Cumulant generating function

Report
Heat flow in chains driven by noise
Hans Fogedby
Aarhus University
and
Niels Bohr Institute
(collaboration with Alberto Imparato, Aarhus)
Outline
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Equilibrium
Non equilibrium
Fluctuation theorem
Driven bound particle
Driven harmonic chain
General fluctuation theorem
Summary
Venice 2012
Heat flow in chains
2
Equilibrium
Single degree of freedom - particle in potential U(x) at temperature T
Static description
B oltzm ann - G ibbs schem e:
exp[-  U ( x )]
P0 ( x ) 
Z ( ) 
Z ( )
 dx
Thermostat
Temperature T
,   1/T
exp[-  U ( x )]
Dynamic description
Q(t)
Particle
U(x)
L angevin schem e:
dx
dt
 
dU ( x )
k
  (t )
dx
 ( t ) ( t '    ( t  t ')
Substrate
  2  T , FD theorem
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x
Heat flow in chains
3
Heat distribution
Fluctuating heat transferred in time t
dU
Force 
, V elocity
dx
dx
dt
dx ( )
 dU 
Q (t )   d  


 dx  x ( ) d 
0
t
x (t )

dU
dx
 U [ x ( t )]  U [ x (0)]
dx
x(0)
Heat distribution function
i
P (Q , t ) 

 i
d
2 i
exp(   Q )  exp(  Q ( t ) 
Characteristic function at long times
exp(  Q (  ) 
 dx
init
dx final P0 ( x init ) P0 ( x final ) exp(  [U ( x final )  U ( x init )])
Heat distribution – General result
i
P (Q , t ) 

 i
d
2 i
exp(   Q )
Z (   )Z (   )
Z ( )
2
Fogedby-Imparato ‘09
Venice 2012
Heat flow in chains
4
Harmonic oscillator
Harmonic potential
U ( x) 
1
kx
2
2
P (Q )
Partition function
Z (  )=  2  /  k 
1/ 2
Heat distribution function
~ log ( Q )
< exp(  Q (  ))> = [  /(    )]
2
2
2
1/ 2
P ( Q )  (  /  ) K (  | Q |)
0
(K 0 B essel function of 2nd kind)
~ exp(   Q )
Properties of P(Q)
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Distribution normalizable
Distribution even in Q
Mean <Q> = 0
Log divergence for small Q
Exponential tails for large Q
Independent of k
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0
Q
Plot of P(Q) vs Q
Heat flow in chains
5
Non equilibrium
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Gibbs/Boltzmann scheme does not exist
Phase space distribution unknown
No free energy
Dynamic description:
Hydrodynamics
Transport equation
Master equation
Langevin/Fokker Planck equations
•
Close to equilibrium (well understood):
Linear response
Fluctuation-dissiption theorem
Kubo formula
Transport coefficients
•
Far from equilibrium (open issues):
Low d model studies
Fourier’s law
Fluctuation theorems
Large deviation functions
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Heat flow in chains
6
Fluctuation theorem
Example: System driven by two heat reservoirs
T1
Q1
System
T2
Q2
E nergy balance:  Q1    Q 2 
H eat flux: 
dQ

dt
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Heat reservoirs drive system
Non equilibrium steady state set up
Transport of heat
Heat is fluctuating
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Fourier law : 
dQ
  k T  k
T 2  T1
dt
L
k : heat conductivity, L : system size
H eat conductivity k :
B allistic:
k  L
D iffusive:
k  const.
A nom alous:
k  L ,  0
L ocalization:
k  L ,  0
Heat flow in chains


7
Heat distribution
Fluctuating heat transferred in time t
Q(t)
Fluctuating heat Q(t)

t
Q (t ) 
H eat
 d  q ( )
0
M ean
Q (t )  t  q 
C um ulant
Q (t )   Q (t )   t
2
2
Heat distribution function
t
t
F(q)
Large deviation function
P ( Q , t )  exp(- tF ( q )), q  Q / t
Large deviation function F(q)
q  ,
F (q)  q
P ( q , t )  exp(  const.  t q ), exponential tail
q  q0
F (q )  (q  q0 )
2
q
q0
P(q)
Heat distribution
G aussian
P ( Q , t )  exp(  const.  t ( q  q 0 ) )
2
P ( Q , t )  exp(  const.  ( Q  Q 0 ) / t )
2
G aussian, random w alk
exponential
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Heat flow in chains
q0
q
8
Gallavotti-Cohen fluctuation theorem (FT)
F ( q )  F (  q )  q (1 / T1  1 / T 2 )

