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*OSP 2013*

The 2013 Open Statistical Physics meeting took place on Wednesday, 27 March 2013. 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 | Penrose | --- | --- |

11:10 | Ford | Schomerus | --- |

11:45 | Cao | Hickey | Mathijssen |

12:05 | Turner | Rahman | Davenport |

12:25 | Lunch break |
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13:45 | Fyodorov | Chleboun | --- |

14:20 | Klages | Nahum | --- |

14:55 | Tea break |
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15:20 | Dettman | Vamai | --- |

15:55 | Mehlig | Mey | --- |

16:30 | Reeks | --- | --- |

17:05 | Close |
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17:15 | Departure |

**Participants**

Jiarui Cao (Warwick)

Paul Chleboun (Warwick)

Anthony Davenport (Open)

Carl Dettman (Bristol)

Ian Ford (University College London)

Yan Fyodorov (Queen Mary, University of London)

Uwe Grimm (Open)

Jim Hague (Open)

James Hickey (Nottingham)

Jonathan Keelan (Open)

Rainer Klages (Queen Mary, University of London)

Arnold J. Th. M. Mathijssen (Oxford)

Bernhard Mehlig (Gothenburg)

Antonia Mey (Free University Berlin)

Mike Maitland (Warwick)

Adam Nahum (Oxford)

Oliver Penrose (Heriot Watt)

Mohammed Rizwanur Rahman (Bristol)

Mike Reeks (Newcastle)

Henning Schomerus (Lancaster)

Robert Turner (Nottingham)

Paul Upton (Open)

Csilla Varnai (Warwick)

Michael Wilkinson (Open)

**Talks and Abstracts**

**Jiarui Cao**(Warwick)

*Condensation in totally asymmetric inclusion process*Inclusion process is an interacting particle system defined on one dimensions lattice where particles perform random walk and interact through 'inclusion' effect. The rate of a particle jumps from site x to site y is $ p(x,y)\eta_x(d+\eta_y) $, where \eta_i is the number of particles on site i and d is diffusion rate. Under the condition of small d, condensation phenomenon (all particles concentrate on one or a few sites) occurs.

We only focus on totally asymmetric case with nearest neighbor jumps, i.e. p(x,y)=1 if y=x+1, p(x,y)=0 otherwise. Our aim is analyzing limiting dynamics of the condensate's emergence. Starting from finite systems, we analyze the asymptotical dynamics in infinite system with fixed average particle density \rho. The whole dynamic could be divided into four regimes: nucleation regime, coarsening regime, saturation regime and stationary regime. Each of them has its special dynamics and can be described by different limiting functions.

This is a joint work with Paul. Chleboun and Stefan Grosskinsky.

**Paul Chleboun**(Warwick)

*Time scale separation and dynamic heterogeneity in the low temperature East model*We consider the non-equilibrium dynamics of the East model, a linear chain of 0-1 spins evolving under a simple Glauber dynamics in the presence of a kinetic constraint which forbids flips of those spins whose left neighbour is 1. We focus on the glassy effects caused by the kinetic constraint as the equilibrium density of 0's tends to zero. Specifically we analyse time scale separation and dynamic heterogeneity, i.e. non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium. For any mesoscopic length scale we show that the characteristic time scales associated with two system sizes are well separated. In particular, the evolution of mesoscopic domains, i.e. maximal blocks of the form $111..10$, occurs on a time scale which depends sharply on the size of the domain, a clear signature of dynamic heterogeneity. Finally we show that no form of time scale separation can occur on the equilibrium scale, contrary to what was previously assumed in the physical literature based on numerical simulations.

**Anthony Davenport**(Open University)

*Bilayer graphene gap enhancement via substrate interactions*We present numerical calculations for electron-phonon interactions in the adiabatic limit showing electron band gap enhancement. A simple non-zero temperature Green's function approach is taken to calculate the energy gap enhancement throughout a parameter space of intra-layer bias, inter-layer bias, temperature and natural phonon frequencies. We find that the electron-phonon effect is significantly smaller than that of the monolayers previously studied, owing to the lack of a charge density wave instability. Enhancement of the electron energy gap peaks at the order of ~10% with moderate electron-phonon coupling strengths. We discuss the implications of such enhancements for graphene electronics.

