APS Journals

In the late 1990s, Andrea Liu and Sid Nagel proposed jamming as a unifying framework for understanding how a freely flowing system becomes rigid. Rather than viewing rigidity solely through the lens of equilibrium thermodynamics, they emphasized that arrest can be achieved through various control parameters, such as volume fraction, temperature, or applied shear. This broader perspective reframed rigidity in amorphous solids as a universal, collective phenomenon that transcends traditional phase transitions.

A first concrete foundation for this program used simulations of purely repulsive spheres to identify a sharp onset of rigidity at a critical packing fraction—point J [Phys. Rev. E 68, 011306 (2003)]. Near this threshold, the system exhibits unusual scaling of the shear and bulk moduli, signaling behavior distinct from conventional solidification. A complementary theoretical interpretation, introducing the notion of marginality in the resulting solid, followed [Phys. Rev. E 72, 051306 (2005)]. It argued that the proliferation of low-frequency vibrational excitations around a jammed state, together with constraint counting and Maxwell isostaticity, underpins the critical properties observed at jamming. These insights established deep connections between jamming, mechanical stability, and constraint satisfaction problems.

Subsequent developments broadened the conceptual reach of the framework. Several critical aspects of jamming were shown to be largely independent of spatial dimension, thereby strengthening prospects for a first-principles theory [Phys. Rev. E 74, 041127 (2006)]. That work further showed that nucleation probabilities are strongly suppressed in high dimensions, suggesting that glass formation may become more generic as dimension increases. In parallel, a powerful set of metrics for characterizing amorphous structure was developed, thus uncovering a subtle form of order in marginally jammed solids [Phys. Rev. E 68, 041113 (2003)]. Although such systems lack conventional translational symmetry breaking, they exhibit suppressed long-wavelength density fluctuations—a property now known as hyperuniformity. This concept has since found applications far beyond sphere packing, influencing fields ranging from photonic materials and disordered solids to tissues in living matter.

Together, these works transformed jamming from a phenomenological observation into a central organizing principle in statistical physics, linking rigidity, disorder, dimensionality, and collective criticality.

A sequence of influential contributions established community structure as a central organizing principle of complex networks. One landmark study introduced edge betweenness as the basis of a community-detection algorithm [Phys. Rev. E 69, 026113 (2004)]. The key insight was that communities correspond to densely connected subgraphs linked by relatively few inter-community edges; by progressively removing edges with high betweenness, the modular structure of the network becomes apparent. This approach catalyzed extensive work on modularity optimization and inspired later algorithmic developments—most notably the Louvain method—which remains among the most widely used tools for community detection.

Assessing the performance of community-detection algorithms quickly emerged as a nontrivial challenge. Early benchmarks relied on synthetic graphs with planted partitions, but these lacked the heterogeneous degree and community-size distributions characteristic of empirical systems. A significant advance came with the introduction of benchmark graphs incorporating such heterogeneity [Phys. Rev. E 78, 046110 (2008)]. These constructions provided a far more stringent testing ground for modularity optimization and Potts-model clustering methods, exposing algorithmic limitations that conventional tests often obscured and establishing a new standard for validation.

Beyond static structure, network growth models also received important refinements. An extension of the Barabási–Albert preferential attachment model [Rev. Mod. Phys. 74, 47 (2002)] introduced a triad-formation step that promotes clustering during network growth [Phys. Rev. E 65, 026107 (2002)]. The resulting networks preserved the defining features of scale-free systems—power-law degree distributions and short path lengths—while achieving tunable, high clustering coefficients. This modification demonstrated how minimal local rules can reconcile empirical clustering with scale-free topology and complemented broader theoretical efforts to generalize mechanisms of network formation.

