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Advances in Nuclear Fuel Chemistry (Woodhead Publishing Series in Energy) 1st Edition Markus H.A. Piro (Editor)
The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.
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Lorenzo Piro
Optimal Navigation in Active Matter
Doctoral Thesis accepted by the Georg August University of Göttingen, Göttingen, Germany
Author
Dr. Lorenzo Piro
Department of Living Matter Physics
Max Planck Institute for Dynamics and Self-Organization (MPI-DS)
Göttingen, Niedersachsen, Germany
ISSN 2190-5053
Springer Theses
ISBN 978-3-031-52576-6
Supervisors
Prof. Ramin Golestanian
Department of Living Matter Physics
Max Planck Institute for Dynamics and Self-Organization (MPI-DS)
Göttingen, Niedersachsen, Germany
Rudolf Peierls Centre for Theoretical Physics
University of Oxford Oxford, UK
Dr. Benoît Mahault
Department of Living Matter Physics
Max Planck Institute for Dynamics and Self-Organization (MPI-DS)
Göttingen, Niedersachsen, Germany
ISSN 2190-5061 (electronic)
ISBN 978-3-031-52577-3 (eBook) https://doi.org/10.1007/978-3-031-52577-3
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Supervisors’ Foreword
Biological microswimmers often live on a limited energy supply, such that the optimization of biological processes offers them with a certain evolutionary advantage. In particular, efficient propulsion and navigation in terms of energy dissipation and travel time is of crucial importance for these microorganisms. The design of optimally navigating artificial swimmers also has potentially valuable applications such as targeted drug delivery, which has become an increasingly realistic perspective due to the recent progress in the experimental realization of controllable microswimmers able to perform complex tasks.
The first optimal navigation problem was formulated in 1931 by the mathematician Ernst Zermelo, who was interested in determining, for a boat sailing in windy conditions, the trajectory ensuring shortest arrival time at a given target. Since this pioneering work, much theoretical progress has been achieved while the field, however, still faces open challenges. For microswimmers navigation is notably much more challenging than at our macroscopic scale, as these microorganisms are subject to additional complexities such as fluctuations or hydrodynamic interactions.
Lorenzo’s Ph.D. work precisely addresses some of these important questions, and provides a significant advance towards real world applications. The material presented in this book indeed offers valuable insights on how efficient navigation may be achieved in the presence of confining forces and flows, or thermal fluctuations. On the more conceptual side, the scientific endeavors of Lorenzo led him to draw interesting connections between the problem of navigation on Riemannian manifolds and the geodesics of Randers spaces that were initially introduced in the context of general relativity. In an attempt to establish more realistic theoretical descriptions, the energetic cost of navigation is finally addressed via a novel formulation of the problem that accounts for the hydrodynamic coupling between the surrounding flow field and the swimmer body geometry.
In addition, this book proposes a pedagogical introduction to the current state of the art in the theoretical approaches of navigation problems. I thus have no doubt that it will be much appreciated by graduate students and more experienced researchers with an interest in this field.
Göttingen, Germany October 2023
Prof. Ramin Golestanian
Dr. Benoît Mahault
Abstract
Motile active matter systems are composed by a collection of agents, each of which extracts energy from the surrounding environment in order to convert it into selfdriven motion. At the microscopic scale, however, directed motion is hindered by both the presence of stochastic fluctuations. Living microorganisms therefore had to develop simple yet effective propulsion and steering mechanisms in order to survive.
We may turn the question of how these processes work in nature around and ask how they should work in order to perform a task in the theoretically optimal way, an issue which falls under the name of the optimal navigation problem. The first formulation of this problem dates back to the seminal work of E. Zermelo in 1931, in which he addressed the question of how to steer a ship in the presence of an external stationary wind so as to reach the destination in the shortest time.
Despite the considerable progress made over the years in this context, however, there are still a number of open challenges. In this thesis, we therefore aim to generalize Zermelo’s solution by adding more and more ingredients in the description of the optimal navigation problem for microscopic active particles.
First, borrowing theoretical tools from differential geometry, we here show how to extend the analytical solution of this problem to when motion occurs on curved surfaces and in the presence of arbitrary flows. Interestingly, we reveal that it can elegantly be solved by finding the geodesics of an asymmetric metric of general relativity, known as the Randers metric.
Then, we study the case in which navigation happens in the presence of strong external forces. In this context, route optimization can be crucial as active particles may encounter trapping regions that would substantially slow-down their progress. Comparing the exploration efficiency of Zermelo’s solution with a more trivial strategy in which the active agent always points in the same direction, here we highlight the importance of the optimal path stability, which turns out to be fundamental in the design of the proper navigation strategy depending on the task at hand.
We then take it a step further and include a key ingredient in the comprehensive study of optimal navigation in active matter, namely stochastic fluctuations. Although methods already exist to obtain both analytically and numerically the optimal strategies even in the presence of noise, their implementation requires the presence of an
external interpreter that takes away the active agent’s autonomy. Inspired by the tactic behaviours observed in nature, we here introduce a whole new class of navigation strategies that allows an active particle to navigate semi-autonomously in a complex and noisy environment. Moreover, our study reveals that the performance of the theoretical optimal strategy can be reproduced starting from some simple principles based on symmetry and stability arguments.
Finally, we lay the ground for moving towards a more realistic description of the problem. In fact, we extend the optimization problem by also considering the energetic costs involved in navigation and how these depend on the shape of the active particle itself. Remarkably, our analysis uncovers the existence of an interesting tradeoff between the minimization of the arrival time at a target and the corresponding energetic cost, which in turn determines the optimal shape of an active particle.
Acknowledgements
During these three years as a Ph.D. student in the department of Living Matter Physics (LMP) in Göttingen, I have been faced with a number of challenges. Nevertheless, through thick and thin, I have had the huge pleasure of getting to know many wonderful people who have contributed, each in their own way and to a different extent, to my achievement of this milestone.
First and foremost, I would like to thank my supervisor Ramin Golestanian for diligently guiding me through this journey with his constant support. I am sincerely grateful to him for always inspiring me to improve my skills as young researcher and to do my best with his valuable advice and critical comments.
This work would never have reached its current form without the careful and critical revisions of my group leader Benoît Mahault. Over all these years, his door has always been open for me, relentlessly willing to provide me with the help I needed to move forward, especially in my moments of greatest frustration. I learnt more from him about this job than I could ever have imagined, and for that (and his endless patience) I am eternally grateful.
I acknowledge our collaborators Evelyn Tang and Andrej Vilfan for their insightful contributions that led to the development of the ideas behind the results presented in Chaps. 2 and 5 of this thesis, respectively.
I also thank the members of my thesis advisory committee (TAC) Jörg Enderlein and Matthias Krüger for their helpful inputs and feedback during our meetings. Furthermore, I would like to thank Stefan Klumpp, Viola Priesemann and Aljaž Godec for showing interest in my work and accepting to be part of the thesis examination board.
One of the challenges in my Ph.D. has been navigating around the hurdles of bureaucracy. I would therefore like to thank Ay¸se Bolik, Viktoryia Novak, Antje Erdmann and Frauke Bergmann for all the administrative and technical support they have tirelessly and patiently given me throughout these years.
Then, I would like to thank all my colleagues in the LMP department. On the one hand, the stimulating and (literally) active environment in the department kept my passion and interest in scientific research alive. On the other hand, I feel blessed to have been able to work with people who are, for the most part, my dear friends. I will
never forget the happy hours, the parties, the volleyball matches (Go Le Rane!), the countless pints of beers we drank and schnitzels we ate together. I would also like to thank all my friends outside the department that I was lucky enough to meet here in Göttingen. Thanks to all of you I could always find a way to relieve the stress from work and recharge the batteries before the start of another week.
A special thanks goes to Vincent Ouazan-Reboul. Together we have been through a lot and, whatever the circumstances, I could always count on him to offer me the back-up I sought both within and beyond work. I could not have wished for a better person to share this whole journey with.
It goes without saying that I thank my parents, Roberto and Roberta, and my brother Daniele for always cheering for me and encouraging me in difficult times. Although from afar, their unconditional love and support have been essential to give me the guts to get over every obstacle.
Last but not least, I would like to thank Martina for consistently backing me up in my decisions and always being there for me, despite the distance. Her extraordinary strength has been my secret weapon during the Ph.D. and I can safely say that this work would never have achieved this form without her constant loving support.
3.4
Chapter 1 Introduction
If I have seen further, it is by standing on the shoulders of Giants.