**Carl Dettman**(Bristol)

*Connectivity of confined random geometric networks*(see J. Stat. Phys. 147, 758-778 (2012))

**Ian Ford**(University College London)

*Jarzynski equality for a system governed by Tsallis statistics*The work performed on a system over the course of an isothermal nonquasistatic process, when suitably averaged, can be related to the free energy difference associated with the change in system Hamiltonian, a celebrated result known as the Jarzynski equality. A requirement is that the system should start out in an equilibrium state described by the Boltzmann distribution corresponding to the initial Hamiltonian at the prevailing temperature. Using a stochastic thermodynamics approach, we study an overdamped particle in a time dependent potential, but in the presence of a spatially varying temperature profile. The initial state is stationary and described by a Tsallis distribution, a result that we show can emerge from a constrained maximisation of Shannon entropy rather than the Tsallis entropy. The corresponding Jarzynski equality for a nonquasistatic process involves q-exponentials rather than the ordinary exponentials characteristic of the isothermal case. We illustrate this result using numerical simulations.

**Yan Fyodorov**(Queen Mary, University of London)

*Fluctuations and extreme values in multifractal patterns*The goal is to understand sample-to-sample fluctuations in disorder-generated multifractal intensity patterns. Arguably the simplest model of that sort is the exponential of an ideal periodic 1/f Gaussian noise. It most naturally emerges in the random matrix theory context, but attracted also an independent interest in statistical mechanics of disordered systems. I will discuss the threshold of extreme values of 1/f noise and provide a rather compelling explanation for the mechanism behind its universality. Revealed mechanisms are conjectured to retain their qualitative validity for a broad class of disorder-generated multifractal fields. The presentation will be mainly based on the joint work Y.V. Fyodorov. P Le Doussal and A Rosso, J Stat Phys: 149 (2012), 898-920.

**Juan P Garrahan**(Nottingham)

*Trajectory phase transitions, Lee Yang zeros, and high-order cumulants in full counting statistics*(see http://arxiv.org/abs/1209.2524)

**James Hickey**(Nottingham)

*Time-Integrated observables and dynamical phases in closed quantum systems*(based on this pre-print: http://arxiv.org/abs/1211.4773)

**Rainer Klages**(QMUL)

*Normal and anomalous fluctuation relations for Gaussian stochastic dynamics*We study transient work fluctuation relations (FRs) for Gaussian stochastic systems generating anomalous diffusion. For this purpose we use a Langevin approach by employing two different types of additive noise: (i) internal noise where the fluctuation-dissipation relation of the second kind (FDR II) holds, and (ii) external noise without FDR II. For internal noise the existence of FDR II implies the existence of the fluctuation-dissipation relation of the first kind (FDR I), which in turn leads to conventional (normal) forms of transient work FRs. For systems driven by external noise we obtain violations of normal FRs, which we call anomalous FRs. We derive them in the long-time limit and demonstrate the existence of logarithmic factors in FRs for intermediate times. We also outline possible experimental verifications.

[1] A.V.Chechkin, F.Lenz, R.Klages, J.Stat.Mech. L11001 (2012)

[2] R.Klages, A.V.Chechkin, P.Dieterich, Anomalous fluctuation relations, book chapter in: R.Klages, W.Just, C.Jarzynski (Editors), Nonequilibrium Statistical Physics of Small Systems, Wiley-VCH (2013)

**Arnold J. Th. M. Mathijssen**(Oxford)

*Fluid Entrainment by a Micro-Swimmer near a Surface*Because of their size bacteria and fabricated micro-swimmers swim at low Reynolds number. As they move they set up velocity fields which stir the surrounding fluid. We have recently shown that as a swimmer moves along an infinite straight trajectory tracer particles far from the swimmer move in closed loops, whereas those close to the swimmer are entrained by its motion1. Most experiments on microswimmers are performed in finite geometries and therefore here we extend these results to describe the tracer trajectories when an individual swimmer is close to a no-slip boundary. Our work is a step towards understanding biofilm formation, and how swimmers enhance diffusion and hence increase nutrient uptake.

