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Discovering fundamental principles governing the dynamics of granular media has been a long-standing challenge. Recent predictions of detailed balance steady states (DBSS), supported by experimental observations in cyclic shear experiments of planar granular systems, called into question the common belief that the detailed balance principle is only a feature of equilibrium. Here, we first show analytically that DBSS in planar granular dynamics arise when a certain conditional cell order distribution is independent of the condition. We then demonstrate that this condition is met in rotational shear experiments, which indeed also give rise to robust DBSS. This suggests that DBSS not only exist but are also quite common. We also show that, when the unconditional cell order distribution maximises the entropy, as has been found recently, then this distribution is determined by a single parameter - the ratio of splitting and merging rates of cells of any arbitrary order. These results simplify the modelling of the complex dynamics of planar granular systems to the solution of recently proposed evolution equations, demonstrating their predictive power., Comment: 5 pages, 2 figures; Supplementary Material: 3 pages, 3 figures. Comments welcome!

Deflection of stray asteroids by low-momentum impacts is an Earth's key defense strategy. Using a proof-of-principle numerical model, we show that even such impacts on rubble pile asteroids (RPAs) may lead to fragmentation because of inherent internal stress chains. Such chains occur in three- and two-dimensional (2D) aggregates and we study this phenomenon in 2D. We first establish the existence of stress chains in our aggregates and then show that the dynamic shear stress and displacements are highest along the chains, increasing significantly the risk of fracturing there. Our simulations suggest that this phenomenon is independent of the aggregate's particle size distribution. We conclude that future studies are needed to quantitatively assess fragmentation risk as a function of stress chain statistics in RPAs.

A basic problem in the science of realistic granular matter is the plethora of heuristic models of the stress field in the absence of a first-principles theory. Such a theory is formulated here, based on the idea that static granular assemblies can be regarded as two-phase composites. A thought experiment is described, demonstrating that the state of such materials can be varied continuously from marginal stability, via a two-phase granular assembly, then porous structure, and finally be made perfectly elastic. For completeness, I review briefly the condition for marginal stability in infinitely large assemblies. The general solution for the stress equations in d=2 is reviewed in detail and shown to be consistent with the two-phase idea. A method for identifying the phases of finite regions in larger systems is constructed, providing a stability parameter that quantifies the `proximity' to the marginally stable state. The difficulty involved in deriving stress fields in such composites is a unique constraint on the boundary between phases and, to highlight it, a simple case of a stack of plates of alternating phase is solved explicitly. An effective medium approximation, which satisfies this constraint, is then developed and analysed in detail. This approach forms a basis for the extension of the stress theory to general granular solids that are not marginally stable or at the yield threshold., Comment: 10 pages, 3 figures, Supplementary material file. To appear in Phys. Rev. E

Understanding the structural evolution of granular systems is a long-standing problem. A recently proposed theory for such dynamics in two dimensions predicts that steady states of very dense systems satisfy detailed-balance. We analyse analytically and numerically the steady states of this theory in systems of arbitrary density and report the following. 1. We discover that all such dynamics almost certainly possess only one physical steady state, which may or may not satisfy detailed balance. 2. We show rigorously that, if a detailed balance solution is possible then it is unique. The above two results correct an erroneous conjecture in the literature. 3. We show rigorously that the detailed-balance solutions in very dense systems are globally stable, extending the local stability found for these solutions in the literature. 4. In view of recent experimental observations of robust detailed balance steady states in very dilute cyclically sheared systems, our results point to a self-organisation of process rates in dynamic granular systems., Comment: 6 pages, 3 figures, supplemental material. To appear in Journal of Physics A: Mathematical and Theoretical