Isaac Newton
1.1 Models of Active Matter Systems
A unifying characteristic of various living organisms is that they can extract and consume energy in order to move, apply mechanical forces or deform their shape [1]. Collections of such self-driven agents fall under the name of active matter systems [2]. Due to the constant energy dissipation at the individual level, these systems are continuously driven far from thermal equilibrium. This characteristic allows them to exhibit a variety of intriguing phenomenons without any equilibrium equivalent, such as the emergence of collective motion [3, 4], self-organized structures [5–8]or activity-induced patterns [9, 10]. A broad class of active matter systems is that characterised by agents capable of moving autonomously in their environment, hence referred to as motile active matter [11] which spans several scales, ranging from bacteria [12], molecular motors [13, 14]orcells [15, 16], to schools of fish, flocks of birds or even human crowds [17, 18].
To this day, there are countless open challenges in this field due to the high degree of complexity of these systems, be they living or man-made [19]. As a result, the study of active matter has recently brought together a growing number of scientists from various disciplines including materials science [20], robotics [21–23], chemical [24], biological [25], soft matter [2, 26] and statistical physics [1]. Over the years, researchers have thus been devoted to the development of new minimal models aimed at rationalizing, predicting and controlling the properties of motile active matter systems. These are typically phenomenological models based on global principles such as conservation laws and symmetries, which help to minimize the number of parameters controlling the dynamics of the system while also making them simple enough to be numerically efficient.
An outstanding example of minimal model inspired by animal behavior [27, 28] is the renowned Vicsek model [29], which was proposed to understand the collective motion of a group of agents. It consists of a set of point-like particles self-propelling in two dimensions and influencing each other’s orientation through local alignment of their velocities. Despite the simplicity of this model, it remarkably allows to study collective motion across many scales, spanning from microtubules [30] to flocks of birds [31]. It represents, in fact, one of the simplest of models showing a transition to collective motion within a population of self-propelling agents and, in the study of active matter, it certainly plays the role of prototype in a way analogous to Ising’s model in ferromagnetism.
Similarly, there has been a growing interest in the modeling of microscopic selfpropelled organisms at the individual level, also known as microswimmers or active particles [32]. The former term refers to force-free and torque-free organisms featuring a hydrodynamic coupling with the fluid they are immersed into, which occurs via the force fields generated by their swimming patterns. The latter defines the broader class of objects moving in an inert medium (typically a viscous fluid) yielding just hydrodynamic drag and a stochastic exchange of momentum.1 In either case, these microorganisms are able to draw energy from their surroundings and then turn it into directed motion by means of a variety of mechanisms, which we shall now briefly review.
1.1.1 Self-propulsion Mechanisms
Directed motion at the microscale is especially problematic because of the combination of two factors: the absence of inertia and the presence of thermal fluctuations. On those scales, microorganisms indeed live in a world where fluid friction and viscosity dominate over inertia, also known as the low Reynolds number (or overdamped )regime[33–36]. The absence of inertia has interesting and counter-intuitive implications, as illustrated by the scallop theorem [36]: a swimmer making a timesymmetric movement cannot achieve a net displacement at low Reynolds numbers. This has therefore led biological microorganisms to evolve propulsion mechanisms that break temporal reversibility.
In general, since microswimmers motion is momentum-conserving,2 the far-field flow they generate can be well mathematically described by a force dipole [34, 35]. As shown schematically in Fig. 1.1a, this characterization splits microswimmers into two classes. Pushers, who move the fluid towards them on the sides of their body and
1 Despite this difference, in the following we will actually not make any distinction between these two terms since their observable behavior—when considered individually—is indistinguishable in a homogeneous environment [32].
2 Due to the absence of inertia at low Reynolds number.
Fig. 1.1 a Schematic representation of two self-propulsion mechanisms of microswimmers at low Reynolds number. On the left, pushers generate the propulsion from the rear end due to the rotation of a bundle of flagella. Pullers (on the right) generate instead the force from the front end thanks to two flagella beating in a breaststroke way. Thanks to these non-reciprocal beating patterns they are able to swim in one direction (from left to right in this figure). b Left panel: a colony of E. Coli bacteria observed at the microscope as an example of pusher microswimmer. Source Wikimedia Commons, Rocky Mountain Laboratories, NIAID, NIH, public domain. Right panel: Chlamydomonas reinhardtii algae under a Scanning Electron Microscope (SEM). A paradigmatic example of pullers. Source Wikimedia Commons, Dartmouth Electron Microscope Facility, Dartmouth College, public domain c Aswarm of Myxococcus xanthus under a SEM, an example of gliding bacteria. Source ETH Zurich/Gregory J. Velicer and Jürgen Berger d Illustration of the self-diffusiophoresis mechanism characterizing the motion of a Janus particle. This colloid is driven by the local concentration gradient of the reaction product (oxygen, O2 ) generated by the catalysis of hydrogen peroxide (H2 O2 ) on the platinum-coated side (blue hemisphere)
away from them at the extremities, and whose forces forming the dipole point away from them. An emblematic example is the Escherichia Coli bacterium, which uses rotating flagella at the back of its body to self-propel [37] (left panel in Fig. 1.1b). In contrast, when the forces forming the dipole point towards the microswimmer, it is called puller : it draws fluid towards itself in the direction of motion and pushes it out from the sides. An example is given by the Chlamydomonas reinhardtii,an alga with two flagella that move collecting fluid from the front as in a breaststroke style [38] (right panel in Fig. 1.1b).
However, there are many situations in which the presence of the fluid can be neglected or taken into account effectively through friction forces. This is generally the case when local interactions dominate the dynamics like, for e.g., in twodimensional systems where the swimmers are in contact with a substrate that acts as a momentum sink and screens hydrodynamic effects. Biological examples are gliding bacteria like the Myxococcus xanthus (shown in Fig. 1.1c) whose motility relies on
specific pili [39], and also crawling eukaryotic cells which self-propel using their actin cytoscheleton [40].
At the same time, a variety of artificial microswimmers have been experimentally realized in order to reproduce the qualitative behavior of motile biological active matter [32, 33]. One of the simplest examples is provided by self-propelling liquid droplets [41]. Their motion is caused by a Marangoni flow [42] generated by a selfsustained gradient in the surface tension of the droplet. Experimentally, this effect has been shown for water droplets containing bromine in an oil suspension [43]. Another paradigmatic example is provided by Janus particles [44–46], spherical colloids with hemispheres of distinct physicochemical properties. These particles are typically half-coated with a layer of platinum (Pt) that catalyzes a chemical reaction in an aqueous solution. As schematically shown in Fig. 1.1d, this in turn breaks the symmetry of the reaction product distribution around the particle, giving rise to phoretic and osmotic forces [47, 48]: a propulsion mechanism known as self-diffusiophoresis.
All in all, a microswimmer can autonomously accomplish directed motion in several ways. We shall now discuss some minimal models that effectively describe the generic features of the motion of such microscopic active particles regardless of their specific propulsion mechanism.
1.1.2 Models of Active Particles
As anticipated at the beginning of the previous section, a particle immersed in a fluid is subject to the stochastic forces resulting from collisions with the fluid molecules. The corresponding motion that these interactions generate is called Brownian motion, named after the botanist Robert Brown who first noticed and described this phenomenon in 1827 [49]. The intensity of these fluctuations can be quantified via the so-called translational diffusion coefficient D which depends both on the fluid temperature T and on the particle mobility μ via the renowned Einstein relation [50]: D = μk B T , with k B being the Boltzmann constant. At the same time, the particle also undergoes rotational diffusion over time scales τr given by the inverse of the corresponding rotational diffusion coefficient Dr .
Thus, we can now write down the overdamped equations of motion of a passive particle immersed in a quiescent fluid in two dimensions as3 :
3 The generalization to the three-dimensional case is straightforward, with the particle position represented by Cartesian coordinates ( x , y , z ) and the orientation by the azimuthal and polar angles (φ, θ )
where r = x ˆ x + y ˆ y is the particle position, θ its orientation angle, while ξ and ξθ stand for independent delta-correlated white noises with unit variance, used to model the randomness of thermal fluctuations. These stochastic dynamics represent a model for what is known as a passive Brownian particle (PBP) whose motion is purely diffusive and each degree of freedom is completely decoupled from the others. The overdamped dynamics in Eq. (1.1) moreover implies that the average particle displacement with respect to its initial position will always vanish. These random fluctuations therefore hinder the ability of a microswimmer to perform an effective directed motion even in a quiescent fluid. It is then quite crucial for an active particle to find a good steering strategy that can overcome thermal fluctuations.
