Heat flow in chains driven by noise

Hans FogedbyAarhus University

andNiels Bohr Institute

(collaboration with Alberto Imparato, Aarhus)

Heat flow in chains 2

Outline

• Equilibrium • Non equilibrium• Fluctuation theorem• Driven bound particle • Driven harmonic chain• General fluctuation theorem• Summary

Venice 2012

Heat flow in chains 3

Equilibrium Single degree of freedom - particle in potential U(x) at temperature T

Particle

ThermostatTemperature T

Substrate

Q(t)

0

Boltzmann - Gibbs scheme:exp[- ( )] ( ) , 1/

( )

( ) exp[- ( )]

U xP x TZ

Z dx U x

Langevin scheme:( ) ( )

( ) ( ' ( ')

2 , FD theorem

dx dU x tdt dxt t t t

T

Static description

Dynamic description

k

U(x)

xVenice 2012

Heat flow in chains 4

( )

( )0 (0)

Force , Velocity

( ) ( ) [ ( )] [ (0)] x tt

x x

dU dxdx dtdU dx dUQ t d dx U x t U xdx d dx

init final 0 init 0 final final init exp( ( ) ( ) ( ) exp( [ ( ) ( )]) Q dx dx P x P x U x U x

Fluctuating heat transferred in time t

Heat distribution

Heat distribution function

Characteristic function at long times

Heat distribution – General result

Fogedby-Imparato ‘09

Venice 2012

( , ) exp( ) exp( ( ) 2

i

i

dP Q t Q Q ti

2

( ) ( ) ( , ) exp( )2 ( )

i

i

d Z ZP Q t Qi Z

Heat flow in chains 5

21 ( ) 2

U x kx

Harmonic oscillator

( )P Q

~log( )Q

~ exp( )Q

Q0• Distribution normalizable• Distribution even in Q• Mean <Q> = 0• Log divergence for small Q• Exponential tails for large Q• Independent of k

2 2 2 1/ 2

0

<exp( Q( ))>=[ /( )] ( ) ( / ) ( | |) 0 (K Bessel function of 2nd kind)

P Q K Q

Harmonic potential

Partition function

Heat distribution function

1/ 2 Z( )= 2 / k

Plot of P(Q) vs Q

Properties of P(Q)

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Heat flow in chains 6

Non equilibrium• 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 7

Fluctuation theorem

SystemT1 Q1 Q2T2

1 2

2 1

Energy balance:

Heat flux:

Fourier law:

: heat conductivity, : system size Heat conductivity : Ballistic: Diffusive: const.

Q QdQdt

T TdQ Tdt L

L

L

k k

kk

kk

Anomalous: , 0

Localization: , 0

L

L

k

k

Example: System driven by two heat reservoirs

• Heat reservoirs drive system • Non equilibrium steady state set up• Transport of heat• Heat is fluctuating

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Heat flow in chains 8

0

2 2

Heat ( ) ( )

Mean ( )

Cumulant ( ) ( )

t

Q t d q

Q t t q

Q t Q t t

( , ) exp(- ( )), / P Q t tF q q Q t

Large deviation function F(q) , ( )

( , ) exp( const. ), exponential tail

q F q q

P q t t q

20 0

20

20

( ) ( )

( , ) exp( const. ( ) )

( , ) exp( const. ( ) / ) Gaussian, random walk

q q F q q q

P Q t t q q

P Q t Q Q t

Large deviation function

Heat distribution

Fluctuating heat Q(t)Q(t)

t

P(q)

q

F(q)

q0q

t

0q

Heat distributionFluctuating heat transferred in time t

Heat distribution function

Gaussian

exponential

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Heat flow in chains 9

Gallavotti-Cohen fluctuation theorem (FT)

1 2

1 2

( ) ( ) (1/ 1/ )

( , ) 1 1exp , ( , )

F q F q q T T

P Q t Q Q qtP Q t T T

0

20

0 2 1

For close to :

( ) ( - ) FT implies 4 (1/ 1/ )

q q

F q A q qAq T T

1 2

For large

( ) for ( ) for

FT implies (1/ 1/ )

q

F q B q qF q B q q

B B T T

• 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

Large deviation function

linearF(q)

