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*OSP 2012*

The 2012 Open Statistical Physics meeting took place on Wednesday 7th March 2012. As in previous years, we had three parallel session of talks.

**Schedule**

Time | Session A | Session B | Session C |

10:30 | Welcome address |
||

10:35 | Ford | Baule | Kühn |

11:10 | Evans | Harris | Giuggioli |

11:45 | Tate | Villavicencio-Sanchez | Lenz |

12:05 | Eden-Jones | Kennard | Challenger |

12:25 | Lunch break |
||

13:45 | Sherkunov | Moura | Chen |

14:20 | --- | Reeks | Hague |

14:55 | Tea break |
||

15:20 | Prellberg | Katzav | Ueltschi |

15:55 | Sala | Hickey | Wild |

16:15 | Nahum | Davenport | --- |

16:35 | Close |
||

17:00 | Departure |

**Participants**

Adrian Baule (QMUL)

Joseph Challenger (Manchester)

Yang Chen (Imperial)

Anthony Davenport (Open)

Kym Eden-Jones (Edinburgh)

Bob Evans (Bristol)

Ian Ford (UCL)

Luca Giuggioli (Bristol)

Uwe Grimm (Open)

Jim Hague (Open)

Rosemary Harris (QMUL)

James Hickey (Nottingham)

Eytan Katzav (King's)

Jonathan Keelan (Open)

Harry Kennard (Open)

Reimer Kühn (King's)

Friedrich Lenz (QMUL)

Alessandro Moura (Aberdeen)

Adam Nahum (Oxford)

Thomas Prellberg (QMUL)

Mike Reeks (Newcastle)

Gabriele Sala (Royal Holloway)

Yury Sherkunov (Lancaster)

Stephen Tate (Warwick)

Daniel Ueltschi (Warwick)

Paul Upton (Open)

Rodrigo Villavicencio-Sanchez (QMUL)

David Wild (Warwick)

Michael Wilkinson (Open)

**Talks and Abstracts**

**Adrian Baule**(QMUL)

*A statistical theory of jammed anisotropic objects*Random packings of elongated anisotropic objects can reach higher packing fractions than 64%, which is the random close packing fraction of spheres. The behaviour of the packing fraction as a function of the objects' aspect ratio is usually non-monotonic and exhibits a peak at an aspect ratio of around 1.3/1.4. In this talk I present a theory for jammed anisotropic objects based on a statistical description of the Voronoi volume. Results for dimer-shaped objects are discussed.

**Joseph Challenger**(Manchester)

*I*

*nvestigating intrinsic fluctuations in biochemical systems*Mathematical models of biochemical reaction systems are usually constructed from determinstic rate equations. However, this approach is not appropriate when the number of molecules involved is low. Here the underlying stochasticity present in the system becomes important. The rate equations treat the molecular concentrations as smoothly varying functions. In this talk, a master equation approach is used, where the system is described by discrete states, namely the molecular populations of the chemical species involved. In the mean field limit the rate equations can be recovered. In addition to this, leading order corrections to the rate equations can be obtained by using the system-size expansion due to van Kampen.

We have incorporated these results into COPASI, a software package designed to simulate and study biochemical reaction systems. This allows the expansion procedure to be automated. Once the reaction system has been described, COPASI can calculate the covariance matrix associated with the fluctuations exhibited by the chemical species present in the system. We give examples of the application of the method to biologically relevant systems.

Some details can be found here: http://arxiv.org/abs/1112.0672

**Yang Chen**(Imperial)

*Painlevé transcendent, Coulomb fluid and the information theory of wireless communications*I will show that the moment generating function of a random variable known as the Shannon Capacity (an entropy) which characterises the upper limits achievable by any transmission scheme, is related to a particular Painlevé V transcendant (PV), in the SINGLE USER case. (A particular PVI in the multi-user scheme).

For large number of channels n, we combine the `Coulomb Fluid Approximation' which involves potential theory and the Painlevé equation to compute moments of the Shannon Capacity and show deviation from Gaussian as P the signal to noise ratio increases.