In this regard, there exists models with various degrees of sophistication to describe active motion [51, 52]. However, here we will ignore the specific selfpropulsion mechanisms and rather focus on a generic description. To this end, selfpropelling particles can be divided into two main classes. On the one hand, the so-called active Brownian particles (ABPs). In this case the microswimmer selfpropels at a constant speed v0 , but its orientation gradually changes due to rotational diffusion, which causes a non-trivial coupling between rotation and translation. The corresponding overdamped equations of motion in 2D are
where ˆ u = cos θ ˆ x + sin θ ˆ y is the unit vector corresponding to the particle intrinsic direction of motion. Note that for active particles, in general, noise could result from sources other than thermal. Activity itself can in fact give rise to stochastic fluctuations. A clear example are the chemical reactions underlying the self-propulsion mechanisms of Janus particles, whose motion observed in experiments is remarkably well reproduced by this simple effective model [44, 53]. In Fig. 1.2a are shown some sample trajectories of ABPs with different self-propelling speeds v0 obtained from numerical simulations of (1.2).
On the other hand, we find the so-called run-and-tumble particles (RTPs). As the name suggests, these microswimmers switch between two states: runs, where they move at constant speed v0 in a straight line, and tumbles, when they abruptly and randomly change their orientation. As a result, RTPs differ from ABPs in their orientation dynamics [54]. Namely, the tumbling events of a RTP are uncorrelated, occur at an average rate α and their number in a certain time window has a Poisson distribution, such that the tumbling probability is given by Ptumble = 1 e α .The orientation dynamics thus becomes [55]
Fig. 1.2 a Four exemplary trajectories of ABPs with different self-propulsion speeds v0 starting from r 0 = 0 and with an initial orientation θ0 = π /4. The translational and rotational diffusion coefficients are here set to D = 0 01 and Dr = 0 1, respectively. b Corresponding curves of the mean square displacement (MSD) obtained from numerical simulations taking an ensemble average over 103 trajectories. You can notice the three different scaling regimes that an ABP undergoes: diffusive at very short times, ballistic when τ /τr ≈ 1 and again diffusive at times τ /τr > 1.Legend and parameters as in a
where the sum runs over all the tumbling events occurred at times τi , while the angle variation ∆θi is drawn from a uniform distribution between 0 and 2π . This model of active particles has been developed to describe the motion of E. Coli bacteria [12] and Chlamydomonas algae [56] whose propulsion mechanisms have been already discussed above.
Although the dynamics at short times depend on the model adopted, both ABPs and RTPs share the same properties at long times. It is possible to quantify this by looking at the mean square displacement (MSD) of an active particle, which measures how much on average it departs from its initial position r 0 within a time τ , i.e. mathematically defined as MSD(τ ) =<( r (τ ) r 0 )2 >, where < > stands for the ensemble average. For an active particle, be it an ABP or a RTP, it is given by [57]
where τr is the typical time scale of rotational diffusion, such that τr = D 1 r or τr = α 1 for ABPs or RTPs, respectively. Depending on the observed time scale, we can distinguish three different regimes, as shown in Fig. 1.2b. In the limit τ < τr , (1.4) reduces to MSD(τ ) = 4 D τ , which is a purely diffusive motion with diffusion constant D and is also the exact solution for the MSD of a PBP. However, at slightly longer times, i.e. when τ ≈ τr , the motion is effectively ballistic since MSD(τ ) = 4 D τ + 2v 2 0 τ 2 ∝ τ 2 . Lastly, at time scales much larger than the characteristic rotational diffusion, i.e. τ > τr , we get MSD(τ ) = (4 D + 2v0 L )τ ∝ τ , such that we recover again a diffusive motion with an enhanced diffusion coefficient Deff = D + v0 L /2. We have here defined L ≡ v0 τr , known as the active particle persistence length, i.e. the length scale over which it travels straight on average before its direction is randomized.
There are, however, limiting regimes in which the impact of noise is negligible. A good measure is provided by the Péclet number (Pe) of the system. This dimensionless parameter quantifies the relative importance between directed motion and diffusive effects, and for an ABP can be defined as [32]
up to a constant numerical prefactor. In cases where Pe is large, the effects of stochastic fluctuations on the motion of the microswimmer can be neglected. These essentially correspond to situations in which the characteristic time scale of diffusion is much larger than the observed time window. In practice, this is the case when, e.g., active motion occurs in a high-viscosity fluid, such as honey, oils or long-chain hydrocarbons [58, 59].
We have thus shown that the motion of an active particle is characterised by both a ballistic regime at relatively short times and a diffusive one at longer times. Moreover, the transition between the two regimes turns out to be essentially governed by the swimmer persistence length, which in turn depends on its intrinsic activity. In practical terms, the ability of tuning this quantity can thus be crucial for the survival of biological microswimmers navigating in a complex environment [12, 60, 61].
1.1.3 Smart Active Particles
Through millions of years of evolution, biological microswimmers had to develop mechanisms to control their own activity in order to accomplish vital tasks. Some examples are the search strategies displayed by bacteria like E. Coli while looking for their nutrient [12, 34] or by spermatozoa trying to locate the egg [62]. Through the comparison of the chemical concentration levels along their path, these microorganisms are capable of altering their tumbling rate over time so as to effectively climb up the concentration gradient. This process is called klinokinesis with adaptation [63] and results in a biased random walk known as chemotaxis [64], which is depicted in Fig. 1.3.4
However, these adaptation mechanisms—first observed over 100 years ago [66 ]— characterize only a specific class of microswimmers since they require sophisticated sensory and motility machinery as well as internal information processing, which in turn involves memory. On the other hand, behaviors observed in nature have been a source of inspiration for the design of artificial self-propelled microswimmers capable of performing autonomously specialized tasks in complex environments [67, 68] in a much simpler fashion.
4 An analogous behavior has also been observed in photosynthetic biological microorganisms in the presence of light gradients and is thus called phototaxis [65].
Fig. 1.3 Graphical representation of bacterial chemotaxis. By self-regulating its own motility apparatus, and thereby its persistence length L , a microswimmer is able to climb up chemical concentration gradients autonomously. Here, the red arrow represents the swimmers heading direction ˆ u while the blue circles stand for the chemical particles, whose concentration grows from right to left as indicated by the bar gradient underneath. Qualitatively, something similar occurs in all other types of taxis, the main difference being in the sensory and motility machinery of the active particle at hand
An example is provided by chemically active particles like, e.g., synthetic phoretic colloids, which can exhibit (anti-)chemotaxis by (anti-)aligning their axis of orientation to the local gradient of (possibly self-produced) chemicals [69–72]. Remarkably, this reproduces well the behavior displayed by individual enzymes [8, 73].
Based on this simple mechanism—known as orthokinesis—it has therefore recently become possible to design synthetic microswimmers capable of responding to a variety of external stimuli such as light [74–76] or viscosity [77] gradients or even gravitational [78] or magnetic [79–81] fields.
Currently, a major challenge in active matter is understanding how such processes can be optimized [82, 83], both at the collective scale and at the individual active particle level.
1.2 Optimal Control Theory
The proper mathematical framework for rigorously studying optimization problems is known as optimal control theory [84]. Building on the pioneering works by Richard Bellman [85] and Lev Pontryagin [86] from last century, this theoretical tool has become very popular also outside the mathematical sciences thanks to its many applications in engineering [87, 88] and finance [89]. Optimal control (OC) is essentially an extension of variational calculus aimed at finding the control strategy that minimizes/maximizes a given objective functional, also known as cost function
Let us now briefly illustrate the main ideas that lead to the formal solution [90]. Consider the deterministic dynamics of an active agent generally described by the following n -dimensional differential equation
where t indicates the time and q represents the state of the agent. The variable c is the so-called control upon which one can act to manipulate the state q and, in fact, it does not have a predefined equation of motion. The system (1.6) is furthermore equipped with the boundary conditions
q (t0 ) = q 0 qi (tf ) = qi , f for i = 1,..., k ,
where k ≤ n , while t0 and tf are the initial and the final times respectively. In the following, we will treat the more general case in which tf is not specified.
