Tricritical Behavior in Dynamical Phase Transitions.

We identify a new scenario for dynamical phase transitions associated with time-integrated observables occurring in diffusive systems described by the macroscopic fluctuation theory. It is characterized by the pairwise meeting of first- and second-order bias-induced phase transition curves at two tricritical points. We formulate a simple, general criterion for its appearance and derive an exact Landau theory for the tricritical behavior. The scenario is demonstrated in three examples: the simple symmetric exclusion process biased by an activity-related structural observable; the Katz-Lebowitz-Spohn lattice gas model biased by its current; and in an active lattice gas biased by its entropy production.

Two-Dimensional Crystals far from Equilibrium.

When driven by nonequilibrium fluctuations, particle systems may display phase transitions and physical behavior with no equilibrium counterpart. We study a two-dimensional particle model initially proposed to describe driven non-Brownian suspensions undergoing nonequilibrium absorbing phase transitions. We show that when the transition occurs at large density, the dynamics produces long-range crystalline order. In the ordered phase, long-range translational order is observed because equipartition of energy is lacking, phonons are suppressed, and density fluctuations are hyperuniform. Our study offers an explicit microscopic model where nonequilibrium violations of the Mermin-Wagner theorem stabilize crystalline order in two dimensions.

Orientation-Dependent Propulsion of Active Brownian Spheres: From Self-Advection to Programmable Cluster Shapes.

Applications of active particles require a method for controlling their dynamics. While this is typically achieved via direct interventions, indirect interventions based, e.g., on an orientation-dependent self-propulsion speed of the particles, become increasingly popular. In this Letter, we investigate systems of interacting active Brownian spheres in two spatial dimensions with orientation-dependent propulsion using analytical modeling and Brownian dynamics simulations. It is found that the orientation dependence leads to self-advection, circulating currents, and programmable cluster shapes.

Rheology of Dense Suspensions under Shear Rotation.

Dense non-Brownian suspensions exhibit a spectacular and abrupt drop in viscosity under change of shear direction, as revealed by shear inversions (reversals) or orthogonal superposition. Here, we introduce an experimental setup to systematically explore their response to shear rotations, where one suddenly rotates the principal axes of shear by an angle θ, and measure the shear stresses with a biaxial force sensor. Our measurements confirm the genericness of the transient decrease of the resistance to shear under unsteady conditions. Moreover, the orthogonal shear stress, which vanishes in steady state, takes non-negligible values with a rich θ dependence, changing qualitatively with solid volume fraction ϕ and resulting in a force that tends to reduce or enhance the direction of flow for small or large ϕ. These experimental findings are confirmed and rationalized by particle-based numerical simulations and a recently proposed constitutive model. We show that the rotation angle dependence of the orthogonal stress results from a ϕ-dependent interplay between hydrodynamic and contact stresses.

Getting the most out of maths: How to coordinate mathematical modelling research to support a pandemic, lessons learnt from three initiatives that were part of the COVID-19 response in the UK.

In March 2020 mathematics became a key part of the scientific advice to the UK government on the pandemic response to COVID-19. Mathematical and statistical modelling provided critical information on the spread of the virus and the potential impact of different interventions. The unprecedented scale of the challenge led the epidemiological modelling community in the UK to be pushed to its limits. At the same time, mathematical modellers across the country were keen to use their knowledge and skills to support the COVID-19 modelling effort. However, this sudden great interest in epidemiological modelling needed to be coordinated to provide much-needed support, and to limit the burden on epidemiological modellers already very stretched for time. In this paper we describe three initiatives set up in the UK in spring 2020 to coordinate the mathematical sciences research community in supporting mathematical modelling of COVID-19. Each initiative had different primary aims and worked to maximise synergies between the various projects. We reflect on the lessons learnt, highlighting the key roles of pre-existing research collaborations and focal centres of coordination in contributing to the success of these initiatives. We conclude with recommendations about important ways in which the scientific research community could be better prepared for future pandemics. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Active Ising Models of flocking: a field-theoretic approach.