The interplay between network topology and dynamics is vividly illustrated by epidemic spreading processes. A seminal study showed that in scale-free networks, uniform random immunization fails to eradicate infections because the heavy-tailed degree distribution eliminates a finite epidemic threshold [Phys. Rev. E 65, 036104 (2002)]. The absence of a characteristic connectivity scale implies that highly connected hubs sustain transmission even at arbitrarily low infection rates. Targeted immunization strategies focusing on these hubs were shown to be dramatically more effective, revealing how structural heterogeneity fundamentally alters dynamical outcomes.

Closely related phenomena arise in cascading failures. In networks that transport redistributable flows, targeted attacks on heavily loaded nodes can trigger sequences of overload failures, potentially leading to large-scale collapse [Phys. Rev. E 66, 065102 (2002)]. This mechanism is particularly relevant for infrastructure systems such as communication networks and power grids, where load is unevenly distributed and small perturbations can propagate nonlocally.

Subsequent work refined this perspective by modeling load through betweenness centrality and assigning node capacity proportional to its initial load [Phys. Rev. E 69, 045104 (2004)]. Within this framework, the removal of a single highly loaded node can drastically reduce global efficiency and initiate systemic breakdown. Strikingly, networks that are robust against random failures may remain extremely fragile under targeted attacks—a structural asymmetry that has become central to understanding resilience in heterogeneous systems.

Empirical social systems provided some of the earliest and most influential applications of statistical physics to network science. One early study demonstrated that social networks differ fundamentally from technological and biological networks by simultaneously exhibiting high clustering and assortative mixing by degree [Phys. Rev. E 68, 036122 (2003)]. Using a simple group-based generative model, it showed that heterogeneity in group sizes naturally produces positive degree correlations consistent with empirical observations. This result illustrated how minimal structural ingredients can reproduce distinctive macroscopic features of social organization.

Community structure soon emerged as another defining property of social networks. A systematic methodology for identifying and characterizing modular organization was introduced and applied to an email communication network within a university [Phys. Rev. E 68, 065103 (2003)]. The analysis revealed a self-similar distribution of community sizes, suggesting that universal mechanisms of self-organization—analogous to those observed in river networks—may govern the formation and evolution of social structures.

Dynamical processes on networks soon became a central focus. Mean-field rate equations were developed to describe rumor spreading on heterogeneous networks, explicitly incorporating variability in node degree [Phys. Rev. E 69, 066130 (2004)]. Stochastic numerical solutions clarified how network topology shapes both the speed and the ultimate reach of information diffusion, with implications for peer-to-peer systems and broader models of social contagion.

More recently, attention has shifted toward multilayer and multiplex networks, which better capture the interconnected nature of real-world systems. A notable study analyzed the competition between epidemic spreading and awareness diffusion across coupled network layers [Phys. Rev. E 90, 012808 (2014)]. It identified a metacritical point at which the epidemic threshold becomes contingent on the state of the awareness process and showed that mass media broadcasting can suppress this transition. These results demonstrated how interacting dynamical processes on multilayer structures generate novel forms of collective behavior and criticality.

The past few years have witnessed impressive progress in artificial intelligence (AI), with AI-based technologies reshaping many aspects of human activity, including the sciences. At the core of this progress lie neural networks, flexible statistical models defined by a (typically) large number of parameters that can be tuned to learn adapted representations from data. While neural networks are now widespread, it is less widely appreciated that physicists have played an active role in their theoretical development since their early emergence in the 1980s.

Among the most influential is a work by Saad and Solla [Phys. Rev. E 52, 4225 (1995)], which provided one of the first theoretical studies of stochastic-gradient dynamics in two-layer neural networks. That work derived what is now called scaling limits for high-dimensional stochastic gradient descent, showing that in this regime the learning trajectories can be characterised by an autonomous system of coupled ordinary differential equations describing a reduced set of macroscopic quantities. Two years later, the generalization properties of ensembles of high-dimensional linear models was considered [Phys. Rev. E 55, 811 (1997)]. Despite its apparent simplicity, this setting sheds light in important properties shared by neural networks and which defy classical statistical thinking – for instance that overfitting is not necessarily at odds with generalisation.

