic energy, noting also that the non Gaussian highly intermittent part of the distribution of the compression is associated with the RUM component.

References
IJzermans, R. H. A. Meneguz, E. & Reeks, M. W. 2010 Segregation of particles in incompressible random flows: singularities, intermittency and random uncorrelated motion (RUM) J. Fluid Mech.653, 99–136.
Fevrier, P Simonin, O. & Squires, K. D. 2005 Partitioning of particle velocity in gas-solid turbulent flows into a continuous field and a spatially uncorrelated random distribution; theoretical formalism and numerical study J. Fluid Mech.553, 1–46.

Antonia Mey (Nottingham)
Dynamical Phase Transitions of Lattice Protein Models
We investigate space-time behaviour in finite size systems, with a first order like thermodynamic transition. In particular we look at the dynamics of a lattice protein model (G¯o model), using Monte Carlo and Transition Path sampling techniques in order to reweigh trajectories according to their native activity using the s-ensemble. We characterise phase space regions in terms of trajectory activity and find dynamic crossovers between active and inactive trajectories. In particular when considering models with homogeneous native interactions a first order like transition between native and non-native states is observed. This transition extends away from the critical temperature Tfold along a first order line of critical s. Furthermore we introduce heterogeneous native interaction energies, for which the dynamic behaviour is very rich. Active and inactive regions are very sequence dependent and generally inactive trajectories are highly native, kinetically trapped states, allowing for a novel transition between active and inactive phases. We chose a representative 48mer sequence, whose inactive trajectories consists of trapping states away from the folding pathway but with similar average nativeness as active trajectories. This is supported by mean first passage time studies to the native state, using representative states from the inactive and active phase space regions. States from active trajectories show a power law distribution, while states from inactive trajectories have an exponential distribution of mean first passage times. It can also be shown that for the topologically.

Loop models and random curves in three dimensions
While the study of loop models and random curves in 2D has reached a high level of sophistication, many qualitative questions remain unanswered in 3D. I will briefly describe derivations of field theories (CP^{n-1} and RP^{n-1} sigma models) for a class 3D loop models, and supporting numerical results. I will also indicate why these results are of more general applicability for random curves in 3D.

This talk is based on joint work with J.T. Chalker, P. Serna, A.M. Somoza and M. Ortuno.

Kenneth O'Neill (Strathclyde)
Capture Zone Distribution in Submonolayer Deposition
We consider the nucleation and growth of islands during the submonolayer deposition of monomers. The theory of such cases are mainly based on the use of mean-field rate equations. Despite the fact that these give excellent predictions of average quantities, they differ from results that are obtained experimentally or by Monte Carlo simulations. The reason is rate equations do not incorporate any spatial fluctuations. Mulheran and Blackman [ref] suggested a capture zone construction as a way of allowing us to go beyond the mean-field approach. Pimpinelli and Einstein [ref] suggested that the Generalised Wigner Surmise may describe the capture zone distribution for critical island size, $i \in \mathbb{Z}^+$ and the dimension, $d=1,2,3$. The cases of $d=1$, $i=1$ and $d=2$, $i=0,1$ have been tested in the literature. In this work we have confirmed the results and extended the range to all other combinations of $i$ and $d$. In the case of one-dimensional point-island model, a point of interest was to determine whether the Generalised Wigner Surmise and the Blackman-Mulheran model [ref] are compatible. By analysing the asymptotics solutions, we proved that this is not the case.

Dino Osmanovic (UCL)
An open and shut case: polymer profiles in the nuclear pore complex
The nuclear pore complex is a hole in a membrane surrounding the nucleus of a cell, filled with polymers that are tethered to the walls of the hole. Its function is to allow certain molecules to pass through whilst blocking the passage of others. The selected molecules are carried through as cargo attached to proteins that seem to have the ability to open a passage through the polymer tangle. We have investigated the behaviour of a interacting polymers in a simplified pore geometry using a novel density function theory. We find that the entropy cost of polymer extension towards the centre of the pore can be balanced by attractive forces between the polymers. The equilibrium profile of polymers in the pore can peak at the centre of the pore or at the outer wall, depending on the details of the model. We speculate that the system has evolved to provide open and shut cases for molecular transport.

Thomas Prellberg (Queen Mary)
Exact solution of two non-crossing partially directed walks with contact interaction
We present the analysis of a model of two non-crossing partially directed walks with attractive contact interactions, which at one end are tethered to a fixed point and at the other end are subjected to pulling forces away from the point of tethering.

Using the kernel method, we derive the full generating function for this model and obtain the phase diagram.

