Abstract:
A field theoretic renormalization group method is presented which is capable of dealing with crossover problems associated with a change in the upper critical dimension. The method leads to flow functions for the parameters and coupling constants of the model which depend on the set of parameters which characterize the fixed point landscape of the underlying problem. Similar to Nelson's trajectory integral method any vertex function can be expressed as a line integral along a renormalization group trajectory, which in the field theoretic formulation are given by the characteristics of the corresponding Callan-Symanzik equation. The field theoretic renormalization automatically leads to a separation of the regular and singular parts of all crossover scaling functions. The method is exemplified for the crossover problem in magnetic phase transitions, percolation problems and quantum phase transitions. The broad applicability of the method is emphasized.

Abstract:
The tube model is a central concept in polymer physics, and allows to reduce the complex many-filament problem of an entangled polymer solution to a single filament description. We investigate the probability distribution function of conformations of confinement tubes and single encaged filaments in entangled semiflexible polymer solution. Computer simulations are developed that mimic the actual dynamics of confined polymers in disordered systems with topological constraints on time scales above local equilibration but well below large scale rearrangement of the network. We observe the statistical distribution of curvatures and compare our results to recent experimental findings. Unexpectedly, the observed distributions show distinctive differences from free polymers even in the absence of excluded volume. Extensive simulations permit to attribute these features to entropic trapping in network void spaces. The transient non-equilibrium distributions are shown to be a generic feature in quenched-disorder systems on intermediate time scales.

Abstract:
We review our current understanding of the critical dynamics of magnets above and below the transition temperature with focus on the effects due to the dipole--dipole interaction present in all real magnets. Significant progress in our understanding of real ferromagnets in the vicinity of the critical point has been made in the last decade through improved experimental techniques and theoretical advances in taking into account realistic spin-spin interactions. We start our review with a discussion of the theoretical results for the critical dynamics based on recent renormalization group, mode coupling and spin wave theories. A detailed comparison is made of the theory with experimental results obtained by different measuring techniques, such as neutron scattering, hyperfine interaction, muon--spin--resonance, electron--spin--resonance, and magnetic relaxation, in various materials. Furthermore we discuss the effects of dipolar interaction on the critical dynamics of three--dimensional isotropic antiferromagnets and uniaxial ferromagnets. Special attention is also paid to a discussion of the consequences of dipolar anisotropies on the existence of magnetic order and the spin--wave spectrum in two--dimensional ferromagnets and antiferromagnets. We close our review with a formulation of critical dynamics in terms of nonlinear Langevin equations.

Abstract:
Inspired by the physics of magnetohydrodynamics (MHD) a simplified coupled Burgers-like model in one dimension (1d), a generalization of the Burgers model to coupled degrees of freedom, is proposed to describe 1dMHD. In addition to MHD, this model serves as a 1d reduced model for driven binary fluid mixtures. Here we have performed a comprehensive study of the universal properties of the generalized d-dimensional version of the reduced model. We employ both analytical and numerical approaches. In particular, we determine the scaling exponents and the amplitude-ratios of the relevant two-point time-dependent correlation functions in the model. We demonstrate that these quantities vary continuously with the amplitude of the noise cross-correlation. Further our numerical studies corroborate the continuous dependence of long wavelength and long time-scale physics of the model on the amplitude of the noise cross-correlations, as found in our analytical studies. We construct and simulate lattice-gas models of coupled degrees of freedom in 1d, belonging to the universality class of our coupled Burgers-like model, which display similar behavior. We use a variety of numerical (Monte-Carlo and Pseudospectral methods) and analytical (Dynamic Renormalization Group, Self-Consistent Mode-Coupling Theory and Functional Renormalization Group) approaches for our work. The results from our different approaches complement one another. Possible realizations of our results in various nonequilibrium models are discussed.

Abstract:
We investigate the viscoelastic properties of entangled networks of semiflexible polymers. At intermediate time scales the elastic response of these networks to shear deformation is described by the plateau modulus $G$. Different scaling laws with polymer concentration $c$ have been proposed based on the assumption that the deformation field is affine on all length scales. We develop a numerical approach that allows to calculate the modulus via free energy changes for both affine and non-affine deformations. The non-affine deformation field is obtained by a free energy minimization. Our findings allow for a confirmation of a power law $G \propto c^{7/5} l_p^{-1/5}$ with polymer concentration $c$ and persistence length $l_p$ and furthermore quantify the systematic deviations due to the affinity assumption.

