Spontaneous vortex formation by microswimmers with retarded attractions.

Collective states of inanimate particles self-assemble through physical interactions and thermal motion. Despite some phenomenological resemblance, including signatures of criticality, the autonomous dynamics that binds motile agents into flocks, herds, or swarms allows for much richer behavior. Low-dimensional models have hinted at the crucial role played in this respect by perceived information, decision-making, and feedback, implying that the corresponding interactions are inevitably retarded. Here we present experiments on spherical Brownian microswimmers with delayed self-propulsion toward a spatially fixed target. We observe a spontaneous symmetry breaking to a transiently chiral dynamical state and concomitant critical behavior that do not rely on many-particle cooperativity. By comparison with the stochastic delay differential equation of motion of a single swimmer, we pinpoint the delay-induced effective synchronization of the swimmers with their own past as the key mechanism. Increasing numbers of swimmers self-organize into layers with pro- and retrograde orbital motion, synchronized and stabilized by steric, phoretic, and hydrodynamic interactions. Our results demonstrate how even most simple retarded interactions can foster emergent complex adaptive behavior in small active-particle ensembles.

Signal propagation and linear response in the delay Vicsek model.

Retardation between sensation and action is an inherent biological trait. Here we study its effect in the Vicsek model, which is a paradigmatic swarm model. We find that (1) a discrete time delay in the orientational interactions diminishes the ability of strongly aligned swarms to follow a leader and, in return, increases their stability against random orientation fluctuations; (2) both longer delays and higher speeds favor ballistic over diffusive spreading of information (orientation) through the swarm; (3) for short delays, the mean change in the total orientation (the order parameter) scales linearly in a small orientational bias of the leaders and inversely in the delay time, while its variance first increases and then saturates with increasing delays; and (4) the linear response breaks down when orientation conservation is broken.

Information conduction and convection in noiseless Vicsek flocks.

Physical interactions generally respect certain symmetries, such as reciprocity and energy conservation, which survive in coarse-grained isothermal descriptions. Active many-body systems usually break such symmetries intrinsically, on the particle level, so that their collective behavior is often more naturally interpreted as a result of information exchange. Here we study numerically how information spreads from a "leader" particle through an initially aligned flock, described by the Vicsek model without noise. In the low-speed limit of a static spin lattice, we find purely conductive spreading, reminiscent of heat transfer. Swarm motility and heterogeneity can break reciprocity and spin conservation. But what seems more consequential for the swarm response is that the dispersion relation acquires a significant convective contribution along the leader's direction of motion.

Maximum efficiency of low-dissipation heat pumps at given heating load.

We derive an analytical expression for maximum efficiency at fixed power of heat pumps operating along a finite-time reverse Carnot cycle under the low-dissipation assumption. The result is cumbersome, but it implies simple formulas for tight upper and lower bounds on the maximum efficiency and various analytically tractable approximations. In general, our results qualitatively agree with those obtained earlier for endoreversible heat pumps. In fact, we identify a special parameter regime when the performance of the low-dissipation and endoreversible devices is the same. At maximum power, heat pumps operate as work to heat converters with efficiency 1. Expressions for maximum efficiency at given power can be helpful in the identification of more practical operation regimes.

Density and polarization of active Brownian particles in curved activity landscapes.

Suspensions of motile active particles with space-dependent activity form characteristic polarization and density patterns. Recent single-particle studies for planar activity landscapes identified several quantities associated with emergent density-polarization patterns that are solely determined by bulk variables. Naive thermodynamic intuition suggests that these results might hold for arbitrary activity landscapes mediating bulk regions, and thus could be used as benchmarks for simulations and theories. However, the considered system operates in a nonequilibrium steady state and we prove by construction that the quantities in question lose their simple form for curved activity landscapes. Specifically, we provide a detailed analytical study of polarization and density profiles induced by radially symmetric activity steps, and of the total polarization for the case of a general radially symmetric activity landscape. While the qualitative picture is similar to the planar case, all the investigated variables depend not only on bulk variables but also comprise geometry-induced contributions. We verified that all our analytical results agree with exact numerical calculations.

Polarization-density patterns of active particles in motility gradients.