The optimization problem can actually be addressed with tools similar to those used in analytical mechanics. Let us therefore consider a cost function of the form
where φ and L are known as the endpoint cost and the running cost respectively, although the latter can be identified as a Lagrangian. The goal is now to determine the history of c(t ) that would minimize/maximize C . To this end, we should first adjoin the equations of motion (1.6) to the Lagrangian introducing a vector of Lagrange multipliers p, whose components are the generalized momenta conjugated to the state variable q :
We may identify the system Hamiltonian as
and then integrate by parts the last term in (1.8), yielding
where we have dropped the explicit dependencies. Let us now compute the variation in the cost C , which reads
where we have used the identity δ q (tf ) = d q (tf ) ˙ q (tf )d tf and placed δ q 0 = 0, since q (t0 ) is fixed. At optimality, the cost function has to be stationary, i.e. δ C = 0.By requiring this to hold for arbitrary variations of q , c and tf , we obtain the necessary conditions for optimality, also known as Pontryagin’s principle [86]:
together with the boundary conditions:
where n is the dimensionality of the system (1.6). These last two systems, along with the equation of motion (1.6), make up the two-point boundary-value problem to be solved in order to obtain the OC strategy.5
1.2.1 Stochastic Optimal Control
The formalism presented in the previous section does not take into account the possible presence of stochastic fluctuations affecting the system’s dynamics. These in fact characterize motion at the microscopic scale and are therefore a key ingredient to be included in a thorough treatment of optimization problems in active matter. To this end, however, one must resort to a slightly different approach, which we illustrate in the following.
Let us consider a system governed by the following stochastic differential equations
with D being a diffusion constant and ξ a n –dimensional white noise delta-correlated process with unit variance. We will hereafter assume that the initial state of the system is fixed q (t0 ) = q 0 and the final time is unconstrained. The aim here is to find the control c(t ) that minimizes
which is analogous to the cost function (1.7) with the only difference that here we have to take an ensemble average over all trajectories starting from q 0 . We can now formally define the optimal cost function, also called value function,simplyas
5 Note that the last condition in system (1.10) is necessary if and only if the final time tf is not specified.
where c(t → tf ) is the control sequence between an intermediate time t and the final time tf . Then, according to Bellman’s principle of optimality [91], this function can be expressed recursively as
The Taylor expansion of the first term on the right reads
up to first order in d t ,wherewehaveusedItô’s lemma [92] <d q 2 >= O (d t ). Plugging then this back into (1.15), dividing by d t both sides and taking the limit d t → 0 gives
which is the stochastic version of the so-called Hamilton-Jacobi-Bellman (HJB) equation [93] with boundary condition C ( q , tf ) = φ( q (tf )). Finally, the minimization with respect to the control c of the expression in square brackets is carried out by imposing the optimality condition [94]
Thus, inserting the expression of the control c obtained from (1.17)into(1.16), we get the n -dimensional partial differential equation to be solved in order to address the stochastic optimal control (SOC) problem.
Note that this approach can also be used to obtain the OC strategy in the deterministic case by simply setting D = 0. This would therefore be an equivalent alternative to employing Pontryagin’s principle (1.10) introduced above. There, one needs instead to solve a set of 2n ordinary differential equations with two-point boundary conditions. Depending on the situation, one shall decide which approach is more convenient on a case-by-case basis.
This equivalence can be readily worked out by identifying the value function gradients, ∇ q C , with the generalized momenta p at D = 0. The system Hamiltonian (1.9) in that case indeed takes the form H = L + F · ∇ q C , such that the first and second necessary conditions of Pontryagin’s principle (1.10) match exactly with (1.16) and (1.17) respectively.6 The third condition in (1.10) can instead be straightforwardly recovered from Eq. (1.16), namely ∂t C =−H, and then considering the boundary condition C ( q , tf ) = φ( q (tf ))
6 Specifically, the first correspondence can be proven by taking the gradients of both sides in (1.16).
We will now show the application of both these equivalent approaches in some practical examples.
1.3 The Problem of Optimal Navigation
Finding the fastest path towards a desired destination can be highly beneficial for living organisms tracking a food source [15, 95, 96], a potential mate [97], or escaping from toxic areas [98] or predators [99]. Moreover, most often their motion happens in the presence of a fluid flow or an external force, which can hinder their navigation. The optimal path is therefore distinct from the shortest one, making this a complex problem in the field of active matter.
At the same time, thanks to the recent theoretical and experimental advances in the design of artificial microswimmers [44, 45, 48, 51, 67, 100], addressing this issue has important technological applications ranging from environmental sustainability and monitoring [101, 102], route planning [103, 104] to targeted cargo delivery at the microscale [105–107].
Hence, it is of no surprise that this problem is one of the oldest known applications of OC theory. Its classical solution can be traced back to Ernst Zermelo’s work in 1931 [108]. There, he addressed the problem of how to steer a vessel navigating at constant speed v0 in the presence of a stationary wind f ( r ) so as to minimize the travel time to reach a given destination. Please refer to Fig. 1.4 for an illustration of the scenario. In this context, the equations governing the motion of the ship are
where r = x
y is the vessel position and
its heading direction.
Fig. 1.4 Diagrammatic illustration of Zermelo’s problem for a ship navigating under constant wind (black arrow). In order to reach the destination in the shortest time, the ship must be oriented (red arrow) so as to compensate for the transversal component of the wind. In this case indeed ZP indicates that θ = θ0 (const), that is, the optimal path is simply straight (green line)
The solution to this problem can be readily found applying Pontryagin’s principle (1.10) from OC theory [90]. Since the goal here is to minimize the total travel time, we may simply place L = 1 and φ = 0 in the cost function (1.7), such that the system Hamiltonian (1.9) takes the form
where the control variable is the ship’s heading angle θ and the state is given by its position vector r . In this specific case, the necessary conditions for optimality (1.10) thus translate into the following system
where the last condition holds at any time since the Hamiltonian does not depend explicitly on time and is thus a conserved quantity. We may then use the third equation to replace p by θ in this system so to obtain
which is the sought optimal steering strategy that, solved together with the equation of motion (1.18), minimizes the travel time. Hereafter, we will refer to it as Zermelo’s Policy (ZP). One then just has to select7 the proper initial orientation θ0 so as to get the minimum time path that reaches the desired destination, which henceforth will be referred to as Zermelo’s path.
For the sake of illustration, let us now consider the case where a self-propelled agent moves in the presence of a shear flow f ( r ) = κ y ˆ x with κ > 0. Starting from the origin, the task is to reach in the shortest time another stationary point r T at distance along the x axis, i.e. r T = ˆ x with > 0. According to Zermelo’s solution, this simply amounts to solving the following set of ordinary differential equations (ODEs)
where we have rescaled space and time as r → r and t → t /κ, such that the dynamics is characterized by only one non-dimensional parameter, namely ζ ≡ v0 /κ .This ODE system can be solved exactly and the corresponding solution reads
7 This can be achieved using a shooting method [109].
Fig. 1.5 Optimal navigation in a shear flow. a The ZP solution (green curve) has been obtained by integrating (1.22)at ζ = 1 for an agent starting from the origin r 0 = 0 (green circle) and aiming at another point downstream r T = ˆ x (magenta circle). The colour here codes for the flow intensity and the black arrows indicate its direction. b Heat map of 103 stochastic trajectories obtained from numerical simulations of OP in the same setup as in a with the diffusivity set to D = 0.01.The white arrows indicate the corresponding optimal control map ˆ u opt (θ ( r ))
x (t ) = x 0 + y0 t + ζ 2 [arcsinh [(t
y (t ) = y0 ζ [ψ (t ) λ]
θ (t ) = arctan (tan θ0 t ), (1.23)
where we have defined λ ≡ √1 + tan 2 θ0 and
) ≡
1 + (
0 )2 , with ( x 0 , y0 ) and θ0 being the initial position and orientation of the active agent, respectively.
In order to solve the full problem, all that is left to determine are the initial angle θ0 and the (optimal) arrival time at the target topt . These depend on the relative position of the target r T and the initial point r 0 . Since in our case these share the same y-coordinate, i.e. yT = y (topt ) = y0 , from (1.23) we obtain topt = 2tan θ0 . (1.24)
Moreover, the target position is such that x T x 0 = x (topt ) x 0 = 1 (in units of ). This condition, together with (1.23) and (1.24), implies arcsinh (2tan θ0 λ) + 2tan θ0 λ = 2/ζ , (1.25)
which is an implicit equation to be solved for θ0 . Hence, the full system of Eqs. (1.23–1.25) finally represents the complete analytical solution to the optimal navigation problem in a shear flow. The resulting Zermelo’s path corresponds to the green curve shown in Fig. 1.5a. Here, the agent exploits the flow to reach the target faster. Nevertheless, it also has to be cautious not to go too far in the vertical direction to avoid being carried away by the flow, making this a non-trivial optimization problem. Zermelo’s approach, however, does not account for thermal fluctuations, which play a prominent role at the micro-scale. In such a case, one must therefore reframe the problem of optimal navigation within the context of SOC theory. To this end,
let us consider a self-propelled particle moving on the plane at fixed speed v0 in presence of a stationary force field f ( r ) and translational diffusion with diffusivity D . Its motion obeys the following stochastic differential equation
which just corresponds to a particular choice for the dynamics in (1.12).