It is accepted that stress and structure self-organize cooperatively during quasi-static dynamics of granular systems, but the consequences of this self-organization are not fully understood. Such an understanding is essential because local structural properties of the settled material are then correlated with the local stress, which calls into question existing linear theories of stress transmission in granular media. A method to quantify the local stress-structure correlations is necessary for addressing this issue and we present here such a method for planar systems. We then use it to analyze numerically several different systems, compressed quasi-statically by two different procedures. We define cells, cell orders, cell orientations, and cell stresses and report the following. 1. The mean ratio of cell principal stresses decreases with cell order and increases with friction. 2. The ratio distributions collapse onto a single curve under a simple scaling, for all packing protocols and friction coefficients. 3. Cells orient along the local stress major principal axes. 4. A simple first-principles model explains the correlations between the local cell and stress principal axis orientations. Our results quantify the cooperative stress-structure self-organization and provide a way to relate quantitatively the stress-structure coupling to different process parameters and particle characteristics. Significantly, the strong stress-structure correlation, driven by structural re-organization upon application of external stress, suggests that current stress theories of granular matter need to be revisited., Comment: 6 pages, submitted

The method, proposed in \cite{Za22} to derive the densest packing fraction of random disc and sphere packings, is shown to yield in two dimensions too high a value that (i) violates the very assumption underlying the method and (ii) corresponds to a high degree of structural order. The claim that the obtained value is supported by a specific simulation is shown to be unfounded. One source of the error is pointed out., Comment: 1 page, supplemental material is referenced, typo corrected

There is no agreement in the literature on the rate of diffusion of a particle in a cooling granular gas. Predictions and model assumptions range from the conventional to very exotic dependence of the mean square distance (MSD) on time. This problem is addressed here by calculating the MSD from first-principles. The calculation is based on random-walking and it circumvents the common use of continuum equations and equations of states, which involve approximations that erode at low gas particle densities. The MSD is found to increase logarithmically with time -- slower than even in anomalous diffusion. This result is consistent with the well-established Haff's law for the decay of the kinetic energy, which is also derived along the way from the same first principles. This derivation also pins down Haff's time constant, alleviating the usual need for a fitting parameter. The diffusion theory is then used to calculate explicitly the time evolution of any initial particle velocity distribution, yielding an unusual functional form. The limitations of the theory are discussed and extensions to it are outlined., Comment: 4 pages, 1 figure, submitted for publication @ New J. Phys

Predicting the densest random disc packing fraction is an unsolved paradigm problem relevant to a number of disciplines and technologies. One difficulty is that it is ill-defined without setting a criterion for the disorder. Another is that the density depends on the packing protocol and the multitude of possible protocol parameters has so far hindered a general solution. A new approach is proposed here. After formulating a well-posed form of the general protocol-independent problem for planar packings of discs, a systematic criterion is proposed to avoid crystalline hexagonal order as well as further topological order. The highest possible random packing fraction is then derived exactly: $\phi_{RCP}=0.852525...$. The solution is based on the cell order distribution that is shown to: (i) yield directly the packing fraction; (ii) parameterise all possible packing protocols; (iii) make it possible to define and limit all topological disorder. The method is further useful for predicting the highest packing fraction in specific protocols, which is illustrated for a family of simply-sheared packings that generate maximum-entropy cell order distributions., Comment: 4 pages, 2 figures, typo corrected in v2

The principle of detailed balance (DB) states that every kinetic transition in a system with many micro-states, $\mu$, is balanced, on average, with the opposite transition, $\mu_i\leftrightharpoons\mu_j$. Since its introduction by Boltzmann, this principle has been used by luminaries, such as Einstein, Eddington, Kramers, Pauli, Ehrenfest, Dirac, Onsager, and many others to derive significant results that underpin much of our scientific understanding. The current belief is that DB is satisfied only in equilibrium systems, while non-equilibrium steady states can only be balanced by cycles, such as $A\to B\to C\to A$. We show here experimentally that DB can exist and is commonly and robustly satisfied in a family of quasi-statically cyclically sheared granular systems. We further study the approach to DB as a function of system size and time. Given the significant impact that this principle has had on equilibrium systems, we believe that this discovery paves the way for better models of the dynamics of non-equilibrium systems., Comment: 5 pages, 4 figures