**Anthony Davenport**(Open)

*Three-dimensional lattice bipolaron*We present analytical calculations and quantum Monte Carlo simulations describing the formation of bipolarons on a three-dimensional cubic lattice. A U − V model is derived from a two particle extended Hubbard–Holstein model with local and near-neighbor electron-phonon interaction, which we solve analytically for high phonon frequency. The extended model is simulated using a continuous-time quantum Monte Carlo code. We compute singlet bipolaron properties including total energy, number of associated phonons, inverse mass and average bipolaron size. We discuss the binding conditions needed to create polaron pairs and demonstrate the qualitative differences between two- and three- dimensional bipolarons. Our key finding is that significantly more attractive force is needed to attain pairing of 3D bipolarons than that needed in 2D. Dramatic changes to the bipolaron properties are also found when inter-site interactions are turned on. The addition of inter-site interaction makes the range of the crossover from polaron to Bipolaron with decreasing Hubbard U much larger, with the crossover starting at lower values of U. On binding the total energy is decreased, and the number of associated phonons is increased. A minimum is found in the effective mass for high coupling constant and large nearest neighbour coupling at the point where on-site bipolarons and inter-site bipolarons are degenerate.

**Kym Eden-Jones**(Edinburgh)

*Kinetic Monte Carlo simulation of auto-catalytic protein aggregation*Filamentous protein aggregates such as amyloid fibrils have been implicated in a wide variety of degenerative diseases. Using kinetic Monte Carlo simulations we compare the lag phase and length distribution characteristics of two auto-catalytic processes which have been proposed as contributing factors in amyloid fibril formation.

**Bob Evans**(Bristol)

*Fluids in confinement: A novel fluctuating phase*Confined fluids (or Ising magnets) exhibit phase behaviour different from and richer than that in bulk. Consider a simple fluid confined between two parallel walls (substrates), separated by a distance L. The walls are either (i) identical or (ii) exert different surface fields so that one wall is attractive and is wet by liquid (solvophilic) while the other is repulsive and may be dry, i.e. wet by gas(solvophobic).Case (i) is well-studied and much is known. In case (ii) a ‘delocalized interface phase’ forms in the temperature range T

_{cb}>T> T

_{w}, where T

_{cb}and T

_{w}are the bulk critical and the wetting transition temperatures, respectively. This phase exhibits a liquid-gas interface near L/2 with pronounced thermal fluctuations. Its properties are investigated using an effective interfacial Hamiltonian approach and a fully microscopic classical density functional theory. For the physically relevant case of a fluid with -r

^{-6}interatomic attraction, we determine the scaling functions of the solvation force (the excess pressure due to confinement), the adsorption and the susceptibility of the confined fluid for temperatures removed from T

_{cb}. We argue that these results remain valid beyond our mean-field treatment. In this temperature range, and for large separations, the solvation force is repulsive and decays as L

^{-3}. This result is compared with the so-called critical Casimir force which pertains for T near T

_{cb}.

**Ian Ford**(UCL)

*Entropy production and fluctuation relations from full phase space stochastic dynamics*The total entropy production of stochastic systems is non-negative, on average, and can be divided in general into three components, but only two of these are never negative on average. The first component corresponds to the so-called excess heat, and is associated with the relaxation of a system towards a stationary state. It is non-negative, on average. The second and third comprise the so-called house-keeping heat, a contribution that represents the dissipation of energy by a system when in a non-equilibrium stationary state, or more specifically, is a measure of irreversibility arising from the breakage of detailed balance in the dynamics. We denote these two components the transient and generalised house-keeping heat and we obtain an integral fluctuation theorem for the latter, valid for all Markovian stochastic dynamics. This means that it, too, is never negative, but in contrast the transient house-keeping heat can take either sign. Previous treatments of entropy production in stochastic systems are obtained when the stationary probability distribution is symmetric for all variables that are odd under time reversal, which includes directional variables such as velocity, but this condition does not always hold. We illustrate the implications of this analysis using simple systems in discrete and continuous dynamical variables, including the familiar and important case of thermal conduction.

**Luca Giuggioli**(Bristol)

*Oscillatory dynamics in the movement of territorial animals*The collective emergence of animal territorial patterns has been shown to result from the movement and interaction of the so-called territorial random walkers, that is animals that move as Brownian walkers depositing short-lived scent in any location they visit and that are excluded from areas marked by the active scent of neighbouring individuals. Here we show that a damped oscillatory dynamics in the mean square displacement (MSD) of an animal may appear depending on the movement statistics.