Since our goal is to minimize the total travel time to reach a target at position r T , we may identify the particle position with the state variable and simply set φ = 0 and L = 1 in the definition of the cost function (1.13), which will thus take the form C ( r ) = <tf t0 > r depending only on the initial position r . As a result, the value function (1.14) can be identified with the mean first-passage time T ( r ) (MFPT) and the corresponding HJB equation (1.16) reads
where the gradients are taken with respect to r . According to the optimality condition (1.17), the optimal choice for the heading direction θ (our control variable) is then obtained by
valid at every point in space. As the desired strategy should minimize T ( r ),we expect the relevant control to point down its gradient and shall thus retain only the solution with a negative sign in (1.28). Inserting this into (1.27), we finally get the final expression for the stochastic HJB equation for the MFPT:
to be solved with the boundary condition T ( r T ) = 0. The system composed by the last two Eqs. (1.28) and (1.29) hence provides the strategy which, by design, leads to the fastest possible trajectories on average. Hereafter, we will thus refer to it as the Optimal Policy (OP).
Figure 1.5b shows an example of the vector field corresponding to the OP control map ˆ u opt (θ ( r )) obtained in the same setup used to illustrate the deterministic case (shown in Fig. 1.5a). Simulations of the Brownian dynamics (1.26) carried out with such control reveal that the stochastic trajectories tend to be symmetrically spread around Zermelo’s path. This is consistent with the fact that Zermelo’s solution is indeed recovered from OP in the limit D → 0, an observation with important implications that will be discussed extensively in Chap. 4
1.3.1 An Alternative Approach: Reinforcement Learning
Living organisms have shaped their behavior, functionality and morphology over millions of years in order to find the right balance between costs, risks and benefits. Evolution is therefore ultimately an optimisation process occurring as a result of the continuous interaction between an organism and its surroundings through what is called a feedback loop. This cycle, shown in Fig. 1.6, is also at the very heart of reinforcement learning (RL), a broad branch of machine learning algorithms that covers all possible decision-making problems [110], with potential applications ranging from engineering [111] and robotics [112, 113] to biology and active matter [23, 114–117].
In a nutshell, RL algorithms formalize the concept of learning how to carry out a task via trials and error. Let us imagine an agent moving in a certain environment (please refer to Fig. 1.6). Its actions lead to a change in the state of the environment (e.g. its position), which are then externally interpreted into rewards and finally fed back into the agent. Through a number of iterations, this agent tries to maximize the cumulative reward given a sequence of actions and, based on the experience gathered, it eventually learns the (ideally) optimal strategy.
In contrast to OC theory, approaches based on RL make it possible to study problems in which the dynamics of the environment is not exactly known a priori [118]. Let us imagine a glider that has to be airborne for as long as possible: it can only measure certain properties of its surroundings such as local temperature, pressure or wind direction, yet the pilot does not have enough information to be able to plan an
Fig. 1.6 Schematic representation of the feedback loop at the basis of both evolution of biological systems and RL algorithms. The agent, be it an artificial active particle, a robot or a living organism, modifies the state of the environment via its actions; it then receives updated information about the state of the environment together with a certain reward and, by gathering experience over time, will thus adapt its actions accordingly
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Hempie had, indeed, taken the news of the Crabapple Blossoms very calmly. It was true she had never cared very much for Prunella, maintaining always that "she was just her mother over again." All the same, Prunella remained Master Nathaniel's daughter and Ranulph's sister, and hence had a certain borrowed preciousness in the eyes of Hempie. Nevertheless she had refused to indulge in lamentations, and had preserved on the subject a rather grim silence.
His eye roved restlessly over the familiar room. It was certainly a pleasant one—fantastic and exquisitely neat. "Neat as a Fairy's parlour"—the old Dorimarite expression came unbidden to his mind.
There was a bowl of autumn roses on the table, faintly scenting the air with the hospitable, poetic perfume that is like a welcome to a little house with green shutters and gay chintzes and lavenderscented sheets. But the host who welcomes you is dead, the house itself no longer stands except in your memory—it is the cry of the cock turned into perfume. Are there bowls of roses in the Fairies' parlours?
"I say, Hempie, these are new, aren't they?" he said, pointing to a case of shells on the chimney-piece—very strange shells, as thin as butterfly's wings and as brightly coloured. And, as well, there were porcelain pots, which looked as if they had been made out of the petals of poppies and orchids, nor could their strange shapes ever have been turned on a potter's wheel in Dorimare.
Then he gave a low whistle, and, pointing to a horse-shoe of pure gold, nailed on to the wall, he added, "And that, too! I'll swear I've never seen it before. Has your ship come in, Hempie?"
The old woman looked up placidly from her darning: "Oh! these came when my poor brother died and the old home was broken up. I'm glad to have them, as I never remember a time when they weren't in the old kitchen at home. I often think it's strange how bits of chiney and brittle stuff like that lives on, long after solid flesh and bone has turned to dust. And it's a queer thing, Master Nat, as one gets old, how one lives among the dumb. Bits of chiney ... and the Silent People," and she wiped a couple of tears from her eyes.
Then she added, "Where these old bits of things came from I never rightly knew. I suppose the horse-shoe's valuable, but even in bad harvests my poor father would never turn it into money. He used to say that it had been above our door in his father's time, and in his grandfather's time, and it had best stay there. I shouldn't wonder if he thought it had been dropped by Duke Aubrey's horse. And as for the shells and pots ... when we were children, we used always to whisper that they came from beyond the hills."
Master Nathaniel gave a start, and stared at her in amazement.
"From beyond the hills?" he repeated, in a low, horrified voice.
"Aye, and why not?" cried Hempie, undaunted. "I was country-bred, Master Nat, and I learned not to mind the smell of a fox or of a civet cat ... or of a Fairy. They're mischievous creatures, I daresay, and best left alone. But though we can't always pick and choose our neighbours, neighbourliness is a virtue all the same. For my part, I'd never have chosen the Fairies for my neighbours—but they were chosen for me. And we must just make the best of them."
"By the Sun, Moon, and Stars, Hempie!" cried Master Nathaniel in a horrified voice, "you don't know what you're talking about, you...."
"Now, Master Nat, don't you try on your hoighty-toighty-his-Worshipthe-Mayor-of-Lud-in-the-Mist-knock-you-down-and-be-thankful-forsmall-mercies ways with me!" cried Hempie, shaking her fist at him.
"I know very well what I'm talking about. Long, long ago I made up my mind about certain things. But a good nurse must keep her mind to herself—if it's not the same as that of her master and mistress. So I never let on to you when you were a little boy, nor to Master Ranulph neither, what I thought about these things. But I've never held with fennel and such like. If folks know they're not wanted, it just makes them all the more anxious to come—be they Fairies or Dorimarites. It's just because we're all so scared of our neighbours that we get bamboozled by them. And I've always held that a healthy stomach could digest anything—even fairy fruit. Look at my boy, now, at Ranulph—young Luke writes he's never looked so bonny. No, fairy fruit nor nothing else can poison a clean stomach."
"I see," said Master Nathaniel drily He was fighting against the sense of comfort that, in spite of himself, her words were giving him. "And are you quite happy, too, about Prunella?"
"Well, and even if I'm not," retorted Hempie, "where's the good of crying, and retching, and belching, all day long, like your lady downstairs? Life has its sad side, and we must take the rough with the smooth. Why, maids have died on their marriage eve, or, what's worse, bringing their first baby into the world, and the world's wagged on all the same. Life's sad enough, in all conscience, but there's nothing to be frightened about in it or to turn one's stomach. I was country-bred, and as my old granny used to say, 'There's no clock like the sun and no calendar like the stars.' And why? Because it gets one used to the look of Time. There's no bogey from over the hills that scares one like Time. But when one's been used all one's life to seeing him naked, as it were, instead of shut up in a clock, like he is in Lud, one learns that he is as quiet and peaceful as an old ox dragging the plough. And to watch Time teaches one to sing. They say the fruit from over the hills makes one sing. I've never tasted so much as a sherd of it, but for all that I can sing."