Gravitational-collapse-based explanations of the cosmic web lead to problems in estimating the total mass in the universe. A first-principles several-scales model is developed here for the structural organisation of cosmic matter in a flat universe, showing that the web formation could be driven by inelastic collisions before gravity took hold, suggesting a possible way to resolve these problems. The following results are derived. (i) The diffusion rate in the particulate gas after recombination is sub-anomalous, with a rapid decay of particle velocities. (ii) The evolution of the particle velocity distribution is calculated explicitly. (iii) The gas density is shown to be unstable, leading to void formation and clusters nucleation. (iv) Rounded clusters are shown to be unstable and tend to elongate. (v) An equation is derived for the growth of long clusters into filaments and solved explicitly. The fast-growing clusters deplete the regions around them and generate large voids, potentially giving rise to the cosmic web before gravity dominated., Comment: 7 pages, 5 figures, submitted

Using cyclic shear to drive a two dimensional granular system, we determine the structural characteristics for different inter-particle friction coefficients. These characteristics are the result of a competition between mechanical stability and entropy, with the latter's effect increasing with friction. We show that a parameter-free maximum-entropy argument alone predicts an exponential cell order distribution, with excellent agreement with the experimental observation. We show that friction only tunes the mean cell order and, consequently, the exponential decay rate and the packing fraction. We further show that cells, which can be very large in such systems, are short-lived, implying that our systems are liquid-like rather than glassy.

An intermediate step to modelling behaviour of active matter is understanding interactions of active objects (AOs) with inanimate matter, which often lead to a range of rich behaviour. We present a range of simulations of the interaction of a self-energised AO with three-dimensional granular matter and develop a first-principles theoretical model to describe the observed phenomena. The AO oscillates horizontally, which causes it to either rise against gravity or sink, depending on the oscillation amplitude and frequency. We identify two competing mechanisms that drive the vertical motion. When the AO moves below a critical speed, $v_c$, it generates a jammed stagnant zone ahead of it, which effects an upward force and leads to the rise. Above $v_c$ and certain kinetic energy, the medium around the AO fluidises and the AO sinks into the layer supporting it. The duration of the rising and sinking phases depend non-trivially on the AO's amplitude and frequency leading to an intricate nonlinear dynamics. We derive the equation of motion for the time-dependent depth from first-principles and show that its solutions agree well with a wide range of computer simulations, which we perform within the range of parameters allowed by the finiteness of the simulated system., Comment: 5 pages, 5 figures, submitted to Phys. Rev. Lett

Isostaticity theory (IT) provides stress field equations for marginally rigid packs of non-cohesive particles, interacting through hard-core forces. Its main advantage over strain-based theories is by closing the stress equations with stress-structure, rather than stress-strain, relations, which enables modelling the stress chains, often observed in experiments and simulations. The usefulness of IT has been argued to extend beyond its applicability at marginal rigidity. It has been shown to be derivable from first principles in two-dimensions, with the structure quantified by a fabric tensor that couples to the stress field. However, upscaling IT to the continuum is done currently empirically by postulating convenient closure equations. The problem is that a volume average of the fabric tensor vanishes in the continuum limit, trivialising the closure equation. This poses an unusual upscaling problem, necessitating a new approach. Such an approach is developed here, resolving the problem for planar granular assemblies. The new method is developed initially for idealised 'unfrustrated' packs by coarse-graining first to the two-grain scale, after which a conventional coarse-graining can be used. It is then extended to general realistic systems, by introducing an intermediate `de-frustration' procedure. The applicability of the method is illustrated with a tractable example., Comment: 6 pages, 5 figures, to appear in Granular Matter, Introductory paragraph in memoriam of the late Bob Behringer

A general theory is developed for the evolution of the cell order (CO) distribution in planar granular systems. Dynamic equations are constructed and solved in closed form for several examples: systems under compression; dilation of very dense systems; and the general approach to steady state. We find that all the steady states are stable and that they satisfy detailed balance-like condition when the CO$\,\leq 6$. Illustrative numerical solutions of the evolution are shown. Our theoretical results are validated against an extensive simulation of a sheared system. The formalism can be readily extended to other structural characteristics, paving the way to a general theory of structural organisation of granular systems., Comment: main text: 5 pages, 5 figures; supplementary material: 4 pages, 4 figures