Using an approach based on Doi-Peliti field theory, we study several different Active Ising Models (AIMs), in each of which collective motion (flocking) of self-propelled particles arises from the spontaneous breaking of a discrete symmetry. We test the predictive power of our field theories by deriving the hydrodynamic equations for the different microscopic choices of aligning processes that define our various models. At deterministic level, the resulting equations largely confirm known results, but our approach has the advantage of allowing systematic generalization to include noise terms. Study of the resulting hydrodynamics allows us to confirm that the various AIMs share the same phenomenology of a first-order transition from isotropic to flocked states whenever the self-propulsion speed is nonzero, with an important exception for the case where particles align only pairwise locally. Remarkably, this variant fails entirely to give flocking-an outcome that was foreseen in previous work, but is confirmed here and explained in terms of the scalings of various terms in the hydrodynamic limit. Finally, we discuss our AIMs in the limit of zero self-propulsion where the ordering transition is continuous. In this limit, each model is still out of equilibrium because the dynamical rules continue to break detailed balance, yet it has been argued that an equilibrium universality class (Model C) prevails. We study field-theoretically the connection between our AIMs and Model C, arguing that these particular models (though not AIMs in general) lie outside the Model C class. We link this to the fact that in our AIMs without self-propulsion, detailed balance is not merely still broken, but replaced by a different dynamical symmetry in which the dynamics of the particle density is independent of the spin state. .

Conching chocolate is a prototypical transition from frictionally jammed solid to flowable suspension with maximal solid content.

The mixing of a powder of 10- to 50-µm primary particles into a liquid to form a dispersion with the highest possible solid content is a common industrial operation. Building on recent advances in the rheology of such "granular dispersions," we study a paradigmatic example of such powder incorporation: the conching of chocolate, in which a homogeneous, flowing suspension is prepared from an inhomogeneous mixture of particulates, triglyceride oil, and dispersants. Studying the rheology of a simplified formulation, we find that the input of mechanical energy and staged addition of surfactants combine to effect a considerable shift in the jamming volume fraction of the system, thus increasing the maximum flowable solid content. We discuss the possible microscopic origins of this shift, and suggest that chocolate conching exemplifies a ubiquitous class of powder-liquid mixing.

Stochastic Hydrodynamics of Complex Fluids: Discretisation and Entropy Production.

Many complex fluids can be described by continuum hydrodynamic field equations, to which noise must be added in order to capture thermal fluctuations. In almost all cases, the resulting coarse-grained stochastic partial differential equations carry a short-scale cutoff, which is also reflected in numerical discretisation schemes. We draw together our recent findings concerning the construction of such schemes and the interpretation of their continuum limits, focusing, for simplicity, on models with a purely diffusive scalar field, such as 'Model B' which describes phase separation in binary fluid mixtures. We address the requirement that the steady-state entropy production rate (EPR) must vanish for any stochastic hydrodynamic model in a thermal equilibrium. Only if this is achieved can the given discretisation scheme be relied upon to correctly calculate the nonvanishing EPR for 'active field theories' in which new terms are deliberately added to the fluctuating hydrodynamic equations that break detailed balance. To compute the correct probabilities of forward and time-reversed paths (whose ratio determines the EPR), we must make a careful treatment of so-called 'spurious drift' and other closely related terms that depend on the discretisation scheme. We show that such subtleties can arise not only in the temporal discretisation (as is well documented for stochastic ODEs with multiplicative noise) but also from spatial discretisation, even when noise is additive, as most active field theories assume. We then review how such noise can become multiplicative via off-diagonal couplings to additional fields that thermodynamically encode the underlying chemical processes responsible for activity. In this case, the spurious drift terms need careful accounting, not just to evaluate correctly the EPR but also to numerically implement the Langevin dynamics itself.

Hierarchical microphase separation in non-conserved active mixtures.

Non-equilibrium phase separating systems with reactions, such as biomolecular condensates and bacteria colonies, can break time-reversal symmetry (TRS) in two distinct ways. Firstly, the conservative and non-conservative sectors of the dynamics can be governed by incompatible free energies; when both sectors are present, this is the leading-order TRS violation, captured in its simplest form by 'Model AB'. Second, the diffusive dynamics can break TRS in its own right. This happens only at higher order in the gradient expansion (but is the leading behaviour without reactions present) and is captured by 'Active Model B+' (AMB+). Each of the two mechanisms can lead to microphase separation, by quite different routes. Here we introduce Model AB+, for which both mechanisms are simultaneously present, and show that for slow reaction rates the system can undergo a new type of hierarchical microphase separation, which we call 'bubbly microphase separation'. In this state, small droplets of one fluid are continuously created and absorbed into large droplets, whose length-scales are controlled by the competing reactive and diffusive dynamics.