Suddenly, all the pent-up misery and fear of the last thirty years seemed to be loosening in Master Nathaniel's heart—he was sobbing, and Hempie, with triumphant tenderness, was stroking his hands and murmuring soothing words, as she had done when he was a little boy.
When his sobs had spent themselves, he sat down on a stool at her feet, and, leaning his head against her knees, said, "Sing to me, Hempie."
"Sing to you, my dear? And what shall I sing to you? My voice isn't what it once was ... well, there's that old song—'Columbine,' I think they call it—that they always seem singing in the streets these days —that's got a pretty tune."
And in a voice, cracked and sweet, like an old spinet, she began to sing:
"When Aubrey did live there lived no poor
The lord and the beggar on roots did dine With lily, germander, and sops in wine. With sweet-brier, And bon-fire, And strawberry-wire, And columbine."
As she sang, Master Nathaniel again heard the Note. But, strange to say, this time it held no menace. It was as quiet as trees and pictures and the past, as soothing as the drip of water, as peaceful as the lowing of cows returning to the byre at sunset.
CHAPTER XI
A STRONGER ANTIDOTE THAN REASON
Master Nathaniel sat at his old nurse's feet for some minutes after she had stopped singing. Both his limbs and his mind seemed to be bathed in a cool, refreshing pool.
So Endymion Leer and Hempie had reached by very different paths the same conclusion—that, after all, there was nothing to be frightened about; that, neither in sky, sea, nor earth was there to be found a cavern dark and sinister enough to serve as a lair for IT—his secret fear.
Yes, but there were facts as well as shadows. Against facts Hempie had given him no charm. Supposing that what had happened to Prunella should happen to Ranulph? That he should vanish for ever across the Debatable Hills.
But it had not happened yet—nor should it happen as long as Ranulph's father had wits and muscles.
He might be a poor, useless creature when menaced by the figments of his own fancy. But, by the Golden Apples of the West, he would no longer sit there shaking at shadows, while, perhaps, realities were mustering their battalions against Ranulph.
It was for him to see that Dorimare became a country that his son could live in in security.
It was as if he had suddenly seen something white and straight—a road or a river—cutting through a sombre, moonlit landscape. And the straight, white thing was his own will to action. He sprang to his feet and took two or three paces up and down the room.
"But I tell you, Hempie," he cried, as if continuing a conversation, "they're all against me. How can I work by myself! They're all against me, I say."
"Get along with you, Master Nat!" jeered Hempie tenderly. "You were always one to think folks were against you. When you were a little boy it was always, 'You're not cross with me, Hempie, are you?' and peering up at me with your little anxious eyes—and there was me with no more idea of being cross with you than of jumping over the moon!"
"But, I tell you, they are all against me," he cried impatiently "They blame me for what has happened, and Ambrose was so insulting that I had to tell him never to put his foot into my house again."
"Well, it isn't the first time you and Master Ambrose have quarrelled —and it won't be the first time you make it up again. It was, 'Hempie, Brosie won't play fair!' or 'Hempie, it's my turn for a ride on the donkey, and Nat won't let me!' And then, in a few minutes, it was all over and forgotten. So you must just step across to Master Ambrose's, and walk in as if nothing had happened, and, you'll see, he'll be as pleased as Punch to see you."
As he listened, he realized that it would be very pleasant to put his pride in his pocket and rush off to Ambrose and say that he was willing to admit anything that Ambrose chose—that he was a hopelessly inefficient Mayor, that his slothfulness during these past months had been criminal—even, if Ambrose insisted, that he was an eater of, and smuggler of, and receiver of, fairy fruit, all rolled into one—if only Ambrose would make friends again.
Pride and resentment are not indigenous to the human heart; and perhaps it is due to the gardener's innate love of the exotic that we take such pains to make them thrive.
But Master Nathaniel was a self-indulgent man, and ever ready to sacrifice both dignity and expediency to the pleasure of yielding to a sentimental velleity.
"By the Golden Apples of the West, Hempie," he cried joyfully, "you're right! I'll dash across to Ambrose's before I'm a minute older,"
and he made eagerly for the door
On the threshold he suddenly remembered how he had seen the door of his chapel ajar, and he paused to ask Hempie if she had been up there recently, and had forgotten to lock it.
But she had not been there since early spring.
"That's odd!" said Master Nathaniel.
And then he dismissed the matter from his mind, in the exhilarating prospect of "making up" with Ambrose.
It is curious what tricks a quarrel, or even a short absence, can play with our mental picture of even our most intimate friends. A few minutes later, as Master Ambrose looked at his old playmate standing at the door, grinning a little sheepishly, he felt as if he had just awakened from a nightmare. This was not "the most criminally negligent Mayor with whom the town of Lud-in-the-Mist had ever been cursed;" still less was it the sinister figure evoked by Endymion Leer. It was just queer old Nat, whom he had known all his life.
Just as on a map of the country round Lud, in the zig-zagging lines he could almost see the fish and rushes of the streams they represented, could almost count the milestones on the straight lines that stood for roads; so, with regard to the face of his old friend— every pucker and wrinkle was so familiar that he felt he could have told you every one of the jokes and little worries of which they were the impress.
Master Nathaniel, still grinning a little sheepishly, stuck out his hand. Master Ambrose frowned, blew his nose, tried to look severe, and then grasped the hand. And they stood there fully two minutes, wringing each other's hand, and laughing and blinking to keep away the tears.
And then Master Ambrose said, "Come into the pipe-room, Nat, and try a glass of my new flower-in-amber. You old rascal, I believe it was that that brought you!"
A little later when Master Ambrose was conducting Master Nathaniel back to his house, his arm linked in his, they happened to pass Endymion Leer.
For a few seconds he stood staring after them as they glimmered down the lane beneath the faint moonlight. And he did not look overjoyed.
That night was filled to the brim for Master Nathaniel with sweet, dreamless sleep. As soon as he laid his head on the pillow he seemed to dive into some pleasant unknown element—fresher than air, more caressing than water; an element in which he had not bathed since he first heard the Note, thirty years ago. And he woke up the next morning light-hearted and eager; so fine a medicine was the will to action.
He had been confirmed in it by his talk the previous evening with Master Ambrose. He had found his old friend by no means crushed by his grief. In fact, his attitude to the loss of Moonlove rather shocked Master Nathaniel, for he had remarked grimly that to have vanished for ever over the hills was perhaps, considering the vice to which she had succumbed, the best thing that could have happened to her. There had always been something rather brutal about Ambrose's common sense.
But he was as anxious as Master Nathaniel himself that drastic measures should immediately be taken for stopping the illicit trade and arresting the smugglers. They had decided what these measures ought to be, and the following days were spent in getting them approved and passed by the Senate.
Though the name of Master Nathaniel stank in the nostrils of his colleagues, their respect for the constitution was too deep seated to permit their opposing the Mayor of Lud-in-the-Mist and High Seneschal of Dorimare; besides, Master Ambrose Honeysuckle was a man of considerable weight in their councils, and they were not
uninfluenced by the fact that he was the seconder of all the Mayor's proposals.
So a couple of Yeomen were placed at each of the gates of Lud, with orders to examine not only the baggage of everyone entering the town, but, as well, to rummage through every waggon of hay, every sack of flour, every frail of fruit or vegetables. As well, the West road was patrolled from Lud to the confines of the Elfin Marches, where a consignment of Yeomanry were sent to camp out, with orders day and night to watch the hills. And the clerk to the Senate was ordered to compile a dossier of every inhabitant of Lud.
The energy displayed by Master Nathaniel in getting these measures passed did a good deal towards restoring his reputation among the townsfolk. Nevertheless that social barometer, Ebeneezor Prim, continued to send his new apprentice, instead of coming himself, to wind his clocks. And the grandfather clock, it would seem, was protesting against the slight. For according to the servants, it would suddenly move its hands rapidly up and down its dial, which made it look like a face, alternating between a smirk and an expression of woe. And one morning Pimple, the little indigo page, ran screaming with terror into the kitchen, for, he vowed, from the orifice at the bottom of the dial, there had suddenly come shooting out a green tongue like a lizard's tail.