[1] Dmitri O. Pushkin, Henry Shum, Julia M. Yeomans, Fluid transport by individual microswimmers, arXiv:1209.3329 (2012).

**Bernhard Mehlig**(Gothenburg)

*Tumbling rates in turbulent and random flows (with K Gustavsson and J Einarsson)*Recently, the tumbling rate of small particles in turbulent flows was investigated experimentally and by means of direct numerical simulations. It was found that disks tumble, on average, at a much higher rate than rods, and this fact was related to the observation that rods tend to preferentially align with the vorticity of the flow.

We have analysed the tumbling of small non-spherical particles in random flows with finite correlation length and time. We compute the orientational dynamics systematically in terms of a perturbation expansion in the Kubo number. This makes it possible to address the following questions. First, how and when do disks and rods tumble differently? How does the nature of the Lagrangian flow statistics influence the tumbling? What is the effect of inertia on the orientational dynamics of small particles? We compare the results to those of recent experimental and numerical studies of non-spherical particles tumbling in turbulent flows.

**Antonia Mey**(Free University Berlin)

*Efficient estimation of equilibrium expectations from generalized-ensemble simulations text*Equilibrium properties of complex systems, such as free energy differences or stationary probabilities, can often not be computed from direct simulation due to the rare event nature of the dynamics.

Hence, one of the most common approaches in computational physics is the use of generalized ensemble methods such as parallel or simulated tempering. These methods are limited by the fact that the simulation time required to generate uncorrelated samples in the replicas of interest may be prohibitively long. This problem grows with the number of replicas used, and thus with the dimension of the simulated system.

Here we propose the Transition-matrix Reweighting Analysis Method (TRAM) estimator which can estimate stationary quantities from generalized ensemble simulations with much less simulation data compared to direct counting. TRAM is applicable to any generalized ensemble simulation data, including parallel and simulation tempering, but also data from independent (non-interchanging) simulations at different thermodynamic conditions.

A number of numerical experiments on bistable model systems as well as a molecular dynamics simulation of terminally blocked alanine dipeptide in explicit solvent are carried out and direct counting methods are compared to the estimator showing that same results can be achieved with about an order of magnitude less simulation data.

**Adam Nahum**(Oxford)

*Polymers, loop models and localisation*The universal behaviour of 2D loop models can change dramatically when loops are allowed to cross (intersect). I will describe new phase transitions in such models and argue that they are driven by unbinding of point defects in an appropriate replica sigma model. I will use results for the loop model to explain the phase diagram of a related model for polymer collapse, and will briefly describe a connection between the loop models and Anderson metal-insulator transitions.

**Oliver Penrose**(Heriot Watt)

*Distinguishing the sheep from the goats, or how irreversible PDEs can arise from apparently reversible microscopic dynamics*Most partial differential equations describing the behaviour of material systems are irreversible, i.e. unsymmetrical under time reversal; but the equations used to describe the microscopic motion of the particles composing the material are symmetrical under time reversal. This apparent inconsistency is known as the paradox of irreversibility.

The reason for the apparent inconsistency is that the microscopic equations of motion provide only partial information about the actual motions that take place. Many of the motions permitted by the equations of motion are virtually impossible. This talk is about how to distinguish a class of `plausible' microscopic motions (ones that might actually occur) from the much larger class comprising all motions that are compatible with the equations of motion.

**Mohammed Rizwanur Rahman**(Bristol)

*Keyhole and reflection contributions to wireless connectivity*In wireless communications, an interesting mathematical and practical aspect is the probability that a network of randomly placed nodes is fully connected. The connectivity can be analysed in terms of nodes connected via Line Of Sight (LOS) connections and/or via reflections with boundaries in the network configuration. Therefore, reflections affect the connection probability of a typical network. Analyses have been carried out for a simple two-dimensional geometry and an easily extendable three-dimensional system. Theoretical results are backed up with simulation data. Various system parameters affect the network connectivity, including the density of nodes, attenuation of signals reflected from boundaries and the dimensions of the channel involved. Interestingly, the analysis of reflections contributing to the connectivity of wireless networks/random geometric graphs is new.

