arXiv subjects

Michael E. Cates

Publications and source records attributed to Michael E. Cates.

At least 37 records · Page 2Linked to original sources

arXiv2025

Entropy production rate in thermodynamically consistent flocks

We study the entropy production rate (EPR) of aligning self-propelled particles which undergo a flocking transition towards a polarized collective motion. In our thermodynamically consistent lattice model, individual self-propulsion is the exclusive source of irreversibility. We derive the fluctuating hydrodynamics for large system sizes using a controlled coarse-graining: our procedure entails an exact correspondence between the EPR evaluated at the hydrodynamic and particle-based levels. We reveal that EPR is maximal when the system adopts a homogeneous configuration, either apolar or polar, and reduced in the non-homogeneous state where a polar band travels in a apolar background due to strong spatial EPR modulations. By analyzing the latter we also show that asymmetric energetic exchanges occur at the trailing and leading edges, which we map into a thermodynamic cycle in density-polarization space. Finally, we demonstrate that the regime of weak self-propulsion features a singular scaling of EPR, and a non-analyticity of the travelling band profiles.

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arXiv2025

Geometric theory of (extended) time-reversal symmetries in stochastic processes -- Part II: field theory

In this article, we study the time-reversal properties of a generic Markovian stochastic field dynamics with Gaussian noise. We introduce a convenient functional geometric formalism that allows us to straightforwardly generalize known results from finite dimensional systems to the case of continuous fields. We give, at field level, full reversibility conditions for three notions of time-reversal defined in the first article of this two-part series, namely T-, MT-, and EMT-reversibility. When the noise correlator is invertible, these reversibility conditions do not make reference to any generically unknown function like the stationary probability, and can thus be verified systematically. Focusing on the simplest of these notions, where only the time variable is flipped upon time reversal, we show that time-reversal symmetry breaking is quantified by a single geometric object: the vorticity two-form, which is a two-form over the functional space $\mathbb{F}$ to which the field belongs. Reversibility then amounts to the cancellation of this vorticity two-form. This condition applies at distributional level and can thus be difficult to use in practice. For fields that are defined on a spatial domain of dimension $d=1$, we overcome this problem by building a basis of the space of two-forms $Ω^2(\mathbb{F})$. Reversibility is then equivalent to the vanishing of the vorticity's coordinates in this basis, a criterion that is readily applicable to concrete examples. Furthermore, we show that this basis provides a natural direct-sum decomposition of $Ω^2(\mathbb{F})$, each subspace of which is associated with a distinctive kind of phenomenology. This decomposition enables a classification of celebrated out-of-equilibrium phenomena, ranging from non-reciprocal (chaser/chased) interactions to the flocking of active agents, dynamical reaction-diffusion patterns, (...)

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arXiv2025

Hamiltonian Heat Baths, Coarse-Graining and Irreversibility: A Microscopic Dynamical Entropy from Classical Mechanics

The Hamiltonian evolution of an isolated classical system is reversible, yet the second law of thermodynamics states that its entropy can only increase. This has confounded attempts to identify a `Microscopic Dynamical Entropy' (MDE), by which we mean an entropy computable from the system's evolving phase-space density $ρ(t)$, that equates {\em quantitatively} to its thermodynamic entropy $S^{\rm th}(t)$, both within and beyond equilibrium. Specifically, under Hamiltonian dynamics the Gibbs entropy of $ρ$ is conserved in time; those of coarse-grained approximants to $ρ$ show a second law but remain quantitatively unrelated to heat flow. Moreover coarse-graining generally destroys the Hamiltonian evolution, giving paradoxical predictions when $ρ(t)$ exactly rewinds, as it does after velocity-reversal. Here we derive the MDE for an isolated system XY in which subsystem Y acts as a heat bath for subsystem X. We allow $ρ_{XY}(t)$ to evolve without coarse-graining, but compute its entropy by disregarding the detailed structure of $ρ_{Y|X}$. The Gibbs entropy of the resulting phase-space density $\tildeρ_{XY}(t)$ comprises the MDE for the purposes of both classical and stochastic thermodynamics. The MDE obeys the second law whenever $ρ_X$ evolves independently of the details of Y, yet correctly rewinds after velocity-reversal of the full XY system.