As none of Master Nathaniel's measures brought to light a single smuggler or a single consignment of fairy fruit, the Senate were beginning to congratulate themselves on having at last destroyed the evil that for centuries had menaced their country, when Mumchance discovered in one day three people clearly under the influence of the mysterious drug and with their mouth and hands stained with strangely coloured juices.
One of them was a pigmy pedlar from the North, and as he scarcely knew a word of Dorimarite no information could be extracted from him as to how he had procured the fruit. Another was a little street urchin who had found some sherds in a dustbin, but was in too dazed a state to remember exactly where. The third was the deaf-
mute known as Bawdy Bess. And, of course, no information could be got from a deaf-mute.
Clearly, then, there was some leakage in the admirable system of the Senate.
As a result, rebellious lampoons against the inefficient Mayor were found nailed to the doors of the Guildhall, and Master Nathaniel received several anonymous letters of a vaguely threatening nature, bidding him to cease to meddle with matters that did not concern him, lest they should prove to concern him but too much.
But so well had the antidote of action been agreeing with his constitution that he merely flung them into the fire with a grim laugh and a vow to redouble his efforts.
CHAPTER XII
DAME
MARIGOLD HEARS
THE TAP OF A WOODPECKER
Miss Primrose Crabapple's trial was still dragging on, clogged by all the foolish complications arising from the legal fiction that had permitted her arrest. If you remember, in the eye of the law fairy fruit was regarded as woven silk, and many days were wasted in a learned discussion of the various characteristics of gold tissues, stick tuftaffities, figured satins, wrought grograines, silk mohair and ferret ribbons.
Urged partly by curiosity and, perhaps, also by a subconscious hope that in the comic light of Miss Primrose's personality recent events might lose something of their sinister horror, one morning Dame Marigold set out to visit her old schoolmistress in her captivity.
It was the first time she had left the house since the tragedy, and, as she walked down the High Street she held her head high and smiled a little scornful smile—just to show the vulgar herd that even the worst disgrace could not break the spirit of a Vigil.
Now, Dame Marigold had very acute senses. Many a time she had astonished Master Nathaniel by her quickness in detecting the faintest whiff of any of the odours she disliked—shag, for instance, or onions.
She was equally quick in psychological matters, and would detect the existence of a quarrel or love affair long before they were known to anyone except the parties concerned. And as she made her way that morning to the Guildhall she became conscious in everything that was going on round her of what one can only call a change of key.
She could have sworn that the baker's boy with the tray of loaves on his head was not whistling, that the maid-servant, leaning out of a window to tend her mistress's pot-flowers, was not humming the same tune that they would have been some months ago.
This, perhaps, was natural enough. Tunes, like fruit, have their seasons, and are, besides, ever forming new species. But even the voices of the hawkers chanting "Yellow Sand!" or "Knives and Scissors!" sounded disconcertingly different.
Instinctively, Dame Marigold's delicate nostrils expanded, and the corners of her mouth turned down in an expression of disgust, as if she had caught a whiff of a disagreeable smell.
On reaching the Guildhall, she carried matters with a high hand. No, no, there was no need whatever to disturb his Worship. He had given her permission to visit the prisoner, so would the guardian take her up immediately to her room.
Dame Marigold was one of those women who, though they walk blindfold through the fields and woods, if you place them between four walls have eyes as sharp as a naturalist's for the objects that surround them. So, in spite of her depression, her eyes were very busy as she followed the guardian up the splendid spiral staircase, and along the panelled corridors, hung, here and there, with beautiful bits of tapestry. She made a mental note to tell Master Nathaniel that the caretaker had not swept the staircase, and that some of the panelling was worm-eaten and should be attended to. And she would pause to finger a corner of the tapestry and wonder if she could find some silk just that powder blue, or just that old rose, for her own embroidery.
"Why, I do declare, this panel is beginning to go too!" she murmured, pausing to tap on the wall.
Then she cried in a voice of surprise, "I do believe it's hollow here!"
The guardian smiled indulgently—"You are just like the doctor, ma'am—Doctor Leer. We used to call him the Woodpecker, when he was studying the Guildhall for his book, for he was for ever hopping about and tapping on the walls. It was almost as if he were looking
for something, we used to say And I'd never be surprised myself to come on a sliding panel. They do say as what those old Dukes were a wild crew, and it might have suited their book very well to have a secret way out of their place!" and he gave a knowing wink.
"Yes, yes, it certainly might," said Dame Marigold, thoughtfully. They had now come to a door padlocked and bolted. "This is where we have put the prisoner, ma'am," said the guardian, unlocking it. And then he ushered her into the presence of her old schoolmistress.
Miss Primrose was sitting bolt upright in a straight backed old fashioned chair, against a background of fine old tapestries, faded to the softest loveliest pastel tints—as incongruous with her grotesque ugliness as had been the fresh prettiness of the Crabapple Blossoms.
Dame Marigold stood staring at her for a few seconds in silent indignation. Then she sank slowly on to a chair, and said sternly, "Well, Miss Primrose? I wonder how you dare sit there so calmly after the appalling thing you have brought about."
But Miss Primrose was in one of her most exalted moods—"On her high hobby-horse," as the Crabapple Blossoms used to call it. So she merely glittered at Dame Marigold contemptuously out of her little eyes, and, with a lordly wave of her hand, as if to sweep away from her all mundane trivialities, she exclaimed pityingly, "My poor blind Marigold! Perhaps of all the pupils who have passed through my hands you are the one who are the least worthy of your noble birthright."
Dame Marigold bit her lip, raised her eyebrows, and said in a low voice of intense irritation, "What do you mean, Miss Primrose?"
Miss Primrose cast her eyes up to the ceiling, and, in her most treacly voice she answered, "The great privilege of having been born a woooman!"
Her pupils always maintained that "woman," as pronounced by Miss Primrose, was the most indecent word in the language.
Dame Marigold's eyes flashed: "I may not be a woman, but, at any rate, I am a mother—which is more than you are!" she retorted.
Then, in a voice that at each word grew more indignant, she said, "And, Miss Primrose, do you consider that you yourself have been 'worthy of your noble birthright' in betraying the trust that has been placed in you? Are vice and horror and disgrace and breaking the hearts of parents 'true womanliness' I should like to know? You are worse than a murderer—ten times worse. And there you sit, gloating over what you have done, as if you were a martyr or a public benefactor—as complacent and smug and misunderstood as a princess from the moon forced to herd goats! I do really believe...."
But Miss Primrose's shrillness screamed down her low-toned indignation: "Shake me! Stick pins in me! Fling me into the Dapple!" she shrieked. "I will bear it all with a smile, and wear my shame like a flower given by him!"
Dame Marigold groaned in exasperation: "Who, on earth, do you mean by 'him', Miss Primrose?"
Then her irrepressible sense of humour broke out in a dimple, and she added: "Duke Aubrey or Endymion Leer?"
For, of course, Prunella had told her all the jokes about the goose and the sage.
At this question Miss Primrose gave an unmistakable start; "Duke Aubrey, of course!" she answered, but the look in her eyes was sly, suspicious, and distinctly scared.
None of this was lost upon Dame Marigold. She looked her slowly up and down with a little mocking smile; and Miss Primrose began to writhe and to gibber.
"Hum!" said Dame Marigold, meditatively.
She had never liked the smell of Endymion Leer's personality
The recent crisis had certainly done him no harm. It had doubled his practice, and trebled his influence.
Besides, it cannot have been Miss Primrose's beauty and charms that had caused him to pay her recently such marked attentions.
At any rate, it could do no harm to draw a bow at a venture.
"I am beginning to understand, Miss Primrose," she said slowly. "Two ... outsiders, have put their heads together to see if they could find a plan for humiliating the stupid, stuck-up, 'so-called old families of Lud!' Oh! don't protest, Miss Primrose. You have never taken any pains to hide your contempt for us. And I have always realized that yours was not a forgiving nature. Nor do I blame you. We have laughed at you unmercifully for years—and you have resented it. All the same I think your revenge has been an unnecessarily violent one; though, I suppose, to 'a true woooman,' nothing is too mean, too spiteful, too base, if it serves the interests of 'him'!"
But Miss Primrose had gone as green as grass, and was gibbering with terror: "Marigold! Marigold!" she cried, wringing her hands, "How can you think such things? The dear, devoted Doctor! The best and kindest man in Lud-in-the-Mist! Nobody was angrier with me over what he called my 'criminal carelessness' in allowing that horrible stuff to be smuggled into my loft, I assure you he is quite rabid on the subject of ... er ... fruit. Why, when he was a young man at the time of the great drought he was working day and night trying to stop it, he...."
