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Active matter
Active matter
Active matter
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A flock of starlings acting as a swarm

Active matter is matter composed of large numbers of active "agents", each of which consumes energy in order to move or to exert mechanical forces.[1][2][3][4] Such systems are intrinsically out of thermal equilibrium. Unlike thermal systems relaxing towards equilibrium and systems with boundary conditions imposing steady currents, active matter systems break time reversal symmetry because energy is being continually dissipated by the individual constituents.[5][6][7] Most examples of active matter are biological in origin and span all the scales of the living, from bacteria and self-organising bio-polymers such as microtubules and actin (both of which are part of the cytoskeleton of living cells), to schools of fish and flocks of birds. However, a great deal of current experimental work is devoted to synthetic systems such as artificial self-propelled particles.[8][9][10] Active matter is a relatively new material classification in soft matter: the most extensively studied model, the Vicsek model, dates from 1995.[11]

Research in active matter combines analytical techniques, numerical simulations and experiments. Notable analytical approaches include hydrodynamics,[12] kinetic theory, and non-equilibrium statistical physics. Numerical studies mainly involve self-propelled-particles models,[13][14] making use of agent-based models such as molecular dynamics algorithms or lattice-gas models,[15] as well as computational studies of hydrodynamic equations of active fluids.[12] Experiments on biological systems extend over a wide range of scales, including animal groups (e.g., bird flocks,[16] mammalian herds, fish schools and insect swarms[17]), bacterial colonies, cellular tissues (e.g. epithelial tissue layers,[18] cancer growth and embryogenesis), cytoskeleton components (e.g., in vitro motility assays, actin-myosin networks and molecular-motor driven filaments[19]). Experiments on synthetic systems include self-propelled colloids (e.g., phoretically propelled particles[8][20]), driven granular matter (e.g. vibrated monolayers[21]), swarming robots,[22] and Quincke rotators.[23]

Concepts in Active matter

Active matter systems

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Active matter

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Active matter refers to a class of non-equilibrium physical systems composed of numerous interacting units, such as particles or agents, that are self-propelled and convert free energy from their environment or internal sources into sustained, directed motion and mechanical work. These systems operate far from , consuming energy to drive persistent activity that leads to emergent collective behaviors not observed in passive matter. Unlike equilibrium systems, active matter exhibits broken time-reversal symmetry and can generate long-range correlations and ordered phases through local energy inputs. In biological contexts, active matter manifests in systems like swarming bacteria (e.g., exhibiting run-and-tumble ), molecular motors such as walking along , and larger-scale phenomena including bird flocks and fish schools, where individual self-propulsion and interactions yield complex patterns like milling or vortex formation. Synthetic realizations include active colloids, such as that propel via asymmetric chemical reactions or phoretic effects, and light- or magnetically driven microswimmers designed in laboratories to mimic biological . These examples span scales from nanometers (e.g., cytoskeletal filaments) to meters (e.g., animal groups), highlighting the universality of active principles across and engineered materials. The physics of active matter draws from , hydrodynamics, and to explain phenomena like phase transitions to ordered states (e.g., in the ), giant number fluctuations in dilute suspensions, and instabilities in active gels where contractile forces lead to spontaneous flow. Key challenges include developing a thermodynamic framework for energy dissipation and in these systems, as well as understanding how activity couples to passive like orientation or deformations. has revealed that active matter can violate and produce novel states, such as active nematics with topological defects driving spontaneous motion. Emerging over the past two decades at the intersection of physics, , and , active matter has spurred applications in micro-robotics for , mimicking biological swarms, and understanding cellular processes like tissue morphogenesis. Ongoing studies explore active matter under extreme conditions, such as microgravity, to probe universal principles of . This field continues to grow, with theoretical models informing experimental designs and revealing how local activity generates global order in living and artificial systems.