The cross-fertilization between physics, statistics, and computer science extended beyond neural network theory. A central example lies in computational complexity, which seeks to characterize which problems can be solved efficiently. Although NP-hard problems are intractable in the worst case, typical instances may remain solvable. Mézard and Zecchina demonstrated that the typical-case hardness of random 𝑘-SAT is closely tied to structural features of the underlying energy landscape, such as metastability and clustering [Phys. Rev. E 66, 056126 (2002)]. This physical perspective provided new conceptual tools for understanding complexity and inspired algorithmic developments that drew sustained interest from the computer science community. Nearly a decade later, similar ideas were applied to the problem of detecting communities in large networks [Phys. Rev. E 84, 066106 (2011)], thus identifying the critical connectivity threshold for reliable inference in the stochastic block model.

These works illustrate a long-standing and productive dialogue between statistical physics and the theory of learning and computation. The current convergence between physics and AI is therefore not merely a byproduct of recent technological success, but the continuation of a deep intellectual partnership. As learning models grow ever more complex and operate in increasingly high-dimensional regimes, the conceptual and analytical tools developed within statistical physics remain indispensable for understanding their behavior.

Thermodynamics was originally invented as a phenomenological theory to describe the average behavior of heat and work. In designing the theory, motivation and inspiration was drawn from steam engines and how to operate them optimally. Quantum thermodynamics is the generalization of this classical theory to the quantum domain accounting explicitly for genuine quantum resources available only to quantum devices. It is important to note that this does not mean an overturning of classical statements, such as formulations of the second law of thermodynamics. Rather, quantum thermodynamics seeks to find new formulations of the same universal principles identifying how classical behavior emerges, and how quantum correlations can be leveraged to improve performance.

A central notion of classical thermodynamics is equilibrium, which can be characterized in terms of a few observables. From statistical mechanics it is known that a thermodynamic system relaxes into a state of equilibrium if its dynamics is mixing, that is chaotic and ergodic. Clearly, notions such as chaos and ergodicity are deeply classical since they rely on well-defined trajectories in phase space. In quantum systems, it is instead the mixing of eigenstate phases that leads to thermal statistics of the energy eigenvalues [Phys. Rev. E 50, 888 (1994)]. This eigenstate thermalization hypothesis has become the foundation of our understanding of the emergence of thermal equilibrium states in isolated quantum systems.

The workhorses of thermodynamics are heat engine cycles, which explore the optimal conversion of uncontrolled energy, that is heat, into useful, controlled energy, namely work. Once it had become clear that quantum systems do reach a state of thermodynamic equilibrium, it was obvious that heat engine cycles needed to be generalized to account for the transitions between such equilibrium states. The systematic study of quantum heat engines and provided conceptually sound identifications of heat and work in quantum processes [Phys. Rev. E 76, 031105 (2007)]. It was then shown that quantum engines at maximal power can outperform classical engines [Phys. Rev. E 81, 041106 (2010)]. The classical efficiency at maximum power, namely the Curzon-Ahlborn efficiency, is then obtained only in the weak dissipation limit.

Why then are quantum engines more efficient than classical devices? The answer lies in the fact that quantum systems have access to genuine quantum correlations that disappear in the semiclassical, high-temperature limit. This insight was worked out explicitly for ensembles of quantum batteries by showing that entanglement can be leveraged as an additional resource to improve extractable work from ensembles of quantum batteries [Phys. Rev. E 87, 042123 (2013)].

Strictly speaking the only processes fully characterized by thermodynamics are infinitely slow, quasistatic successions of equilibrium states. For finite-time processes, conventional thermodynamics typically only comprises inequalities that express that some amount of energy is irretrievably lost into the environment and that irreversible entropy is produced. Stochastic thermodynamics was formulated in the late 1990s to address this dichotomy. It generalizes and extends thermodynamics to small systems that may operate arbitrarily far from thermal equilibrium, notably through fluctuation theorems.

