arXiv subjects

Connor Roberts

Publications and source records attributed to Connor Roberts.

7 recordsLinked to original sources

arXiv2025

Exact results and instabilities in the harmonic approximation of active crystals

Condensates of active particles such as cells form almost-crystalline lattices which play a central role in many biological systems. Typically, their properties have been determined merely by analogy to the rather trivial one-dimensional case, leaving a gap between experimentally accessible observables and suitable theoretical models. Within a harmonic approximation, we characterise analytically a two-dimensional triangular lattice of active particles that interact with their nearest neighbours through a general pair potential, obtaining exact expressions for the correlators. We study this "active crystal" as a means of characterising active matter in the dense phase. Our treatment correctly approximates arbitrary pair potentials, rather than demanding an unphysical non-singular bilinear form. We retain "off-diagonal" terms that are routinely neglected despite quantifying the anisotropy of the particles' local potential. From the exact expressions for the correlation matrices, we derive exact results that shed light on the presence (or absence) of crystalline order. We further calculate the mean-squared particle separation, energy, entropy production rate and the onset of a pressure-induced instability resulting in the breakdown of the harmonic approximation. The entropy production rate is found to have a general form that is valid for generic active particles and lattice geometries, while resembling that of non-interacting "active modes".

cond-mat.soft↗

arXiv2025

Conditional splitting probabilities for hidden-state inference in drift-diffusive processes

Splitting probabilities quantify the likelihood of particular outcomes out of a set of mutually-exclusive possibilities for stochastic processes and play a central role in first-passage problems. For two-dimensional Markov processes $\{X(t),Y(t)\}_{t\in T}$, a joint analogue of the splitting probabilities can be defined, which captures the likelihood that the variable $X(t)$, having been initialised at $x_0 \in \mathbb{L}$, exits $\mathbb{L}$ for the first time via either of the interval boundaries \emph{and} that the variable $Y(t)$, initialised at $y_0$, is given by $y_{\rm exit}$ at the time of exit. We compute such joint splitting probabilities for two classes of processes: processes where $X(t)$ is Brownian motion and $Y(t)$ is a decoupled internal state, and unidirectionally coupled processes where $X(t)$ is drift-diffusive and depends on $Y(t)$, while $Y(t)$ evolves independently. For the first class we obtain generic expressions in terms of the eigensystem of the Fokker-Planck operator for the $Y$ dynamics, while for the second we carry out explicit derivations for three paradigmatic cases (run-and-tumble motion, diffusion in an intermittent piecewise-linear potential and diffusion with stochastic resetting). Drawing on Bayes' theorem, we subsequently introduce the related notion of conditional splitting probabilities, defined as the posterior likelihoods of the internal state $Y$ \emph{given} that the observable degree of freedom $X$ has undergone a specific exit event. After computing these conditional splitting probabilities, we propose a simple scheme that leverages them to partially infer the assumedly hidden state $Y(t)$ from point-wise detection events.

math-ph↗

arXiv2024

Stochastic cloaking: concealing a region from diffusive particles

We introduce "stochastic cloaking," where a region of space is concealed from an ensemble of diffusing particles whose individual trajectories are governed by a stochastic (Langevin) equation. Our simulations reveal how different interpretations of the Langevin equation affect the cloaking performance of an annular single-layer invisibility cloak of smoothly varying diffusivity in two dimensions. Near-perfect cloaking is achieved under the Ito convention, indicated by the cloak preventing particles from accessing an inner core without disturbing the particle density outside the cloak. The cloak's performance can be further improved by regularising its singular behaviour. We believe our demonstration of stochastic cloaking is a significant milestone, comparable to earlier developments that extended cloaking from optics and acoustics to thermodynamics.

cond-mat.mtrl-sci↗

arXiv2024

Ratchet-mediated resetting: Current, efficiency, and exact solution

We model an overdamped Brownian particle that is subject to resetting facilitated by a ratchet potential on a spatially periodic domain. This asymmetric potential switches on with a constant rate, but switches off again only upon the particle's first passage to a resetting point at the minimum of the potential. Repeating this cycle sustains a non-equilibrium steady-state, as well as a directed steady-state current which can be harnessed to perform useful work. We derive exact analytic expressions for the probability densities of the free-diffusion and resetting phases, the associated currents for each phase, and an efficiency parameter that quantifies the return in current for given power input. These expressions allow us to fully characterise the system and obtain experimentally relevant results such as the optimal current and efficiency. Our results are corroborated by simulations, and have implications for experimentally viable finite-time resetting protocols.

cond-mat.stat-mech↗

arXiv2023

Run-and-tumble motion in a linear ratchet potential: Analytic solution, power extraction, and first-passage properties

We explore the properties of run-and-tumble particles moving in a piecewise-linear "ratchet" potential by deriving analytic results for the system's steady-state probability density, current, entropy production rate, extractable power, and thermodynamic efficiency. The ratchet's broken spatial symmetry rectifies the particles' self-propelled motion, resulting in a positive current that peaks at finite values of the diffusion strength, ratchet height, and particle self-propulsion speed. Similar nonmonotonic behaviour is also observed for the extractable power and efficiency. We find the optimal apex position for generating maximum current varies with diffusion, and that entropy production can have nonmonotonic dependence on diffusion. In particular, for vanishing diffusion, entropy production remains finite when particle self-propulsion is weaker than the ratchet force. Furthermore, power extraction with near-perfect efficiency is achievable in certain parameter regimes due to the simplifications afforded by modelling "dry" active particles. In the final part, we derive mean first-passage times and splitting probabilities for different boundary and initial conditions. This work connects the study of work extraction from active matter with exactly solvable active particle models and will therefore facilitate the design of active engines through these analytic results.

cond-mat.stat-mech↗

arXiv2022

Optimal power extraction from active particles with hidden states

We identify generic protocols achieving optimal power extraction from a single active particle subject to continuous feedback control under the assumption that its spatial trajectory, but not its instantaneous self-propulsion force, is accessible to direct observation. Our Bayesian approach draws on the Onsager-Machlup path integral formalism and is exemplified in the cases of free run-and-tumble and active Ornstein-Uhlenbeck dynamics in one dimension. Such optimal protocols extract positive work even in models characterised by time-symmetric positional trajectories and thus vanishing informational entropy production rates. We argue that the theoretical bounds derived in this work are those against which the performance of realistic active matter engines should be compared.

cond-mat.stat-mech↗

arXiv2022

Exact solution of a boundary tumbling particle system in one dimension

We derive the fully time-dependent solution to a run-and-tumble model for a particle which has tumbling restricted to the boundaries of a one-dimensional interval. This is achieved through a field-theoretic perturbative framework by exploiting an elegant underlying structure of the perturbation theory. We calculate the particle densities, currents and variance as well as characteristics of the boundary tumbling. The analytical findings, in agreement with Monte-Carlo simulations, show how the particle densities are linked to the scale of diffusive fluctuations at the boundaries. The generality of our approach suggests it could be readily applied to similar problems described by Fokker-Planck equations containing localised reaction terms.

cond-mat.stat-mech↗