Turbulence is ubiquitous, from oceanic currents to small-scale biological and quantum systems. Self-sustained turbulent motion in microbial suspensions presents an intriguing example of collective dynamical behavior among the simplest forms of life and is important for fluid mixing and molecular transport on the microscale. The mathematical characterization of turbulence phenomena in active nonequilibrium fluids proves even more difficult than for conventional liquids or gases. It is not known which features of turbulent phases in living matter are universal or system-specific or which generalizations of the Navier-Stokes equations are able to describe them adequately. Here, we combine experiments, particle simulations, and continuum theory to identify the statistical properties of self-sustained meso-scale turbulence in active systems. To study how dimensionality and boundary conditions affect collective bacterial dynamics, we measured energy spectra and structure functions in dense Bacillus subtilis suspensions in quasi-2D and 3D geometries. Our experimental results for the bacterial flow statistics agree well with predictions from a minimal model for self-propelled rods, suggesting that at high concentrations the collective motion of the bacteria is dominated by short-range interactions. To provide a basis for future theoretical studies, we propose a minimal continuum model for incompressible bacterial flow. A detailed numerical analysis of the 2D case shows that this theory can reproduce many of the experimentally observed features of self-sustained active turbulence.low Reynolds number swimming | velocity increment distributions | scaling S imple forms of life, like amoebae or bacteria, self-organize into remarkable macroscopic patterns (1, 2), ranging from extended networks (3, 4) to complex vortices (5-10) and swarms (11). These structures often bear a striking resemblance to assemblies of higher organisms [e.g., flocks of birds (12) or schools of fish (13,14)] and present important biological model systems to study nonequilibrium phases and their transitions (15)(16)(17). A particularly interesting manifestation of collective behavior in microbial suspensions is the emergence of meso-scale turbulent motion (7,8,18,19). Driven by the microorganisms' self-propulsion and their mutual interactions, such self-sustained "active turbulence" can have profound effects on nutrient mixing and molecular transport in microbiological systems (2,(20)(21)(22). However, in spite of recent progress (19,(23)(24)(25), the phenomenology of turbulent bacterial dynamics is scarcely understood and a commonly accepted theoretical description is lacking (2,16,26). The latter fact may not be surprising given that a comprehensive mathematical characterization of turbulence in conventional fluids has remained elusive after more than a century of intense research (27).In view of the various physical and chemical pathways through which bacteria may communicate (1, 11, 28), a basic yet unsolved problem is to identify those interactions...

Bacterial processes ranging from gene expression to motility and biofilm formation are constantly challenged by internal and external noise. While the importance of stochastic fluctuations has been appreciated for chemotaxis, it is currently believed that deterministic long-range fluid dynamical effects govern cell-cell and cellsurface scattering-the elementary events that lead to swarming and collective swimming in active suspensions and to the formation of biofilms. Here, we report direct measurements of the bacterial flow field generated by individual swimming Escherichia coli both far from and near to a solid surface. These experiments allowed us to examine the relative importance of fluid dynamics and rotational diffusion for bacteria. For cell-cell interactions it is shown that thermal and intrinsic stochasticity drown the effects of long-range fluid dynamics, implying that physical interactions between bacteria are determined by steric collisions and near-field lubrication forces. This dominance of short-range forces closely links collective motion in bacterial suspensions to self-organization in driven granular systems, assemblages of biofilaments, and animal flocks. For the scattering of bacteria with surfaces, long-range fluid dynamical interactions are also shown to be negligible before collisions; however, once the bacterium swims along the surface within a few microns after an aligning collision, hydrodynamic effects can contribute to the experimentally observed, long residence times. Because these results are based on purely mechanical properties, they apply to a wide range of microorganisms.low Reynolds number hydrodynamics | microswimmers C ollective behavior of bacteria, such as biofilm formation (1), swarming (2), and turbulence-like motion in concentrated suspensions (3, 4), has profound effects on foraging, signaling, and transport of metabolites (5, 6), and can be of great biomedical importance (7,8). Large-scale coherence in bacterial systems typically arises from a combination of biochemical signaling (9) and physical interactions. Recent theoretical models that focus on physical aspects of bacterial dynamics identify pairwise long-range hydrodynamic interactions (10-16) as a key ingredient for collective swimming. Such ''microscopic" approaches underpin continuum theories that aim to describe the phenomenology of microbial suspensions (17-23). An assumption underlying many of these theories is that a self-propelled bacterium can be modeled as a force dipole; its body exerts a drag force F on the fluid that is balanced by the rearward flagellar thrust −F. The leading-order fluid velocity field at distance r is therefore a dipolar ''pusher" flow of magnitude u ∝ Fℓ∕ηr 2 (see streamlines in Fig. 1B), where η is the viscosity, and ℓ the distance between the forces (24, 25). While higher order corrections may be due to force-quadrupole contributions (26), the hypothesis that the leading-order flow field around a bacterium is dipolar has not yet been verified experimentally.A closely related, c...

Self-sustained turbulent structures have been observed in a wide range of living fluids, yet no quantitative theory exists to explain their properties. We report experiments on active turbulence in highly concentrated 3D suspensions of Bacillus subtilis and compare them with a minimal fourth-order vector-field theory for incompressible bacterial dynamics. Velocimetry of bacteria and surrounding fluid, determined by imaging cells and tracking colloidal tracers, yields consistent results for velocity statistics and correlations over 2 orders of magnitude in kinetic energy, revealing a decrease of fluid memory with increasing swimming activity and linear scaling between kinetic energy and enstrophy. The best-fit model allows for quantitative agreement with experimental data.

