Abstracts

# Abstracts

**Artem Aerov (MPI, Stuttgart). Supercapacitor on base of electrolyte and carbon nanomateria**.

If concentration of ions in a mixture is large, the common Poisson-Boltzmann approach for detemining the equilibrium concentration profiles is not applicable due to excluded volume of ions. Additional complication is produced by dispersion interactions of ions with each other and with the surrounding solvent, which also can not be neglected in the case of high concentration. We aim to model the electrolytic capacitor with the largest possible energy capacity pro volume. It is clear, that such a supercapacitor can contain a very concentrated electrolyte (ionic liquid). Besides, even a weak electrolyte can produce large ion concentrations at the surface of electrodes, when the voltage is applied. Our method of determining concentration profiles in the case of high ion concentrations will be explained, and some results obtained by means of it will be presented.

**Viktoria Blavatska (ICMP, Lviv). Conformational transitions in random heteropolymer models.**

The conformational transitions in heteropolymers are analyzed within the frames of a lattice model containing two types of monomers A and B. Such a model can describe in particular the sequences of hydrophobic and hydrophilic residues in proteins (K.F. Lau and K.A. Dill, Macromolecules **22**, 3986 (1989)) and polyampholytes with oppositely charged groups (Y. Kantor and M. Kardar, Europhys. Lett. **28**, 169 (1994)). The model is generalized by introducing various types of short-range monomer-monomer interactions. Applying the pruned-enriched Rosenbluth chain-growth algorithm (PERM), the peculiarities of transitions from extended into compact states as function of the fraction of A and B monomers along the heteropolymer chain are studied numerically.

**Anna Bodrova (MSU, Moscow). Theory of granular gases and its application to planetary rings.**

We develop a kinetic theory, describing the behaviour of a mixture of granular particles. We observe breakage of energy equipartition due to inelastic collisions of particles. In a binary mixture larger and heavier bodies obey anomalous diffusion. Taking into account processes of aggregation and fragmentation, we construct steady-state size distribution of particles. The application of our theory to planetary rings is discussed.

**Iddo Eliazar (HIT, Holon), Morrel Cohen (Princeton & Rutgers). A Langevin approach to the distribution of wealth.**

The distribution of wealth in human populations displays a Log-Gauss-Pareto composite statistical structure: its density is Log-Gauss in its central body, and follows power-law decay in its tails. This composite statistical structure is further observed in other complex systems, and on a logarithmic scale it displays a Gauss-Exponential structure: its density is Gauss in its central body, and follows exponential decay in its tails. In this talk we establish an equilibrium Langevin explanation for this statistical phenomenon, and show that: (i) the stationary distributions of Langevin dynamics with sigmoidal force functions display a Gauss-Exponential composite statistical structure; (ii) the stationary distributions of geometric Langevin dynamics with sigmoidal force functions display a Log-Gauss-Pareto composite statistical structure. This equilibrium Langevin explanation is universal -- as it is invariant with respect to the specific details of the sigmoidal force functions applied, and as it is invariant with respect to the specific microscopic statistics of the underlying noise.

**Aleksei V. Chechkin (IPA, Potsdam). First arrival and search efficiency of Levy flights. **

We compare first passage and arrival properties of Brownian motion and Levy flights, that is of Markovian random processes with independent stationary increments distributed with the Gaussian and Levy stable laws. We also present very recent analytical and numerical results on the efficiency of Brownian and Levy search strategies. In particular, we demonstrate utility of Levy flight strategy in case of unfavourable search conditions.

**Victor Dotsenko (LPTMC, Paris). Universal Randomness.**

During last two decades it has been discovered that the statistical properties of a number of microscopically rather different random systems at the macroscopic level are described by the same universal probability distribution function which is called the Tracy-Widom (TW) distribution. Among these systems we find both purely mathematical problems, such as the longest increasing subsequences in random permutations, and quite physical ones, such as directed polymers in random media or polynuclear crystal growth. In this talk I discuss these various random systems and explain what the universal TW function is. Next, I give the main line of the derivation of the TW distribution in one-dimensional directed polymers in random potential.

