Response characteristics of sparse (glassy) networks

Doron Cohen

Ben-Gurion University

dp

dt=[Wβ +Wν

]p wnm = wβnm + νgnm

Daniel Hurowitz (BGU) [1,2]

Yaron de Leeuw (BGU) [3]

Saar Rahav (Technion) [2]

Alex Stotland (BGU) [former works]

Φ

[1] D. Hurowitz, D. Cohen, EPL (2011).

[2] D. Hurowitz, S. Rahav, D. Cohen, EPL (2012).

[3] Y. de Leeuw, D. Cohen, PRE (2012).

http://www.bgu.ac.il/∼dcohen

$DIP, $BSF, $ISF

“Sparsity” of a constructed toy model

dp

dt=[Wβ +Wν

]p

wνn+1,n = gnν

n

n+1

Histogram of the couplings

−4 −2 0 2 40

5

10

15

20

25

log(g)

His

togr

am←− few decades −→

“sparsity” = log wide distribution of couplings

[Hurowitz, Rahav, Cohen (2012)]

“Sparsity” in the Mott hopping model

wnm = w0 e−εnm exp

[−rnm

ξ

]

ρ(r, ε)drdε =Ωd r

d−1dr

rd0f(ε)dε, Ωd = 2, 2π, 4π

f(ε) = 1 Mott hopping model

f(ε) = δ(ε) Degenerate hopping model

Definition of “sparsity” parameter:

s ≡ξ

r0s 1 means “sparse”

For the non-degenerate Mott hopping model:

rd0 =

(∆ξ

T

)ξd ; s =

(T

∆ξ

)1/d

[Miller, Abrahams (1960), Ambegaokar, Halperin, Langer (1971), Pollak (1972), Aharony et al (1991),...,

Amir, Oreg, Imry (2010), de Leeuw, Cohen (2012), Amir, Krich, Vitelli, Oreg, Imry (2012)]

“Sparsity” of a weakly chaotic driven system

Htotal = diagEn − f(t)Vnm

R

LyLx

wνn,m = ν |Vnm|2

[not a Gaussian matrix...]

10-12

10-9

10-6

10-3

x10

-4

10-3

10-2

10-1

100

u = 10-2

u = 10-3

u = 10-4

u = 10-8

u = 10-12

[log-wide distribution]

0 1 2 3 4 5 6ω [scaled]

10-3

10-2

10-1

100

101

X [

scal

ed]

classicalQM - EW1 - meanQM - EW2 - meanQM - EW1 - medianQM - EW2 - median

[median mean]

Stotland,

Budoyo,

Peer,

Kottos,

Cohen

(2008),

Stotland,

Cohen,

Davidson

(2009),

Stotland,

Kottos

Cohen

(2010),

Stotland,

Pecora,

Cohen

(2010,2011)

The definition of “sparse” random network

Random site model (1D/2D): sparsity affected by ξ

wnm = w0 e−εnm e−|xn−xm|/ξ

xn = random locations

εnm = random couplings or random activation potential

Banded lattice model (quasi-1D): sparsity affected by b

wnm = w0 e−εnm B (En − Em)

The models are characterized by two functions:

w(r, ε) = dependence of the rates on r and ε

ρ(r, ε) = distribution of “distances” in (r, ε) space

Remark: For a 1D chain with n.n. transitions, random r is like random ε,

resulting in rates that have the distribution [Alexander, Bernasconi, Schneider, Orbach (1981)]

f(w) ∝ ws−1, (w < w0), s ≡ ξ/r0

Response characteristics

wnm = wβnm + wνnm = wβnm + νgnm

wβnm

wβmn= exp

[−En−Em

TB

], gnm = gmn

wν by themselves - induces diffusion / ergodization

wβ by themselves - leads to equilibrium

Combined - leads to NESS

Linear response and traditional FD: ν × g wβ

Glassy response and Sinai physics: [within a wide crossover regime]

Semi-linear response and Saturation: ν × g wβ

Example: current versus driving

Driving ; Stochastic Motive Force ; Current

Regimes: LRT regime, Sinai regime, Saturation regime

10−10

10−5

100

105

10−6

10−4

10−2

100

|Cur

rent

|, |S

MF

|

ε2

SMFCurrentEq. AEq. B

ν

wn+1,n = wβ + gnν

I ∼ 1

Nw exp

[−E∩

2

]2 sinh

(E2

)

The energy absorption rate: the LRT to SLRT crossover

DrivingSystem

Work (W)Heat (Q)

BathSB(ω) SA(ω)

En

Bath Driving

ther

mal

bat

h

wnm = wβnm + wνnm

ν ≡ Driving intensity

TB ≡ Bath temperature

DLRT = wn

DSLRT =[1/wn

]−1

Expressions above assume

near-neighbor transitions only.

