Time correlations in complex quantum systems - Out of Equilibrium at

E(k) = ∨. 1. 0dx√. 1 − k2x2. 1 − x2. K(k) = ∨. 1. 0dxs. 1. (1 − k2x2)(1 − x2). S is analytic around zero frequency, and has branch points at thresholds ±ωmax ...
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Time correlations in complex quantum systems Sudhir R. Jain Nuclear Physics Division, Bhabha Atomic Research Centre Mumbai 400085, India

(Collaborator : Pierre Gaspard ULB Bruxelles, Belgium )

Instiut Henri Poincare, Paris, France

Statement of the problem H

+ λ V

C ( { t i} ) ρ

= =

t r ρ V ( t n) · · · V ( t 1) ex p (− β H ) tr ex p (− β H )

Random matrix models 2-point function C(t) = tr ρV (t)V (0) (Hi , Vi , Hi ) Htotal = Htotal =

PN

i=1

PN

i=1

Hi Hi

Htotal = Hi hC(t)i =

1 N

tr ρtotal U(−t)Vtotal U(t)Vtotal

= htr ρV (t)V (0)

Gaussian ensembles for averaging

Gaussian ensembles ¡ δ ¢ 2 P (H) = CN δ exp − 2 aH tr H

δ = 1(O), 2(U ), 4(S); similarly, P (V ) q q 2 σ(E) = aπH 2N |E| < 2N aH − E , aH

Nearest neighbour spacing distributions P (S) = = =

µ ¶ πS 2 πS exp − 2 4 ¶ µ 2 2 32S 4S − exp π2 π ¶ µ 18 4 2 2 S 64S exp − 36 π 3 9π

Orthogonal Unitary Symplectic

GOE ´Q ³ P N N P(E1 , · · · , EN ) = C exp − 12 j=1 Ej2 I