arXiv subjects

Raphael Blumenfeld

Publications and source records attributed to Raphael Blumenfeld.

At least 19 recordsLinked to original sources

arXiv2025

Random close packing fraction of bidisperse discs: theoretical derivation

Predicting theoretically the highest density, which a disordered packing of discs can achieve, has been a long-standing unresolved problem. Such predictions are hindered by two difficulties - the dependence of the density on the packing procedure and ensuring disorder. A theory that overcomes these difficulties has been developed recently for mono-disperse disc packing~\cite{Bl21}. However, to minimise order, experiments and numerical simulations often use two-size discs and a prediction of the highest possible packing fraction, $\phi_{RCP}$, for these packings is arguably more useful. This problem is more complex because in such packings, $\phi_{RCP}$ is not a number but a function of the sizes ratio, $D$, and concentrations, $p$, of the disc types. A disorder-guaranteeing theory is formulated here to derive $\phi_{RCP}(p,D)$ under some assumptions, using the concept of the cell order distribution. Exact upper and lower bounds on the densest disordered packing fraction are also derived.

cond-mat.soft↗

arXiv2025

Self-organisation -- the underlying principle and a general formalism

It is proposed that self-organisation (SO) in non-equilibrium systems is governed by a general principle: it emerges when a minute subset of system configurations are exceptionally stable and long-lived to survive the noise generated by the driving and environmental constraints. Guided by this principle, a statistical mechanics-like model is formulated for general SO and its application is illustrated for two example systems: self-organised steady states of quasi-statically driven granular systems in two dimensions and crowd laning, for which illustrative explicit results are derived. In this formalism, maximising a survivability function of the exceptionally few stable configurations is the equivalent of minimising the free energy in traditional statistical mechanics. Parallels with equilibrium statistical mechanics and differences from it are discussed, which provides useful insight to assist in modelling SO in general out-of-equilibrium systems. Similarities and differences between SO in passive and biological systems are also pointed out, suggesting potential extension of this approach in this direction, albeit to very simple systems.

cond-mat.stat-mech↗

arXiv2025

Self-organization, detailed balance, and stress-structure correlations in 2D granular dynamics

We argue that a number of recent experimental and numerical observations point to an ongoing cooperative stress-structure self-organisation (SO) in quasi-static granular dynamics. These observations include: a) detail-insensitive collapses of certain quantities; b) correlations between stress and structure and evidence of entropy-stability competition in settled packings, which cast doubt on most linear stress theories of granular materials; c) detailed balanced steady states, which seem contradictory to the common belief that only systems in thermal equilibrium satisfy detailed balance, but are not, as we explain. We then propose a new statistical mechanical formulation that takes into account the cooperative SO.

cond-mat.stat-mech↗

arXiv2025

Effects of particle angularity on granular self-organization

Recent studies of two-dimensional poly-disperse disc systems revealed a coordinated self-organisation of cell stresses and shapes, with certain distributions collapsing onto a master form for many processes, size distributions, friction coefficients, and cell orders. Here we examine the effects of grain angularity on the indicators of self-organisation, using simulations of bi-disperse regular $N$-polygons and varying $N$ systematically. We find that: the strong correlation between local cell stresses and orientations, as well as the collapses of the conditional distributions of scaled cell stress ratios to a master Weibull form for all cell orders $k$, are independent of angularity and friction coefficient. In contrast, increasing angularity makes the collapses of the conditional distributions sensitive to changes in the friction coefficient.

cond-mat.soft↗

arXiv2024

Detailed balance in non-equilibrium dynamics of granular matter: derivation and implications

Modelling the dynamics of dense granular media is a long standing challenge and essential to many natural phenomena and technological applications. Here, we trace back puzzling experimental observation of detailed-balanced steady states to self-organisation of the neighbour probability distribution. The emergence of detailed balance in non-equilibrium granular dynamics could constitute a major step toward better models of granular media, as well as provide more insight into non-equilibrium processes in general. We show analytically that DBSS emerges when a certain neighbour probability is uniform across the system. This condition leads to a conditional cell order distribution being independent of the condition. We then carry out rotational shear experiments, in which this condition is satisfied, and show that they give rise to robust detailed-balanced steady states. We also show that, when the unconditional cell order distribution maximises the entropy, it is determined by a single constant parameter that is characteristic of all cell transitions. These results illustrate the predictive power of recently proposed evolution equations, which pave the way to simpler models of the dynamics of planar granular systems.

