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Order and disorder
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In physics, the terms order and disorder designate the presence or absence of some symmetry or correlation in a many-particle system.[citation needed]

In condensed matter physics, systems typically are ordered at low temperatures; upon heating, they undergo one or several phase transitions into less ordered states. Examples for such an order-disorder transition are:

The degree of freedom that is ordered or disordered can be translational (crystalline ordering), rotational (ferroelectric ordering), or a spin state (magnetic ordering).

The order can consist either in a full crystalline space group symmetry, or in a correlation. Depending on how the correlations decay with distance, one speaks of long range order or short range order.

If a disordered state is not in thermodynamic equilibrium, one speaks of quenched disorder. For instance, a glass is obtained by quenching (supercooling) a liquid. By extension, other quenched states are called spin glass, orientational glass. In some contexts, the opposite of quenched disorder is annealed disorder.

Characterizing order

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Lattice periodicity and X-ray crystallinity

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The strictest form of order in a solid is lattice periodicity: a certain pattern (the arrangement of atoms in a unit cell) is repeated again and again to form a translationally invariant tiling of space. This is the defining property of a crystal. Possible symmetries have been classified in 14 Bravais lattices and 230 space groups.

Lattice periodicity implies long-range order:[1] if only one unit cell is known, then by virtue of the translational symmetry it is possible to accurately predict all atomic positions at arbitrary distances. During much of the 20th century, the converse was also taken for granted – until the discovery of quasicrystals in 1982 showed that there are perfectly deterministic tilings that do not possess lattice periodicity.

Besides structural order, one may consider charge ordering, spin ordering, magnetic ordering, and compositional ordering. Magnetic ordering is observable in neutron diffraction.

It is a thermodynamic entropy concept often displayed by a second-order phase transition. Generally speaking, high thermal energy is associated with disorder and low thermal energy with ordering, although there have been violations of this. Ordering peaks become apparent in diffraction experiments at low energy.

Long-range order

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Long-range order characterizes physical systems in which remote portions of the same sample exhibit correlated behavior.

This can be expressed as a correlation function, namely the spin-spin correlation function:

where s is the spin quantum number and x is the distance function within the particular system.

This function is equal to unity when and decreases as the distance increases. Typically, it decays exponentially to zero at large distances, and the system is considered to be disordered. But if the correlation function decays to a constant value at large then the system is said to possess long-range order. If it decays to zero as a power of the distance then it is called quasi-long-range order (for details see Chapter 11 in the textbook cited below. See also Berezinskii–Kosterlitz–Thouless transition). Note that what constitutes a large value of is understood in the sense of asymptotics.

Quenched disorder

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In statistical physics, a system is said to present quenched disorder when some parameters defining its behavior are random variables which do not evolve with time. These parameters are said to be quenched or frozen. Spin glasses are a typical example. Quenched disorder is contrasted with annealed disorder in which the parameters are allowed to evolve themselves.

Mathematically, quenched disorder is more difficult to analyze than its annealed counterpart as averages over thermal noise and quenched disorder play distinct roles. Few techniques to approach each are known, most of which rely on approximations. Common techniques used to analyze systems with quenched disorder include the replica trick, based on analytic continuation, and the cavity method, where a system's response to the perturbation due to an added constituent is analyzed. While these methods yield results agreeing with experiments in many systems, the procedures have not been formally mathematically justified. Recently, rigorous methods have shown that in the Sherrington-Kirkpatrick model, an archetypal spin glass model, the replica-based solution is exact. The generating functional formalism, which relies on the computation of path integrals, is a fully exact method but is more difficult to apply than the replica or cavity procedures in practice.

Transition from disordered (left) to ordered (right) states

Annealed disorder

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A system is said to present annealed disorder when some parameters entering its definition are random variables, but whose evolution is related to that of the degrees of freedom defining the system. It is defined in opposition to quenched disorder, where the random variables may not change their values.

Systems with annealed disorder are usually considered to be easier to deal with mathematically, since the average on the disorder and the thermal average may be treated on the same footing.

