Order, Disorder and Criticality

The following sections are included:

• Preface
• Contents

This review is devoted to detailed consideration of the universal statistical properties of one-dimensional directed polymers in a random potential. In terms of the replica Bethe ansatz technique we derive several exact results for different types of the free energy probability distribution functions. In the second part of the review we discuss the problems which are still waiting for solutions.

I describe the manifestation of the non-Euclidean geometry in the behavior of collective observables of some complex physical systems. Specifically, I consider the formation of equilibrium shapes of plants and statistics of sparse random graphs. For these systems I discuss the following interlinked questions: (i) the optimal embedding of plant leaves in the three-dimensional space, (ii) the spectral statistics of sparse random matrix ensembles.

In this chapter, we review concepts and theories of polymer dynamics. We think of it as an introduction to the topic for scientists specializing in other subfields of statistical mechanics and condensed matter theory, so, for the readers reference, we start with a short review of the equilibrium static properties of polymer systems. Most attention is paid to the dynamics of unentangled polymer systems, where apart from classical Rouse and Zimm models we review some recent scaling and analytical generalizations. The dynamics of systems with entanglements is also briefly reviewed. Special attention is paid to the discussion of comparatively weakly understood topological states of polymer systems and possible approaches to the description of their dynamics.

In the last two decades computer simulations in generalized ensembles based on Markov chain Monte Carlo sampling such as the multicanonical, Wang-Landau, and parallel tempering (or replica exchange) methods have emerged as a strong numerical tool for investigations of the statistical physics of macromolecular systems. Many studies have focused on coarse-grained models of polymers on the lattice and in the continuum. Phase diagrams of polymer chains in bulk and in interaction with surfaces were extensively studied. Also, the aggregation behavior in solution has been investigated.

In this chapter, first the theoretical background for these simulations will be described, the employed algorithms explained and their performance assessed. Implementations of these algorithms on parallel computers are also briefly discussed. As an illustration of these concepts, an overview of polymer systems investigated with multicanonical and parallel tempering simulations will be given, focusing on our own recent studies of coarse-grained models.

Macromolecular networks are of interest from both the technological point of view and as a chemical realisation of a network structure. In this chapter we review available experimental data on photo-controllable variants of such networks realised either via chemical bonding or as the result of a self-assembly. Due to a wide range of available systems, the scope is restricted to two chief cases here: the systems based on a self-assembly of decorated gold nanoparticles, and composite materials displaying photoswitchable conductivity. Computer simulations can be seen as a valuable tool for prediction of new effects and features in these systems but are rather scarce or not existent so far. In our view, the coarse-grained type of modelling is the most appropriate approach here, due to large time- and lengthscale of relevant processes. An example of such simulation is provided for the experimentally observed cyclic photo-switchable behaviour of the aggregates formed by azobenzene-modified gold nanoparticles in a pore filled by a solvent.

Applications that require substantial computational resources today cannot avoid the use of heavily parallel machines. Embracing the opportunities of parallel computing and especially the possibilities provided by a new generation of massively parallel accelerator devices such as GPUs, Intel’s Xeon Phi or even FPGAs enables applications and studies that are inaccessible to serial programs. Here we outline the opportunities and challenges of massively parallel computing for Monte Carlo simulations in statistical physics, with a focus on the simulation of systems exhibiting phase transitions and critical phenomena. This covers a range of canonical ensemble Markov chain techniques as well as generalized ensembles such as multicanonical simulations and population annealing. While the examples discussed are for simulations of spin systems, many of the methods are more general and moderate modifications allow them to be applied to other lattice and off-lattice problems including polymers and particle systems. We discuss important algorithmic requirements for such highly parallel simulations, such as the challenges of random-number generation for such cases, and outline a number of general design principles for parallel Monte Carlo codes to perform well.

Electric grids, telecommunication networks, railways, healthcare systems, financial circuits, etc. are infrastructures that are critical for functioning and the welfare of our countries. Most of such infrastructures have been developed and designed according to engineering paradigms that are starting to become inadequate to cope with their increasing complexity. Much of this complexity is simply due to increased system size: as statistical physics teaches us, collection of interacting objects exhibits emergent phenomena (like phase transitions) that goes beyond the properties of the single objects and have peculiar characteristics in the infinite size limit. Moreover, the increase of interdependencies among the infrastructures (think as an example of the interdependence among communication networks and electric grids) is adding a further element of complexity. Hence, the statistical physics’ approach can enlarge the understanding of the fragilities and vulnerabilities of such critical infrastructures. In this chapter we will cover some of the current models of infrastructural grids – both isolated and coupled – hinting out the possible and needed development of the field.

The following section is included:

• Subject Index