Jarzynski first showed that in isothermal processes the free-energy difference between two equilibrium states is related to the exponential average of the work performed over nonequilibrium realizations of the process [Phys. Rev. E 56, 5018 (1997)]. This Jarzynski equality has become the most important example of an integral fluctuation theorem. This was soon complemented by the Crooks fluctuation theorem [Phys. Rev. E 60, 2721 (1999)], which reveals that negative excursions of the entropy production can occur in nonequilibrium processes, but that such negative excursions are exponentially rare. Commonly, this fluctuation theorem is expressed as a ratio of the probability to measure positive excess work and the probability to measure negative excess work, which is equal to the exponential of the excess work.

The situation becomes more complicated in quantum systems, where classical trajectories in phase space lose their distinct meaning. In such situations, thermodynamic notions that are classically defined as functionals along trajectories need to be carefully re-considered. Thermodynamic work was hence later shown not to be a usual hermitian quantum observable, but rather given by a two-time correlation function [Phys. Rev. E 75, 050102(R) (2007)].

Active matter has emerged over the past two decades as a central paradigm for understanding how order, transport, and mechanical response arise in systems that are intrinsically out of equilibrium. Unlike traditional condensed matter systems, where energy is injected at boundaries or through external fields, active systems are driven locally: each constituent consumes energy at the microscopic scale and converts it into motion or mechanical stress. This persistent, internal energy injection fundamentally alters the organizing principles of collective behavior.

In physics, the field first took shape with the agent-based Vicsek model demonstrating a genuinely new kind of nonequilibrium phase transition: a transition not controlled by thermodynamic fields in the conventional sense, but generated by the collective motion of self-propelled entities [Phys. Rev. Lett. 75, 1226 (1995)]. The key novelty was the realization that symmetry breaking can arise endogenously, through persistent self-propulsion tied to an internal degree of freedom with its own dynamics—for example, a bacterium swimming along its body axis for a finite persistence time before reorienting. This coupling between internal dynamics, motion and a modest coupling to neighbors produces long-range correlations and collective order in regimes where equilibrium statistical mechanics predicts that long-range orientational order should be impossible. This behavior directly challenges equilibrium intuition, particularly in low dimensions. In a bird flock, for example, birds only need to sense and respond to the motion of their nearest neighbors to generate highly coordinated flocks, and how these nonlinearities—absent in equilibrium systems—generate qualitatively new scaling behavior and fluctuation properties.

A hydrodynamic explanation of the resulting critical behavior was quickly sought and provided in the form of continuum theories of flocking, most notably by Toner and Tu, which revealed how broken rotational symmetry, conservation laws, and convective nonlinearities conspire to stabilize ordered phases even in low dimensions [Phys. Rev. E 58, 4828 (1998)].

As the field developed, it became clear that activity can break symmetries in multiple, qualitatively distinct ways: active units may move along a direction (polar order), along an axis without head–tail distinction (nematic order), or rotate persistently around a preferred axis (chiral activity). An independent distinction emerged based on how momentum is handled in the environment: in “dry” active matter momentum is absorbed by a substrate [Phys. Rev. E 84, 040301 (2011)], while in “wet” systems momentum is conserved and transmitted through a surrounding fluid [Phys. Rev. E 76, 031921 (2007)]. These two axes—symmetry and momentum conservation—define qualitatively distinct universality classes and instabilities.

Today, theoretical and experimental advances developed within active matter constitute a versatile toolbox for describing a vast range of biological systems across scales, from microtubule assemblies and bacterial suspensions to immune cells, tissues, and bird flocks. Active matter research also inspired a broad range of novel synthetic systems, such as vibrated granular media, catalytically or electrohydrodynamically driven colloids, and bacterial suspensions. Across these systems, an expanding catalogue of nonequilibrium phenomena has been uncovered, from flocking and swarming to motility-induced phase separation, coherent cluster rotation, and novel acoustic modes [Rev. Mod. Phys. 85, 1143 (2013)].