**Yurij Holovatch (ICMP, Lviv). Disorder effects on the shapes of linear and star branched polymers. **

Flexible polymer macromolecules in dilute solutions form crumpled coils with a global shape, which greatly differs from spherical symmetry and is surprisingly anisotropic, as it has been found experimentally and confirmed in many analytical and numerical investigations. In this talk, I will discuss, how the shape can be quantified within universal characteristics and how to calculate these characteristics analytically. Using the field theoretical renormalization group approach the effect of structural disorder in the environment on the universal properties of chain and star polymer macromolecules will be analyzed. The talk is based on recent work performed in collaboration with V. Blavatska and C. von Ferber: Phys. Lett. A **374** (2010) 2861; Condens. Matter Phys. **14** (2011) 33701; *ibid.* **15** (2012) 33603

**Dragi Karevski (CNRS, Nancy). Exact matrix product solution for the boundary-driven Lindblad XXZ-chain**.

We demonstrate that the exact non-equilibrium steady state of the one-dimensional Heisenberg

*XXZ*spin chain driven by boundary Lindblad operators can be constructed explicitly with a matrix product ansatz for the non-equilibrium density matrix where the matrices satisfy a

*quadratic algebra*. This algebra turns out to be related to the quantum algebra

*U_q[SU(2)]*. Coherent state techniques are introduced for the exact solution of the isotropic Heisenberg chain with and without quantum boundary fields and Lindblad terms that correspond to two different completely polarized boundary states. We show that

*unlike in previously considered scenarios*this boundary twist leads to non-vanishing stationary currents of all spin components. Our results suggest that the matrix product ansatz can be extended to more general quantum systems kept far from equilibrium by Lindblad boundary terms.

**Marjana Krasnytska (ICMP, Lviv). Phase transitions for the Potts model on complex networks.**

The Potts model is one of the most popular spin models of statistical physics. Prevailing majority of the work done so far corresponds to the lattice version of the model. However, many natural or man-made systems are much better described by the topology of a network or a random graph. We consider the q-state Potts model on a complex network for which the node-degree distribution manifests a power-law decay governed by the exponent λ. We work within the mean-field approximation, since for systems on random uncorrelated scale-free networks (where the very notion of a space dimension is ill-defined) this method is known often to give asymptotically exact results. Depending on particular values of q and λ one observes either the first-order or the second-order phase transition or the system is ordered at any temperature. In a case study, we consider the limit q →1 (percolation) and find a correspondence between the magnetic exponents and those describing percolation on a scale-free network. Interestingly, logarithmic corrections to scaling appear at λ=4 in this case.

**Ralf Metzler (IPA, Potsdam). Correlated random walk processes.**

In standard Scher-Montroll continuous time random walk (CTRW) processes after each jump a new waiting time and jump length are chosen randomly, independent of previous results. This renewal property may not always be justified, for instance, in inhomogeneous media. I will discuss correlated CTRW processes, in which successive waiting times and/or jump lengths are explicitly dependent on their previous values, and changed only by an incremental value. The resulting random walk in the spaces of waiting times and/or jump lengths gives rise to strong modifications with respect to renewal CTRW theory. I will present results for the mean squared displacements and discuss the weakly non-ergodic properties of correlated CTRW processes.

**Dominique Mouhanna (LPTMC, Paris). Renormalization group approach to frustrated magnets.**

Frustrated magnets have been one of the most studied system of Statistical Physics. They exhibit a very unusual critical behaviour: they display scaling laws accompanied by varying critical exponents. From the Renormalization Group point of view this behaviour has been explained in two different ways according to the technique used: i) weak first-order behaviours with scaling are predicted from both nonperturbative approaches and weak-coupling-epsilon expansion analysis ii) a nonuniversal second-order behaviour is predicted from weak-coupling analysis performed at fixed dimension, i.e. without epsilon-expansion. I review these different techniques and propose an explanation to the discrepency encoutered that provides a unified picture of all theoretical approaches to frustrated magnets.

**Olesya Mryglod (ICMP, Lviv). Temporal characteristics of human dynamics in a virtual world.**

Modern computational social science incorporates new approaches for studying society [1]. Using the so-called digital footprints (recorded data about human actions: e-mail activity, mobile calls, purchases with credit cards etc.), it is possible to quantitatively analyse the individual and collective behavior patterns in order to get some new knowledge about the general `rules' which govern the society. Playing online games offer an example of well-documented collective human activities [2]. Here, we analyse the temporal characteristics of players behaviour in the free massive multiplayer online browser game. Having the data for more than 20,000 players, we build and analyse the distributions of interevent times between their consecutive actions on the amalgamated collective and individual levels. Some features of these distributions reflect the well-known real-life phenomena, such as circadian and weekly cycles or bursts of activity which follows the important events in the world. The peculiarities of different kinds of actions cause the differences between the corresponding interevent time distributions. The distinctions between dynamics of highly active players and those with just small number of actions are shown as well.

[1] Lazer D. et al., Computational Social Science, SCIENCE, 2009, vol. 323, 721-723.

[2] Szell M., Thurner S., Measuring social dynamics in a massive multiplayer online game, Social Networks, 2012, vol. 32, 313-329.

