Aging and Glassy Dynamics in Complex Systems: Some Theoretical Ideas

1994 ◽  
Vol 367 ◽  
Author(s):  
Jean-Philippe Bouchaud
Keyword(s):  
Complex Systems ◽  
Power Law ◽  
Spin Glasses ◽  
A Priori ◽  
Mean Field ◽  
Observation Time ◽  
Spin Glass Phase ◽  
Power Law Decay ◽  
Small Times ◽  
Keywords Aging

AbstractWe discuss some recent experimental results on the non-stationary dynamics of spin-glasses, which serves as an excellent laboratory for other complex systems. Inspired from Parisi's mean-field solution, we propose that the dynamics of these systems can be though of as a random walk in phase space, between traps characterized by trapping time distribution decaying as a power law. The average exploration time diverges in the spin-glass phase, naturally leading to time-dependent dynamics with a charateristic time scale fixed by the observation time tw itself (aging). By the same token, we find that the correlation function (or the magnetization) decays as a stretched exponential at small times t ≪ tw crossing over to power-law decay at large times t ≫ tw. Finally, we discuss recent speculations on the relevance of these concepts to real glasses, where quenched disorder is a priori absent. Keywords: Aging, slow dynamics, spin-glasses, glasses.

scholarly journals Anomalous Decay and Decoherence in Atomic Gases

2019 ◽  
Vol 26 (02) ◽  
pp. 1950008
Author(s):  
A. Tsabary ◽  
O. Kenneth ◽  
J.E. Avron
Keyword(s):  
Initial Data ◽  
Power Law ◽  
Mean Field ◽  
Quadratic Nonlinearity ◽  
Atomic Gases ◽  
Field Type ◽  
Power Law Decay ◽  
Single Qudit ◽  
Range Of Values ◽  
Continuous Range

Pair collisions in atomic gases lead to decoherence and decay. Assuming that all the atoms in the gas are equally likely to collide one is led to consider Lindbladian of mean field type where the evolution in the limit of many atoms reduces to a single qudit Lindbladian with quadratic nonlinearity. We describe three smoking guns for nonlinear evolutions: power law decay and dephasing rates; dephasing rates that take a continuous range of values depending on the initial data and finally, anomalous flow of the Bloch ball towards a hemisphere.

Spin glasses: recent advances in mean-field theory

1987 ◽  
Vol 65 (10) ◽  
pp. 1245-1250 ◽  
Author(s):  
B. W. Southern
Keyword(s):  
Field Theory ◽  
Spin Glasses ◽  
Glass Phase ◽  
Mean Field Theory ◽  
Mean Field ◽  
Three Dimensional ◽  
Spin Glass Phase ◽  
Ising Spin ◽  
Recent Advances ◽  
The Mean

A survey of recent advances in the mean-field theory of Ising spin glasses is presented. The physical picture of the spin-glass phase predicted by this theory is described, and its relationship to real three-dimensional systems is discussed.

Scaling

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Distribution Function ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Power Law ◽  
Scaling Laws ◽  
Power Laws ◽  
Distribution Functions ◽  
Temporal Process ◽  
Approximate Power

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power laws. The scaling object can be a function, a structure, a physical law, or a distribution function that describes the statistics of a system or a temporal process. We focus on scaling laws that appear in the statistical description of stochastic complex systems, where scaling appears in the distribution functions of observable quantities of dynamical systems or processes. The distribution functions exhibit power laws, approximate power laws, or fat-tailed distributions. Understanding their origin and how power law exponents can be related to the particular nature of a system, is one of the aims of the book.We comment on fitting power laws.

scholarly journals Non-Mean-Field Behavior of Realistic Spin Glasses

1996 ◽  
Vol 76 (3) ◽  
pp. 515-518 ◽  
Author(s):  
C. M. Newman ◽  
D. L. Stein
Keyword(s):  
Spin Glasses ◽  
Mean Field ◽  
Field Behavior

scholarly journals Power-law bounds for critical long-range percolation below the upper-critical dimension

2021 ◽  
Author(s):  
Tom Hutchcroft
Keyword(s):  
Long Range ◽  
Power Law ◽  
Upper Bound ◽  
Mean Field ◽  
Maximum Size ◽  
Finite Graph ◽  
Critical Dimension ◽  
Point Function ◽  
Upper Critical Dimension ◽  
Infinite Cluster

AbstractWe study long-range Bernoulli percolation on $${\mathbb {Z}}^d$$ Z d in which each two vertices x and y are connected by an edge with probability $$1-\exp (-\beta \Vert x-y\Vert ^{-d-\alpha })$$ 1 - exp ( - β ‖ x - y ‖ - d - α ) . It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if $$0<\alpha <d$$ 0 < α < d then there is no infinite cluster at the critical parameter $$\beta _c$$ β c . We give a new, quantitative proof of this theorem establishing the power-law upper bound $$\begin{aligned} {\mathbf {P}}_{\beta _c}\bigl (|K|\ge n\bigr ) \le C n^{-(d-\alpha )/(2d+\alpha )} \end{aligned}$$ P β c ( | K | ≥ n ) ≤ C n - ( d - α ) / ( 2 d + α ) for every $$n\ge 1$$ n ≥ 1 , where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality $$(2-\eta )(\delta +1)\le d(\delta -1)$$ ( 2 - η ) ( δ + 1 ) ≤ d ( δ - 1 ) relating the cluster-volume exponent $$\delta $$ δ and two-point function exponent $$\eta $$ η .

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