See also

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Further reading

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Order and disorder

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from Grokipedia
In physics and , order and disorder describe the contrasting states of and within a , where order refers to structured, low- arrangements of particles or , and disorder corresponds to higher-, more probabilistic configurations. This is central to the second law of , which posits that in an , —a quantitative measure of disorder—tends to increase over time, driving spontaneous processes toward states of maximum multiplicity or disorganization. For instance, a neatly stacked pile of bricks represents low (order) with few possible configurations, while a scattered pile embodies high (disorder) with many more arrangements. The association of with disorder originated in the through the work of and , who formalized as S=kln⁡WS = k \ln W, where kk is Boltzmann's constant and WW is the number of microstates corresponding to a macrostate, linking molecular chaos to macroscopic randomness. However, this interpretation has faced critique, as disorder is not an intrinsic driver but a of free dissipation; systems evolve to minimize available free energy in the shortest time, yielding either ordered structures (like crystals) or disordered ones depending on conditions. In closed systems, the second law ensures progression to equilibrium at maximum , but open systems—such as those exchanging with their environment—can sustain or generate local order, as seen in self-organizing patterns like convection cells or fractal formations. Beyond physics, order and disorder permeate and , where maintain (order) amid universal increase by importing low- , such as , ensuring the overall of the rises while local biological order persists. This balance is evident in evolutionary processes, which do not violate thermodynamic laws but exploit open-system dynamics to produce adaptive structures from , countering the misconception that evolution requires decreasing . In physiological contexts, optimal function arises from a between order (stable control) and disorder (flexible variability), as excessive rigidity leads to fragility and unchecked chaos to inefficiency. These principles underscore how order emerges not in opposition to disorder but through its interplay, informing fields from cosmology to .

Fundamental Concepts

Definition of Order

Order in physical, chemical, and informational s is characterized by spatial, temporal, or configurational regularity, wherein the components of a —such as particles, molecules, or bits—follow predictable and repeatable patterns that enable structured arrangements. This regularity manifests as aligned orientations, periodic repetitions, or synchronized behaviors, allowing for the anticipation of system states based on local configurations. Perfect order represents an idealized scenario, exemplified by an ideal crystal lattice where atoms or molecules occupy precise positions in a repeating three-dimensional array without defects, vacancies, or distortions. In contrast, partial order prevails in real-world systems, where imperfections like thermal vibrations or impurities introduce deviations from this ideal, yet maintain overall structural coherence over finite scales. The conceptual foundations of order trace back to , where laid groundwork in 1850 through his formulation of the second law, emphasizing processes that proceed toward states of increasing unavailability of energy for work, implicitly linking to ordered configurations. This idea extended into in the 1870s via , who interpreted order probabilistically as configurations with fewer accessible microstates, providing a bridge from macroscopic regularity to microscopic ensembles. Representative examples include the ordered phase in ferromagnets, where atomic magnetic moments align parallel over macroscopic distances, yielding . Similarly, liquids display short-range molecular order due to intermolecular forces that promote packing, in contrast to the diffuse, unordered distribution in gases where molecules move freely without fixed positions. Long-range order quantifies such extended correlations, as explored in subsequent characterizations.

Definition of Disorder

Disorder in physical systems refers to the absence or disruption of regularity, characterized by and lack of predictability, serving as the direct counterpart to ordered states where correlations and symmetries dominate. This manifests as , compositional, or positional , in which system components exhibit minimal or no spatial or temporal correlations, resulting in increased variability and reduced predictability of macroscopic properties. provides a quantitative measure of this disorder, reflecting the number of accessible microscopic configurations. At a high level, disorder can be classified into static and dynamic forms. Static disorder, also known as positional disorder, arises when particles occupy multiple fixed positions without regular motion, leading to inherent structural irregularities. Dynamic disorder, conversely, involves vibrational motions, such as anharmonic thermal vibrations that introduce time-dependent fluctuations in particle positions. In contrast to ordered phases, disorder is associated with the preservation of full in many-particle systems, while the emergence of order involves . This concept was introduced by in his theory of phase transitions during , where the disordered phase corresponds to a zero value of the order , maintaining the system's original , whereas the ordered phase features a non-zero order that selects a particular symmetry-broken state. Disorder plays a fundamental role in non-equilibrium systems, such as and amorphous materials, where particles are arranged in a jammed, disordered configuration far from , exhibiting solid-like rigidity despite the lack of long-range order. These systems retain liquid-like structural randomness but are trapped in metastable states due to kinetic barriers, highlighting disorder's persistence beyond equilibrium conditions.