But not for nothing was Dame Marigold descended from generations of judges. Quick as lightning, she turned on her: "The great drought? But that must be forty years ago ... long before Endymion Leer came to Dorimare."
"Yes, yes, dear ... of course ... quite so ... I was thinking of what another doctor had told me ... since all this trouble my poor head gets quite muddled," gibbered Miss Primrose. And she was shaking from head to foot.
Dame Marigold rose from her chair, and stood looking down on her in silence for a few seconds, under half-closed lids, with a rather cruel little smile.
Then she said, "Good-bye, Miss Primrose. You have provided me with most interesting food for thought."
And then she left her, sitting there with frightened face against the faded tapestry.
That same day, Master Nathaniel received a letter from Luke Hempen that both perplexed and alarmed him. It was as follows:
Your Worship,—I'd be glad if you'd take Master Ranulph away from this farm, because the widow's up to mischief, I'm sure of that, and some of the folks about here say as what in years gone by she murdered her husband, and she and somebody else, though I don't know who, seem to have a grudge against Master Ranulph, and, if I might take the liberty, I'll just tell your Worship what I heard. It was this way—one night, I don't know how it was, but I couldn't get to sleep, and thinking that a bite, may be, of something would send me off, towards midnight I got up from my bed to go and look in the kitchen for a bit of bread. And half-way down the stairs I heard the sound of low voices, and someone said, "I fear the Chanticleers," so I stood still where I was, and listened. And I peeped down and the kitchen fire was nearly out, but there was enough left for me to see the widow, and a man wrapped up in a cloak, sitting opposite to her with his back to the stairs, so I couldn't see his face. Their talk was low and at first I could only hear words here and there, but they kept making mention of the Chanticleers, and the man said something like keeping the Chanticleers and Master Ambrose Honeysuckle apart, because Master Ambrose had had a vision of Duke Aubrey. And if I hadn't known the widow and how she was a deep one and as fly as you make them, I'd have thought they were two poor daft old gossips, whose talk had turned wild and nasty with old age. And then the man laid his hand on her knee, and his voice was low, but this time it was so clear that I could
hear it all, and I think I can remember every word of it, so I'll write it down for your Worship: "I fear counter orders. You know the Chief and his ways—at any moment he might betray his agents. Willy Wisp gave young Chanticleer fruit without my knowledge. And I told you how he and that doitered old weaver of yours have been putting their heads together, and that's what has frightened me most."
And then his voice became too low for me to hear, till he said, "Those who go by the Milky Way often leave footprints. So let him go by the other."
And then he got up to go, and I crept back to my room. But not a wink of sleep did I get that night for thinking over what I had heard. For though it seemed gibberish, it gave me the shivers, and that's a fact. And mad folks are often as dangerous as bad ones, so I hope your Worship will excuse me writing like this, and that you'll favour me with an answer by return, and take Master Ranulph away, for I don't like the look in the widow's eye when she looks at him, that I don't.
And hoping this finds your Worship well as it leaves me,—I am, Your Worship's humble obedient servant,
LUKE HEMPEN.
How Master Nathaniel longed to jump on to his horse and ride posthaste to the farm! But that was impossible. Instead, he immediately despatched a groom with orders to ride day and night and deliver a letter to Luke Hempen, which bade him instantly take Ranulph to the farm near Moongrass (a village that lay some fifteen miles north of Swan-on-the-Dapple) from which for years he had got his cheeses.
Then he sat down and tried to find some meaning in the mysterious conversation Luke had overheard.
Ambrose seeing a vision! An unknown Chief! Footprints on the Milky Way!
Reality was beginning to become very shadowy and menacing.
He must find out something about this widow. Had she not once appeared in the law-courts? He decided he must look her up without a moment's delay.
He had inherited from his father a fine legal library, and the bookshelves in his pipe-room were packed with volumes bound in vellum and old calf of edicts, codes, and trials. Some of them belonged to the days before printing had been introduced into Dorimare, and were written in the crabbed hand of old town-clerks. It made the past very real, and threw a friendly, humourous light upon the dead, to come upon, when turning those yellow parchment pages, some personal touch of the old scribe's, such as a sententious or facetious insertion of his own—for instance, "The Law bides her Time, but my Dinner doesn't!" or the caricature in the margin of some forgotten judge. It was just as if one of the grotesque plaster heads on the old houses were to give you, suddenly, a sly wink.
But it was the criminal trials that, in the past, had given Master Nathaniel the keenest pleasure. The dry style of the Law was such a magnificent medium for narrative. And the little details of every-day life, the humble objects of daily use, became so startlingly vivid, when, like scarlet geraniums breaking through a thick autumn mist, they blazed out from that grey style so vivid, and, often, fraught with such tragic consequences.
Great was his astonishment when he discovered from the index that it was among the criminal trials that he must look for the widow Gibberty's. What was more, it was a trial for murder.
Surely Endymion Leer had told him, when he was urging him to send Ranulph to the farm, that it had been a quite trivial case, concerning an arrear of wages, or something, due to a discharged servant?
As a matter of fact, the plaintiff, a labourer of the name of Diggory Carp, had been discharged from the service of the late Farmer Gibberty. But the accusation he brought against the widow was that she had poisoned her husband with the sap of osiers.
However, when he had finished the trial, Master Nathaniel found himself in complete sympathy with the judge's pronouncement that the widow was innocent, and with his severe reprimand to the plaintiff, for having brought such a serious charge against a worthy woman on such slender grounds.
But he could not get Luke's letter out of his head, and he felt that he would not have a moment's peace till the groom returned with news from the farm.
As he sat that evening by the parlour fire, wondering for the hundredth time who the mysterious cloaked stranger could have been whose back had been seen by Luke, Dame Marigold suddenly broke the silence by saying, "What do you know about Endymion Leer, Nat?"
"What do I know of Endymion Leer?" he repeated absently. "Why, that he's a very good leach, with very poor taste in cravats, and, if possible, worse taste in jokes. And that, for some unknown reason, he has a spite against me...."
He broke off in the middle of his sentence, and muttered beneath his breath, "By the Sun, Moon and Stars! Supposing it should be...."
Luke's stranger had said he feared the Chanticleers.
A strange fellow, Leer! The Note had once sounded in his voice. Where did he come from? Who was he? Nobody knew in Lud-in-theMist.
And, then, there were his antiquarian tastes. They were generally regarded as a harmless, unprofitable hobby And yet the past was dim and evil, a heap of rotting leaves. The past was silent and belonged to the Silent People.... But Dame Marigold was asking another question, a question that had no apparent connection with the previous one: "What was the year of the great drought?"
Master Nathaniel answered that it was exactly forty years ago, and added quizzically, "Why this sudden interest in history, Marigold?"
Again she answered by asking him a question. "And when did Endymion Leer first arrive in Dorimare?"
Master Nathaniel began to be interested. "Let me see," he said thoughtfully. "It was certainly long before we married. Yes, I remember, we called him in to a consultation when my mother had pleurisy, and that was shortly after his arrival, for he could still only speak broken Dorimarite ... it must be thirty years ago."
"I see," said Dame Marigold drily. "But I happen to know that he was already in Dorimare at the time of the drought." And she proceeded to repeat to him her conversation that morning with Miss Primrose.
"And," she added, "I've got another idea," and she told him about the panel in the Guildhall that sounded hollow and what the guardian had said about the woodpecker ways of Endymion Leer. "And if, partly for revenge for our coldness to him, and partly from a love of power," she went on, "it is he who has been behind this terrible affair, a secret passage would be very useful in smuggling, and would explain how all your precautions have been useless. And who would be more likely to know about a secret passage in the Guildhall than Endymion Leer!"
"By the Sun, Moon and Stars!" exclaimed Master Nathaniel excitedly, "I shouldn't be surprised if you were right, Marigold. You've got a head on your shoulders with something in it more useful than porridge!"
And Dame Marigold gave a little complacent smile.
Then he sprang from his chair, "I'm off to tell Ambrose!" he cried eagerly.
But would he be able to convince the slow and obstinate mind of Master Ambrose? Mere suspicions are hard to communicate. They are rather like the wines that will not travel, and have to be drunk on the spot.
At any rate, he could but try.
"Have you ever had a vision of Duke Aubrey, Ambrose?" he cried, bursting into his friend's pipe-room.
Master Ambrose frowned with annoyance. "What are you driving at, Nat?" he said, huffily.