0q q

linear

Evans et al. ‘93 Gallavotti et al. ‘95

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Heat flow in chains 10

FT for generating function

( , ) exp( ) exp( ( ) 2

i

i

dP Q t Q Q ti

Distribution and characteristic function

Cumulant generating function m exp( ( ) exp( ( ))Q t t m

Legendre transform (steepest descent)'( ) ( )

( ) ( )

q q

F q q

m

m

Fluctuation theorem (FT)

1 2( ) (1/ 1/ )T Tm m

Cumulant generating function

normalization: (0) 0 branch points:

m

1 2

21 2

1 2

2 2

21 2

Close to equilibrium , expand ( )

( ) ( )( ) ( ) '(0) / = ( )( ), ( ) response

''(0) / 2 ( ), C( ) fluctuation

FT: ( ) (1/ 1/ ) implies ( ) (

T T T

T T T C TQ t q T T T T

Q Q t C T t

T T C T T T

m

m m

m

m m

) FD theorem states that C( ) ( ), QED T T

Fluctuation – dissipation theorem

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1 21/ 1/- T T

m

Heat flow in chains 11

1 2 1 2

1 1 1 1

2 2 2 2

// ( )( ) (0) 2 ( )( ) (0) 2 ( )

du dt pdp dt p ut T tt T t

k

1 1 1

2 2 2

/ ( ) / ( ) dQ dt p pdQ dt p p

Equations of motion

Heat exchange

Driven bound particle

exp( ( ) exp( ( ))Q t t m

Derrida-Brunet ‘05, Visco ’06, Fogedby-Imparato ’11, Sabhapandit ‘11

Characteristic function

m is the cumulant generating function

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Heat flow in chains 12

2 2 21 2 1 2 1 2 1 2 1 2

1( ) 2 (1 2 2 2 )2

T T TTm

Cumulant generating function (CGF) m

1 21 2 1 2

2 21 2

1 2 1 2 1 2 1 2

( ) ( )( ) 2

1 1 1 1 1 2

TT

T T T T TT

m

Branch points Cumulant generating function

Properties

2

1 2

2 21 2 1 2 1 2

( ) independent of (spring constant) One reservoir - equilibrium =0 = 0, CGF vanishes Equal temperatures T =T

1 ( ) 42

CGF symmetric Gallavotti

m k

m

m

T

1 2

-Cohen symmetry ( ) (1/ 1/ )m m T T

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m

1 21/ 1/- T T

Heat flow in chains 13

Numerical simulations

1 2 1 2

5max

1, 2, 1, 2

100, 10 independent trajectories

T T

t

maxP(Q/t )

maxQ/t-2 0.5

2.5

0

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Heat flow in chains 14

Driven harmonic chain Hamiltonian

12 2 2 2

1 11 1

1 ( ) ( )2 2 2

N N

n n n Nn n

H p u u u uk k

Equations of motion

1 1

12 1 1 1

1

1 1 1

( 2 )

( 2 )

( 2 )

( ) ( ') 2 ( ')( ) ( ') 2 ( ')

nn

nn n n

NN N N N

N N N

dup

dtdp u u udtdp u u pdtdp u u pdtt t T t tt t T t t

k

k

k

1 1 1 10

( ) ( )[ ( ) ( )]t

Q t d p p

Heat exchange

1exp( ( ) exp( ( )), Q t t Q Q m

Characteristic function

Cumulant generating function

1( ) lim ln exp( ( )t

Q tt

m

Saito-Dhar ‘11, Kundu et al. ‘11, Fogedby-Imparato ‘12

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Heat flow in chains 15

Heat exchange1

1

11

22

12

21

( ')' ( ) ( ')( ( ) ( )) ( , ') ( ')2 2

( , ') ' ( ) ( ') (1/ 2)( ( ) ( ') )

( , ') ' ( ) ( ')

( , ') ' ( ) ( ') (1/ 2) ( ')