Semi-linear response

DrivingSystem

Work (W)Heat (Q)

BathSB(ω) SA(ω)

Semi-linear response:

D[λw] = λD[w]

D[wa + wb] > D[wa] +D[wb]

D(ν) exhibits LRT to SLRT crossover.

SLRT requires resistor network calculation

DLRT = wn

DSLRT =[1/wn

]−1

Experimental implication:

D[λS(ω)] = λ D[S(ω)]

D[S(1)+S(2)] > D[S(1)] +D[S(2)]

The generalized Fluctuation-Dissipation phenomenology

wnm = wβnm + wνnm

W = rate of heating =D(ν)

Tsystem

Q = rate of cooling =DB

TB−

DB

Tsystem

DrivingSystem

Work (W)Heat (Q)

BathSB(ω) SA(ω)

Hence at the NESS:

Tsystem =

(1 +

D(ν)

DB

)TB

Q = W =1/TB

D−1B +D(ν)−1

Experimental way to extract response:

D(ν) =Q(ν)

Q(∞)− Q(ν)DB

D(ν) exhibits LRT to SLRT crossover

D(ν) =

[(wn

wβ + wn

)][(1

wβ + wn

)]−1

D[LRT] = gn ν [weak driving]

D[SLRT] = [1/gn]−1ν [strong driving]

Expressions above assume n.n. transitions only.

Digression - derivation of the cooling rate formula

Q = cooling rate = −∑n,m

(En − Em) wβnm pm

pn − pm = occupation imbalance =

[2 tanh

(−En − Em

2Tnm

)]pnm

wβnm − wβmn = up/down transitions imbalance =

[2 tanh

(−En − Em

2TB

)]wβnm

Q =1

2

∑n,m

(En−Em)2 wβnm

TBpnm −

1

2

∑n,m

(En−Em)2 wβnm

Tnmpnm =

DB

TB−

DB

Tsystem

definition of the diffusion coefficient: DB ≡[

1

2

∑n

(En−Em)2 wβnm

]

definition of effective system temperature:1

Tsystem≡

[1

Tnm

]

The determination of D

w = symmetric matrix that represents a “sparse” network

D = [[w]] (resistor network calculation)

Can be determined via numerical solution of a circuit equation GV = I

D is reflected in:

S(t) =⟨r2(t)

⟩∼ Dt long time spreading

P(t) ∼ 1

(Dt)d/2long time survival probability

N (λ) ∼[λ

D

]d/2spectral function

Can be determined via numerical diagonalization, small λ asymptotics

Random site model, the 1D case

The resistor network calculation can be done analytically for a chain with

near neighbor transitions [Alexander, Bernasconi, Schneider, Orbach (1981)]

f(w) ∝ ws−1, (w < w0), s ≡ ξ/r0

For s > 1 we get diffusion:

D =

(1

N

∑n

1

wn

)−1

=s− 1

sw0,

For 0 < s < 1 we get sub-diffusion (D = 0) with:

S(t) ∼ t2s/(1+s), P(t) ∼ t−s/(1+s), N (λ) ∼ λs/(1+s)

This is reflected in the spectral analysis.

Random site model, 2D case vs 1D case

0 1 2 3 4 5s

0.0

0.2

0.4

0.6

0.8

1.0

D,α

α [S(t) ∝tα ]D

−20 −15 −10 −5 0

X =−1s

10−10

10−5

100

105

D

ResNet

Spectrallinear

nc =4.5

DERH = EXPd+2

(1

sc

)e−1/sc Dlinear sc =

(d

Ωc

nc

)−1/dξ

r0

Spectral signature: 2D vs 1D

−1410−1310−1210−1110−1010−910−810

−710−610

−510−410−310−210−1100

λ

10−2

10−1

100

N(λ)

1D

s=0.2

s=0.4

s=2

s=5

10−1610−1410−1210−1010−810−610

−410−210

λ

2D

s=0.05

s=0.1

s=0.15

10−12 10−8 10−4 100100

101

102

103

PN

10−12 10−8 10−4 100

λ λ

Solid lines - RG theory prediction [Amir, Oreg, Imry (2010)].

Dashed line - Resistor network theory prediction [YdL, DC (2012)].

Linear and ERH estimates

Dlinear =1

2d

∫ ∫w(r, ε)r2ρ(r, ε)dεdr

This will not work for “sparse” systems:

only percolating paths contribute to the transport.

