There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This review summarizes theoretical progress in the field of active matter, placing it in the context of recent experiments. This approach offers a unified framework for the mechanical and statistical properties of living matter: biofilaments and molecular motors in vitro or in vivo, collections of motile microorganisms, animal flocks, and chemical or mechanical imitations. A major goal of this review is to integrate several approaches proposed in the literature, from semimicroscopic to phenomenological. In particular, first considered are ``dry'' systems, defined as those where momentum is not conserved due to friction with a substrate or an embedding porous medium. The differences and similarities between two types of orientationally ordered states, the nematic and the polar, are clarified. Next, the active hydrodynamics of suspensions or ``wet'' systems is discussed and the relation with and difference from the dry case, as well as various large-scale instabilities of these nonequilibrium states of matter, are highlighted. Further highlighted are various large-scale instabilities of these nonequilibrium states of matter. Various semimicroscopic derivations of the continuum theory are discussed and connected, highlighting the unifying and generic nature of the continuum model. Throughout the review, the experimental relevance of these theories for describing bacterial swarms and suspensions, the cytoskeleton of living cells, and vibrated granular material is discussed. Promising extensions toward greater realism in specific contexts from cell biology to animal behavior are suggested, and remarks are given on some exotic active-matter analogs. Last, the outlook for a quantitative understanding of active matter, through the interplay of detailed theory with controlled experiments on simplified systems, with living or artificial constituents, is summarized.

Hydrodynamics of soft active matter

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Statistical physics of vehicular traffic and some related systems

Annual review of condensed matter physics

Microcrystalline particles of Fe2O3 having different sizes (varying between 70 and 5 nm) have been synthesised using a novel three-component micro-emulsion technique. A succession of crystal-size-induced structural transitions was observed. While alpha -Fe2O3 was found to nucleate for a particle size above 30 nm, gamma -Fe2O3 was preferentially formed for a size below 30 nm, whereas amorphous Fe2O3 was formed at a particle size of 5 nm. These structural transformations have been related to the increase in the unit-cell volume that occurs as the particle size is decreased. The size dependence of the lattice parameter is shown to arise from a coupling of the surface energy to the dilatational lattice mode. A model Hamiltonian which incorporates this interaction and displays size-induced phase transitions is defined. The Mossbauer hyperfine field in the microcrystalline samples at 4.2 K was found to be substantially smaller than in the 'bulk'. The hyperfine parameters of the amorphous sample were found to be similar to those pertaining to samples prepared by conventional techniques such as melt quenching. A large anisotropy in the ionic vibrational amplitudes was detected in samples with particles smaller than about 10 nm.

https://iopscience.iop.org/article/10.1088/0022-3719/21/11/014/pdf

Size-induced structural phase transitions and hyperfine properties of microcrystalline Fe2O3

We study the effect of quenched spatial disorder on the current-carrying steady states of driven stochastic systems of particles interacting through hard-core exclusion. Two sorts of models are studied: disordered drop-push processes and their generalizations, and the disordered asymmetric simple exclusion process. Quenched disorder enters through spatially random microscopic transition probabilities and the drive is modeled by asymmetry in transition probabilities between sites. Exact steady-state measures are obtained for the drop-push and the generalized drop-push dynamics in $d$ dimensions for arbitrary disorder. This allows us to compute closed form expressions for the steady-state current and site-dependent densities. The steady state of the asymmetric exclusion process with disordered bond strengths is studied in one dimension by numerical simulation and by a mean-field approximation that allows for density variations from site to site. In the totally asymmetric case, we present strong numerical evidence that the current is invariant under reflection. We show that disorder can induce phase separation into macroscopic regions of different densities. We propose approximations, supported by direct numerical simulations, to describe these phenomena and the phase diagram of the model in the current-density plane in terms of macroscopic parameters of the model. We also study the effect of making the direction of easy flow in each bond a random variable and find that the current decreases with system size in this case. We conclude that there are three distinct regimes in disordered driven diffusive systems in one dimension: a homogeneous regime in which the state of the system is characterized by a single macroscopic density and a nonzero current; a segregated-density regime, where the state of the system is characterized by two distinct phase-separated values of density and a nonzero current; a vanishing-current regime, where the state of the system is characterized by two distinct values of the density and the current decreases as the system size increases and vanishes in the thermodynamic limit. Using a mapping from lattice gases to interfaces, these regimes translate into distinct regimes of interface growth in the presence of columnar disorder.

Driven lattice gases with quenched disorder: Exact results and different macroscopic regimes

We study nonequilibrium phase transitions in a mass-aggregation model which allows for diffusion, aggregation on contact, dissociation, adsorption, and desorption of unit masses. We analyze two limits explicitly. In the first case mass is locally conserved, whereas in the second case local conservation is violated. In both cases the system undergoes a dynamical phase transition in all dimensions. In the first case, the steady state mass distribution decays exponentially for large mass in one phase, and develops an infinite aggregate in addition to a power-law mass decay in the other phase. In the second case, the transition is similar except that the infinite aggregate is missing.

/pdf/nonequilibrium-phase-transitions-in-models-of-aggregation-1rytyuz380.pdf

Nonequilibrium phase transitions in models of aggregation, adsorption, and dissociation

We study driven lattice gas systems with quenched spatial randomness: the disordered drop-push process, for which the steady state is shown to have inhomogeneous product measure form, and the disordered asymmetric exclusion process. We conjecture that time-dependent correlation functions which monitor dissipation of kinematic waves behave as in the pure system if the wave speed is nonzero, and support this with simulations. This speed vanishes close to half filling in the disordered exclusion process where macroscopic regions with different densities coexist.

Steady state and dynamics of driven diffusive systems with quenched disorder

The authors study the effects of a bias-producing external field on a random walk on the infinite cluster in the percolation problem. There are two competing physical effects (drift and trapping) which result in a drift velocity nu which rises and then falls as the field increases. They study these effects on a one-dimensional lattice with random-length branches, and on the diluted Bethe lattice. They calculate nu in the first model with a maximum allowed length for each branch. In the limit that this cutoff length becomes infinite, they find that the velocity vanishes identically above a finite critical value of the field. For the Bethe lattice, they derive an upper bound on the critical field, which varies as (p-pc)1/2 as the percolation concentration pc is approached. In the one-dimensional model, they also investigate the anomalous regime in which the velocity vanishes. They discuss the distribution of steady state times required to traverse N sites, and find that it can be described in terms of a stable distribution of index x with superimposed oscillations. The index x of the stable distribution is given by L/ xi where xi is a characteristic branch length and L is a bias-induced length which describes the exponential buildup of the steady state density of particles towards the end of a branch.

https://iopscience.iop.org/article/10.1088/0305-4470/17/15/017/pdf

Mustansir Barma

Papers

Size-induced structural phase transitions and hyperfine properties of microcrystalline Fe2O3

Driven lattice gases with quenched disorder: Exact results and different macroscopic regimes

Nonequilibrium phase transitions in models of aggregation, adsorption, and dissociation

Steady state and dynamics of driven diffusive systems with quenched disorder

Field-induced drift and trapping in percolation networks