When amorphous solids and supercooled liquids are subjected to shear, their mechanical response organizes in rich and highly structured ways across space and time. A sequence of influential studies in the late 1990s and early 2000s laid the foundation for our contemporary understanding of driven disordered materials.

Molecular-dynamics simulations of a two-dimensional binary Lennard-Jones glass under imposed shear at low temperature revealed that plastic deformation localizes into compact regions—typically only a few particle diameters across [Phys. Rev. E 57, 7192 (1998)]. This work introduced the shear transformation zone (STZ) framework, describing plastic flow in terms of the creation and evolution of localized rearrangements, and provided a microscopic basis for constitutive modeling of amorphous plasticity.

A complementary perspective emerged from simulations of sheared wet foams using the bubble model [Phys. Rev. Lett. 75, 4780 (1995)]. At low driving rates, the response was shown to be intermittent and burst-like, revealing connections to avalanche dynamics in other slowly driven, far-from-equilibrium systems.

Studies of sheared supercooled liquids further demonstrated that, in the athermal and strongly non-Newtonian regime, particle rearrangements become strongly spatially correlated [Phys. Rev. E 58, 3515 (1998)]. The associated correlation length was found to increase as the shear rate decreased, signaling the emergence of increasingly collective dynamics in the quasistatic limit.

Subsequent work using athermal quasistatic algorithms showed that plastic events organize into system-spanning, line-like avalanches whose extent is limited only by system size [Phys. Rev. E 74, 016118 (2006)]. This finding crystallized the connection between slowly driven amorphous materials and other classes of driven nonequilibrium systems exhibiting scale-free behavior.

Taken together, these studies established the modern picture of shear in amorphous matter: plasticity originates from localized particle rearrangements that interact elastically and self-organize into collective structures. This interplay between localization and long-range interactions provides the conceptual framework for understanding yielding, flow heterogeneity, and intermittent dynamics in disordered solids.

The physics of powders and granular materials has long posed formidable challenges in engineering and geophysics. The field took on renewed vitality when physicists recognized the remarkable range of collective phenomena these systems display [Rev. Mod. Phys. 68, 1259 (1996)]. Beneath practical concerns such as mixing, segregation, flow control, clogging, shear banding, packing, and stress propagation lie deep questions in continuum mechanics and statistical physics. These industrial problems also have clear geophysical analogues—creep, slip, subsidence, sound transmission, stratification, and pattern formation—highlighting the broad relevance of granular physics.

A central question concerns the mechanisms by which a static granular assembly can be fluidized under external driving. Early studies revealed unconventional rheological behavior arising from the limited set of dimensional control parameters available in assemblies of hard grains [Phys. Rev. E 64, 051302 (2001); Phys. Rev. E 72, 021309 (2005)]. These insights later evolved into unified frameworks bridging slow, quasistatic flows and inertia-dominated regimes.

Granular materials also became paradigmatic systems for new theoretical concepts, including self-organized criticality, the Edwards ensemble, and the jamming transition (see above). They provided a fertile testing ground for exploring athermal phenomena in disordered media, such as glassy relaxation, hysteresis, and large fluctuations, in both annealed and quenched settings [Phys. Rev. E 57, 1971 (1998)].

At the microscopic level, granular systems introduce ingredients absent from conventional materials: frictional contacts, inelastic collisions, and the inability to sustain tensile stress. These features generate striking macroscopic consequences tied directly to contact-level physics. One prominent manifestation is the emergence of highly heterogeneous stress networks [Phys. Rev. E 57, 3164 (1998)]. These were addressed by novel statistical models of force distributions [Phys. Rev. E 53, 4673 (1996)] that in turn inspired numerous theoretical extensions. Direct measurements of force heterogeneity produced the now iconic images of force chains [Phys. Rev. Lett. 82, 5241 (1999)], whose rearrangement and failure govern the unusual mechanical properties of granular assemblies.

The conceptual tools developed in granular physics have since influenced a wide range of soft matter systems, extending well beyond their original engineering and geophysical motivations.