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arXiv2024

Hyperuniformity in phase ordering: the roles of activity, noise, and non-constant mobility

Hyperuniformity emerges generically in the coarsening regime of phase-separating fluids. Numerical studies of active and passive systems have shown that the structure factor $S(q)$ behaves as $q^ς$ for $q\to 0$, with hyperuniformity exponent $ς= 4$. For passive systems, this result was explained in 1991 by a qualitative scaling analysis of Tomita, exploiting isotropy at scales much larger than the coarsening length $\ell$. Here we reconsider and extend Tomita's argument to address cases of active phase separation and of non-constant mobility, again finding $ς=4$. We further show that dynamical noise of variance $D$ creates a transient $ς= 2$ regime for $\hat q\ll \hat{q}_\ast \sim \sqrt{D} t^{[1-(d+2)ν]/2}$, crossing over to $ς= 4$ at larger $\hat{q}$. Here, $ν$ is the coarsening exponent, with $\ell\sim t^ν$, and $\hat{q} \propto q \ell$ is the rescaled wavenumber. In diffusive coarsening, $ν=1/3$, so the rescaled crossover wavevector $\hat{q}_\ast$ vanishes at large times when $d\geq 2$. The slowness of this decay suggests a natural explanation for experiments that observe a long-lived $ς= 2$ scaling in phase-separating active fluids (where noise is typically large). Conversely, in $d=1$, we demonstrate that with noise the $ς= 2$ regime survives as $t\to\infty$, with $\hat{q}_\ast\sim D^{5/6}$. (The structure factor is not then determined by the zero-temperature fixed point.) We confirm our analytical predictions by numerical simulations of active and passive continuum theories in the deterministic case and of Model B for the stochastic case. We also compare them with related findings for a system near an absorbing-state transition rather than undergoing phase separation. A central role is played throughout by the presence or absence of a conservation law for the centre of mass position of the order parameter field.

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arXiv2024

Geometric theory of (extended) time-reversal symmetries in stochastic processes -- Part I: finite dimension

In this article, we analyze three classes of time-reversal of a Markov process with Gaussian noise on a manifold. We first unveil a commutativity constraint for the most general of these time-reversals to be well defined. Then we give a triad of necessary and sufficient conditions for the stochastic process to be time-reversible. While most reversibility conditions in the literature require knowledge of the stationary probability, our conditions do not, and therefore can be analytically checked in a systematic way. We then show that the mathematical objects whose cancellation is required by our reversibility conditions play the role of independent sources of entropy production. Furthermore, we give a geometric interpretation of the so-called irreversible cycle-affinity as the vorticity of a certain vector field for a Riemannian geometry given by the diffusion tensor. We also discuss the relation between the time-reversability of the stochastic process and that of an associated deterministic dynamics: its Stratonovitch average. Finally, we show that a suitable choice of a reference measure - that can be considered as a prior or a gauge, depending on the context - allows to study a stochastic process in a way that is both coordinate-free and independent of the prescription used to define stochastic integrals. When this reference measure plays the role of a gauge choice, we interpret our previous results through the lens of gauge theory and prove them to be gauge-invariant.

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arXiv2024

Thermodynamically consistent flocking: From discontinuous to continuous transitions

We introduce a family of lattice-gas models of flocking, whose thermodynamically consistent dynamics admits a proper equilibrium limit at vanishing self-propulsion. These models are amenable to an exact coarse-graining which allows us to study their hydrodynamic behavior analytically. We show that the equilibrium limit here belongs to the universality class of Model C, and that it generically exhibits tricritical behavior. Self-propulsion has a non-perturbative effect on the phase diagram, yielding novel phase behaviors depending on the type of aligning interactions. For aligning interactions that increase monotonically with the density, the tricritical point diverges to infinite density reproducing the standard scenario of a discontinuous flocking transition accompanied by traveling bands. In contrast, for models where the aligning interaction is non-monotonic in density, the system can exhibit either (the nonequilibrium counterpart of) an azeotropic point, associated with a continuous flocking transition, or a state with counterpropagating bands.