 1
1 
 exp   Q  
  , Q  qt

P (Q , t )
 T1 T 2  

P (Q , t )
•
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Evans et al. ‘93
Gallavotti et al. ‘95
FT holds far from equilibrium
FT yields fundamental symmetry for large deviation function
FT demonstrated under general conditions
FT generalizes ordinary FD theorem close to equilibrium
For q close to q 0 :
F (q )  A(q - q0 )
F(q)
2
Large deviation function
linear
FT im plies 4 A q 0  (1 / T 2  1 / T1 )
For large q
linear
F ( q )  B  q for q   
F ( q )  B  q for q   
FT im plies B   B   (1 / T1  1 / T 2 )
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Heat flow in chains
q0
q
9
FT for generating function
Distribution and characteristic function
i
P (Q , t ) 

 i
d
2 i
Cumulant generating function
exp(   Q )  exp(  Q ( t ) 
Cumulant generating function m
 exp(  Q ( t )   exp( t m (  ))

norm alization: m (0)  0

branch points:  
m
1 / T1 - 1 / T 2


C lose to equilibrium T1  T 2  T , expand m (  )

F (q )   q  m ( )
m (  )   (T )(T1  T 2 )   C (T ) 
Fluctuation theorem (FT)
m (  )  m (1 / T1  1 / T 2   )
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
Fluctuation – dissipation theorem

m '(  )  q     ( q )



Legendre transform (steepest descent)
2
m '(0)   Q  / t =  q    (T )(T1  T 2 ),  (T ) response
m ''(0)    Q    Q 
2
2
 / t  2 C (T ),
C ( t ) fluctuation
FT : m (  )  m (1 / T1  1 / T 2   ) im plies C ( T )  T  ( T )
2
FD theorem states that C ( T )   ( T ), Q E D
Heat flow in chains
10
Driven bound particle
Equations of motion
du / dt  p
dp / dt   (  1   2 ) p  k u   1   2
  1 ( t ) 1 (0)   2  1T1 ( t )
  2 ( t ) 2 (0)   2  2 T 2  ( t )
Heat exchange
dQ1 / dt  (   1 p   1 ) p
dQ 2 / dt  (   2 p   2 ) p
Characteristic function
 exp(  Q ( t )   exp( t m (  ))
m is the cumulant generating function
Derrida-Brunet ‘05, Visco ’06, Fogedby-Imparato ’11, Sabhapandit ‘11
Venice 2012
Heat flow in chains
11
Cumulant generating function (CGF) m
m ( ) 
1
1   2 
2
 1   2  2  1  2 (1  2  T1  2  T 2  2  T1T 2 ) 

2
2
2
Cumulant generating function
Branch points
m ( ) 
1   2

2

11
1
 


2  T1 T 2

(     )(     )
 1  2 T1T 2
2
2 
 1
 1   2  
1 


 
T
T
 1  2 T1T 2 
 1
2 

1 / T1 - 1 / T2
m
Properties
 m (  ) independent of k (spring constant)
 O ne reservoir - equilibrium  2 = 0
m = 0, C G F vanishes

 E qual tem peratures T1 = T 2
m ( ) 
1
1   2 
2 
 1   2 
2


2
 4  1 2T  

C G F sym m etric
 G allavotti -C ohen sym m etry
m (  )  m (1 / T1  1 / T 2   )
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Heat flow in chains
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2.5
Numerical simulations
P (Q /t m ax )
0
-2
Q /t m ax
0.5
 1  1,  2  2, T1  1, T 2  2
t m ax  100, 10 independent trajectories
5
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Heat flow in chains
13
Driven harmonic chain
Hamiltonian
H 
1
N

2
n 1
p 
2
n
k
N 1

2
( u n  u n 1 ) 
2
n 1
k
2
(u1  u N )
2
2
Equations of motion
du n
dt
dp n
dt
dp1
dt
dp N
dt
Heat exchange
 pn
t
 k ( u n  1  u n 1  2 u n )
 k ( u 2  2 u 1 )   p1   1
N
 d p
1
( )[   p1 ( )  1 ( )]
0
Characteristic function
 exp(  Q ( t )   exp( t m (  )), Q  Q 1
 k ( u N 1  2 u N )   p N   N
  1 ( t ) 1 ( t ')   2  T1 ( t  t ')

Q1 ( t ) 
( t )
N
( t ')   2  T N  ( t  t ')
Cumulant generating function
1
m (  )  lim ln  exp(  Q ( t ) 
t 
t
Saito-Dhar ‘11, Kundu et al. ‘11, Fogedby-Imparato ‘12
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Heat flow in chains
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Mathematical details
Solution
u n ( )

G n 1 ( ) 
G n N ( ) 
Dispersion law

G n 1 ( ) 1 ( )  G nN ( ) N ( )
2k1/2
 sin( N  n ) p  k sin( N  n  1) p
 sin( N  1) p  2 k  sin( N  2) p  k sin( N  3) p
2
2
 sin( n  1) p  k sin( n  2) p
acoustic
 sin( N  1) p  2 k  sin( N  2) p  k sin( N  3) p
2
2
 ( )     2 k  i   , coupling to hea t baths
2
acoustic
  4 k sin ( p / 2), phonon dispersion law
2
2
Heat exchange
Q (t ) 

