arXiv subjects

Kevin A. Mitchell

Publications and source records attributed to Kevin A. Mitchell.

At least 19 recordsLinked to original sources

arXiv2025

Modeling active nematics via the nematic locking principle

Active nematic systems consist of rod-like internally driven subunits that interact with one another to form large-scale coherent flows. They are important examples of far-from-equilibrium fluids, which exhibit a wealth of nonlinear behavior. This includes active turbulence, in which topological defects braid around one another in a chaotic fashion. One of the most studied examples of active nematics is a dense two-dimensional layer of microtubules, crosslinked by kinesin molecular motors that inject extensile deformations into the fluid. Though numerous studies have modeled microtubule-based active nematics, no consensus has emerged on how to fully capture the features of the experimental system. To better understand the foundations for modeling this system, we propose a fundamental principle we call the nematic locking principle: individual microtubules cannot rotate without all neighboring microtubules also rotating. Physically, this is justified by the high density of the microtubules, their elongated nature, and their corresponding steric interactions. We assert that nematic locking holds throughout the majority of the material but breaks down in the neighborhood of topological defects and other regions of low density. We derive the most general nematic transport equation consistent with this principle and also derive the most general term that violates it. We examine the standard Beris-Edwards approach used to model this system and show that it violates nematic locking throughout the majority of the material. We then propose a modification to the Beris-Edwards model that enforces nematic locking nearly everywhere. This modification shuts off fracturing except in regions where the order parameter is reduced. The resulting simulations show strong nematic locking throughout the bulk of the material, consistent with experimental observation.

cond-mat.soft↗

arXiv2025

Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen

When placed in parallel magnetic and electric fields, the electron trajectories of a classical hydrogen atom are chaotic. The classical escape rate of such a system can be computed with classical trajectory Monte Carlo techniques, but these computations require enormous numbers of trajectories, provide little understanding of the dynamical mechanisms involved, and must be completely rerun for any change of system parameter, no matter how small. We demonstrate an alternative technique to classical trajectory Monte Carlo computations, based on classical periodic orbit theory. In this technique, escape rates are computed from a relatively modest number (a few thousand) of periodic orbits of the system. One only needs the orbits' periods and stability eigenvalues. A major advantage of this approach is that one does not need to repeat the entire analysis from scratch as system parameters are varied; one can numerically continue the periodic orbits instead. We demonstrate the periodic orbit technique for the ionization of a hydrogen atom in applied parallel electric and magnetic fields. Using fundamental theories of phase space geometry, we also show how to generate nontrivial symbolic dynamics for acquiring periodic orbits in physical systems. A detailed analysis of heteroclinic tangles and how they relate to bifurcations in periodic orbits is also presented.

nlin.CD↗

arXiv2025

Chaos-generating periodic orbits of topological defects in confined active nematics

Active nematics in two dimensions stir themselves efficiently through internally generated chaotic flows, largely driven by motile $+1/2$ disclinations. We investigate how this tendency toward chaotic fluid stirring can, counterintuitively, produce certain ordered, periodic flows in confinement, characterized by stable periodic orbits of $+1/2$ disclinations. We computationally study two-dimensional active nematics in systems with boundary conditions requiring a prescribed number $n$ of excess $+1/2$ disclinations, using Beris-Edwards nematohydrodynamics simulations alongside an agent-based simulation approach. We find that when confinement is sufficiently strong to prevent defect pair-nucleation, but not strong enough to arrest all flow, then $n=3$ defects generically follow a "golden braid" orbit as observed recently in experiments, and we predict a "silver braid" orbit of $n=4$ defects. For these results and for greater numbers of defects, we show that the periodic or chaotic nature of the dynamics is determined by a balance between the number of defects and the number of vortices in the flow field, suggesting a new design criterion for ordered flows in active nematics.

cond-mat.soft↗

arXiv2024

Strain stiffening due to stretching of entangled particles in random packings of granular materials

