Seminar on the Theory of Complex Systems  Summer Term 2013

Strongly correlated and complex random geometries abound in natural porous media. In fact, porous media are interface dominated materials exhibiting strongly correlated random geometries over many decades in length. The talk will give an overview of methods for geometric characterization and analysis of physical properties for random geometries as they occurr in natural porous media [1,2].
Recently, stochastic multiscale media have attracted considerable interest. A method to generate stochastic morphologies for multiscale media was introduced and applied to carbonates [3,4]. The method is particularly suited for modeling carbonate rocks occurring in petroleum reservoirs that exhibit porosity and grain structure covering several decades in length scales [5]. The mathematical model reproduces correlations with primordial depositional textures, scale dependent intergranular porosity over several decades, vuggy porosity, a percolating pore space, a percolating matrix space, and strong resolution dependence of both physical and morphological descriptors such as permeability or Minkowski functionals. The continuum based model allows discretization at arbitrary resolution and provides synthetic micro-CT images for resolution dependent simulations, morphological analysis or tests of multiscale models and methods. It has recently been used to provide free access to the worldwide largest threedimensional fully threedimensional calibrated porous microstructures [6].

1. R. Hilfer, in: Statistical Physics and Spatial Statistics (Lecture Notes in Physics, vol554), Springer, Berlin, p. 203 (2000)
2. C. Lang et al., Journal of Microscopy 203, 303 (2001)
3. B. Biswal et al., Phys.Rev.E, vol 75, 061303 (2007)
4. B. Biswal et al., Phys.Rev.E, vol 80, 041301 (2009)
5. S. Roth et al., AAPG vol 95, p. 925 (2011)
6. R. Hilfer and T. Zauner, Phys.Rev.E, vol 84 062301 (2011)

The pillars of all transport processes have been established in the molecular-kinetic interpretation of diffusion by Einstein and Smoluchowski. In modern terms the central limit theorem applies whenever dynamical correlations decay quickly. Yet, already the preeiminent dutch physicist Hendrik Antoon Lorentz noted that this theoretical framework fails to account for subtle effects in Brownian motion and has to be completed. In the presentation I will introduce several model systems where persistent correlations emerge with macroscopic measurable consequences.
First, I discuss the Brownian motion of a suspended mesosized particle in a simple liquid and in particular the emergence of hydrodynamic memory via the coupling to the Navier-Stokes equations. High precision experiments have confirmed these effects for the first time recently and suggest to develop new ultrasensitive biophysical tools.
Even more drastic persistent correlations appear in the Lorentz model, originally introduced to describe electronic transport in crystals, nowadays the reference model for porous media and cellular crowding. There a particle explores a disordered matrix of frozen obstacles, such that at high scattering density a localization transition emerges. This phenomenon goes in hand with anomalous subdiffusive transport which can rationalized as a critical phenomenon. We analyze the behavior in terms of simulations, scaling theory, and a newly analytically developed theory based on a low-density expansion. Applying an external bias force, the system can be driven far from equilibrium such that linear response is no longer applicable. The results demonstrate the breakdown of the fluctuation-dissipation theorem at arbitrarily low fields. In particular the transport coefficients are no longer nonanalytic in the frequency, yet a new singular dependence on the field strength arises.

Plant cells typically elongate along a single growth axis. In order to sustain this anisotropy the cell requires spatially extended structures that encode the proper geometrical constraints. The most prominent of these structures is the so-called interphase cortical array. Its components are microtubules: long filamentous protein aggregates that exhibit an interesting intrinsic dynamics, in which they stochastically switch between periods of growth and shrinkage. In the cortical array the microtubules are attached to the inner side of the plasmamembrane, effectively creating a 2D system, in which the only motion is caused by (de)polymerization. Because of the reduced dimensionality growing microtubules can now collide with pre-existing ones, giving rise to an angle dependent scattering events, in which the the colliding microtubule can either alter its course to grow along side the other microtubule, switch to the shrinking state, or simply slip over the obstacle. We address the question whether these interactions are sufficient to explain to explain the high degree of orientational alignment found in the cortical array, using event-driven stochastic simulations and a coarse-grained dynamical model. We then how how this model helps to explain two recent observations: nucleation-driven prepatterning in the buildup of the ordered array and division-plane decisions in developing plant roots.