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arXiv2024

Effects of phase separation on extinction times in population models

We study the effect of phase separating diffusive dynamics on the mean time to extinction in several reaction-diffusion models with slow reactions. We consider a continuum theory similar to model AB, and a simple model where individual particles on two sites undergo on-site reactions and hopping between the sites. In the slow-reaction limit, we project the models' dynamics onto suitable one-dimensional reaction coordinates, which allows derivation of quasi-equilibrium effective free energies. For weak noise, this enables characterisation of the mean time to extinction. This time can be enhanced or suppressed by the addition of phase separation, compared with homogeneous reference cases. We also discuss how Allee effects can be affected by phase separation.

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arXiv2024

Sojourn probabilities in tubes and pathwise irreversibility for Itô processes

The sojourn probability of an Itô diffusion process, i.e. its probability to remain in the tubular neighborhood of a smooth path, is a central quantity in the study of path probabilities. For $N$-dimensional Itô processes with state-dependent full-rank diffusion tensor, we derive a general expression for the sojourn probability in tubes whose radii are small but finite, and fixed by the metric of the ambient Euclidean space. The central quantity in our study is the exit rate at which trajectories leave the tube for the first time. This has an interpretation as a Lagrangian and can be measured directly in experiment, unlike previously defined sojourn probabilities which depend on prior knowledge of the state-dependent diffusivity. We find that while in the limit of vanishing tube radius the ratio of sojourn probabilities for a pair of distinct paths is in general divergent, the same for a path and its time-reversal is always convergent and finite. This provides a pathwise definition of irreversibility for Itô processes that is agnostic to the state-dependence of the diffusivity. For one-dimensional systems we derive an explicit expression for our Lagrangian in terms of the drift and diffusivity, and find that our result differs from previously reported multiplicative-noise Lagrangians. We confirm our result by comparing to numerical simulations, and relate our theory to the Stratonovich Lagrangian for multiplicative noise. For one-dimensional systems, we discuss under which conditions the vanishing-radius limiting ratio of sojourn probabilities for a pair of forward and backward paths recovers the established pathwise entropy production. Finally, we demonstrate for our one-dimensional example system that the most probable tube for a barrier crossing depends sensitively on the tube radius, and hence on the tolerated amount of fluctuations around the smooth reference path.

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arXiv2024

Resolution dependence of most probable pathways with state-dependent diffusivity

Recent experiments have probed the relative likelihoods of trajectories in stochastic systems by observing survival probabilities within a tube of radius $R$ in spacetime. We measure such probabilities here for a colloidal particle in a corrugated channel, corresponding to a bistable potential with state-dependent diffusivity. In contrast to previous findings for state-independent noise, we find that the most probable pathway changes qualitatively as the tube radius $R$ is altered. We explain this by computing the survival probabilities predicted by overdamped Langevin dynamics. At high enough resolution (small enough $R$), survival probabilities depend solely on diffusivity variations, independent of deterministic forces; finite $R$ corrections yield a generalization of the Onsager-Machlup action. As corollary, ratios of survival probabilities are singular as $R \to 0$, but become regular, and described by the classical Onsager-Machlup action, only in the special case of state-independent noise.

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arXiv2023

Theory of Hyperuniformity at the Absorbing State Transition

Hyperuniformity, whereby the static structure factor (or density correlator) obeys $S(q)\sim q^ς$ with $ς> 0$, emerges at criticality in systems having multiple absorbing states, such as periodically sheared suspensions. These lie in the conserved directed percolation (C-DP) universality class, for which analytic results for $ς$ are lacking. Specifically, $ς$ appears inaccessible within an exact `interfacial mapping' that yields other C-DP exponents via functional renormalization group (FRG). Here, using Doi-Peliti field theory for interacting particles and perturbative RG about a Gaussian model, we find $ς= 0^+$ and $ς= 2ε/9 + O(ε^2)$ in dimension $d>4$ and $d=4-ε$ respectively. The latter disproves a previously conjectured scaling relation for $ς$. We show how hyperuniformity emerges from anticorrelation of strongly fluctuating active and passive densities. Our calculations also yield the remaining C-DP exponents without recourse to functional RG methods.