Stress-strain relations for random packings of entangling chains under triaxial compression can exhibit strain stiffening and sustain stresses several orders-of-magnitude beyond typical granular materials. X-ray tomography reveals the transition to this strong strain stiffening occurs when chains are long enough to entangle an average of about one chain each, which results in system-filling clusters of entangled chains. The number of entanglements is nearly proportional to the area surrounded by entangling particles with an excluded volume effect. A tendency was found for chain links to stretch when the packing was strained. The slope of the stress-strain relation of the packing can be calculated from a mean-field model consisting of the product of the effective extensional modulus of the chain, packing fraction, probability of stretched links, and the ratio of strain of stretched links to packing strain. The stress-strain model requires as input measurements of the ratio between local particle deformation and global average strain, and the probability of stretching for non-rigid particles. This results in a quadratic prediction for the stress-strain curve, with a curvature that agrees with experiments within the model uncertainties. This model explains that the strength of these packings comes from stretching of the links of chains, but only when the system-filling network of entanglements provides constraints that prevents failure by shear banding, so that particles must be deformed to move further under strain. In this model, the increasing slope of the stress-strain curve is mainly due to the fraction of stretched links increasing with strain. This model for the stress-strain relation is shown to be generalizable to different shapes of entangling particles by applying it to staples.

cond-mat.soft↗

arXiv2024

Controlling chaos: Periodic defect braiding in active nematics confined to a cardioid

This work examines self-mixing in active nematics, a class of fluids in which mobile topological defects drive chaotic flows in a system comprised of biological filaments and molecular motors. We present experiments that demonstrate how geometrical confinement can influence the braiding dynamics of the defects. Notably, we show that confinement in cardioid-shaped wells leads to realization of the golden braid, a maximally efficient mixing state of exactly three defects with no defect creation or annihilation. We characterize the golden braid state using different measures of topological entropy and the Lyapunov exponent. In particular, topological entropy measured from the stretching rate of material lines agrees well with an analytical computation from braid theory. Increasing the size of the confining cardioid produces a transition from the golden braid, to the fully chaotic active turbulent state.

cond-mat.soft↗

arXiv2023

Maximally mixing active nematics

Active nematics are an important new paradigm in soft condensed matter systems. They consist of rod-like components with an internal driving force pushing them out of equilibrium. The resulting fluid motion exhibits chaotic advection, in which a small patch of fluid is stretched exponentially in length. Using simulation, this Letter shows that this system can exhibit stable periodic motion when sufficiently confined to a square with periodic boundary conditions. Moreover, employing tools from braid theory, we show that this motion is maximally mixing, in that it optimizes the (dimensionless) ``topological entropy'' -- the exponential stretching rate of a material line advected by the fluid. That is, this periodic motion of the defects, counterintuitively, produces more chaotic mixing than chaotic motion of the defects. We also explore the stability of the periodic state. Importantly, we show how to stabilize this orbit into a larger periodic tiling, a critical necessity for it to be seen in future experiments.

cond-mat.soft↗

arXiv2021

Swimmer dynamics in externally-driven fluid flows: The role of noise

We theoretically investigate the effect of random fluctuations on the motion of elongated microswimmers near hydrodynamic transport barriers in externally-driven fluid flows. Focusing on the two-dimensional hyperbolic flow, we consider the effects of translational and rotational diffusion as well as tumbling, i.e. sudden jumps in the swimmer orientation. Regardless of whether diffusion or tumbling are the primary source of fluctuations, we find that noise significantly increases the probability that a swimmer crosses one-way barriers in the flow, which block the swimmer from returning to its initial position. We employ an asymptotic method for calculating the probability density of noisy swimmer trajectories in a given fluid flow, which produces solutions to the time-dependent Fokker-Planck equation in the weak-noise limit. This procedure mirrors the semiclassical approximation in quantum mechanics and similarly involves calculating the least-action paths of a Hamiltonian system derived from the swimmer's Fokker-Planck equation. Using the semiclassical technique, we compute (i) the steady-state orientation distribution of swimmers with rotational diffusion and tumbling and (ii) the probability that a diffusive swimmer crosses a one-way barrier. The semiclassical results compare favorably with Monte Carlo calculations.