  1 (   ') 
F (   ')( 1 ( ) N ( )) M ( ,  ') 

2 2
  N (   ') 
d d '


M 1 1 ( ,  ')      ' A ( ) A ( ')  (1 / 2)( A (  )  A (  ') )
M 2 2 ( ,  ')      ' B ( ) B ( ')


M 1 2 ( ,  ')      ' A ( ) B ( ')  (1 / 2) B (  ')

 i t / 2
B (  )   i G 1 N (  )
sin( t / 2)

d d '
 2
2

1
 ( )  (   ') (   ') 

  ( 1 ,  N )
, F (0)  t , | F (  ) |  2  t  (  )
2
Identities (Gaussian path integral)
 exp(  (1 / 2) B )   det( I   B )
1 / 2
Cumulant generating function
m ( )  
1
T r ln( I  2   F M )
2t
det( A )  exp(T r ln( A ))
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 1
P ( )  exp  
 2
1
0 
1  T1
 (   ') 

  (   ')
1 
2  0
T2 
M 2 1 ( ,  ')      ' B ( ) A ( ')  (1 / 2) B ( )
F ( )  2 e
Noise distribution
1

A ( )   i G 11 ( )
p
0
Heat flow in chains
15
Cumulant generating function
1
d

2 2
m ( )  
ln[1  4 
2
B ( ) |
2
Cumulant generating function
f (  )]
f (  )  T1T 2  (1 / T1  1 / T 2   )
G 1 N ( ) 
k sin( p )
m
B (  )   i G 1 N (  )
, end-to-end G reen's function
D( p)
D ( p )   sin( N  1) p  2 k  sin( N  2) p  k sin( N  3) p
2
2



B( p)
2
T1=10, T2 =12, 2, k1, N=10
Oscillating amplitude
Large deviation function
0.06
0
0
Venice 2012
T1=10, T2 =12, 2, k1, N=10

Heat flow in chains
p
T1=10, T2 =12, 2, k1, N=10 16
Exponential tails
Cumulant generating function
m ( )  
1
d

2 2
ln[1  4 
f (  )  T1T 2  (1 / T1  1 / T 2
2
B ( ) |
2
f (  )]
Ln[..] singular for |B|2 f() =-1/42
yields branch points 1/T1 and 1/T2
Linear tails in F(q)
 )
Exponential tails in P(q)
Large deviation function
slope ~ 1/T 2
slope ~ -1/T1
T1=10, T2 =12, 2, k1, N=10
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Heat flow in chains
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Large N approximation
B( p)
2
Cumulant generating function
m ( )
Oscillating amplitude
N=2
N = 10
T1  T 2  1,   2, k  1
T1  T 2  1,   2, k  1
blue: exact, red: approx
blue: exact, red: approx


0.06
0

0
p
Large N approximation
| B |approx 
2
m ( )   
2
sin( p / 2) sin( p )
 k 1  4(  / k ) sin ( p / 2)
2
dp
2
2
1 / 2

8 k
sin( p / 2) sin( p ) f (  ) 
k cos( p / 2) ln 1 

2
2
1

4(

/
k
)
sin
(
p
/
2)


f (  )  T1T 2  (1 / T1  1 / T 2   )
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Heat flow in chains
18
General fluctuation theorem
 D efine β nk = β n -β k ,β n = 1/T n
 Fix k and define L({ λ n })= exp(β k H )T L({λ n })T exp(-β k H )
-1
 λ n + λ n = β nk

 L({ λ n })= L ({ λ n })
 m ({ λ n })= m ({ λ n })
P N (Q 1 ,Q 2 ,...,Q N )
P N (-Q 1 ,-Q 2 ,...,-Q N ) n
N
= exp(-  β nk Q n ), k= 1,...,N
n= 1
P N (Q 1 ,Q 2 ,...,Q N )=  (Q 1 + Q 2 + ...+ Q N )P N -1 (Q 1 ,Q 2 ,Q k-1 ,Q k+ 1 ,...,Q N )
N = 2, Pres 1 (Q )= Pres 2 (-Q )
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Summary
• Analysis of cumulant generating function (CGF) for single
particle model and harmonic chain
• Gallavotti - Cohen fluctuation theorem (FT) shown
numerically (Evans et al. ’93) and theoretically under general
assumptions (Gallavotti et al. ’95)
• FT holds for bound particle model and for harmonic chain
• Large N approximation for harmonic chain
• General fluctuation theorem
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Heat flow in chains
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