Granular materials can exist in infinitely many configurations, but under well defined external influences and conditions can exhibit perfectly reproducable behaviour, and therefore must be possible to describe by statistical mechanical laws. These must be quite different then the traditional, Hamiltonian, statistical mechanics since the dynamics involved in changing from one configuration to another is dominated by friction. Thus the ergodic, self-sustained equilibration of of conventional statistical mechanics is replaced by externally induced changes from one jammed configuration to another. Several questions arise, which we will attempt to answer here:

\item{1.} How does one specify a reproducible state of a granular system to which it can return after disturbance, by repeating the history of its formation?

\item{2.} Stress due to external fields or boundary forces will propagate through the granular medium even if the grains are perfectly rigid and cannot be strained. This means that the stress is not related to the strain and stress-strain constitutive information is redundant. What are then the constitutive equations required to determine the stress?

\item{3.} It appears that fluctuations are important to the stress equations. What exactly is the key quantity which fluctuates and how can the distribution of the fluctuations be found?

\item{4.} The three points above are theoretical in nature, but they must be supported experimentally. It is not difficult to generate spectacular but incomprehensible effects in granular materials, but there are experiments that really test basic fundamental concepts and these will be described.

The isostaticity theory for stress transmission in macroscopic planar particulate assemblies is extended here to non-rigid particles. It is shown that, provided that the mean coordination number in $d$ dimensions is $d+1$, macroscopic systems can be mapped onto equivalent assemblies of perectly rigid particles that support the same stress field. The error in the stress field that the compliance introduces for finite systems is shown to decay with size as a power law. This leads to the conclusion that the isostatic state is not limited to infinitely rigid particles both in two and in three dimensions, and paves the way to an application of isostaticity theory to more general systems.

Progress is reported on several questions that bedevil understanding of granular systems: (i) are the stress equations elliptic, parabolic or hyperbolic? (ii) how can the often-observed force chains be predicted from a first-principles continuous theory? (iii) How do we relate insight from isostatic systems to general packings? Explicit equations are derived for the stress components in two dimensions including the dependence on the local structure. The equations are shown to be hyperbolic and their general solutions, as well as the Green function, are found. It is shown that the solutions give rise to force chains and the explicit dependence of the force chains trajectories and magnitudes on the local geometry is predicted. Direct experimental tests of the predictions are proposed. Finally, a framework is proposed to relate the analysis to non-isostatic and more realistic granular assemblies.

A recent theory for stress transmission in isostatic planar granular assemblies and trivalent cellular solids predicts a constitutive equation that couples the stress field to the local microstructure [R. C. Ball and R. Blumenfeld, Phys. Rev. Lett. 88, 115505 (2002)]. The theory has been difficult to apply to macroscopic systems because the constitutive equation becomes trivial when coarse-grained by a simple area-average. This problem is resolved here for arbitrary planar topologies. The solution is based on the observation that a staggered order makes it possible to write the equation in terms of a reduced geometric tensor that can be upscaled. The method proposed here makes it possible to apply this idea to realistic systems whose staggered order is generally 'frustrated'. The method consists of a systematic renormalization, which removes the frustration and enables the use of the reduced geometric tensor. A calculation of the stress due to a defect in a honeycomb cellular system is presented as an example.

The marginally rigid state is a candidate paradigm for what makes granular material a state of matter distinct from both liquid and solid. The coordination number is identified as a discriminating characteristic, and for rough-surfaced particles we show that the low values predicted are indeed approached in simple two-dimensional experiments. We show calculations of the stress transmission, suggesting that this is governed by local linear equations of constraint between the stress components. These constraints can in turn be related to the generalized forces conjugate to the motion of grains rolling over each other. The lack of a spatially coherent way of imposing a sign convention on these motions is a problem for scaling up the equations, which leads us to attempt a renormalization-group calculation. Finally, we discuss how perturbations propagate through such systems, suggesting a distinction between the behaviour of rough and smooth grains.

This paper proposes a new volume function for calculation of the entropy of planar granular assemblies. This function is extracted from the antisymmetric part of a new geometric tensor and is rigorously additive when summed over grains. It leads to the identification of a conveniently small phase space. The utility of the volume function is demonstrated on several case studies, for which we calculate explicitly the mean volume and the volume fluctuations.

Stress transmission in planar open-cell cellular solids is analysed using a recent theory developed for marginally rigid granular assemblies. This is made possible by constructing a one-to-one mapping between the two systems. General trivalent networks are mapped onto assemblies of rough grains, while networks where Plateau rules are observed, are mapped onto assemblies of smooth grains. The constitutive part of the stress transmission equations couples the stress directly to the local rotational disorder of the cellular structure via a new fabric tensor. An intriguing consequence of the analysis is that the stress field can be determined in terms of the microstructure alone independent of stress-strain information. This redefines the problem of structure-property relationship in these materials and poses questions on the relations between this formalism and elasticity theory. The deviation of the stress transmission equations from those of conventional solids has been interpreted in the context of granular assemblies as a new state of solid matter and the relevance of this interpretation to the state of matter of cellular solids is discussed.