Abstract:
We study the elasticity of random stiff fiber networks. The elastic response of the fibers is characterized by a central force stretching stiffness as well as a bending stiffness that acts transverse to the fiber contour. Previous studies have shown that this model displays an anomalous elastic regime where the stretching mode is fully frozen out and the elastic energy is completely dominated by the bending mode. We demonstrate by simulations and scaling arguments that, in contrast to the bending dominated \emph{elastic energy}, the equally important \emph{elastic forces} are to a large extent stretching dominated. By characterizing these forces on microscopic, mesoscopic and macroscopic scales we find two mechanisms of how forces are transmitted in the network. While forces smaller than a threshold $F_c$ are effectively balanced by a homogeneous background medium, forces larger than $F_c$ are found to be heterogeneously distributed throughout the sample, giving rise to highly localized force-chains known from granular media.

Abstract:
We study the elasticity of cross-linked networks of thermally fluctuating stiff polymers. As compared to their purely mechanical counterparts, it is shown that these thermal networks have a qualitatively different elastic response. By accounting for the entropic origin of the single-polymer elasticity, the networks acquire a strong susceptibility to polydispersity and structural randomness that is completely absent in athermal models. In extensive numerical studies we systematically vary the architecture of the networks and identify a wealth of phenomena that clearly show the strong dependence of the emergent macroscopic moduli on the underlying mesoscopic network structure. In particular, we highlight the importance of the full polymer length that to a large extent controls the elastic response of the network, surprisingly, even in parameter regions where it does not enter the macroscopic moduli explicitly. We provide theoretical scaling arguments to relate the observed macroscopic elasticity to the physical mechanisms on the microscopic and the mesoscopic scale.

Abstract:
We calculate the distribution function of the end--to--end distance of a semiflexible polymer with large bending rigidity. This quantity is directly observable in experiments on single semiflexible polymers (e.g., DNA, actin) and relevant to their interpretation. It is also an important starting point for analyzing the behavior of more complex systems such as networks and solutions of semiflexible polymers. To estimate the validity of the obtained analytical expressions, we also determine the distribution function numerically using Monte Carlo simulation and find good quantitative agreement.

Abstract:
We study the phase transitions of interacting bosons at zero temperature between superfluid (SF) and supersolid (SS) states. The latter are characterized by simultaneous off-diagonal long-range order and broken translational symmetry. The critical phenomena is described by a long-wavelength effective action, derived on symmetry grounds and verified by explicit calculation. We consider two types of supersolid ordering: checkerboard (X) and collinear (C), which are the simplest cases arising in two dimensions on a square lattice. We find that the SF--CSS transition is in the three-dimensional XY universality class. The SF--XSS transition exhibits non-trivial new critical behavior, and appears, within a $d=3-\epsilon$ expansion to be driven generically first order by fluctuations. However, within a one--loop calculation directly in $d=2$ a strong coupling fixed point with striking ``non-Bose liquid'' behavior is found. At special isolated multi-critical points of particle-hole symmetry, the system falls into the 3d Ising universality class.

Abstract:
The dynamic structure factor of semiflexible polymers in solution is derived from the wormlike chain model. Special attention is paid to the rigid constraint of an inextensible contour and to the hydrodynamic interactions. For the cases of dilute and semidilute solutions exact expressions for the initial slope are obtained. When the hydrodynamic interaction is treated on the level of a renormalized friction coefficient, the decay of the structure factor due to the structural relaxation obeys a stretched exponential law in agreement with experiments on actin. We show how the characteristic parameters of the system (the persistence length \ell_p, the lateral diameter a of the molecules and the mesh size \xi_m of the network) are readily determined by a single scattering experiment with scattering wavelength \lambda obeying a << \lambda <<\ell_p and \lambda<\xi_m. We also find an exact explicit expression for the effective (wave vector dependent) dynamic exponent z(k)<3 for semiflexible polymers and thus an enlightening explanation for a longstanding puzzle in polymer physics.