This non-monotonicity in the MSD is shown to depend on one dimensionless parameter, given by the ratio of the correlation distance between successive steps to the size of the territory. As that parameter increases, the time dependence of an animal’s MSD displays a transition from monotonic, characteristic of Brownian walks, to non-monotonic, characteristic of highly correlated walks. These results represent a novel way of determining the degree of persistence in animal movement processes within conﬁned regions.

**Jim Hague**(Open)

*Quantum simulators for electron-phonon interactions in deformable materials and unconventional superconductors*We propose two approaches for quantum simulation of electron-phonon interactions using Rydberg states of cold atoms and ions. In the first, we show how systems of cold atoms and ions can be mapped onto electron-phonon systems of the Su–Schrieﬀer–Heeger type, which is appropriate for strongly deformable materials such as polymers. In the second, we discuss how a bilayer system of cold Rydberg fermions can be used as a quantum simulator for the effects of electron-phonon interactions in unconventional superconductors. We use quantum Monte Carlo simulations to show that the bilayer Rydberg system is practically identical to a Hubbard-Froehlich model (which has been proposed as a model for cuprate superconductors). The figure shows the results of QMC simulations for the Hubbard-Froehlich model and the bilayer Rydberg systems (here a is the lattice spacing, b the interlayer distance, E the energy of a 1D bipolaron and Rsc the Froehlich interaction screening radius), and the agreement is excellent (the inset shows the residual between the Froehlich Rsc=0.332a and Rydberg b=a/2 curves). For both quantum simulators, we discuss how properties of the simulated Hamiltonian can be tuned and how to read physically relevant properties from the simulator.

**Rosemary Harris**(QMUL)

*Effect of memory on current fluctuations in interacting particle systems*I will give an overview of some recent results on current fluctuations in interacting particle systems. In particular, I will discuss how long-range memory dependence can modify the current large deviation principle, leading for example to a superdiffusive regime in the phase diagram of the well-known asymmetric simple exclusion process.

**James Hickey**(Nottingham)

*A stochastic viewpoint of non-equilibrium s-ensemble quantum master equations: From jump trajectories to homodyne detection*Recent investigations into the dynamics of quantum jump trajectories revealed that the statistics of the trajectories admitted a thermodynamic formalism and the dynamics of several systems possessed space-time phase transitions. We reformulate the application of the s-ensemble, the ensemble of biased trajectories, in examining the statistics of such trajectories in terms of reduced characteristic operators and their generalized master equations. We then employ this approach in the study of quadrature trajectories, whose typical X-quadrature trajectory statistics are related to homodyne detection. We illustrate our approach with two simple examples: a driven 2-level system and a blinking 3-level system. We find that the statistics of the Y-quadrature is related to the dynamics of the quantum jump trajectories in these systems and explore the typical quadrature trajectories of biased quantum jump trajectories.

**Eytan Katzav**(King's)

*Stability and roughness of crack paths in 2D heterogeneous brittle materials*

*(together with M. Adda-bedia and B. Derrida)*I will present a recent study on the stability of propagating cracks in heterogeneous two-dimensional brittle materials and on the roughness of the surfaces created by this irreversible process. A stochastic model describing the propagation of the crack tip based on an elastostatic description of crack growth in the framework of linear elastic fracture mechanics will be introduced. The model recovers the stability of straight cracks (better known as the T-criterion) and allows for the study of the roughening of fracture surfaces. We show that in a certain limit, the problem becomes exactly solvable and yields analytic predictions for the power spectrum of the paths. This result suggests a surprising alternative to the conventional power law analysis often used in the analysis of experimental data and thus calls for a revised interpretation of the experimental results.

**Harry Kennard**(Open)

*Anisotropic covering of fractal sets*The anisotropy of fractal sets is an important factor in determining their scattering properties. In order to scrutinise this anisotropy, we consider the optimal covering of fractal sets in a two-dimensional space using ellipses which become increasingly anisotropic as their size is reduced. If the minor axis of an ellipse is \epsilon and the major axis is \delta, we set \delta=\epsilon^\alpha, where 0< \alpha < 1 is an exponent characterising the anisotropy of the covers. We find that the number of points N covered by an ellipse centred on any given point follows a power law: its expectation value is <N> ~ \epsilon^\beta, where \beta is a generalised dimension. We present numerical work, using a generalisation of the Sierpinski Carpet, which demonstrates that \beta(\alpha) may be different for fractal sets of the same fractal dimension. Results for \beta(\alpha) of the distribution of particles suspended in a random flow will also be presented.