( , ') ' ( ) ( ') (

NN

d dQ t F M

M A A A A

M B B

M A B B

M B A

11 1

/ 2 2

1/ 2) ( ) ( ) ( ) ( ) ( )

sin( / 2) ( ) 2 , (0) , | ( ) | 2 ( )

N

i t

BA i G B i G

tF e F t F t

Noise distribution1

1

111

12

1 ' ( ) exp ( ) ( ') ( ') 2 2 2

( , )

01 ( ') ( ')2 0

N

d dP

T

T

Identities (Gaussian path integral)1/ 2 exp( (1/ 2) ) det( )

det( ) exp(Tr ln( ))B I B

A A

Cumulant generating function1 ( ) Tr ln( 2 ) 2

I FMt

m

Solution1 1

1 2 2

2 2

2

( ) ( ) ( ) ( ) ( )sin( ) sin( 1) ( )

sin( 1) 2 sin( 2) sin( 3)sin( 1) sin( 2) ( )

sin( 1) 2 sin( 2) sin( 3)

( ) 2 , coupling to hea

n n nN N

n

nN

u G GN n p N n pG

N p N p N pn p n pG

N p N p N p

i

k

k kk

k k

k

2 2

t baths

4 sin ( / 2), phonon dispersion lawp k

p

Dispersion law

acoustic

0

2k1/2

Mathematical details

acoustic

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Heat flow in chains 16

Cumulant generating function2 2

1 2 1 2

1

1

2 2

1 ( ) ln[1 4 ( ) | ( )]2 2

( ) (1/ 1/ ) ( ) ( )

sin( ) ( ) , end-to-end Green's function( )

( ) sin( 1) 2 sin( 2) sin( 3)

N

N

d B f

f TT T TB i G

pGD p

D p N p N p N p

m

k

k k

Oscillating amplitudeLarge deviation function

Cumulant generating function

T1=10, T2 =12, 2, k1, N=10

T1=10, T2 =12, 2, k1, N=10

T1=10, T2 =12, 2, k1, N=10p

2( )B p

0

0.06

0

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m

Heat flow in chains 17

2 2

1 2 1 2

1( ) ln[1 4 ( ) | ( )]2 2

( ) (1/ 1/ )

d B f

f TT T T

m

Ln[..] singular for |B|2 f() =-1/42 yields branch points 1/T1 and 1/T2

Linear tails in F(q)Exponential tails in P(q)

T1=10, T2 =12, 2, k1, N=10

1slope ~ -1/T

Large deviation function

Exponential tails

Cumulant generating function

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2slope ~ 1/T

Heat flow in chains 18

Large N approximation

Oscillating amplitude

p0

0.06

0

2approx 2 2

1/ 2

2 2

1 2 1 2

2 sin( / 2)sin( ) | |1 4( / )sin ( / 2)

8 sin( / 2)sin( ) ( ) ( ) cos( / 2) ln 1 2 1 4( / )sin ( / 2)

( ) (1/ 1/ )

p pBp

dp p p fpp

f TT T T

kk

k m k k

2( )B p( )m

N=2

N=10

1 2T T 1, 2, 1blue: exact, red: approx

k 1 2T T 1, 2, 1blue: exact, red: approx

k

Cumulant generating function

Large N approximation

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Heat flow in chains 19Venice 2012

General fluctuation theorem

nk n k n n

-1n k n k

n n nk

n n

n n

Define β = β -β ,β =1/T

Fix k and define L({λ })=exp(β H)TL({λ })T exp(-β H)

λ +λ =β

L({λ })=L ({λ })

({λ })= ({λ })m m

NN 1 2 N

nk nn=1N 1 2 N n

N 1 2 N 1 2 N N-1 1 2 k-1 k+1 N

res 1 res 2

P (Q ,Q ,...,Q ) =exp(- β Q ), k=1,...,NP (-Q ,-Q ,...,-Q )

P (Q ,Q ,...,Q )= (Q +Q +...+Q )P (Q ,Q ,Q ,Q ,...,Q ) N=2, P (Q)=P (-Q)

Heat flow in chains 20

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

Venice 2012