The colocalization of density modulations and particle polarization is a characteristic emergent feature of motile active matter in activity gradients. We employ the active-Brownian-particle model to derive precise analytical expressions for the density and polarization profiles of a single Janus-type swimmer in the vicinity of an abrupt activity step. Our analysis allows for an optional (but not necessary) orientation-dependent propulsion speed, as often employed in force-free particle steering. The results agree well with measurement data for a thermophoretic microswimmer presented in the companion paper [Söker et al., Phys. Rev. Lett. 126, 228001 (2021)10.1103/PhysRevLett.126.228001], and they can serve as a template for more complex applications, e.g., to motility-induced phase separation or studies of physical boundaries. The essential physics behind our formal results is robustly captured and elucidated by a schematic two-species "run-and-tumble" model.

Maximum efficiency of absorption refrigerators at arbitrary cooling power.

We consider absorption refrigerators consisting of simultaneously operating Carnot-type heat engine and refrigerator. Their maximum efficiency at given power (MEGP) is given by the product of MEGPs for the internal engine and refrigerator. The only subtlety of the derivation lies in the fact that the maximum cooling power of the absorption refrigerator is not limited just by the maximum power of the internal refrigerator, but, due to the first law, also by that of the internal engine. As a specific example, we consider the simultaneous absorption refrigerators composed of low-dissipation (LD) heat engines and refrigerators, for which the expressions for MEGPs are known. The derived expression for maximum efficiency implies bounds on the MEGP of LD absorption refrigerators. It also implies that a slight decrease in power of the absorption refrigerator from its maximum value results in a large nonlinear increase in efficiency, observed in heat engines, whenever the ratio of maximum powers of the internal engine and the refrigerator does not diverge. Otherwise, the increase in efficiency is linear as observed in LD refrigerators. Thus, in all practical situations, the efficiency of LD absorption refrigerators significantly increases when their cooling power is slightly decreased from its maximum.

How Activity Landscapes Polarize Microswimmers without Alignment Forces.

Active-particle suspensions exhibit distinct polarization-density patterns in activity landscapes, even without anisotropic particle interactions. Such polarization without alignment forces is at work in motility-induced phase separation and betrays intrinsic microscopic activity to mesoscale observers. Using stable long-term confinement of a single thermophoretic microswimmer in a dedicated force-free particle trap, we examine the polarized interfacial layer at a motility step and confirm that it does not exert pressure onto the bulk. Our observations are quantitatively explained by an analytical theory that can also guide the analysis of more complex geometries and many-body effects.

Fully Steerable Symmetric Thermoplasmonic Microswimmers.

A cornerstone of the directed motion of microscopic self-propelling particles is an asymmetric particle structure defining a polarity axis along which these tiny machines move. This structural asymmetry ties the orientational Brownian motion to the microswimmers directional motion, limiting their persistence and making the long time motion effectively diffusive. Here, we demonstrate a completely symmetric thermoplasmonic microswimmer, which is propelled by laser-induced self-thermophoresis. The propulsion direction is imprinted externally to the particle by the heating laser position. The orientational Brownian motion, thus, becomes irrelevant for the propulsion, allowing enhanced control over the particles dynamics with almost arbitrary steering capability. We characterize the particle motion in experiments and simulations and also theoretically. The analysis reveals additional noise appearing in these systems, which is conjectured to be relevant for biological systems. Our experimental results show that even very small particles can be precisely controlled, enabling more advanced applications of these micromachines.

Finite-Size Scaling at the Edge of Disorder in a Time-Delay Vicsek Model.

Living many-body systems often exhibit scale-free collective behavior reminiscent of thermal critical phenomena. But their mutual interactions are inevitably retarded due to information processing and delayed actuation. We numerically investigate the consequences for the finite-size scaling in the Vicsek model of motile active matter. A growing delay time initially facilitates but ultimately impedes collective ordering and turns the dynamical scaling from diffusive to ballistic. It provides an alternative explanation of swarm traits previously attributed to inertia.

Maximum efficiency of low-dissipation refrigerators at arbitrary cooling power.