Characterizing Order

Lattice Periodicity and X-ray Crystallinity

Lattice periodicity refers to the regular, repeating arrangement of atoms or molecules within a , defined by a fundamental that tiles space through . This repetition occurs along three dimensions, with common types including cubic lattices, as seen in , and hexagonal lattices, as in , ensuring that the structure remains invariant under discrete translations by lattice vectors./01:_Translational_Symmetry) Such periodicity is the defining feature of crystalline order, enabling uniform physical properties like isotropic in highly symmetric lattices. The detection of lattice periodicity relies on X-ray diffraction (XRD), which exploits the wave nature of X-rays to probe atomic-scale arrangements. In 1912, Max von Laue, along with Walter Friedrich and Paul Knipping, conducted the first experiments demonstrating that X-rays passing through a produce interference patterns, confirming both the electromagnetic wave character of X-rays and the periodic atomic structure of crystals. This discovery laid the foundation for , revolutionizing the study of ordered phases. Shortly thereafter, in 1913, William Lawrence Bragg derived the condition for constructive interference in these patterns, known as : nλ=2dsin⁡θn\lambda = 2d \sin\theta where nn is an integer (the order of diffraction), λ\lambda is the X-ray wavelength, dd is the interplanar spacing, and θ\theta is the incidence angle. The derivation arises from the path length difference between X-rays reflected from adjacent crystal planes: for rays incident at angle θ\theta, the extra path length is 2dsin⁡θ2d \sin\theta, which must equal nλn\lambda for in-phase reinforcement due to the periodic scattering centers. Sharp diffraction peaks in XRD patterns thus directly evidence the long-range translational periodicity, distinguishing crystalline order from amorphous disorder. To quantify the degree of order in partially crystalline materials, the crystallinity index is calculated from XRD data as the ratio of the integrated intensity of crystalline peaks to the total scattered intensity, often assessed by the sharpness and resolution of peaks relative to broad amorphous halos. This metric, typically expressed as a , reflects the proportion of material exhibiting periodic lattice . Historically, XRD has been instrumental in identifying ordered phases in metals, such as the face-centered cubic arrangement in alloys, and in minerals, like the trigonal of , enabling precise characterization of their structural integrity and phase purity.

Long-Range Order

Long-range order refers to the persistence of correlations in the properties of a system over arbitrarily large distances, distinguishing ordered phases from disordered ones where such correlations decay rapidly. In , this is quantified through the decay of s, such as the pair correlation function g(r)g(r), which measures the probability of finding two particles at separation rr relative to a random distribution. In systems with long-range order, g(r)g(r) approaches 1 as r→∞r \to \infty with sustained oscillations or slow algebraic decay, reflecting global structural coherence; in contrast, disordered systems exhibit of g(r)−1g(r) - 1 to zero, indicating only short-range correlations. Mathematically, long-range order is often characterized by a non-zero order parameter ψ=lim⁡∣i−j∣→∞⟨σiσj⟩\psi = \lim_{|i-j| \to \infty} \langle \sigma_i \sigma_j \rangle, where σk\sigma_k represents local variables (e.g., spin components) at sites ii and jj. This average captures the alignment or coherence between distant parts of the system, remaining finite in ordered phases due to spontaneous symmetry breaking. In crystalline solids, long-range translational order manifests in the Fourier transform of the electron density ρ(r)\rho(\mathbf{r}), yielding the structure factor S(k)S(\mathbf{k}) with sharp delta-function peaks at reciprocal lattice vectors G\mathbf{G}, signaling perfect periodicity over infinite distances. This builds briefly on the concept of lattice periodicity as the foundational periodic arrangement enabling such infinite-range density correlations. Representative examples illustrate long-range order in diverse systems. In ferromagnets below the TCT_C, arises from aligned spins extending across the entire sample, establishing ψ≠0\psi \neq 0 via exchange interactions. In superconductors, off-diagonal long-range order (ODLRO) emerges through formation, where the two-particle correlation function ⟨ψ†(r)ψ†(0)⟩\langle \psi^\dagger(\mathbf{r}) \psi^\dagger(0) \rangle remains finite at large r\mathbf{r}, with the ξ\xi quantifying the spatial extent of this phase coherence. However, long-range order is not universal and breaks down in low-dimensional systems due to thermal fluctuations. The Mermin-Wagner theorem proves that continuous symmetries cannot be spontaneously broken at finite temperatures in one- or two-dimensional lattices with short-range interactions, as demonstrated in the Ising and Heisenberg models, where correlations decay algebraically rather than sustaining a finite order parameter.