Solid-oxide fuel cells produce electric current from energy released by a spontaneous electrochemical reaction. The efficiency of these devices depends crucially on the microstructure of their electrodes and, in particular, on the three-phase boundary (TPB) length, along which the energy-producing reaction occurs. We present a systematic maximisation of the TPB length as a function of four readily-controllable microstructural parameters, for any given mean hydraulic radius, which is a conventional measure of the permeability to gas flow. We identify the maximising parameters and show that the TPB length can be increased by a factor of over 300% compared to current common practices. We support this result by calculating the TPB of several numerically simulated structures. We also compare four models for a single intergranular contact in the sintered electrode and show that the model commonly used in the literature is oversimplified and unphysical. We then propose two alternatives., Comment: 11 pages, 9 figures

We identify two orthogonal sources of structural entropy in rattler-free granular systems - affine, involving structural changes that only deform the contact network, and topological, corresponding to different topologies of the contact network. We show that a recently developed connectivity-based granular statistical mechanics separates the two naturally by identifying the structural degrees of freedom with spanning trees on the graph of the contact network. We extend the connectivity-based formalism to include constraints on, and correlations between, degrees of freedom as interactions between branches of the spanning tree. We then use the statistical mechanics formalism to calculate the partition function generally and the different entropies in the high-angoricity limit. We also calculate the degeneracy of the affine entropy and a number of expectation values. From the latter, we derive an equipartition principle and an equation of state relating the macroscopic volume and boundary stress to the analogue of the temperature, the contactivity., Comment: 14 pages, 9 figures

We use a DEM simulation and analytical considerations to study the dynamics of a self-energised object, modelled as a disc, oscillating horizontally within a two-dimesional bed of denser and smaller particles. We find that, for given material parameters, the immersed object (IO) may rise, sink or not change depth, depending on the oscillation amplitude and frequency, as well as on the initial depth. With time, the IO settles at a specific depth that depends on the oscillation parameters. We construct a phase diagram of this behaviour in the oscillation frequency and velocity amplitude variable space. We explain the observed rich behaviour by two competing effects: climbing on particles, which fill voids opening under the disc, and sinking due to bed fluidisation. We present a cavity model that allows us to derive analytically general results, which agree very well with the observations and explain quantitatively the phase diagram. Our specific analytical results are the following. (i) Derivation of a critical frequency, $f_c$, above which the IO cannot float up against gravity. We show that this frequency depends only on the gravitational acceleration and the IO size. (ii) Derivation of a minimal amplitude, $A_{min}$, below which the IO cannot rise even if the frequency is below $f_c$. We show that this amplitude also depends only on the gravitational acceleration and the IO size. (iii) Derivation of a critical value, $g_c$, of the IO's acceleration amplitude, below which the IO cannot sink. We show that the value of $g_c$ depends on the characteristics of both the IO and the granular bed,as well as on the initial IO's depth., Comment: 7 pages, 10 figures (some with subfigures), to appear in Phys. Rev. E

We derive the macroscopic stress tensor for plug-free dense granular flow, using a first-principles coarse-graining of the inter-granular forces. The derivation is based on the assumption, which defines the da Vinci Fluid model, that the inter-granular interactions are dominated by normal contact forces and solid friction. An explicit form for the stress -- strain rate relation is obtained, providing closure for the flow equations of dense granular fluids and improves on existing models, which are also based on solid friction as the dominant drag mechanism.

We analyse the general solutions for the stress field in planar annuli of isostatic media, a model often used for marginally rigid granular materials in Couette cells. We demonstrate that these solutions are much richer than in rectangular symmetries. Even for uniform media, stress chains are found to curve, broaden away from the stress source, attenuate and leak stress into a cone of influence. Most spectacularly, stress chains may bend back and transmit forces oppositely to the original direction. None of these phenomena arises in solutions for uniform media in Cartesian coordinates. We further analyse non-uniform media, which exhibit chain branching and stress leakage from the chains. These results are directly relevant to the many experiments on granular materials, carried out in Couette cells. They also shed light on, and are supported by, hitherto unexplained experimental observations of curved and back-bending chains, which we point out. In particular, we use our results to provide a new interpretation for the pattern of slip lines observed experimentally., Comment: 12 pages, 6 figures (12, including all sub-figures), submitted to Granular Matter