**Mike Reeks**(Newcastle)

*Particle drift in turbulent flows: the influence of local structure and inhomogeneity*The way particles interact with turbulent structures, particularly in regions of high vorticity and strain rate, has been investigated in simulations of homogeneous turbulence and in simple flows which have a periodic or persistent structure e.g. separating flows and mixing layers. The influence on both settling under gravity and diffusion has been reported and the divergence (compressibility) of the underlying particle velocity field along a particle trajectory has been recognized as an important quantity in quantifying these features. This paper shows how these features can be incorporated in a formal way into a two-fluid model of the dispersed particle phase. In particular the PDF equation for the particle velocity and position is formerly derived on the basis of a stochastic process that involves the statistics of both the particle velocity and local compressibility along particle trajectories. The PDF equation gives rise to contributions to both the drift and particle diffusion coefficient that depend upon the correlation of these quantities with the local carrier flow velocity.

**Henning Schomerus**(Lancaster)

*Random matrix theory of transport through Majorana nanowires*Majorana zero-modes correspond to quasiparticles that are their own antiparticles. These modes constitute a promising platform for a wide range of applications, including topological quantum computation.

Ground-breaking experiments in 2012 detected the first signatures of Majorana zero modes in topological superconductors. To rule out alternative explanations of the available data additional signatures need to be identified. In particular, this requires to understand the effects of disorder. Here I focus on the mathematical properties of appropriate random-matrix ensembles related to two physical effects, weak antilocalization (arXiv:1206.6687) and the 4pi-periodic Josephson effect (arXiv:1210.5412).

**Robert Turner**(Nottingham)

*Generalised ensembles of non-equilibrium dynamics in open quantum systems*We present an extension of the "thermodynamics of quantum jump trajectories" method for studying the dynamics of quantum open system (a method for obtaining the full counting statistics of quantum jumps at long times based on large-deviations). The method introduced in [1] allows for an analysis of ensembles of dynamical trajectories for long but fixed times. One can think of it as a dynamical analog of a fixed volume ensemble in equilibrium statistical mechanics. Here we show how to consider generalised ensembles, such as those of trajectories where total time fluctuates [2], an analog of a constant pressure ensemble for configurations. Among other things this allows us to develop new efficient numerical approaches to access rare non-equilibrium trajectories in these systems. We illustrate our ideas with simple but non trivial models displaying metastability such as dissipative few level systems and the micromaser.

[1] J.P. Garrahan and I. Lesanovsky, Phys. Rev. Lett. 104, 160601 (2010).

[2] R.M. Turner, A.A. Budini and J.P. Garrahan, to be published.

**Csilla Varnai**(Warwick)

*Statistical machine learning analysis of the van der Waals potential and steric effects in proteins*The packing of polypeptide chains is partly driven by steric effects, generally modelled by a penalty term for overlapping atoms, incorporated into the van der Waals (vdW) potential term of the energy function. We use Contrastive Divergence (CD), a novel statistical machine learning technique, to estimate the parameters of the energy function of a coarse grained protein model. Two vdW potentials are considered: a hard cutoff model, and a Lennard-Jones (LJ) potential, the latter with parameters either inferred by CD or adopted from the CHARMM or AMBER force fields. During parameter estimation, we find a trade-off in the Monte Carlo step size between local and efficient exploration of the rugged energy landscape, and a systematic error in the inferred parameter values when the training set of the inference does not follow the Boltzmann distribution.

Nested Sampling and Monte Carlo simulations of polyalanine peptides and protein G demonstrate the suitability of our method, with the native conformation of protein G only found correctly using the LJ potential, owing to its better modelling of the long range packing of protein side chains. Moreover, the conformational distributions observed experimentally and suggested by rigidity analysis are only correctly modelled using the inferred parameter values. Heat capacity curves are also calculated, and the LJ potential with inferred parameters is shown to compare favourably to experimental results.