Types of Disorder

Quenched Disorder

Quenched disorder refers to static, frozen randomness in physical systems, where random heterogeneities such as defects or impurities are fixed in place and do not evolve with time, often resulting from non-equilibrium processes like rapid cooling or irradiation that lock in configurations far from thermodynamic equilibrium. This type of disorder is characterized by its non-ergodic nature, meaning the system cannot sample all possible configurations due to the immobility of the random variables, in contrast to annealed disorder where such variables can equilibrate thermally. Common mechanisms include the introduction of vacancies or dopants in semiconductors, where high-temperature processing followed by quenching prevents atomic diffusion and fixes the positions of these impurities. A prominent of quenched disorder is , a phenomenon where disorder-induced random potentials cause electron wavefunctions to become exponentially localized, rendering electrons immobile and leading to insulating behavior even at zero temperature. First described by in 1958, this occurs in disordered lattices where scattering from random site potentials prevents diffusive transport. Representative examples of quenched disorder include diluted magnets, where non-magnetic impurities randomly occupy lattice sites, creating random bond disorder that disrupts spin alignments and leads to frustrated magnetic states. Another key example is amorphous semiconductors such as hydrogenated (a-Si:H), produced by or from the melt, which exhibits structural disorder from irregular atomic bonding and dangling bonds that localize charge carriers. Theoretical modeling of quenched disorder often employs the random field , which captures the impact of random external fields on spin systems. The Hamiltonian for this model is given by H=−J∑⟨ij⟩σiσj−∑ihiσi,H = -J \sum_{\langle i j \rangle} \sigma_i \sigma_j - \sum_i h_i \sigma_i, where J>0J > 0 is the ferromagnetic coupling between nearest-neighbor spins σi=±1\sigma_i = \pm 1, and hih_i are quenched random fields drawn from a distribution (e.g., Gaussian) representing local heterogeneities like magnetic impurities. This model illustrates how quenched randomness can destroy long-range order, particularly in low dimensions, by favoring disordered ground states through domain formation.

Annealed Disorder

Annealed disorder refers to a type of structural in physical systems where the positions of atoms, , or other microscopic elements fluctuate dynamically due to , resulting in a time-averaged random configuration that is in . This contrasts with static forms of disorder, as the fluctuations allow for and reconfiguration on experimental timescales, typically observed in high-temperature phases of materials like solid solutions. The statistical foundation of annealed disorder lies in the maximization of configurational entropy, which quantifies the number of accessible microstates for the random arrangements. For a system with site occupations governed by probabilities pip_i, the configurational entropy is given by S=−k∑ipiln⁡pi,S = -k \sum_i p_i \ln p_i, where kk is Boltzmann's constant. In binary alloys, this manifests as the random mixing entropy for components with concentrations xx and 1−x1-x: ΔSmix=−kN[xln⁡x+(1−x)ln⁡(1−x)],\Delta S_{\text{mix}} = -k N [x \ln x + (1-x) \ln (1-x)], where NN is the number of sites; this entropy term stabilizes the disordered state at elevated temperatures by favoring random distributions over ordered ones. Representative examples include binary alloys such as Cu-Zn β-brass, where at high temperatures above approximately 470°C, the atoms occupy lattice sites randomly, forming a disordered in . Similarly, in spin glasses with frustrated interactions, fluctuate thermally to reach equilibrium configurations, averaging to a random state despite underlying in couplings, though the disorder itself remains fixed. Annealed disorder differs from ordered states through deviations in local correlations, quantified by short-range order parameters αij=1−Pijxj\alpha_{ij} = 1 - \frac{P_{ij}}{x_j}, where PijP_{ij} is the of finding atom type jj adjacent to type ii, and xjx_j is the average concentration of jj. In fully random annealed configurations, αij=0\alpha_{ij} = 0, but partial ordering at intermediate temperatures yields nonzero values, indicating local preferences without long-range periodicity. This parameter highlights how thermal averaging in annealed systems can produce effective randomness even when instantaneous snapshots show correlations.