A new first-principles statistical mechanics formulation is proposed to describe slow and dilated granular fluids, where prolonged intergranular contacts vitiate collision theory. The contacts, where all the important physics takes place, are regarded as quasi-particles that can appear and disappear. A contact potential, $\nu$, is defined as a measure of the fluctuations and the mean coordination number per particle and its fluctuations are calculated as a function of it. This formulation extends the Edwards statistical mechanics to slow dynamic systems and converges to it when the motion stops. The theory is applied to a model system of a simply sheared granular material in the limit of small confining stress. The dependence of the contact potential on the shear rate is derived, making it possible to calibrate $\nu$ experimentally and predict the coordination number distribution as a function of the shear rate. Setting next $\nu=0$ as the jamming point, where the critical mean coordination number is $z_c$, a finite value is found for the shear rate there, $\dot{\gamma}_c$. A simple mean field theory then yields the scaling of the mean coordination number and its fluctuations near $\dot{\gamma}_c$. Existing results in the literature appear to support the predictions., Comment: 5 pages, 6 figures, submitted to Phys. Rev. Lett

We first show that the currently accepted statistical mechanics for granular matter is flawed. The reason is that it is based on the volume function, which depends only on a minute fraction of all the structural degrees of freedom and is unaffected by most of the configurational microstates. Consequently, the commonly used partition function underestimates the entropy severely. We then propose a new formulation, replacing the volume function with a ${\it connectivity}$ function that depends on all the structural degrees of freedom and accounts correctly for the entire entropy. We discuss the advantages of the new formalism and derive explicit results for two- and three-dimensional systems. We test the formalism by calculating the entropy of an experimental two-dimensional system, as a function of system size, and showing that it is an extensive variable., Comment: 5 pages, 3 figures

The continuous-time random walk (CTRW) model is useful for alleviating the computational burden of simulating diffusion in actual media. In principle, isotropic CTRW only requires knowledge of the step-size, $P_l$, and waiting-time, $P_t$, distributions of the random walk in the medium and it then generates presumably equivalent walks in free space, which are much faster. Here we test the usefulness of CTRW to modelling diffusion of finite-size particles in porous medium generated by loose granular packs. This is done by first simulating the diffusion process in a model porous medium of mean coordination number, which corresponds to marginal rigidity (the loosest possible structure), computing the resulting distributions $P_l$ and $P_t$ as functions of the particle size, and then using these as input for a free space CTRW. The CTRW walks are then compared to the ones simulated in the actual media. In particular, we study the normal-to-anomalous transition of the diffusion as a function of increasing particle size. We find that, given the same $P_l$ and $P_t$ for the simulation and the CTRW, the latter predicts incorrectly the size at which the transition occurs. We show that the discrepancy is related to the dependence of the effective connectivity of the porous media on the diffusing particle size, which is not captured simply by these distributions. We propose a correcting modification to the CTRW model -- adding anisotropy -- and show that it yields good agreement with the simulated diffusion process. We also present a method to obtain $P_l$ and $P_t$ directly from the porous sample, without having to simulate an actual diffusion process. This extends the use of CTRW, with all its advantages, to modelling diffusion processes of finite-size particles in such confined geometries., Comment: 9 pages, 7 figures

A first-principles theory is developed for the general evolution of a key structural characteristic of planar granular systems - the cell order distribution. The dynamic equations are constructed and solved in closed form for a number of examples: dense systems undergoing progressive compaction; initial dilation of very dense systems; and the approach to steady state of general systems. It is shown that the convergence to steady state is exponential, except when contacts are only broken and no new contacts are made, in which case the approach is algebraic in time. Where no closed form solutions are possible, illustrative numerical solutions of the evolution are shown. These show that the dynamics are sensitive to the cell event rates, which are process dependent. The formalism can be extended to other structural characteristics, paving the way to a general theory of structural organisation of granular systems, parameterised by the contact event rates., Comment: 4 pages, 3 figures, of which two with subfigures, submitted to Phys. Rev. Lett, minor editing in version 2