We define a percolation threshold by:∫ ∫w(r,ε)>wc

ρ(r, ε)drdε = nc

nc = 2 for 1D networks

nc ≈ 4.2 for the random site model in 2D

Effective range hopping (ERH) estimate:

DERH =1

2d

∫ ∫minw(r, ε), wcr2ρ(r, ε)dεdr

gs = D/Dlinear

0 4 8 12 16

σ

10203040

b

0.0

0.5

1.0

0 10 20 30 40 50 60 70 80σ

10−4

10−3

10−2

10−1

100

g s

Resistor Network

DERH [nc =2]

ERH vs VRH

ERH threshold wc is defined via∫ ∫w(r,ε)>wc

ρ(r, ε)drdε = nc

VRH trade-off is along∫ ε

0

∫ r

0

ρ(r′, ε′)dr′dε′ = n∗

VRH optimum is the (r∗, ε∗) point

where w(r, ε) is maximal along this curve.

VRH estimate is D ∼ w(r∗, ε∗)× (r∗)2

ε

r

ERH thresholdVRH trade−offVRH optimum

const "w" contours

consistency if n∗ = nc/d

Application: rate of heating

Htotal = En − f(t)Vnm

gs ≡〈〈|Vnm|2〉〉s〈〈|Vnm|2〉〉a

f(t) = low freq noisy driving

; diffusion in energy space:

DSLRT = gsDLRT

; energy absorption:

E = (particles/energy)×D

DLRT = gc4

3π

m2v3E

L2Lwall f 2

[The “Wall Formula”]

Φ

DLRT = gc

( eL

)2

vELfree Φ2

[The “Drude Formula”]

Some numerical results

Beyond the “Wall Formula”

[ Dependence on deformation U ]

10-4

10-3

10-2

10-1

u

10-8

10-6

10-4

10-2

100

g

LRT (quantum)SLRT (untextured)SLRT

Beyond the “Drude Formula”

[ Dependence on disorder W ]

10-3

10-2

10-1

100

101

w

10-8

10-6

10-4

10-2

100

GΦ

Conclusions

1. “hopping random-site mode” =⇒ “sparse” matrices

2. “weak quantum chaos” =⇒ “sparse” matrices

3. Applications: beyond the “Drude formula” and beyond the “Wall formula”.

4. Resistors network calculation to get the response coefficient.

5. Effective Range Hopping approximation scheme.

6. Semi-linear response if the driving is stronger then the background relaxation.

7. “Glassy NESS” featuring wide distribution of microscopic temperatures.

8. Definition of effective NESS temperature, and extension of the F-D phenomenology.

9. For very strong driving - quantum saturation of the NESS temperature (T → T∞).

10. Topological aspects: The emergence of the Sinai regime.

11. Topological term in the formula for the heating rate.

Perspective and references

The classical LRT approach: Ott, Brown, Grebogi, Wilkinson, Jarzynski, Robbins, Berry, Cohen

The Wall formula (I): Blocki, Boneh, Nix, Randrup, Robel, Sierk, Swiatecki, Koonin

The Wall formula (II): Barnett, Cohen, Heller [1] - regarding gc

Semi Linear response theory: Cohen, Kottos, Schanz... [2-6]

Billiards with vibrating walls: Stotland, Cohen, Davidson, Pecora [7,8] - regarding gs

Sparsity: Austin, Wilkinson, Prosen, Robnik, Alhassid, Levine, Fyodorov, Chubykalo, Izrailev, Casati

Random networks: Mott; Miller, Abrahams; Ambegaokar, Halperin, Langer; Pollak [...]

Random site model: Alexander, Bernasconi, Schneider, Orbach; Amir, Oreg, Imry [...]

Extensions related to: Sinai; Derrida,Pomeau; Burlatsky, Oshanin, Mogutov, Moreau, [...]

[1] A. Barnett, D. Cohen, E.J. Heller (PRL 2000, JPA 2000)

[2] D. Cohen, T. Kottos, H. Schanz (JPA 2006)

[3] S. Bandopadhyay, Y. Etzioni, D. Cohen (EPL 2006)

[4] M. Wilkinson, B. Mehlig, D. Cohen (EPL 2006)

[5] A. Stotland, R. Budoyo, T. Peer, T. Kottos, D. Cohen (JPA/FTC 2008)

[6] A. Stotland, T. Kottos, D. Cohen (PRB 2010)

[7] A. Stotland, D. Cohen, N. Davidson (EPL 2009)

[8] A. Stotland, L.M. Pecora, D. Cohen (EPL 2010, PRE 2011)

[9] D. Hurowitz, D. Cohen (EPL 2011)

[10] Yaron de Leeuw, D. Cohen (PRE 2012)