Definition and Fundamentals

Definition

Active matter refers to collections of self-propelled agents that consume energy from their environment to generate systematic motion or mechanical forces, thereby maintaining the system in a state far from . These agents, which can include biological entities like or synthetic particles, operate through internal mechanisms that drive persistent, directed movement without relying on external concentration gradients. In contrast to passive matter, which obeys and time-reversal symmetry in equilibrium conditions, active matter inherently breaks these symmetries due to continuous local , enabling phenomena such as spontaneous organization and nontrivial steady-state currents. This nonequilibrium character arises from the unidirectional conversion of into work at the individual agent level, distinguishing active systems from passive ones where motion ceases without external driving. Active matter manifests across a wide range of scales, from molecular levels—such as motor proteins that propel along cytoskeletal filaments—to macroscopic assemblies like flocks of birds or schools of fish. At these scales, the agents convert environmental energy sources into mechanical work; in biological contexts, this often involves from , while synthetic realizations may harness external inputs like or to induce propulsion.

Key Characteristics

Active matter systems are distinguished by the persistent, directed motion of their constituent agents, which arises from dissipation rather than external forces. This persistence is quantified by the persistence length lp=v0τl_p = v_0 \tau, where v0v_0 is the characteristic self-propulsion speed and τ\tau is the reorientation time, representing the distance an agent travels before significantly changing direction. In models of active Brownian particles, this length scale emerges from the competition between directed propulsion and , leading to superdiffusive trajectories over intermediate times. A central metric for activity in these systems is the Pe=v0L/DPe = v_0 L / D, which compares advective transport due to self-propulsion over a system size LL to diffusive spreading governed by the diffusion coefficient DD. High PePe values indicate regimes where directed motion dominates, enabling phenomena absent in passive systems. This parameter is particularly relevant in dilute suspensions, where it controls the onset of instabilities and clustering. Unlike equilibrium fluids, active matter exhibits giant number fluctuations, where the fluctuations scale as ⟨(δn)2⟩/⟨n⟩2∼N2/d−1\langle (\delta n)^2 \rangle / \langle n \rangle^2 \sim N^{2/d - 1} in dd dimensions (e.g., independent of subsystem size NN in 2D), contrasting with the equilibrium scaling ∼1/N\sim 1/N. These anomalous fluctuations, first predicted in models of oriented active suspensions, arise from the between and orientational order, amplified by activity. Experimental observations in bacterial suspensions and synthetic colloids confirm this enhanced variability, signaling nonequilibrium long-range correlations. Activity also drives the emergence of orientational order and correlations, even in low dimensions where thermal fluctuations would destroy it in equilibrium systems. In two dimensions, polar flocks develop long-range orientational order, as described by the Vicsek model, due to velocity alignment interactions sustained by persistent motion. Nematic order similarly persists, with correlations extending over macroscopic scales, fostering collective flows and defects.

Historical Development

Early Biological Observations

The earliest scientific observations of active matter phenomena in biological systems date back to the late 19th century, when botanist Wilhelm Pfeffer described directed movements of microorganisms in response to chemical gradients in 1884, coining the term "chemotaxis" based on experiments with bacteria and sperm cells. Pfeffer's work highlighted how these self-propelled motions allowed organisms to navigate environments without external forces, laying foundational empirical evidence for nonequilibrium dynamics in living systems. In the mid-20th century, these observations were extended to specific bacterial behaviors, with Julius Adler's 1966 studies on Escherichia coli demonstrating chemotactic responses to nutrients like sugars and amino acids, revealing a bias in random motion toward favorable conditions. Further insights into bacterial motility emerged from three-dimensional tracking experiments, which visualized the "run-and-tumble" pattern in E. coli, where cells alternate straight swimming runs with random reorientations via tumbling, enabling efficient exploration and chemotaxis. Concurrently, observations of intracellular dynamics revealed active contractions driving cell shape changes, as seen in amoeboid movement studied in the early 20th century. Microscopic examinations of protozoa like Amoeba proteus showed rhythmic contractions of cytoplasmic fibrils, suggesting a contractile apparatus akin to muscle, with later biochemical identification of actin-like proteins in non-muscle cells, such as in the slime mold Physarum polycephalum, confirming myosin-actin interactions powered by ATP hydrolysis. These findings underscored self-driven cytoskeletal remodeling as a key mechanism for cellular motility, dissipating metabolic energy to maintain far-from-equilibrium states. Collective behaviors in multicellular organisms provided additional early examples of coordinated active motion without central control. In , C.M. Breder's 1976 analysis of schooling patterns described how groups maintain parallel orientation and spacing through local interactions, optimizing hydrodynamic efficiency and predator avoidance in species like and sardines. Similarly, insect swarms, such as those of midges (), were observed in the mid-20th century to exhibit synchronized hovering and directional shifts, driven by visual cues and wind, as documented in field studies of lekking behaviors where males form dynamic aerial displays. These pre-1990s empirical accounts recognized and phototaxis—light-directed motility in like —as inherently active processes fueled by continuous energy input from , distinguishing them from passive and foreshadowing the dissipative nature of .