Stealth Entropy Production in Active Field Theories near Ising Critical Points.

We address the steady-state entropy production rate (EPR) of active scalar ϕ^{4} theories, which lack time-reversal symmetry, close to a phase-separation critical point. We consider both nonconserved (model A) and conserved (model B) dynamics at Gaussian level, and also address the former at leading order in ε=4-d. In each case, activity is irrelevant in the RG sense: the active model lies in the same (dynamic Ising) universality class as its time-reversible counterpart. Hence one might expect that activity brings no new critical behavior. Here we show instead that, on approach to criticality in these models, the singular part of the EPR per (diverging) spacetime correlation volume either remains finite or itself diverges. A nontrivial critical scaling for entropy production thus ranks among universal dynamic Ising-class properties.

Absorbing-State Transitions in Granular Materials Close to Jamming.

We consider a model for driven particulate matter in which absorbing states can be reached both by particle isolation and by particle caging. The model predicts a nonequilibrium phase diagram in which analogs of hydrodynamic and elastic reversibility emerge at low and high volume fractions respectively, partially separated by a diffusive, nonabsorbing region. We thus find a single phase boundary that spans the onset of chaos in sheared suspensions to the onset of yielding in jammed packings. This boundary has the properties of a nonequilibrium second order phase transition, leading us to write a Manna-like mean field description that captures the model predictions. Dependent on contact details, jamming marks either a direct transition between the two absorbing states, or occurs within the diffusive region.

Contractile and chiral activities codetermine the helicity of swimming droplet trajectories.

Active fluids are a class of nonequilibrium systems where energy is injected into the system continuously by the constituent particles themselves. Many examples, such as bacterial suspensions and actomyosin networks, are intrinsically chiral at a local scale, so that their activity involves torque dipoles alongside the force dipoles usually considered. Although many aspects of active fluids have been studied, the effects of chirality on them are much less known. Here, we study by computer simulation the dynamics of an unstructured droplet of chiral active fluid in three dimensions. Our model considers only the simplest possible combination of chiral and achiral active stresses, yet this leads to an unprecedented range of complex motilities, including oscillatory swimming, helical swimming, and run-and-tumble motion. Strikingly, whereas the chirality of helical swimming is the same as the microscopic chirality of torque dipoles in one regime, the two are opposite in another. Some of the features of these motility modes resemble those of some single-celled protozoa, suggesting that underlying mechanisms may be shared by some biological systems and synthetic active droplets.

Shear-Induced First-Order Transition in Polar Liquid Crystals.

The hydrodynamic theory of polar liquid crystals is widely used to describe biological active fluids as well as passive molecular materials. Depending on the "shear-alignment parameter", in passive or weakly active polar fluids under external shear, the polar order parameter p is either inclined to the flow at a fixed (Leslie) angle, or rotates continuously. Here, we study the role of an additional "shear-elongation parameter" that has been neglected in the recent literature and causes |p| to change under flow. We show that this effect can give rise to a shear-induced first-order phase transition from isotropic to polar, and significantly change the rheological properties of both active and passive polar fluids.

Tunable shear thickening in suspensions.

Shear thickening, an increase of viscosity with shear rate, is a ubiquitous phenomenon in suspended materials that has implications for broad technological applications. Controlling this thickening behavior remains a major challenge and has led to empirical strategies ranging from altering the particle surfaces and shape to modifying the solvent properties. However, none of these methods allows for tuning of flow properties during shear itself. Here, we demonstrate that by strategic imposition of a high-frequency and low-amplitude shear perturbation orthogonal to the primary shearing flow, we can largely eradicate shear thickening. The orthogonal shear effectively becomes a regulator for controlling thickening in the suspension, allowing the viscosity to be reduced by up to 2 decades on demand. In a separate setup, we show that such effects can be induced by simply agitating the sample transversely to the primary shear direction. Overall, the ability of in situ manipulation of shear thickening paves a route toward creating materials whose mechanical properties can be controlled.

Strong Coupling in Conserved Surface Roughening: A New Universality Class?