We analytically derive maximum efficiency at given cooling power for Carnot-type low-dissipation refrigerators. The corresponding optimal cycle duration depends on a single parameter, which is a specific combination of irreversibility parameters and bath temperatures. For a slight decrease in power with respect to its maximum value, the maximum efficiency exhibits an infinitely fast nonlinear increase, which is standard in heat engines, only for a limited range of parameters. Otherwise, it increases only linearly with the slope given by ratio of irreversibility parameters. This behavior can be traced to the fact that maximum power is attained for vanishing duration of the hot isotherm. Due to the lengthiness of the full solution for the maximum efficiency, we discuss and demonstrate these results using simple approximations valid for parameters yielding the two different qualitative behaviors. We also discuss relation of our findings to those obtained for minimally nonlinear irreversible refrigerators.

Underdamped active Brownian heat engine.

Active Brownian engines rectify energy from reservoirs composed of self-propelling nonequilibrium molecules into work. We consider a class of such engines based on an underdamped Brownian particle trapped in a power-law potential. The energy they transform has thermodynamic properties of heat only if the nonequilibrium reservoir can be assigned a suitable effective temperature consistent with the second law and thus yielding an upper bound on the engine efficiency. The effective temperature exists if the total force exerted on the particle by the bath is not correlated with the particle position. In general, this occurs if the noise autocorrelation function and the friction kernel are proportional as in the fluctuation-dissipation theorem. But even if the proportionality is broken, the effective temperature can be defined in restricted, fine-tuned, parameter regimes, as we demonstrate on a specific example with harmonic potential.

Effects of noise-induced coherence on the fluctuations of current in quantum absorption refrigerators.

We investigate the effects of noise-induced coherence on average current and current fluctuations in a simple model of a quantum absorption refrigerator with degenerate energy levels. We describe and explain the differences and similarities between the system behavior when it operates in the classical regime, where the populations and coherences in the corresponding quantum optical master equation decouple in a suitably chosen basis, and in the quantum regime, where such a transformation does not exist. The differences between the quantum and the classical cases are observable only close to the maximum current regime, where the system steady-state becomes nonunique. This allows us to approximate the system dynamics by an analytical model based on a dichotomous process that explains the behavior of the average current both in the classical and in the quantum cases. Due to the nonuniqueness, the scaled cumulant generating function for the current at the vicinity of the critical point exhibits behavior reminiscent of the dynamical first-order phase transition. Unless the system parameters are fine-tuned to a single point in the parameter space, the corresponding current fluctuations are moderate in the quantum case and large in the classical case.

Physically consistent numerical solver for time-dependent Fokker-Planck equations.

We present a simple thermodynamically consistent method for solving time-dependent Fokker-Planck equations (FPE) for overdamped stochastic processes, also known as Smoluchowski equations. It yields both transition and steady-state behavior and allows for computations of moment-generating and large-deviation functions of observables defined along stochastic trajectories, such as the fluctuating particle current, heat, and work. The key strategy is to approximate the FPE by a master equation with transition rates in configuration space that obey a local detailed balance condition for arbitrary discretization. Its time-dependent solution is obtained by a direct computation of the time-ordered exponential, representing the propagator of the FPE, by summing over all possible paths in the discretized space. The method thus not only preserves positivity and normalization of the solutions but also yields a physically reasonable total entropy production, regardless of the discretization. To demonstrate the validity of the method and to exemplify its potential for applications, we compare it against Brownian-dynamics simulations of a heat engine based on an active Brownian particle trapped in a time-dependent quartic potential.

Diffusing up the Hill: Dynamics and Equipartition in Highly Unstable Systems.

Stochastic motion of particles in a highly unstable potential generates a number of diverging trajectories leading to undefined statistical moments of the particle position. This makes experiments challenging and breaks down a standard statistical analysis of unstable mechanical processes and their applications. A newly proposed approach takes advantage of the local characteristics of the most probable particle motion instead of the divergent averages. We experimentally verify its theoretical predictions for a Brownian particle moving near an inflection in a highly unstable cubic optical potential. The most likely position of the particle atypically shifts against the force, despite the trajectories diverging in the opposite direction. The local uncertainty around the most likely position saturates even for strong diffusion and enables well-resolved position detection. Remarkably, the measured particle distribution quickly converges to a quasistationary one with the same atypical shift for different initial particle positions. The demonstrated experimental confirmation of the theoretical predictions approves the utility of local characteristics for highly unstable systems which can be exploited in thermodynamic processes to uncover energetics of unstable systems.