We present a first-principles theory for plug-free dense granular flow. This is done by coarse-graining directly the microscopic dynamics and deriving an explicit relation between the macroscopic stress and strain rate tensors. The newly derived relation not only differs significantly from that of the existing empirical models for such flows but it also provides a novel understanding of the effect of rigid-like rotational regions in the flow., Comment: 5 pages, 1 figure, submitted to Physical Review Letters

Dependence of structural self-organization of granular materials on preparation and grain parameters is key to predictive modelling. We study 60 different mechanically equilibrated polydisperse disc packs, generated numerically by two protocols. We show that, for same-variance disc size distributions (DSDs): 1. the mean coordination number of rattler-free packs vs. the packing fraction is a function independent of initial conditions, friction and the DSD: 2. all quadron volume and cell order distributions collapse to universal forms, also independent of the above. We conclude that, contrary to common wisdom, equilibrated granular structures are determined mainly by the packing protocol and higher moments of the DSD., Comment: 4 pages, 7 figures

We identify the fundamental factors determining the microstructural (MS) statistics of granular systems, using numerical experiments on 2D assemblies of polydisperse frictional discs and studying the emergent properties of quadrons (Qs), the basic structural elements of granular solids, whose statistics are universal-like[1]. The dependence of the structures and of the packing fraction on friction and initial state are analysed and we report the following. 1. Derivation of an analytical formula for the mean Q volume in terms of: the mean coordination number, the packing fraction and the rattlers fraction. 2. Derivation of a unique, initial-state-independent, relation between the mean coordination number and the rattler-free packing fraction. We support the relation with data for a range of different systems. 3. Collapse of the Q volume distributions of all systems onto one curve, having an exponential tail. 4. Decomposition of the Q volumes distribution into conditional distributions, each of these also collapsing onto a single curve. 5. The mean Q volume decreases with increasing inter-granular friction coefficients, an effect prominent in high order cells. We argue that this phenomenon is due to an increased probability of stable irregularly-shaped cells and test this by a herewith developed free cell analytical model. We conclude that, in principle, the MS characteristics are governed mainly by the packing procedure, while effects of friction and initial states can be scaled away. Yet, mechanical constraints limit slightly occurrence of small Q volumes in large cells, which does depend on friction. We quantify this deviation from exact collapse for these cells. 6. We argue that our results support the view that ensemble granular statistical mechanics does not satisfy the uniform measure assumption of conventional statistical mechanics., Comment: 12 pages, 21 figures, submitted to Physical Review E

We discuss the statistical mechanics of granular matter and derive several significant results. First, we show that, contrary to common belief, the volume and stress ensembles are inter-dependent, necessitating the use of both. We use the combined ensemble to calculate explicitly expectation values of structural and stress-related quantities for two-dimensional systems. We thence demonstrate that structural properties may depend on the angoricity tensor and that stress-based quantities may depend on the compactivity. This calls into question previous statistical mechanical analyses of static granular systems and related derivations of expectation values. Second, we establish the existence of an intriguing equipartition principle - the total volume is shared equally amongst both structural and stress-related degrees of freedom. Third, we derive an expression for the compactivity that makes it possible to quantify it from macroscopic measurements., Comment: 5 pages, including 2 figures, To appear in Phys. Rev. Lett

Auxetic materials, or negative-Poisson's-ratio materials, are important technologically and fascinating theoretically. When loaded by external stresses, their internal strains are governed by correlated motion of internal structural degrees of freedom. The modelling of such materials is mainly based on ordered structures, despite existence of auxetic behaviour in disordered structures and the advantage in manufacturing disordered structures for most applications. We describe here a first-principles expression for strains in disordered such materials, based on insight from a family of 'iso-auxetic' structures. These are structures, consisting of internal structural elements, which we name 'auxetons', whose inter-element forces can be computed from statics alone. Iso-auxetic structures make it possible not only to identify the mechanisms that give rise to auxeticity, but also to write down the explicit dependence of the strain rate on the local structure, which is valid to all auxetic materials. It is argued that stresses give rise to strains via two mechanisms: auxeton rotations and auxeton expansion / contraction. The former depends on the stress via a local fabric tensor, which we define explicitly for 2D systems. The latter depends on the stress via an expansion tensor. Whether a material exhibits auxetic behaviour or not depends on the interplay between these two fields. This description has two major advantages: it applies to any auxeton-based system, however disordered, and it goes beyond conventional elasticity theory, providing an explicit expression for general auxetic strains and outlining the relevant equations., Comment: 7 pages, 4 figures, Appeared in Journal of Superconductivity and Novel Magnetism Volume 25, 565-571 (2012)