Mohammed Rizwanur Rahman (Bristol)
Statistics and Its Application in Point Pattern Analysis

Ekaterina Roberts (Kings College)
Tailored graph ensembles as proxies or null models for real networks II: results on directed graphs
There is a great demand, especially in cellular biology, for precise mathematical approaches to studying the observed topology of networks. We generate new tools with which to quantify the macroscopic topological structure of large directed networks, via a statistical mechanical analysis of constrained maximum entropy ensembles of directed random graphs. We look at prescribed joint distributions for in- and out-degrees and prescribed degree-degree correlation functions. We follow the approach introduced in an earlier paper for undirected networks. Applications of these tools include: comparing networks; distinguishing between meaningful and random structural features; and, defining and generating tailored random graphs as null models. We calculate exact and explicit formulae for the leading orders in the system size of the Shannon entropies of these ensembles. The results are applied to data on gene regulation networks.

References:
Roberts, E S , Coolen, A C C, Schlitt, T, Tailored graph ensembles as proxies or null models for real networks II: results on directed graphs, http://arxiv.org/abs/1101.6022 (2011).
Annibale, A , Coolen, A C C , Fernandes, L P , Fraternali, F and Kleinjung, J, Tailored graph ensembles as proxies or null models for real networks I: tools for quantifying structure J. Phys. A, 42 (48):485001, (2009).

Dom Russell (Edinburgh)
Noise-induced dynamical transition in systems with two symmetric absorbing states
We investigate the effect of noise strength on the macroscopic ordering dynamics of systems with symmetric absorbing states. Using an explicit stochastic microscopic model, we present evidence for a phase transition in the coarsening dynamics, from an Ising-like to a voter-like behavior, as the noise strength is increased past a nontrivial critical value. By mapping to a thermal diffusion process, we argue that the transition arises due to locally-absorbing states being entered more readily in the high-noise regime, which in turn prevents surface tension from driving the ordering process.

Yury Sherkunov (Lancaster)
Magnetic ordering of adatoms in graphene
We investigate magnetic properties of a dilute ensemble of magnetic adatoms on graphene. The present study is focused on the adatoms residing at the centers of the graphene lattice hexagons. The electron-mediated (RKKY) interaction in an undoped graphene depends on the adatoms positions on the three sublattices of the p3×p3 superlattice formed by intervalley scattering induced Friedel oscillations and it has 1/R3 dependence. This leads to an ordering transition of adatoms into an antifferimagnetic state, with a typically antiferromagnetic temperature dependence of magnetic susceptibility. The phase transition in the ensemble of adatoms also manifests itself as a maximum, in the vicinity of the transition temperature, of resistivity as a function of temperature

Hemant Tailor (UCL)
Quasistatic tearing of the collagen triple helix
We employ a transfer matrix approach to calculate the free energy of a triple stranded linear polymer and use it to determine the quasistatic force extension curve for the axial extraction
of a single strand from the triple stranded collagen molecule. We use a polymer cross-linking potential that breaks at a given strain, such that the molecule tears as the extension increases.
The approach describes quasistatic extension, but we explore the possibility that the tearing might in fact require nucleation.

Hugo Touchette (Queen Mary)
Equivalence of ensembles for general many-body systems
It has been proved for a class of mean-field-type systems that the concavity of the thermodynamic entropy determines the equivalence or nonequivalence of the microcanonical and canonical ensembles at the equilibrium-state and macrostate levels. Here we show that this correspondence is actually a general result of statistical mechanics: it holds not just for mean-field systems, but for any many-body system for which equilibrium states can be defined and in principle calculated.
Similar equivalence results also hold for the canonical and grand-canonical ensembles and other dual ensembles.

Christos Vassilicos (Imperial)
Dissipative evolution of homogeneous turbulence
The von K´arman-Howarth equation implies an infinity of invariants corresponding to an infinity of different asymptotic behaviours of the double and triple velocity correlation functions at infinite separations. Given an asymptotic behaviour at infinity for which the Birkhoff-Saffman invariant is not infinite, there are either none, or only one or only two finite invariants. If there are two, one of them is the Loitsyansky invariant and the decay of large eddies cannot be self-similar. We examine the consequences of this infinity of invariants on a particular family of exact solutions of the von K´arman-Howarth equation.

Elena Vedmedenko (Hamburg)
Critical Temperatures of Finite Samples at Finite Observation Times
Using exact diagonalization, Monte-Carlo and mean-field techniques, characteristic temperature scales for ferromagnetic order are discussed for the Ising and the classical anisotropic Heisenberg model on finite lattices in one and two dimensions. The interplay between nearest-neighbor exchange, anisotropy and the presence of surfaces leads, as a function of temperature, to a complex behavior of the distance-dependent spin-spin correlation function. A finite experimental observation time is considered in addition which is simulated within the Monte-Carlo approach by an incomplete statistical average. We find strong surface effects for small nanoparticles which cannot be explained within a simple Landau or mean-field concept and which give rise to characteristic trends of the spin-correlation function in different temperature regimes. Thereby, it is shown that the crossover temperatures for transitions from paramagnetic to superparamagnetic and from superparamagnetic to ferromagnetic behavior can be defined and extracted reliably from the spin-correlation function.