**Reimer Kühn**(King's)

*Spectra of sample auto-covariance matrices derived from time series*We compute spectra of sample auto-covariance matrices of second order stationary stochastic processes. We look at a limit in which both the matrix dimension N and the sample size M used to define empirical averages diverge, with their ratio \alpha=N/M kept fixed. We find a remarkable scaling relation which expresses the spectral density \rho(\lambda) of sample auto-covariance matrices for processes

*with*dynamical correlations as a continuous superposition of appropriately rescaled copies of the spectral density \rho^{(0)}_\alpha(\lambda) for a sequence of

*uncorrelated*random variables. The rescaling factors are given by the Fourier transform \hat C(q) of the auto-covariance function of the stochastic process. We also obtain a closed-form approximation for the scaling function \rho^{(0)}_\alpha (\lambda). This depends on the shape parameter \alpha, but is otherwise universal: it is independent of the details of the underlying random variables, provided only they have finite variance. Our results are corroborated by numerical simulations using auto-regressive processes.

**Friedrich Lenz**(QMUL)

*A stochastic model for the dynamics of foraging bumblebees*In a laboratory experiment by Ings and Chittka [1], the foraging of bumblebees in an artificial flower carpet was studied. In our work we construct a mathematical model for the foraging dynamics based on statistical data analysis of the recorded bumblebee trajectories. We start with a stochastic reorientation model, which is defined in terms of turning angles and speed changes. We amend this model in form of a generalized Langevin equation and estimate the parameters for the velocity distributions and velocity autocorrelations from the data. We numerically generate data from this model and compare it to the experimental results. Within this framework, we can also analyze the effect of predation risk on the bumblebee dynamics.

[1] Thomas C. Ings and Lars Chittka. Current Biology, 18(19):1520-15 24 (2008)

**Alessandro Moura**(Aberdeen)

*Colliding particles in open chaotic flows*We study the collision probability p of particles advected by open flows with chaotic advection. We show that p scales with the particle size (or, alternatively, reaction distance) as a power law whose coefficient is determined by the fractal dimensions of the invariant sets defined by the advection dynamics. We also argue that this same scaling also holds for the reaction rate of active particles in the low-density regime. These analytical results are compared to numerical simulations, and they are found to agree very well.

**Adam Nahum**(Oxford)

*Lattice gauge theory and the universal statistics of line defects*Vortex lines are a feature of many random or disordered three-dimensional systems. They show universal statistical properties on long length scales, and geometrical phase transitions analogous to percolation transitions but in distinct universality classes. I will argue that the universal behaviour can be understood by mapping a lattice model for the vortex problem to the replica limit of a lattice gauge theory, and that this leads to a simple field-theoretic classification of the possible kinds of critical behaviour.

**Thomas Prellberg**(QMUL)

*Anomalous critical behaviour in the polymer collapse transition of three-dimensional lattice trails*Trails (bond-avoiding walks) provide an alternative lattice model of polymers to self-avoiding walks, and adding self-interaction at multiply visited sites gives a model of polymer collapse. Recently, a two-dimensional model (triangular lattice) where doubly and triply visited sites are given different weights was shown to display a rich phase diagram with first and second order collapse separated by a multi-critical point. A kinetic growth process of trails (KGT) was conjectured to map precisely to this multi-critical point. Two types of low temperature phases, globule and crystal-like, were encountered. Here, we investigate the collapse properties of a similar extended model of interacting lattice trails on the simple cubic lattice with separate weights for doubly and triply visited sites. Again we find first and second order collapse transitions dependent on the relative sizes of the doubly and triply visited energies. However we find no evidence of a low temperature crystal-like phase with only the globular phase in existence.

Intriguingly, when the ratio of the energies is precisely that which separates the first order from the second-order regions anomalous finite-sized scaling appears. At the finite size location of the rounded transition clear evidence exists for a first order transition that persists in the thermodynamic limit. This location moves as the length increases, with its limit apparently at the point that maps to a KGT. However, if one fixes the temperature to sit at exactly this KGT point then only a critical point can be deduced from the data. The resolution of this apparent contradiction lies in the breaking of crossover scaling and the difference in the shift and transition width (crossover) exponents.