A new, da Vinci, fluid is described as a model for flow of dense granular matter. We postulate local properties of the fluid, which are generically different from ordinary fluids in that energy is dissipated by solid friction. We present the equation of flow of such a fluid and show that it gives rise to formation and growth of plug flow regions, which is characteristic of flow of granular matter. Simple explicit examples are presented to illustrate the evolution of plug flow regions., Comment: 5 pages, 1 figure, submitted to Phys. Rev. Lett., Corrected reference

We introduce and study a da Vinci Fluid, a fluid whose dissipation is dominated by solid friction. We analyse the flow rheology of a discrete model and then coarse-grain it to the continuum. We find that the model gives rise to behaviour that is characteristic of dense granular fluids. In particular, it leads to plug flow. We analyse the nucleation mechanism of plugs and their development. We find that plug boundaries generically expand and we calculate the growth rate of plug regions. In systems whose internal effective friction coefficient is relatively uniform we find that the linear size of plug regions grows as (time)$^{1/3}$. The suitability of the model to granular materials is discussed., Comment: 5 pages, 3 figures, edited for clarifications and added references

A method is proposed for the characterisation of the entropy of cellular structures, based on the compactivity concept for granular packings. Hamiltonian-like volume functions are constructed both in two and in three dimensions, enabling the identification of a phase space and making it possible to take account of geometrical correlations systematically. Case studies are presented for which explicit calculations of the mean vertex density and porosity fluctuations are given as functions of compactivity. The formalism applies equally well to two- and three-dimensional granular assemblies., Comment: 14 pages, 4 figures, to appear in The European Physical Journal E - Soft Matter

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., Comment: 6 pages, 3 figures included, to appear in "IMA Volume in Mathematics and its Applications"

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 to 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., Comment: 4 pages, 2 figures, Corrected typos and clkearer text, submitted to Phys. Rev. Lett

A recent theory for stress transmission in isostatic granular and cellular systems predicts a constitutive equation that couples the stress field to the local microstructure. The theory could not be applied to macroscopic systems because the constitutive equation becomes trivial upon straightforward coarse-graining. This problem is resolved here for arbitrary planar structures. The solution is based on the observation that staggered order makes it possible to couple the stress to a reduced geometric tensor that can be coarse-grained. The method proposed here makes it possible to apply this idea to realistic systems whose staggered order is generally 'frustrated'. This is achieved by a renormalization procedure which removes the frustration and enables the use of the upscalable reduced tensor. As an example we calculate the stress due to a defect in a periodic solid foam.

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., Comment: To appear in Physical Review Letters

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., Comment: 8 pages, figures included. To appear in J. Phys. A: Math. Gen

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., Comment: 7 pages, with 4 figures

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., Comment: 4 pages, 2 figures included

We propose a minimal nonlinear model of brittle crack propagation by considering only the motion of the crack-tip atom. The model captures many essential features of steady-state crack velocity and is in excellent quantitative agreement with many-body dynamical simulations. The model exhibits lattice-trapping. For loads just above this, the crack velocity rises sharply, reaching a limiting value well below that predicted by elastic continuum theory. We trace the origin of the low limiting velocity to the anharmonicity of the potential well experienced by the crack-tip atom., Comment: To appear in Physical Review Letters, 9 pages, self-contained TeX file, email to rafi@xi.com or blh@lanl.gov for two postscript figures