cond-mat.soft↗

arXiv2024

Force chains bias the dynamic response to impacts in rubble-pile asteroids

The impact response of rubble-pile asteroids is essential for both elucidating their formation and evolution history and evaluating the efficacy of impact defense strategies. Although state-of-the-art numerical simulations have allowed for the replication of many macroscopic impact characteristics consistent with observations, the understanding of dynamics and response mechanisms within rubble-pile structures remains incomplete and requires further in-depth investigation. Such understanding is critical for assessing the effects and safety of impact defense missions. The loose structure of rubble-pile asteroids affects inhomogeneous internal stress propagation via inherent force chains, which may lead to structural fracturing. We demonstrate this phenomenon here, using a proof-of-principle two-dimensional model of granular aggregates. We find that the velocity response front to impact disturbances preferentially propagates along pre-existing force chains, with particles not in chains responding more slowly. The sites within the response zone where high dynamic stresses manifest are strongly correlated with these initial force chains, and the damages that result are predominantly located within areas enclosed by these chains. The strong correlation between pre-existing force chains and dynamic response is independent of the location, magnitude, direction of the disturbance velocity, or the aggregate's particle size distribution. All evidence suggests that the core reasons for this propagation preference lie in the structural heterogeneity of granular aggregates and the resulting differences in mechanical wave propagation. This investigation provides guidance for future research aimed at quantitatively assessing fragmentation risks based on the statistical properties of force chains.

astro-ph.EP↗

arXiv2023

Granular solids transmit stress as two-phase composites

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.

cond-mat.soft↗

arXiv2023

Steady states of two-dimensional granular systems are unique, stable, and sometimes satisfy detailed balance

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.

cond-mat.soft↗

arXiv2022

Experimental evidence of detailed balance in granular systems

The principle of detailed balance (DB) states that every kinetic transition in a system with many micro-states, $μ$, is balanced, on average, with the opposite transition, $μ_i\leftrightharpoonsμ_j$. 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.

cond-mat.soft↗

arXiv2022

Coordinated Stress-Structure Self-Organization in Granular Packing

During quasi-static dynamics of granular systems, the stress and structure self-organise, but there is currently no quantitative measure or understanding of this phenomenon. 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 results. 1. Cells orient along the local stress major principal axes. 2. The mean ratio of cell principal stresses decreases with cell order and increases with friction. 3. The ratio distributions collapse onto a single curve under a simple scaling, for all packing protocols and friction coefficients. 4. A constructed model explains the correlations between the local cell and stress principal axis orientations. 5. The collapse of the stress ratios onto a Weibull distribution is explained theoretically. 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.

cond-mat.soft↗

arXiv2022

Comment on "Explicit Analytical Solution for Random Close Packing in d=2 and d=3", Physical Review Letters {\bf 128}, 028002 (2022)

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.

cond-mat.dis-nn↗

arXiv2021

Sub-anomalous diffusion and unusual velocity distribution evolution in cooling granular gases: theory

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.

cond-mat.stat-mech↗

arXiv2021

The disc random packing problem: a disorder criterion and an explicit solution

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: $ϕ_{RCP}=0.852525...$. 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.

cond-mat.stat-mech↗

arXiv2021

Collision-driven emergence of the cosmic web

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.

astro-ph.CO↗

arXiv2020

Friction-controlled entropy-stability competition in granular systems

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.

cond-mat.soft↗

arXiv2020

Sink-rise dynamics of horizontally oscillating active matter in granular media: Theory

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.

cond-mat.soft↗

arXiv2020

The unusual problem of upscaling isostaticity theory for granular matter

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.

cond-mat.soft↗

arXiv2019

Structural Evolution of Granular Systems: Theory

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.

cond-mat.soft↗