Anomalous Decay and Decoherence in Atomic Gases

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.

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Aging and Glassy Dynamics in Complex Systems: Some Theoretical Ideas

MRS Proceedings ◽

10.1557/proc-367-347 ◽

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.

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Mean-field type forward-backward doubly stochastic differential equations and related stochastic differential games

Frontiers of Mathematics in China ◽

10.1007/s11464-020-0889-y ◽

2020 ◽

Vol 15 (6) ◽

pp. 1307-1326

Author(s):

Qingfeng Zhu ◽

Lijiao Su ◽

Fuguo Liu ◽

Yufeng Shi ◽

Yong’ao Shen ◽

...

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Differential Games ◽

Mean Field ◽

Stochastic Differential Games ◽

Field Type ◽

Doubly Stochastic

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

Probability Theory and Related Fields ◽

10.1007/s00440-021-01043-7 ◽

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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Crackling noise and avalanches in minerals

Physics and Chemistry of Minerals ◽

10.1007/s00269-021-01138-6 ◽

2021 ◽

Vol 48 (5) ◽

Author(s):

Ekhard K. H. Salje ◽

Xiang Jiang

Keyword(s):

Electric Fields ◽

Power Law ◽

Mean Field Theory ◽

Mean Field ◽

Fractal Dimensions ◽

Similar Time ◽

Scaling Properties ◽

Pattern Formations ◽

Observation Method ◽

Great Complexity

AbstractThe non-smooth, jerky movements of microstructures under external forcing in minerals are explained by avalanche theory in this review. External stress or internal deformations by impurities and electric fields modify microstructures by typical pattern formations. Very common are the collapse of holes, the movement of twin boundaries and the crushing of biominerals. These three cases are used to demonstrate that they follow very similar time dependences, as predicted by avalanche theories. The experimental observation method described in this review is the acoustic emission spectroscopy (AE) although other methods are referenced. The overarching properties in these studies is that the probability to observe an avalanche jerk J is a power law distributed P(J) ~ J−ε where ε is the energy exponent (in simple mean field theory: ε = 1.33 or ε = 1.66). This power law implies that the dynamic pattern formation covers a large range (several decades) of energies, lengths and times. Other scaling properties are briefly discussed. The generated patterns have high fractal dimensions and display great complexity.

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