physics.flu-dyn↗

arXiv2021

Noise-induced aggregation of swimmers in the Kolmogorov flow

We investigate a model for the dynamics of ellipsoidal microswimmers in an externally imposed, laminar Kolmogorov flow. Through a phase-space analysis of the dynamics without noise, we find that swimmers favor either cross-stream or rotational drift, depending on their swimming speed and aspect ratio. When including noise, i.e. rotational diffusion, we find that swimmers are driven into certain parts of phase space, leading to a nonuniform steady-state distribution. This distribution exhibits a transition from swimmer aggregation in low-shear regions of the flow to aggregation in high-shear regions as the swimmer's speed, aspect ratio, and rotational diffusivity are varied. To explain the nonuniform phase-space distribution of swimmers, we apply a weak-noise averaging principle that produces a reduced description of the stochastic swimmer dynamics. Using this technique, we find that certain swimmer trajectories are more favorable than others in the presence of weak rotational diffusion. By combing this information with the phase-space speed of swimmers along each trajectory, we predict the regions of phase space where swimmers tend to accumulate. The results of the averaging technique are in good agreement with direct calculations of the steady-state distributions of swimmers. In particular, our analysis explains the transition from low-shear to high-shear aggregation.

cond-mat.soft↗

arXiv2021

Fractal generation in a two-dimensional active-nematic fluid

Active fluids, composed of individual self-propelled agents, can generate complex large-scale coherent flows. A particularly important laboratory realization of such an active fluid is a system composed of microtubules, aligned in a quasi-two-dimensional (2D) nematic phase, and driven by ATP-fueled kinesin motor proteins. This system exhibits robust chaotic advection and gives rise to a pronounced fractal structure in the nematic contours. We characterize such experimentally derived fractals using the power spectrum and discover that the power spectrum decays as $k^{-β}$ for large wavenumbers $k$. The parameter $β$ is measured for several experimental realizations. Though $β$ is effectively constant in time, it does vary with experimental parameters, indicating differences in the scale-free behavior of the microtubule-based active nematic. Though the fractal patterns generated in this active system are reminiscent of passively advected dye in 2D chaotic flows, the underlying mechanism for fractal generation is more subtle. We provide a simple, physically inspired mathematical model of fractal generation in this system that relies on the material being locally compressible, though the total area of the material is conserved globally. The model also requires that large-scale density variations be injected into the material periodically. The model reproduces the power spectrum decay $k^{-β}$ seen in experiments. Linearizing the model of fractal generation about the equilibrium density, we derive an analytic relationship between $β$ and a single dimensionless quantity $r$, which characterizes the compressibility.

cond-mat.soft↗

arXiv2020

Topological Dynamics of Volume-Preserving Maps Without an Equatorial Heteroclinic Curve

Understanding the topological structure of phase space for dynamical systems in higher dimensions is critical for numerous applications, including the computation of chemical reaction rates and transport of objects in the solar system. Many topological techniques have been developed to study maps of two-dimensional (2D) phase spaces, but extending these techniques to higher dimensions is often a major challenge or even impossible. Previously, one such technique, homotopic lobe dynamics (HLD), was generalized to analyze the stable and unstable manifolds of hyperbolic fixed points for volume-preserving maps in three dimensions. This prior work assumed the existence of an equatorial heteroclinic intersection curve, which was the natural generalization of the 2D case. The present work extends the previous analysis to the case where no such equatorial curve exists, but where intersection curves, connecting fixed points may exist. In order to extend HLD to this case, we shift our perspective from the invariant manifolds of the fixed points to the invariant manifolds of the invariant circle formed by the fixed-point-to-fixed-point intersections. The output of the HLD technique is a symbolic description of the minimal underlying topology of the invariant manifolds. We demonstrate this approach through a series of examples.

