Far-From-Equilibrium SRO States

• Far-from-equilibrium SRO states are configurations with pronounced short-range correlations that arise in systems driven away from thermal equilibrium.
• Advanced imaging, scattering methods, and nonlinear simulations are employed to probe these states, highlighting their dynamic and pathway-dependent features.
• This local order underpins macroscopic phenomena such as jamming, rigidity, and memory effects, offering insights for designing novel non-equilibrium materials.

Far-from-equilibrium short-range order (SRO) states denote configurations in which systems driven or prepared away from thermal equilibrium develop pronounced short-range correlations without attaining global or long-range periodic order. Such SRO states are characteristic of diverse non-equilibrium materials and physical regimes, including glasses, rapidly quenched alloys, granular assemblies, colloidal suspensions, and various amorphous solids. The distinctive features of far-from-equilibrium SRO are fundamentally tied to the system's dynamical evolution, external driving, and the inability of standard equilibrium statistical mechanics—usually predicated on ergodicity and free-energy minimization—to fully describe their structure or response.

1. Fundamental Characteristics of Far-From-Equilibrium Systems

Far-from-equilibrium systems are not governed by a universal extremality principle such as free energy minimization, which structures equilibrium phases. Instead, such systems are characterized by persistent energy flux, strong nonlinear responses, nonergodicity (the system does not explore its full configuration space even on long timescales), and dynamical processes that span broad length and time scales. Notable governing equations include the nonlinear Navier–Stokes equation for incompressible flows, which underpins the physics of turbulence, and low-dimensional models such as the Lorenz equations that display chaotic and far-from-equilibrium dynamics.

A key distinction is that, in these systems, local ordering—defining SRO—may reflect a memory of the preparation or processing history, with the system becoming "trapped" far from global equilibrium. The resulting states display local environments and pairwise correlations distinct from both equilibrium liquids and crystals, frequently featuring dynamic arrest, glassy relaxation, or intermittent rearrangements (Jaeger et al., 2010).

2. Manifestations and Materials with SRO under Far-from-Equilibrium Conditions

Multiple classes of materials and phenomena exemplify the paradigm of far-from-equilibrium SRO:

• Glasses and Amorphous Materials: Rapid cooling (quenching) prevents full crystallization, freezing local arrangements while suppressing global order. SRO governs their characteristic rigidity and mechanical anomalies.
• Granular Assemblies: Under shear, vibration, or slow compaction, clusters and force chains form, exhibiting strong correlations at the nearest-neighbor scale. These features dictate the macroscopic response, including jamming, yield, and avalanche statistics.
• Colloidal Suspensions and Foams: At high density, geometrical constraints lead to SRO apparent in particle packing and cage dynamics, while the bulk remains globally disordered.
• Metallic Glasses and Rapidly Processed Alloys: Non-equilibrium processing routes, including melt spinning and splat cooling, force the system into disordered yet locally correlated configurations; this enhances strength and toughness beyond equilibrium crystalline phases.

Key is the persistence of short-range correlations—detectable in pair correlation functions, scattering structure factors, and direct imaging—but an absence of Bragg peaks or periodicity. These local motifs (e.g., clusters, chains, cages, or tiles) often serve as seeds or nuclei for structural relaxation or incipient crystallization upon subsequent annealing, and their statistics heavily influence macroscopic observables (Jaeger et al., 2010).

3. Methodologies for Probing and Characterizing SRO in Non-Equilibrium Regimes

Analytical, experimental, and computational methodologies have been adapted or developed to paper far-from-equilibrium SRO:

• Advanced Imaging and Particle Tracking: High-speed video microscopy or X-ray/neutron free electron laser techniques resolve individual particles/granular constituents, enabling direct quantification of short-range correlations, force distributions, and dynamic cooperative rearrangements.
• Scattering Approaches: Dynamic light scattering and X-ray scattering, extended to non-equilibrium protocols, recover pair correlation functions and allow time-resolved tracking of local order parameters.
• Effective Temperature Concepts: Out-of-equilibrium systems often lack a well-defined thermodynamic temperature. Instead, generalized "effective temperatures"—often extracted via local fluctuation-dissipation ratios—serve to relate fluctuations and linear response, even for SRO in jammed or arrested configurations.
• Nonlinear and Hydrodynamic Simulations: Hydrodynamic models (e.g., for turbulence) and numerical solutions to nonlinear equations (e.g., driven lattice models or molecular dynamics) enable the exploration of emergent SRO and its relaxation under diverse forcing conditions (Jaeger et al., 2010).

These techniques, combined with direct computation of correlation functions and response to perturbation, have unveiled the dynamic evolution of SRO under various external drives (shearing, quenching, or vibration), and also facilitate prediction of macroscopic properties from microstructural statistics.

4. Absence of Universal Free-Energy Minimization and Pathway Dependence

A central feature of far-from-equilibrium SRO is its strong dependence on the history of system preparation. The microstructure and local statistical arrangement are sensitive to details such as quench rate, protocol sequence, applied stress, and mechanical agitation. In the absence of a universal minimization principle, these pathway-dependent structures persist and determine properties such as mechanical rigidity, yielding, and memory effects.

The lack of ergodicity implies that two samples of the same material, prepared by different non-equilibrium routes, may exhibit identical compositions and average densities but diverge in local SRO and thus in macroscopic properties. This pathway-dependency is seen, for example, in glasses produced by melt-quenching versus vapor deposition, or in granular materials compacted by tapping versus shearing. SRO thus encodes information about the nonequilibrium protocol at the microscopic level (Jaeger et al., 2010).

5. SRO and the Onset of Macroscopic Phenomena: Jamming, Rigidity, and Yielding

SRO in far-from-equilibrium systems is not a mere local curiosity but underpins the emergence of major macroscopic phenomena:

• Jamming Transitions: The formation of locally dense clusters or force chains provides the microscopic basis for rigidity in granular or colloidal systems, marking the crossover from flowing to arrested (jammed) states.
• Memory and Aging: SRO arrangements may retain a record of earlier structural configurations, affecting relaxation dynamics, aging rates, and the onset of failure or yielding under stress.
• Mechanical Response: Force distributions and spatial correlations within SRO domains are key determinants of the initiation and propagation of fractures or shear bands.
• Dynamic Heterogeneity: Heterogeneity in local relaxation and rearrangement timescales arises from the heterogeneous SRO environments, resulting in non-Gaussian statistics and dynamic facilitation phenomena observed in glass-formers and jammed systems (Jaeger et al., 2010).

6. Prospects, Control, and Theoretical Advancements

Understanding and controlling SRO under far-from-equilibrium conditions has direct implications for materials engineering, such as the design of high-strength alloys, composite polymers, and advanced amorphous materials. Theoretical frameworks—ranging from generalized statistical mechanics and effective thermodynamics to coarse-grained hydrodynamic and kinetic models—are under active development to capture the unique physics of these non-equilibrium assemblies.

Future directions include:

• The inversion of structural data to identify hidden or emergent SRO motifs using machine learning and data-driven algorithms.
• Development of processing protocols (controlled quenching, cyclic driving, annealing) that exploit pathway dependence of SRO for targeted material properties.
• Exploration of universal properties and scaling regimes in SRO-dominated phases across different classes of far-from-equilibrium matter.

Experimental advances in in-situ probing and rapid imaging are expected to deepen understanding, while cross-disciplinary links—to biological assembly, soft matter physics, and nonequilibrium statistical mechanics—will drive further insights into the organizing principles underlying SRO in far-from-equilibrium systems.

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By integrating nonlinear, non-ergodic, and multiscale dynamical features, the paper of SRO in far-from-equilibrium settings provides a route to rationalize and predict emergent behavior across a vast range of material and physical systems where traditional equilibrium paradigms break down (Jaeger et al., 2010).