A minimal model is constructed for two-dimensional fracture propagation. The heterogeneous process zone is presumed to suppress stress relaxation rate, leading to non-quasistatic behavior. Using the Yoffe solution, I construct and solve a dynamical equation for the tip stress. I discuss a generic tip velocity response to local stress and find that noise-free propagation is either at steady state or oscillatory, depending only on one material parameter. Noise gives rise to intermittency and quasi-periodicity. The theory explains the velocity oscillations and the complicated behavior seen in polymeric and amorphous brittle materials. I suggest experimental verifications and new connections between velocity measurements and material properties., Comment: To appear in Phys. Rev. Lett., 6 pages, self-contained TeX file, 3 postscript figures upon request from author at rafi@cnls.lanl.gov or rafi@xi.com, http://cnls-www.lanl.gov/homepages/rafi/rafindex.html

A first-principles statistical theory is constructed for the evolution of two dimensional interfaces in Laplacian fields. The aim is to predict the pattern that the growth evolves into, whether it becomes fractal and if so the characteristics of the fractal pattern. Using a time dependent map the growing region is conformally mapped onto the unit disk and the problem is converted to the dynamics of a many-body system. The evolution is argued to be Hamiltonian, and the Hamiltonian is shown to be the conjugate function of the real potential field. Without surface effects the problem is ill-posed, but the Hamiltonian structure of the dynamics allows introduction of surface effects as a repulsive potential between the particles and the interface. This further leads to a field representation of the problem, where the field's vacuum harbours the zeros and the poles of the conformal map as particles and antiparticles. These can be excited from the vacuum either by fluctuations or by surface effects. Creation and annihilation of particles is shown to be consistent with the formalism and lead to tip-splitting and side-branching. The Hamiltonian further allows to make use of statistical mechanical tools to analyse the statistics of the many-body system. I outline the way to convert the distribution of the particles into the morphology of the interface. In particular, I relate the particles statistics to the distributions of the curvature and the growth probability along the physical interface and to the fractal dimension. If the pattern turns fractal the latter distribution gives rise to a multifractal spectrum, which can be explicitly calculated for a given particles distribution. A `dilute boundary layer approximation' is discussed, which allows explicit calculations and shows emergence of an algebraically long tail, Comment: 14 pages in plain TeX. CNLS Newsletter #112

A statistical theory of two-dimensional Laplacian growths is formulated from first-principles. First the area enclosed by the growing surface is mapped conformally to the interior of the unit circle, generating a set of dynamically evolving quasi-particles. Then it is shown that the evolution of a surface-tension-free growing surface is Hamiltonian. The Hamiltonian formulation allows a natural extension of the formalism to growths with either isotropic or anisotropic surface tension. It is shown that the curvature term can be included as a surface energy in the Hamiltonian that gives rise to repulsion between the quasi-particles and the surface. This repulsion prevents cusp singularities from forming along the surface at any finite time and regularizes the growth. An explicit example is computed to demonstrate the regularizing effect. Noise is then introduced as in traditional statistical mechanical formalism and a measure is defined that allows analysis of the spatial distribution of the quasi-particles. Finally, a relation is derived between this distribution and the growth probability along the growing surface. Since the spatial distribution of quasi-particles flows to a stable limiting form, this immediately translates into predictability of the asymptotic morphology of the surface. An exactly-solvable class of arbitrary initial conditions is analysed explicitly., Comment: 28 pages, plain TeX, 2 postscript figure (not included) available upon request. To appear in Physical Review E. Report LA-UR-93-4338. email: rafi@goshawk.lanl.gov

I show that the evolution of a two dimensional surface in a Laplacian field can be described by Hamiltonian dynamics. First the growing region is mapped conformally to the interior of the unit circle, creating in the process a set of mathematical zeros and poles that evolve dynamically as the surface grows. Then the dynamics of these quasi-particles is analysed. A class of arbitrary initial conditions is discussed explicitly, where the surface-tension-free Laplacian growth process is integrable. This formulation holds only as long as the singularities of the map are confined to within the unit circle. But the Hamiltonian structure further allows for surface tension to be introduced as an energetic term that effects repulsion between the quasi-particles and the surface. These results are used to formulate a first-principles statistical theory of pattern formation in stochastic growth, where noise is a key player., Comment: 17 pages, tex, 1 postscript figure not included (available upon request), (This version has less than 80 characters per line)