nlin.CD↗

arXiv2017

Mode-locking in advection-reaction-diffusion systems: an invariant manifold perspective

Fronts propagating in two-dimensional advection-reaction-diffusion (ARD) systems exhibit rich topological structure. When the underlying fluid flow is periodic in space and time, the reaction front can lock to the driving frequency. We explain this mode-locking phenomenon using so-called burning invariant manifolds (BIMs). In fact, the mode-locked profile is delineated by a BIM attached to a relative periodic orbit (RPO) of the front element dynamics. Changes in the type (and loss) of mode-locking can be understood in terms of local and global bifurcations of the RPOs and their BIMs. We illustrate these concepts numerically using a chain of alternating vortices in a channel geometry.

nlin.PS↗

arXiv2017

Topological Chaos in a Three-Dimensional Spherical Fluid Vortex

In chaotic deterministic systems, seemingly stochastic behavior is generated by relatively simple, though hidden, organizing rules and structures. Prominent among the tools used to characterize this complexity in 1D and 2D systems are techniques which exploit the topology of dynamically invariant structures. However, the path to extending many such topological techniques to three dimensions is filled with roadblocks that prevent their application to a wider variety of physical systems. Here, we overcome these roadblocks and successfully analyze a realistic model of 3D fluid advection, by extending the homotopic lobe dynamics (HLD) technique, previously developed for 2D area-preserving dynamics, to 3D volume-preserving dynamics. We start with numerically-generated finite-time chaotic-scattering data for particles entrained in a spherical fluid vortex, and use this data to build a symbolic representation of the dynamics. We then use this symbolic representation to explain and predict the self-similar fractal structure of the scattering data, to compute bounds on the topological entropy, a fundamental measure of mixing, and to discover two different mixing mechanisms, which stretch 2D material surfaces and 1D material curves in distinct ways.

physics.flu-dyn↗

arXiv2016

Using invariant manifolds to construct symbolic dynamics for three-dimensional volume-preserving maps

Topological techniques are powerful tools for characterizing the complexity of many dynamical systems, including the commonly studied area-preserving maps of the plane. However, the extension of many topological techniques to higher dimensions is filled with roadblocks preventing their application. This article shows how to extend the homotopic lobe dynamics (HLD) technique, previously developed for 2D maps, to volume-preserving maps of a three-dimensional phase space. Such maps are physically relevant to particle transport by incompressible fluid flows or by magnetic field lines. Specifically, this manuscript shows how to utilize two-dimensional stable and unstable invariant manifolds, intersecting in a heteroclinic tangle, to construct a symbolic representation of the topological dynamics of the map. This symbolic representation can be used to classify system trajectories and to compute topological entropy. We illustrate the salient ideas through a series of examples with increasing complexity. These examples highlight new features of the HLD technique in 3D. Ultimately, in the final example, our technique detects a difference between the 2D stretching rate of surfaces and the 1D stretching rate of curves, illustrating the truly 3D nature of our approach.

nlin.CD↗

arXiv2015

Matter, Energy, and Heat Transfer in a Classical Ballistic Atom Pump

A ballistic atom pump is a system containing two reservoirs of neutral atoms or molecules and a junction connecting them containing a localized time-varying potential. Atoms move through the pump as independent particles. Under certain conditions, these pumps can create net transport of atoms from one reservoir to the other. While such systems are sometimes called "quantum pumps," they are also models of classical chaotic transport, and their quantum behavior cannot be understood without study of the corresponding classical behavior. Here we examine classically such a pump's effect on energy and temperature in the reservoirs, in addition to net particle transport. We show that the changes in particle number, of energy in each reservoir, and of temperature in each reservoir vary in unexpected ways as the incident particle energy is varied.