Interactions between swimming cells and surfaces are essential to many microbiological processes, from bacterial biofilm formation to human fertilization. However, despite their fundamental importance, relatively little is known about the physical mechanisms that govern the scattering of flagellated or ciliated cells from solid surfaces. A more detailed understanding of these interactions promises not only new biological insights into structure and dynamics of flagella and cilia but may also lead to new microfluidic techniques for controlling cell motility and microbial locomotion, with potential applications ranging from diagnostic tools to therapeutic protein synthesis and photosynthetic biofuel production. Due to fundamental differences in physiology and swimming strategies, it is an open question of whether microfluidic transport and rectification schemes that have recently been demonstrated for pusher-type microswimmers such as bacteria and sperm cells, can be transferred to pullertype algae and other motile eukaryotes, because it is not known whether long-range hydrodynamic or short-range mechanical forces dominate the surface interactions of these microorganisms. Here, using high-speed microscopic imaging, we present direct experimental evidence that the surface scattering of both mammalian sperm cells and unicellular green algae is primarily governed by direct ciliary contact interactions. Building on this insight, we predict and experimentally verify the existence of optimal microfluidic ratchets that maximize rectification of initially uniform Chlamydomonas reinhardtii suspensions. Because mechano-elastic properties of cilia are conserved across eukaryotic species, we expect that our results apply to a wide range of swimming microorganisms.algal surface accumulation | swimming rectification S urface interactions of motile cells play crucial roles in a wide range of microbiological phenomena, perhaps most prominently in the formation of biofilms (1) and during the fertilization of mammalian ova (2). However, despite their widely recognized importance, the basic physical mechanisms that govern the response of swimming bacteria, algae, or spermatozoa to solid surfaces have remained unclear. This predicament is exemplified by the current debate (3-6) about the relevance of hydrodynamic long-range forces and steric short-range interactions for the accumulation of flagellated cells at liquid-solid interfaces. From a general perspective, improving our understanding of cell surface scattering processes promises not only new insights into structure, dynamics, and biological functions of flagella and cilia, it will also help to advance microfluidic techniques for controlling microbial locomotion (7,8), with potential applications in diagnostics (9), therapeutic protein synthesis (10), and photosynthetic biofuel production (11-14). That microfluidic circuits provide an excellent test bed for developing and assessing new strategies for the control of cell motility was recently demonstrated by the rectification of ...

Confining surfaces play crucial roles in dynamics, transport and order in many physical systems, but their effects on active matter, a broad class of dynamically self-organizing systems, are poorly understood. We investigate here the influence of global confinement and surface curvature on collective motion by studying the flow and orientational order within small droplets of a dense bacterial suspension. The competition between radial confinement, self-propulsion, steric interactions and hydrodynamics robustly induces an intriguing steady single-vortex state, in which cells align in inwardly-spiralling patterns accompanied by a thin counterrotating boundary layer. A minimal continuum model is shown to be in good agreement with these observations.Comment: 5 pages, 4 figure

A considerable body of experimental and theoretical work claims the existence of negative absolute temperatures in spin systems and ultra-cold quantum gases. Here, we clarify that such findings can be attributed to the use of a popular yet inconsistent entropy definition, which violates fundamental thermodynamic relations and fails to produce sensible results for simple analytically tractable classical and quantum systems. Within a mathematically consistent thermodynamic formalism, based on an entropy concept originally derived by Gibbs, absolute temperature remains positive even for systems with bounded spectrum. We address spurious arguments against the Gibbs formalism and comment briefly on heat engines with efficiencies greater than one.

The recent experimental realization of exotic matter states in isolated quantum systems and the ensuing controversy about the existence of negative absolute temperatures demand a careful analysis of the conceptual foundations underlying microcanonical thermostatistics. Here we provide a detailed comparison of the most commonly considered microcanonical entropy definitions, focusing specifically on whether they satisfy or violate the zeroth, first, and second laws of thermodynamics. Our analysis shows that, for a broad class of systems that includes all standard classical Hamiltonian systems, only the Gibbs volume entropy fulfills all three laws simultaneously. To avoid ambiguities, the discussion is restricted to exact results and analytically tractable examples.

Surface-attached bacterial biofilms are self-replicating active liquid crystals and the dominant form of bacterial life on earth ( 1 – 4 ). In conventional liquid crystals and solid-state materials, the interaction potentials between the molecules that comprise the system determine the material properties. However, for growth-active biofilms it is unclear whether potential-based descriptions can account for the experimentally observed morphologies, and which potentials would be relevant. Here, we overcame previous limitations of single-cell imaging techniques ( 5 , 6 ) to reconstruct and track all individual cells inside growing three-dimensional (3D) biofilms with up to 10,000 individuals. Based on these data, we identify, constrain, and provide a microscopic basis for an effective cell-cell interaction potential, which captures and predicts the growth dynamics, emergent architecture, and local liquid crystalline order of Vibrio cholerae biofilms. Furthermore, we show how external fluid flows control the microscopic structure and 3D morphology of biofilms. Our analysis implies that local cellular order and global biofilm architecture in these active bacterial communities can arise from mechanical cell-cell interactions, which cells can modulate by regulating the production of particular matrix components. These results establish an experimentally validated foundation for improved continuum theories of active matter and thereby contribute to solving the important problem of controlling biofilm growth.