**Gleb Oshanin (LPTMC, Paris). Random walks and patterns generated by random permutations of integers.**

We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time n, whose moves to the right or to the left are prescribed by the rise- and-descent sequence associated with a given random permutation. We determine exactly the probability of finding the trajectory of such a permutation-generated random walk at site X at time n, obtain the probability measure of different excursions and define the asymptotic distribution of the number of U-turns of the trajectories - permutation peaks and trough. In the second part, we focus on some statistical properties of surfaces obtained by randomly placing integers 1,2,3,...,L on sites of a 1d or 2d lattices containing L sites. We calculate the distribution function of the number of local peaks - sites the number at which is larger than the numbers appearing at nearest-neighboring sites - and discuss surprising collective behavior emerging in this model.

**Julian Talbot (LPTMC, Paris). Stochastic models of blocking in particulate flows.**

Vehicular traffic on single-track roads in remote areas, filtration of suspensions of solid particles and macromolecular flow through artificial or biological micro-channels provide diverse examples of processes subject to blocking. To provide a theoretical approach to these situations, we formulate stochastic models where particles enter a narrow channel randomly at a given average rate, either in the same direction - the concurrent flow model, or in opposite directions - the counter-current flow model. Without interference, the particles exit the channel after a given transit time. However, if two particles are simultaneously presentin the channel (opposing particles in the counterflow model), the flow is instantaneously interrupted. We obtain exact solutions for the survival probability that no blockage occurs up to time t, the mean survival time, as well as the number and type (in the case of mixtures) of particles that successfully exit the channel before blocking occurs.The models can be adapted to account for clustering of the particulate streams, multiple parallel channels and reversible blocking.

** Mikhail Tamm (MSU, Moscow). Dynamics of a crumpled globule and diffusion-limited reactions in gene regulation.**

In the recent years there has been a renewed interest in the concept of so-called crumpled or fractal globule first suggested as an intermediate stage in the process of polymer coil collapse by Grosberg, Nechaev and Shakhnovich in the 1980s [1]. The reason behind this interest is that the recent Hi-C experiments [2] show that the large-scale folding of the chromatin (i.e., the matter containing genetic material) in the nuclei of eukaryotic cells seems to bear the exactly same statistical properties as the crumpled globule. In my talk, I will discuss the dynamics of the crumpled globule state, and argue that it should be of crucial importance for gene regulation (in particular, for the promoter-enhancer interactions). I will present a simple scaling theory of the crumpled globule dynamics, and show that it is characterized by anomalously fast first passage times with respect to cyclization reactions. Finally, I will show the preliminary results of DPD computer simulations of the crumpled globule in support of the scaling theory presented.

1. A. Grosberg, S. Nechaev, E. Shakhnovich, J. Phys. France, 49, 2095 (1988).

2. E. Lieberman-Aiden et al., Science, 326:289-93 (2009))

**Mykola Tasinkevych (MPI, Stuttgart). Nematic colloids: topological defects, effective interactions, and
structure formation**.

Liquid crystals are known for their anisotropic mechanical and optical properties which originate from the long-range orientational molecular ordering. If a liquid crystal is used as a host liquid in a colloidal suspension, this ordering gives rise to an additional long-range interaction between the colloidal particles. The type of the interaction is controlled by the presence and symmetry of topological defects of the director field. Particle clustering, formation of superstructures, and even new phases are immediate consequences of these anisotropic interactions. For these liquid crystalline systems molecular structure of the host liquid is usually not important, and often a continuum description of liquid crystals is used. Several continuum models, each characterized by its own order parameter, such as the director eld, tensorial order parameter, or particle density exist and describe phenomena occurring on a particular length-scale. Here we use Landau-de Gennes model with tensorial order parameter. This formalism is applicable for intermediate distances between colloidal particles, where nonlinear effects become important. We use adaptive nite elements methods in order to minimize the corresponding Landau-de Gennes free energy functional. With this technique at hand we first discuss the case of a single colloidal particle suspended in a nematic host. We consider colloidal particles which impose either normal, or degenerate tangential anchoring on the director eld. We analise relative stability of different types of topological defects, and show how the core structure of so-called "hedgehog" or "boojum" topological defects changes with temperature, particle size or elastic anisotropy.Then we discuss nematic mediated effective interaction between two colloidal particles and the role played by the topological defects at the intermediate particle-particle separations. Finally, some ideas of how to use an interplay between nematic elasticity and surface patterning, in order to promote a grows of large scale colloidal crystals, will be presented.

**Oleg Vasilyev (MPI, Stuttgart). Numerical simulation method for particles with complex interactions.**

The modification of Molecular Dynamics method is proposed for simulation of systems consisting of particles with complex interactions. These particles are interact via patches (short range attractive spots on a surface). A motion of such patchy particle is represented as a translational motion and a rotation of a rigid body. The rotational orientation of a particle is described by a quaternion. This method lets to simulate systems with smooth interaction potentials as well as driven systems.