The equation of motion of twists on classical antiferromagnetic Heisenberg spin chains are derived. It is shown that twists interact via position- and velocity-dependent long-range two-body forces. A quiescent regime is identified wherein the system conserves momentum. With increasing kinetic energy the system exits this regime and momentum conservation is violated due to walls annihilation. A bitwist system is shown to be integrable and its exact solution is analysed. Many-twist systems are discussed and novel periodic twist lattice solutions are found on closed chains. The stability of these solutions is discussed.

The transmission of stress through a marginally stable granular pile in two dimensions is exactly formulated in terms of a vector field of loop forces, and thence in terms of a single scalar potential. This leads to a local constitutive equation coupling the stress tensor to fluctuations in the local geometry. For a disordered pile of rough grains this means the stress tensor components are coupled in a frustrated manner. In piles of rough grains with long range staggered order, frustration is avoided and a simple linear theory follows. We show that piles of smooth grains can be mapped onto a pile of unfrustrated rough grains, indicating that the problems of rough and smooth grains may be fundamentally distinct.

This paper examines the effect of cooling on disentanglement forces in polymers and the implications for both single chain pullout and polymer dynamics. I derive the explicit dependence of the distribution of these forces on temperature, which is found to exhibit a rich behaviour. Most significantly, it is shown to be dominated by large fluctuations up to a certain temperature $T_0$ that can be determined from molecular parameters. The effects of these fluctuations on chain friction are analysed and they are argued to undermine the traditional melt-based models that rely on a typical chain friction coefficient. A direct implication for first principles calculation of viscosity is discussed. This quantifies the limit of validity of such descriptions, such as Rouse dynamics and the Tube model, and pave the way to model polymer dynamics around the glass transition temperature.

This paper addressses the kinetics and dynamics of a family of domain wall solutions along classical antiferromagnetic Heisenberg spin chains at low energies. The equation of motion is derived and found to have long range position- and velocity-dependent two-body forces. A 'quiescent' regime is identified where the forces between walls are all repelling. Outside this regime some of the interactions are attractive, giving rise to wall collisions whereupon the colliding walls annihilate. The momentum of the system is found to be conserved in the quiescent regime and to suffer discontinuous jumps upon annihilation. The dynamics are illustrated by an exact solution for a double wall system and a numerical solution for a many-wall system. On circular chains the equations support stable periodic stripes that can rotate as a rigid body. It is found that the stripes are more stable the faster they rotate. The periodic structure can be destabilised by perturbing the walls' angular velocity in which case there is a transition to another periodic structure, possibly via a cascade of annihilation events.

The Belavin-Polyakov equation t u = t × t s with t 2 = 1 has been shown recently to describe the evolution of a wide class of space curves, and has several physical applications. Here, we obtain a hierarchy of exact multi-twist solutions for this nonlinear system. As an illustration, we apply our results to continuum magnetic models. When u and s denote temporal and spatial variables respectively, these twists describe very low-energy domain walls travelling along an antiferromagnetic spin chain. When they denote independent spatial variables, the solutions represent twists in the static configuration (texture) of a two-dimensional ferromagnet.

Recent experiments on the pullout of single polymer chains have revealed a complex behavior of the force fluctuations. This paper analyzes the pullout process theoretically and numerically and shows that these fluctuations can be made to shed light on disentanglement dynamics. To facilitate the analysis, I first derive the probability density function of the threshold force needed to disentangle one entanglement point. This function is found to be dominated by large fluctuations, which bears directly on the observed statistics. The average and variance of the force are calculated, and a numerical investigation of the dynamics is carried out to check the results. Finally, applications to deformations in several macroscopic systems are discussed.

I discuss thin magnetic layers in the context of a two-dimensional ferromagnetic Heisenberg spin system. In the low energy regime it is shown that the system follows a Belavin-Polyakov type of equation. It is argued that unlike in traditional systems where these equations occur only under uniform boundary conditions, the boundary conditions in this case are less restrictive, allowing for a new family of solutions. These solutions consist of magnetic domains of spins that are oriented at relative opposite directions. The boundaries between regions are sharp on the continuous scale but within a domain wall the magnetization changes orientation continuously from one ground state to another. All the magnetic energy in the system is shown to concentrate along the domain walls. It is therefore argued that most favorable for the wails is to rearrange in a hierarchical or fractal fashion because such an arrangement lowers the overall energy density. It is suggested that this hierarchical structure of magnetic domain boundaries should be observable by magnetic force microscopy. Recent results suggest that such configurations may also dominate the structure of domain walls in magnetostrictive materials and magnetic nanotubes.

Recent years witnessed an explosion of research activity in single-wall nanotubes (SWNT). Their phenomenal strength is accompanied by a relative softness to radial deformations, which we propose to exploit for actively controlling the shape of SWNT-based composite materials. We show that if the SWNT are supplemented with magneto-elastic interactions then magnetic domain walls can form on the curved surface. The magneto-elastic coupling leads to surface wrinkles along the domain walls. We suggest that, by applying an external magnetic field, one can control the number and location of the wrinkles, thus magneto-actively controlling the material properties.