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arXiv2023

Active-passive mixtures with bulk loading: a minimal active engine in one dimension

We study a one-dimensional mixture of active (run-and-tumble) particles and passive (Brownian) particles, with single-file constraint, in a sawtooth potential. The active particles experience a ratchet effect: this generates a current, which can push passive particles against an applied load. The resulting system operates as an active engine. Using numerical simulations, we analyse the efficiency of this engine, and we discuss how it can be optimised. Efficient operation occurs when the active particles self-organise into teams, which can push the passive ones against large loads by leveraging collective behaviour. We discuss how the particle arrangement, conserved under the single-file constraint, affects the engine efficiency. We also show that relaxing this constraint still allows the engine to operate effectively.

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arXiv2023

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.

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arXiv2023

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 bi-axial 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.

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arXiv2023

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.

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arXiv2023

Two-dimensional crystals far from equilibrium

When driven by nonequilibrium fluctuations, particle systems may display phase transitions and physical behaviour 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.

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arXiv2023

Towards a liquid-state theory for active matter

In equilibrium, the collective behaviour of particles interacting via steep, short-ranged potentials is well captured by the virial expansion of the free energy at low density. Here, we extend this approach beyond equilibrium to the case of active matter with self-propelled particles. Given that active systems do not admit any free-energy description in general, our aim is to build the dynamics of the coarse-grained density from first principles without any equilibrium assumption. Starting from microscopic equations of motion, we obtain the hierarchy of density correlations, which we close with an ansatz for the two-point density valid in the dilute regime at small activity. This closure yields the nonlinear dynamics of the one-point density, with hydrodynamic coefficients depending explicitly on microscopic interactions, by analogy with the equilibrium virial expansion. This dynamics admits a spinodal instability for purely repulsive interactions, a signature of motility-induced phase separation. Therefore, although our approach should be restricted to dilute, weakly-active systems a priori, it actually captures the features of a broader class of active matter.

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arXiv2023

Interface roughening in nonequilibrium phase-separated systems

Interfaces of phase-separated systems roughen in time due to capillary waves. Because of fluxes in the bulk, their dynamics is nonlocal in real space and is not described by the Edwards-Wilkinson or Kardar-Parisi-Zhang (KPZ) equations, nor their conserved counterparts. We show that in the absence of detailed balance, the phase-separated interface is described by a new universality class that we term |q|KPZ. We compute the associated critical exponents via one-loop renormalization group, and corroborate the results by numerical integration of the |q|KPZ equation. Deriving the effective interface dynamics from a minimal field theory of active phase separation, we finally argue that the |q|KPZ universality class generically describes liquid-vapor interfaces in active systems.

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arXiv2022

Role of initial conditions in $1D$ diffusive systems: compressibility, hyperuniformity and long-term memory

We analyse the long-lasting effects of initial conditions on fluctuations in one-dimensional diffusive systems. We consider both the fluctuations of current for non-interacting diffusive particles starting from a step-like initial density profile, and the mean-square displacement of tracers in homogeneous systems with single-file diffusion. For these two cases, we show analytically (via the propagator and Macroscopic Fluctuation Theory, respectively) that the long-term memory of initial conditions is mediated by a single static quantity: a generalized compressibility that quantifies the density fluctuations of the initial state. We thereby identify a universality class of hyperuniform initial states whose dynamical variances coincide with the `quenched' cases studied previously; we also describe a continuous family of other classes among which equilibrated (or `annealed') initial conditions are but one family member. We verify our predictions through extensive Monte Carlo simulations.

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