Editing Fluctuation–dissipation theorem (section)

==Violations in glassy systems==
While the fluctuation–dissipation theorem provides a general relation between the response of systems obeying [[detailed balance]], when detailed balance is violated comparison of fluctuations to dissipation is more complex. Below the so called [[glass temperature]] <math>T_\text{g}</math>, [[Spin glass|glassy systems]] are  not equilibrated, and slowly approach their equilibrium state. This slow approach to equilibrium is synonymous with the violation of detailed balance. Thus these systems require large time-scales to be studied while they slowly move toward equilibrium.

<!-- Deleted image removed: [[File:Violation of FDT1.png|600px|thumb|Violation of the fluctuation-dissipation theorem (FDT) in the <math>D=3</math> Edwards-Anderson system with temperature <math>T</math>, magnetic susceptibility <math>\chi</math>, and spin-temporal correlation function <math>C</math>. Figure replotted from <ref name="Parisi2017">{{cite journal |title=A statics-dynamics equivalence through the fluctuation–dissipation ratio provides a window into the spin-glass phase from nonequilibrium measurements |doi=10.1073/pnas.1621242114 |url=https://www.pnas.org/content/114/8/1838/tab-article-info | volume=114 |year=2017 |journal=Proceedings of the National Academy of Sciences |pages=1838–1843 |author=Baity-Jesi Marco, Calore Enrico, Cruz Andres, Antonio Fernandez Luis, Miguel Gil-Narvión José, Gordillo-Guerrero Antonio, Iñiguez David, Maiorano Andrea, Marinari Enzo, Martin-Mayor Victor, Monforte-Garcia Jorge, Muñoz Sudupe Antonio, Navarro Denis, Parisi Giorgio, Perez-Gaviro Sergio, Ricci-Tersenghi Federico, Jesus Ruiz-Lorenzo Juan, Fabio Schifano Sebastiano, Seoane Beatriz, Tarancón Alfonso, Tripiccione Raffaele, Yllanes David}}</ref>.]] -->
To study the violation of the fluctuation-dissipation relation in glassy systems, particularly [[spin glasses]], researchers have performed numerical simulations of macroscopic systems (i.e. large compared to their correlation lengths) described by the three-dimensional [[Edwards-Anderson model]] using supercomputers.<ref name="Parisi2017">{{cite journal |title=A statics-dynamics equivalence through the fluctuation–dissipation ratio provides a window into the spin-glass phase from nonequilibrium measurements | doi=10.1073/pnas.1621242114 | volume=114 | year=2017 |journal=Proceedings of the National Academy of Sciences |pages=1838–1843 |author=Baity-Jesi Marco, Calore Enrico, Cruz Andres, Antonio Fernandez Luis, Miguel Gil-Narvión José, Gordillo-Guerrero Antonio, Iñiguez David, Maiorano Andrea, Marinari Enzo, Martin-Mayor Victor, Monforte-Garcia Jorge, Muñoz Sudupe Antonio, Navarro Denis, Parisi Giorgio, Perez-Gaviro Sergio, Ricci-Tersenghi Federico, Jesus Ruiz-Lorenzo Juan, Fabio Schifano Sebastiano, Seoane Beatriz, Tarancón Alfonso, Tripiccione Raffaele, Yllanes David|issue=8 |pmid=28174274 |pmc=5338409 |arxiv=1610.01418 |bibcode=2017PNAS..114.1838B |doi-access=free }}</ref> In their simulations, the system is initially prepared at a high temperature, rapidly cooled to a temperature <math>T=0.64 T_\text{g}</math> below the glass temperature <math>T_\text{g}</math>, and left to equilibrate for a very long time <math>t_\text{w}</math> under a magnetic field <math>H</math>. Then, at a later time <math>t + t_\text{w}</math>, two dynamical observables are probed, namely the [[response function]]
<math display="block">\chi(t+t_\text{w},t_\text{w})\equiv\left.\frac{\partial m(t+t_\text{w})}{\partial H}\right|_{H=0}</math>
and the spin-temporal [[correlation function]]
<math display="block">C(t+t_\text{w},t_\text{w})\equiv \frac{1}{V}\left.\sum_{x}\langle S_x(t_\text{w}) S_x(t+t_\text{w})\rangle\right|_{H=0}</math>
where <math>S_x = \pm 1</math> is the spin living on the node <math>x</math> of the cubic lattice of volume <math>V</math>, and <math display="inline">m(t) \equiv \frac{1}{V} \sum_{x} \langle S_{x}(t) \rangle</math> is the magnetization density. The fluctuation-dissipation relation in this system can be written in terms of these observables as
<math display="block">T\chi(t+t_\text{w}, t_\text{w}) = 1-C(t+t_\text{w}, t_\text{w})</math>

Their results confirm the expectation that as the system is left to equilibrate for longer times, the fluctuation-dissipation relation is closer to be satisfied.

In the mid-1990s, in the study of dynamics of spin glass models, a generalization of the fluctuation–dissipation theorem was discovered  that holds for asymptotic non-stationary states, where the temperature appearing in the equilibrium relation is substituted by an effective temperature with a non-trivial dependence on the time scales.<ref>{{cite journal | title=Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model |doi=10.1103/PhysRevLett.71.173 | volume=71 |year=1993 |journal=Physical Review Letters |pages=173–176 |author=Cugliandolo L. F.|authorlink= Leticia Cugliandolo|author2= Kurchan J.|issue=1 |pmid=10054401 |arxiv=cond-mat/9303036 |bibcode=1993PhRvL..71..173C |s2cid=8591240 }}</ref> This relation is proposed to hold in glassy systems beyond the models for which it was initially found.