Emergence of Theoretical Models

The emergence of theoretical models for active matter in the mid-1990s marked a shift from qualitative biological descriptions to quantitative physics, inspired by observations of collective animal behaviors such as bird flocking and fish schooling. A foundational contribution was the , introduced in 1995, which simulates that align their velocities with neighbors within a fixed while subject to random , revealing a to collective ordered motion above a critical threshold. This agent-based approach demonstrated how local alignment rules could lead to global coherence without centralized control, laying the groundwork for studying emergent order in driven systems. Concurrently, John Toner and Yuhai Tu developed a hydrodynamic theory for , starting with a 1995 paper that proposed a dynamical XY model for polar-ordered flocks and extended it through 1998 to derive continuum equations capturing long-range order and giant fluctuations in two dimensions. Their framework includes the momentum equation ∂tv+(v⋅∇)v=−∇P+ν∇2v+f,\partial_t \mathbf{v} + (\mathbf{v} \cdot \nabla) \mathbf{v} = -\nabla P + \nu \nabla^2 \mathbf{v} + \mathbf{f}, where v\mathbf{v} is the velocity field, PP is , ν\nu is , and f\mathbf{f} represents active and terms, highlighting anomalous scaling behaviors distinct from equilibrium systems. In the early 2000s, Sriram Ramaswamy and collaborators advanced the field by exploring active nematics, systems of rod-like particles with orientational order but no polar axis, showing in 2003 that such models on substrates exhibit giant density fluctuations and long-time correlations due to activity-induced instabilities. Ramaswamy's work also introduced the classification of active matter into "dry" systems, where momentum is not conserved due to substrate friction, and "wet" systems, which include hydrodynamic interactions in a fluid medium, providing a conceptual dichotomy for modeling diverse active phenomena. By 2013, M. C. Marchetti and colleagues synthesized these developments into a unified hydrodynamic framework for soft active matter, integrating microscopic origins with continuum descriptions to predict generic instabilities, phase transitions, and nonequilibrium patterns across biological and synthetic systems. This review emphasized the role of symmetry and activity in driving collective behaviors, establishing active matter as a distinct branch of nonequilibrium statistical physics.

Theoretical Frameworks

Microscopic Models

Microscopic models in active matter focus on the dynamics of individual , capturing their motion and interactions through discrete agent-based rules or differential equations, which enable simulations of emergent phenomena at the particle level. These approaches emphasize bottom-up descriptions, where active forces and orientational persistence drive behaviors without deriving coarse-grained fields. Seminal formulations include run-and-tumble dynamics for abrupt reorientations and persistent random walks for smoother turning, often extended with alignment rules for collective effects. Run-and-tumble particles (RTPs) represent a key microscopic model for organisms exhibiting ballistic motion interrupted by random reorientations, such as flagellated . In this framework, each particle ii propels at constant speed v0v_0 along its orientation u^i\hat{\mathbf{u}}_i during a "run" phase, with position updating as r˙i=v0u^i\dot{\mathbf{r}}_i = v_0 \hat{\mathbf{u}}_i. Tumbles occur stochastically at rate 1/τ1/\tau, instantaneously randomizing the direction u^i\hat{\mathbf{u}}_i uniformly on the unit circle (in 2D) or sphere (in 3D). This model captures persistence over run length v0τv_0 \tau and has been pivotal in studying motility-induced , where interactions like volume exclusion lead to clustering despite no explicit attraction. Active Brownian particles (ABPs) provide another foundational microscopic description, modeling overdamped swimmers with continuous rotational diffusion, suitable for colloidal or eukaryotic systems. The dynamics follow the Langevin equations r˙=v0u^+2Dξ\dot{\mathbf{r}} = v_0 \hat{\mathbf{u}} + \sqrt{2D} \boldsymbol{\xi}
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