Cycling Tames Power Fluctuations near Optimum Efficiency.

According to the laws of thermodynamics, no heat engine can beat the efficiency of a Carnot cycle. This efficiency traditionally comes with vanishing power output and practical designs, optimized for power, generally achieve far less. Recently, various strategies to obtain Carnot's efficiency at large power were proposed. However, a thermodynamic uncertainty relation implies that steady-state heat engines can operate in this regime only at the cost of large fluctuations that render them immensely unreliable. Here, we demonstrate that this unfortunate trade-off can be overcome by designs operating cyclically under quasistatic conditions. The experimentally relevant yet exactly solvable model of an overdamped Brownian heat engine is used to illustrate the formal result. Our study highlights that work in cyclic heat engines and that in quasistatic ones are different stochastic processes.

Active particles bound by information flows.

Self-organization is the generation of order out of local interactions. It is deeply connected to many fields of science from physics, chemistry to biology, all based on physical interactions. The emergence of collective animal behavior is the result of self-organization processes as well, though they involve abstract interactions arising from sensory inputs, information processing, storage, and feedback. Resulting collective behaviors are found, for example, in crowds of people, flocks of birds, and swarms of bacteria. Here we introduce interactions between active microparticles which are based on the information about other particle positions. A real-time feedback of multiple active particle positions is the information source for the propulsion direction of these particles. The emerging structures require continuous information flows. They reveal frustrated geometries due to confinement to two dimensions and internal dynamical degrees of freedom that are reminiscent of physically bound systems, though they exist only as nonequilibrium structures.

Brownian motion surviving in the unstable cubic potential and the role of Maxwell's demon.

The trajectories of an overdamped particle in a highly unstable potential diverge so rapidly, that the variance of position grows much faster than its mean. A description of the dynamics by moments is therefore not informative. Instead, we propose and analyze local directly measurable characteristics, which overcome this limitation. We discuss the most probable particle position (position of the maximum of the probability density) and the local uncertainty in an unstable cubic potential, V(x)∼x^{3}, both in the transient regime and in the long-time limit. The maximum shifts against the acting force as a function of time and temperature. Simultaneously, the local uncertainty does not increase faster than the observable shift. In the long-time limit, the probability density naturally attains a quasistationary form. We interpret this process as a stabilization via the measurement-feedback mechanism, the Maxwell demon, which works as an entropy pump. The rules for measurement and feedback naturally arise from the basic properties of the unstable dynamics. All reported effects are inherent in any unstable system. Their detailed understanding will stimulate the development of stochastic engines and amplifiers and, later, their quantum counterparts.

Work and power fluctuations in a critical heat engine.

We investigate fluctuations of output work for a class of Stirling heat engines with working fluid composed of interacting units and compare these fluctuations to an average work output. In particular, we focus on engine performance close to a critical point where Carnot's efficiency may be attained at a finite power as reported by M. Campisi and R. Fazio [Nat. Commun. 7, 11895 (2016)2041-172310.1038/ncomms11895]. We show that the variance of work output per cycle scales with the same critical exponent as the heat capacity of the working fluid. As a consequence, the relative work fluctuation diverges unless the output work obeys a rather strict scaling condition, which would be very hard to fulfill in practice. Even under this condition, the fluctuations of work and power do not vanish in the infinite system size limit. Large fluctuations of output work thus constitute inseparable and dominant element in performance of the macroscopic heat engines close to a critical point.

Diverging, but negligible power at Carnot efficiency: Theory and experiment.

We discuss the possibility of reaching the Carnot efficiency by heat engines (HEs) out of quasistatic conditions at nonzero power output. We focus on several models widely used to describe the performance of actual HEs. These models comprise quantum thermoelectric devices, linear irreversible HEs, minimally nonlinear irreversible HEs, HEs working in the regime of low-dissipation, overdamped stochastic HEs and an underdamped stochastic HE. Although some of these HEs can reach the Carnot efficiency at nonzero and even diverging power, the magnitude of this power is always negligible compared to the maximum power attainable in these systems. We provide conditions for attaining the Carnot efficiency in the individual models and explain practical aspects connected with reaching the Carnot efficiency at large power output. Furthermore, we show how our findings can be tested in practice using a standard Brownian HE realizable with available micromanipulation techniques.