**Mike Reeks**(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:

[1] IJzermans, R. H. A. Meneguz, E. & Reeks, M. W. 2010 Segregation of particles in incompressible random flows: singularities, intermittency and random uncorrelated motion, J. Fluid Mech.653, 99–136.

[2] 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

**Gabriele Sala (Royal Holloway)**

*A field trip through spin ice*Fractionalisation - the breaking up of an apparently indivisible microscopic degree of freedom - is one of the most counterintuitive phenomena in many-body physics. Here we study its most fundamental manifestation in spin ice, the only known fractionalised magnetic compound in 3D: we directly visualise the 1/r^2 magnetic Coulomb field of monopoles which emerge as the atomic magnetic dipoles fractionalise. We analyse the internal magnetic field distribution, relevant for local experimental probes. In particular, we present new zero-field NMR measurements which exhibit excellent agreement with the calculated line shapes, noting that this experimental technique can in principle measure directly the monopole density in spin ice. The distribution of field strengths is captured by a simple analytical form which exhibits a low density of low-field sites - in apparent disagreement with reported \mu SR results. Counterintuitively, the density of low-field locations decreases as the local ferromagnetic correlations imposed by the ice rules weaken.

http://arxiv.org/abs/1112.3363

**Yury Sherkunov**(Lancaster)

*Phase transitions in dipolar gases in optical lattices*We study the phase diagrams of two-dimensional lattice dipole systems with variable geometry. For bipartite square and triangular lattices with tunable vertical sublattice separation, we find rich phase diagrams featuring a sequence of easy-plane magnetically ordered phases separated by incommensurate spin-wave states.

**Stephen Tate**(Warwick)

*The virial expansion and combinatorics*I will present on some combinatorial identities and different approaches to understanding the Virial Expansion and contrast how they all work and produce the bounds they do on the radius of convergence. I will go into depth on the development of the combinatorial technique, including the basic theory of Combinatorial Species of Structure and how various weight structures work. I will explain how the two main approaches (the algebraic of Ruelle and that of combinatorics) differ and where we can see some similarity. I will state the best known bounds on the radius of convergence.

**Daniel Ueltschi**(Warwick)

*Random cycles in statistical physics*Certain quantum statistical systems, such as the Bose gas and the Heisenberg model, can be studied with the help of stochastic representations that involve random cycles. Phase transitions are characterised by the occurrence of infinite cycles. I will present numerical results and heuristics that show that the joint distribution of cycle lengths is given by Poisson-Dirichlet. (Collaborations with Goldschmidt, Grosskinsky, Lovisolo, and Windridge.)

**Rodrigo Villavicencio-Sanchez**(QMUL)

*Current fluctuations in the two-dimensional zero-range process*We study the zero-range process on a diamond lattice and show it allows the formation of current loops. These have been related to the breakdown of the Gallavotti-Cohen fluctuation relation. Additionally, we study a two-dimensional square lattice with open boundaries in the x-direction and periodic boundary conditions in the y-direction. We use both a microscopic and a hydrodynamic approach to calculate the rate function of particle current fluctuations. For large system sizes, our analysis lets us test the convergence to the recently proposed isometric fluctuation relation [Hurtado et al., PNAS 108, 7704 (2011)].

**David Wild**(Warwick)

*Exploring the energy landscapes of protein folding simulations with Bayesian computation*Nested sampling is a Bayesian sampling technique developed to explore probability distributions localised in an exponentially small area of the parameter space. The algorithm provides both posterior samples and an estimate of the evidence (marginal likelihood) of the model. The nested sampling algorithm also provides an efficient way to calculate free energies and the expectation value of thermodynamic observables at any temperature, through a simple post-processing of the output. Previous applications of the algorithm have yielded large efficiency gains over other sampling techniques, including parallel tempering (replica exchange). In this paper we describe a parallel implementation of the nested sampling algorithm and its application to the problem of protein folding in a Go-type force field of empirical potentials that were designed to stabilize secondary structure elements in room-temperature simulations. We demonstrate the method by conducting folding simulations on a number of small proteins which are commonly used for testing protein folding procedures: protein G, the SH3 domain of Src tyrosine kinase and chymotrypsin inhibitor 2. A topological analysis of the posterior samples is performed to produce energy landscape charts, which give a high level description of the potential energy surface for the protein folding simulations. These charts provide qualitative insights into both the folding process and the nature of the model and force field used.

Preprint of the work at http://arxiv.org/abs/1010.4735.