In my talk I will consider the non-equilibrium dynamics of observables for finite one-dimensional quantum systems. By calculating the dependence of time and statistical averages on system size the role of the local and the exponentially many non-local conserved quantities for the thermalization of a part of a closed quantum system will be investigated. As a main result, a condition for thermalization is derived which is more general than the often used eigenstate thermalization hypothesis. The idea that a subsystem of a closed quantum system can act as effective bath for the rest of the system is then further investigated by numerical density matrix renormalization group calculations for a specific model consisting of a noninteracting fermionic chain with each site of the chain coupled to an additional bath site. In addition to the question of thermalization in the long-time limit, also the intermediate time dynamics is studied analytically for this model in limiting cases.

1. J. Sirker, N.P. Konstantinidis, N. Sedlmayr, arXiv: 1303.3064 (2013)
2. N. Sedlmayr, J. Ren, F. Gebhard, J. Sirker, PRL 110, 100406 (2013)

Membrane adhesion plays a key role in a number of biological processes and is typically mediated by domains consisting of a large number of ligand-receptor bonds. The dynamics of domain growth has been studied on cells and vesicles over the last two decades, whereby a number of different growth behaviors and a variety of domain morphologies have been characterized. However, a comprehensive theoretical framework accounting for these observations is largely missing. The difficulty is that in all these processes, there are three omnipresent stochastic elements that are intimately connected and span several time and length scales. These are (i) membrane deformations and fluctuations, (ii) protein binding and unbinding, and (iii) the diffusion of binders. I will present a model for each of these elements and provide means of their coupling. Finally, I will show how to successfully coarse grain the problem, and build a Monte Carlo scheme that is accurate from the nucleation to the late stages of domain growth including the diffusion limited dynamics.

A light impurity in a Fermi sea undergoes a transition from a polaron to a molecular bound state for increasing interaction. We present a method to compute the spectral functions of the polaron and molecule in a unified framework based on the functional renormalization group with full self-energy feedback. We discuss the energy spectra and decay widths of the attractive and repulsive polaron branches as well as the molecular bound state, and we compare our results to recent experiments with ultracold gases.

Biological materials possess some remarkable mechanical properties. Cells and tissues can adjust, remodel, stiffen, soften, in some cases even pack up and leave when circumstances require action. Surprisingly, most systems that exhibit this stunningly complex response, such as the cytoskeleton inside cells and the extracellular matrix, share a common design: under a microscope, they are crosslinked, hierarchical networks of biological polymers. Even more surprisingly, many of the in vivo behaviors can be reproduced in vitro in reconstituted proteinaceous polymer gels. Many of these systems, most notably collagen, play a purely structural role in living organisms. In other words, their function _is_ their mechanical response. Biopolymer networks are therefore particularly suited to begin to understand the complex relationship between structural design and functionality in living systems.
In this seminar, I recall the general theory of nonlinear elasticity, and I will discuss our efforts to bridge the gap from microscopic structure to macroscopic mechanical response of such nonlinear systems using collagen as an example. Time permitting, I will discuss our first steps towards controlling the mechanical properties of biomimetic synthetics.

The notion of chaos is frequently invoked in the foundations of quantum statistical physics. Yet, the definition of quantum chaos for many-particle systems is still not fully understood. We show that nonintegrable lattices of spins 1/2, which are often considered to be chaotic, do not exhibit the basic property of classical chaotic systems, namely, exponential sensitivity to small perturbations. We compare the re- sponses of chaotic lattices of classical spins and nonintegrable lattices of spins 1/2 to imperfect reversal of spin dynamics known as Loschmidt echo. In the classical case, Loschmidt echoes exhibit exponential sensitivity to small perturbations characterized by twice the value of the largest Lyapunov exponent of the system. In the case of spins 1/2, Loschmidt echoes are only power-law sensitive to small perturbations. Our findings imply that it is impossible to define Lyapunov exponents for lattices of spins 1/2 even in the macroscopic limit. At the same time, the above absence of exponential sensitivity to small perturbations is an encouraging news for the efforts to create quantum simulators. The power-law sensitivity of spin 1/2 lattices to small perturbations is predicted to be measurable in nuclear magnetic resonance experiments.
Reference: B.V. Fine, T. A. Elsayed, C. M. Kropf and A. S. de Wijn, arXiv:1305.2817