Thermodynamic Aspects

Order-Disorder Transitions

Order-disorder transitions represent phase changes in physical systems where an ordered state, characterized by structured arrangements such as aligned or periodic atomic lattices, gives way to a disordered state as or external fields vary. These transitions are classified using the Ehrenfest scheme, which distinguishes orders based on discontinuities in derivatives of the ; first-order transitions exhibit discontinuities in the first derivative (e.g., or ), accompanied by , while second-order (continuous) transitions show discontinuities in higher derivatives like specific heat, without release. In , developed by , the transition is modeled through an expansion of the free energy in powers of the order parameter ψ, which quantifies the degree of order (e.g., in magnets). The Landau free energy functional takes the form F=F0+a(T−Tc)ψ2+bψ4,F = F_0 + a(T - T_c) \psi^2 + b \psi^4, where F_0 is the free energy of the disordered phase, a and b are positive coefficients, T is , and T_c is the critical temperature. Minimizing F with respect to ψ yields ψ = 0 above T_c (disordered state) and ψ ∝ (T_c - T)^{1/2} below T_c (ordered state), predicting a second-order transition where the order parameter vanishes continuously at T_c. A classic example is the Curie point in ferromagnets, where below T_c (e.g., 1043 K for iron), align spontaneously to form ferromagnetic order, but thermal agitation randomizes them into above T_c, marking a second-order transition. In binary alloys like CuAu, the transition involves atomic site ordering into an L1_0 structure below ~670 K, forming antiphase domains, which disorder into a random face-centered cubic lattice upon heating, often as a process near . Near T_c, order-disorder transitions exhibit governed by universality classes, where systems with similar symmetries (e.g., the 3D Ising class for uniaxial magnets or Ising-like alloys) share independent of microscopic details. The correlation length ξ, measuring the spatial extent of order fluctuations, diverges as ξ ∝ |T - T_c|^{-ν}, with ν ≈ 0.630 in the 3D Ising model, signaling the onset of long-range correlations that vanish at the transition.

Role of Entropy in Disorder

Entropy plays a pivotal role in by driving systems toward disordered states in . The second law of thermodynamics states that for any in an , the change in ΔS is greater than or equal to zero, ensuring that the total of the system and its surroundings increases or remains constant. This law reflects the natural tendency of systems to evolve toward configurations with greater disorder because such states correspond to a larger number of accessible microstates, as captured by the statistical definition of . In essence, disorder maximizes the multiplicity Ω of microscopic arrangements compatible with the observed macroscopic properties, thereby elevating the S = k \ln \Omega, where k is Boltzmann's constant. In solid materials, configurational entropy specifically measures the disorder due to the random distribution of atomic species on lattice sites. For an ideal binary solid solution with N_A atoms of species A and N_B atoms of species B distributed over N = N_A + N_B sites, the number of distinct arrangements, or multiplicity Ω, is given by the combinatorial expression: Ω=(NA+NB)!NA! NB!\Omega = \frac{(N_A + N_B)!}{N_A! \, N_B!} The associated configurational entropy is then S = k \ln \Omega, which quantifies the increased uncertainty and randomness in the atomic positions compared to a perfectly ordered crystal. This entropy contribution arises purely from the positional disorder and is significant in systems where atoms can interchange without energetic barriers, such as in annealed equilibrium states. The balance between order and disorder is governed by the G = H - T S, where H represents the , T the absolute temperature, and S the . Ordered configurations typically exhibit lower H owing to optimized interactions and bonding, but they possess lower S due to restricted microstates. At low temperatures, the term -T S is small, so minimizing G favors the low- ordered state. However, as temperature rises, the -T S contribution grows in magnitude, outweighing any enthalpic disadvantage and stabilizing the higher- disordered state, thereby explaining why disorder predominates at elevated temperatures. Prominent examples illustrate entropy's role in promoting disorder. During melting, the transformation from crystalline order to liquid disorder yields an entropy increase ΔS_fus that drives the phase change; for many metals, this adheres to Richard's rule, with ΔS_fus ≈ R (the , approximately 8.3 J/mol·K), reflecting the proliferation of microstates in the . Similarly, the represents a kinetic wherein rapid cooling traps the material in a highly disordered, non-crystalline state, preserving the high configurational entropy of the supercooled without allowing reorganization into an ordered . This frozen disorder maintains elevated entropy relative to the equilibrium , highlighting entropy's influence even in kinetically constrained systems.

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