Download The Non Equilibrium Steady State of Sparse Daniel Hurowitz H

The Non Equilibrium Steady State of Sparse
Systems with Non Trivial Topology
Daniel Hurowitz
Department of Physics, Ben Gurion University, Beer Sheva
Htotal = diag{En } − f (t){Vnm } + F · {Wnm } + HBath
I
Doron Cohen (BGU) [1,2]
I
Saar Rahav (Technion)[2]
1. D. Hurowitz and D. Cohen, Europhysics Letters 93, 60002 (2011)
2. D. Hurowitz, S. Rahav and D. Cohen, Europhysics Letters 98, 20002 (2012).
“Sparsity”
Htotal = diag{En } − f (t){Vnm } + F · {Wnm } + HBath
Histogram of couplings
n
wn,n+1
Histogram
25
20
n+1
15
10
5
0
−4
−2
0
log(g)
2
←− few decades −→
4
wn+1,n
= gn 2
gn = |Vn,n+1 |2 = couplings
“sparsity” = log wide distribution of couplings
Current vs. driving
Driving ; Stochastic Motive Force
; Current
Regimes: LRT regime, Sinai regime, Saturation regime
|Current|, SMF
0
10
−2
10
−4
10
SMF
|Current|
~est
lrt/sat
−6
10
−10
−5
10
I
10
∼
0
10
5
10
ε2
1
E∩
E
w exp −
2 sinh
N
2
2
Extent of the “Sinai regime” is determined by width of distribution of rates
Master equation description of dynamics
Htotal = diag{En } − f (t){Vnm } + F · {Wnm } + HBath
Quantum master equation for the reduced probability matrix:
dρ
2
= −i[H0 , ρ] − [V, [V, ρ]] + W β ρ ≡ Wρ
dt
2
Stochastic rate equation:
X
dpn
=
wnm pm − wmn pn
dt
m
The transition rates:
wnm
=
β
wnm
+ wnm
wnm
=
wmn
= gnm 2
=
En −Em
exp −
TB
Steady state equation:
ρ̇ = Wρ = 0
β
wnm
β
wmn
The Stochastic Motive Force (SMF)
If we had only a bath
wnm
En −Em
= exp −
wmn
TB
ε2=0.002
2.5
ε2=0.195
SMF
potential
2
We define a “field”
wnm
E(x) ≡ ln
wmn
1.5
1
0.5
0
−0.5
x
−1
0
200
400
600
800
and “potentials”
E(x1 ; x2 ) =
Z
x2
E(x)dx
[potential variation]
x1
n
o
E∩ ≡ maximum |E(x1 ; x2 )|
[activation barrier]
I
E ≡ E(x)dx if no driving = 0 [SMF]
With driving, E 6= 0. This means
Q
n
wn,n+1 6=
Q
n
wn+1,n .
1000
Current vs. driving
Driving ; Stochastic Motive Force
; Current
Regimes: LRT regime, Sinai regime, Saturation regime
|Current|, SMF
0
10
−2
10
−4
10
SMF
|Current|
~est
lrt/sat
−6
10
−10
−5
10
I
10
∼
0
10
5
10
ε2
1
E∩
E
w exp −
2 sinh
N
2
2
Extent of the “Sinai regime” is determined by width of distribution of rates
Emergence of the “Sinai regime”
Sinai [1982]: Transport in a chain with random transition rates.
Assume transition rates are uncorrelated.√
; build up of a potential barrier E∩ ∝ N .
; exponentially small current.
But... we have telescopic correlations: En,n+1 ∼ ∆n ≡ (En −En+1 )
Yet... we have sparsely distributed couplings: wn,n+1
= gn 2
E ≈ −
X
n

1
∆n
1 
∼
1 + gn 2 TB
TB 
2 ,
2
1/
√ ,
[±] N ∆,
Build up may occur if gn are from a log-wide distribution.
I
∼
1
E∩
E
w exp −
2 sinh
N
2
2
2 < 1/gmax
2 > 1/gmin
otherwise
Beyond fluctutation dissipation phenomenology:
Topological term in EAR formula
Q̇ =
Xh
i
β
β
w←
p
−
w
p
∆n
−
−
→
n n
n n−1
−5
10
DB
DB
− (0) + TB E I
TB
T
i
DB h
(gn 2 ) − (gn 2 )2 + Var(g)4
TB
≈
≈
EAR
n
−10
10
total EAR
excess EAR
Topological term
Current
−15
10
−10
10
The EAR is correlated with the current.
0
10
ε2
The quantum mechanical steady state
=
X
In→m
=
wmn pn − wnm pm ≡ tr(În→m ρ)
wnm pm − wmn pn
m
În→m
= |niwmn
hn| − |miwnm
hm|
β
β
β
În→m
= |niwmn
hn| − |miwmn
hm|
Quantum
dpn
dt
Classical
Stochastic
0.070
0.069
0.068
0.067
0.066
0.065
0.064
0.070
0.069
0.068
0.067
0.066
0.065
0.064
-5
0
5
Ε2
Quantum
dρ
dt
În→m
Jˆnm
= −i[H0 , ρ] − [V, [V, ρ]] + W β ρ
h
i2
2 ˆnm
= i J , V̂
= i |miVmn hn| − |niVnm hm|
Current
0.1
2
0.001
10-5
10-7
10-8 10-5 0.01
10
Ε2
104
107
Summary of main results
1. Due to the sparsity of the perturbation matrix, the NESS is of
glassy nature [1].
2. An extension of the Fluctuation-Dissipation phenomenology
has been proposed [1].
3. A log-wide distribution of couplings is required in order to
have a Sinai regime.
4. The topological term in the EAR is correlated with the current
but sub-linear in driving intensity.
5. Novel saturation effect in the quantum model.
6. The quantum current operator in the reduced description
includes off diagonal elements of the probability matrix.
[1] D. Hurowitz and D. Cohen, Europhysics Letters 93, 60002 (2011).
References and Acknowledgements
1. D. Hurowitz and D. Cohen, Europhysics Letters 93, 60002 (2011).
2. D. Hurowitz, S. Rahav and D. Cohen, Europhysics Letters 98, 20002
(2012).
I Sparsity: Austin, Wilkinson, Prosen, Robnik, Alhassid, Levine, Fyodorov,
Chubykalo, Izrailev, Casati
I Energy absorption by sparse systems: Cohen, Kottos, Schanz, Wilkinson,
Mehlig
I Network theory: Schnakenberg, Zia, Hill
I Sinai physics: Sinai, Derrida, Pomeau, Burlatsky, Oshanin, Mogutov,
Moreau, Bouchard
Acknowledgement: Bernard Derrida