The Kardar-Parisi-Zhang (KPZ) equation defines the main universality class for nonlinear growth and roughening of surfaces. But under certain conditions, a conserved KPZ equation (CKPZ) is thought to set the universality class instead. This has non-mean-field behavior only in spatial dimension d<2. We point out here that CKPZ is incomplete: It omits a symmetry-allowed nonlinear gradient term of the same order as the one retained. Adding this term, we find a parameter regime where the one-loop renormalization group flow diverges. This suggests a phase transition to a new growth phase, possibly ruled by a strong-coupling fixed point and thus described by a new universality class, for any d>1. In this phase, numerical integration of the model in d=2 gives clear evidence of non-mean-field behavior.

Does a Growing Static Length Scale Control the Glass Transition?

Several theories of the glass transition propose that the structural relaxation time τ_{α} is controlled by a growing static length scale ξ that is determined by the free energy landscape but not by the local dynamic rules governing its exploration. We argue, based on recent simulations using particle-radius-swap dynamics, that only a modest factor in the increase in τ_{α} on approach to the glass transition may stem from the growth of a static length, with a vastly larger contribution attributable, instead, to a slowdown of local dynamics. This reinforces arguments that we base on the observed strong coupling of particle diffusion and density fluctuations in real glasses.

Phoretic Interactions Generically Induce Dynamic Clusters and Wave Patterns in Active Colloids.

We introduce a representative minimal model for phoretically interacting active colloids. Combining kinetic theory, linear stability analyses, and a general relation between self-propulsion and phoretic interactions in autodiffusiophoretic and autothermophoretic Janus colloids collapses the parameter space from six to two dimensionless parameters: area fraction and Péclet number. This collapse arises when the lifetime of the self-generated phoretic fields is not too short, and leads to a universal phase diagram showing that phoretic interactions generically induce pattern formation in typical Janus colloids, even at very low density. The resulting patterns include waves and dynamic aggregates closely resembling the living clusters found in experiments on dilute suspension of Janus colloids.

Spatiotemporal Self-Organization of Fluctuating Bacterial Colonies.

We model an enclosed system of bacteria, whose motility-induced phase separation is coupled to slow population dynamics. Without noise, the system shows both static phase separation and a limit cycle, in which a rising global population causes a dense bacterial colony to form, which then declines by local cell death, before dispersing to reinitiate the cycle. Adding fluctuations, we find that static colonies are now metastable, moving between spatial locations via rare and strongly nonequilibrium pathways, whereas the limit cycle becomes almost periodic such that after each redispersion event the next colony forms in a random location. These results, which hint at some aspects of the biofilm-planktonic life cycle, can be explained by combining tools from large deviation theory with a bifurcation analysis in which the global population density plays the role of control parameter.

Avalanches mediate crystallization in a hard-sphere glass.

By molecular-dynamics simulations, we have studied the devitrification (or crystallization) of aged hard-sphere glasses. First, we find that the dynamics of the particles are intermittent: Quiescent periods, when the particles simply "rattle" in their nearest-neighbor cages, are interrupted by abrupt "avalanches," where a subset of particles undergo large rearrangements. Second, we find that crystallization is associated with these avalanches but that the connection is not straightforward. The amount of crystal in the system increases during an avalanche, but most of the particles that become crystalline are different from those involved in the avalanche. Third, the occurrence of the avalanches is a largely stochastic process. Randomizing the velocities of the particles at any time during the simulation leads to a different subsequent series of avalanches. The spatial distribution of avalanching particles appears random, although correlations are found among avalanche initiation events. By contrast, we find that crystallization tends to take place in regions that already show incipient local order.

How Far from Equilibrium Is Active Matter?

Active matter systems are driven out of thermal equilibrium by a lack of generalized Stokes-Einstein relation between injection and dissipation of energy at the microscopic scale. We consider such a system of interacting particles, propelled by persistent noises, and show that, at small but finite persistence time, their dynamics still satisfy a time-reversal symmetry. To do so, we compute perturbatively their steady-state measure and show that, for short persistent times, the entropy production rate vanishes. This endows such systems with an effective fluctuation-dissipation theorem akin to that of thermal equilibrium systems. Last, we show how interacting particle systems with viscous drags and correlated noises can be seen as in equilibrium with a viscoelastic bath but driven out of equilibrium by nonconservative forces, hence providing energetic insight into the departure of active systems from equilibrium.