cond-mat.quant-gas↗

arXiv2015

Ballistic atom pumps

We examine a classically-chaotic system consisting of two reservoirs of particles connected by a channel containing oscillating potential-energy barriers. We investigate whether such a system can preferentially pump particles from one reservoir to the other, a process often called "quantum pumping." We show how to make a "particle diode" which under specified conditions permits net particle pumping in only one direction. Then we examine systems having symmetric barriers. We find that if all initial particle energies are considered, a system with symmetric barriers cannot preferentially pump particles. However, if only finite initial energy bands are considered, the system can create net particle transport in either direction. We study the system classically, semiclassically, and quantum mechanically, and find that the quantum description cannot be fully understood without the insight gained from classical and semiclassical analysis.

cond-mat.quant-gas↗

arXiv2015

Finite-time barriers to front propagation in two-dimensional fluid flows

Recent theoretical and experimental investigations have demonstrated the role of certain invariant manifolds, termed burning invariant manifolds (BIMs), as one-way dynamical barriers to reaction fronts propagating within a flowing fluid. These barriers form one-dimensional curves in a two-dimensional fluid flow. In prior studies, the fluid velocity field was required to be either time-independent or time-periodic. In the present study, we develop an approach to identify prominent one-way barriers based only on fluid velocity data over a finite time interval, which may have arbitrary time-dependence. We call such a barrier a burning Lagrangian coherent structure (bLCS) in analogy to Lagrangian coherent structures (LCSs) commonly used in passive advection. Our approach is based on the variational formulation of LCSs using curves of stationary "Lagrangian shear", introduced by Farazmand, Blazevski, and Haller [Physica D 278-279, 44 (2014)] in the context of passive advection. We numerically validate our technique by demonstrating that the bLCS closely tracks the BIM for a time-independent, double-vortex channel flow with an opposing "wind".

nlin.PS↗

arXiv2015

Frozen reaction fronts in steady flows: a burning-invariant-manifold perspective

The dynamics of fronts, such as chemical reaction fronts, propagating in two-dimensional fluid flows can be remarkably rich and varied. For time-invariant flows, the front dynamics may simplify, settling in to a steady state in which the reacted domain is static, and the front appears "frozen". Our central result is that these frozen fronts in the two-dimensional fluid are composed of segments of burning invariant manifolds---invariant manifolds of front-element dynamics in $xyθ$-space, where $θ$ is the front orientation. Burning invariant manifolds (BIMs) have been identified previously as important local barriers to front propagation in fluid flows. The relevance of BIMs for frozen fronts rests in their ability, under appropriate conditions, to form global barriers, separating reacted domains from nonreacted domains for all time. The second main result of this paper is an understanding of bifurcations that lead from a nonfrozen state to a frozen state, as well as bifurcations that change the topological structure of the frozen front. Though the primary results of this study apply to general fluid flows, our analysis focuses on a chain of vortices in a channel flow with an imposed wind. For this system, we present both experimental and numerical studies that support the theoretical analysis developed here.

nlin.PS↗

arXiv2013

A turnstile mechanism for fronts propagating in fluid flows

We consider the propagation of fronts in a periodically driven flowing medium. It is shown that the progress of fronts in these systems may be mediated by a turnstile mechanism akin to that found in chaotic advection. We first define the modified ("active") turnstile lobes according to the evolution of point sources across a transport boundary. We then show that the lobe boundaries may be constructed from stable and unstable \emph{burning invariant manifolds}---one-way barriers to front propagation analogous to traditional invariant manifolds for passive advection. Because the burning invariant manifolds (BIMs) are one-dimensional curves in a three-dimensional ($xyθ$) phase space, their projection into $xy$-space exhibits several key differences from their advective counterparts: (lobe) areas are not preserved, BIMs may self-intersect, and an intersection between stable and unstable BIMs does not map to another such intersection. These differences must be accommodated in the correct construction of the new turnstile. As an application, we consider a lobe-based treatment protocol for protecting an ocean bay from an invading algae bloom.

nlin.CD↗