1. Heat flow in chains driven by noise Hans Fogedby Aarhus University and Niels Bohr Institute (collaboration with Alberto Imparato, Aarhus)

2. Outline Equilibrium Non equilibrium Fluctuation theorem Driven bound particle Driven harmonic chain General fluctuation theorem Summary Venice 2012Heat flow in chains2

3. Equilibrium Single degree of freedom - particle in potential U(x) at temperature T Particle Thermostat Temperature T Substrate Q(t) Static description Dynamic description  U(x) x Venice 2012Heat flow in chains3

4. Fluctuating heat transferred in time t Heat distribution Heat distribution function Characteristic function at long times Heat distribution – General result Fogedby-Imparato ‘09 Venice 2012Heat flow in chains4

5. Harmonic oscillator Distribution normalizable Distribution even in Q Mean = 0 Log divergence for small Q Exponential tails for large Q Independent of k Harmonic potential Partition function Heat distribution function Plot of P(Q) vs Q Properties of P(Q) Venice 2012Heat flow in chains5

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 Venice 2012Heat flow in chains6

7. Fluctuation theorem SystemT1T1 Q1Q1 Q2Q2 T2T2 Example: System driven by two heat reservoirs Heat reservoirs drive system Non equilibrium steady state set up Transport of heat Heat is fluctuating Venice 2012Heat flow in chains7

8. Large deviation function F(q) Large deviation function Heat distribution Fluctuating heat Q(t) Q(t) t P(q) q F(q) q Heat distribution Fluctuating heat transferred in time t Heat distribution function Venice 2012Heat flow in chains8

9. Gallavotti-Cohen fluctuation theorem (FT) 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 F(q) q Evans et al. ‘93 Gallavotti et al. ‘95 Venice 2012Heat flow in chains9

10. FT for generating function Distribution and characteristic function Cumulant generating function  Legendre transform (steepest descent) Fluctuation theorem (FT) Cumulant generating function Fluctuation – dissipation theorem Venice 2012Heat flow in chains10   

11. Equations of motion Heat exchange Driven bound particle Derrida-Brunet ‘05, Visco ’06, Fogedby-Imparato ’11, Sabhapandit ‘11 Characteristic function  is the cumulant generating function Venice 2012Heat flow in chains11

12. Cumulant generating function (CGF)  Branch points Cumulant generating function Properties Venice 2012Heat flow in chains12   

13. Numerical simulations Venice 2012Heat flow in chains13

14. Driven harmonic chain Hamiltonian Equations of motion Heat exchange Characteristic function Cumulant generating function Saito-Dhar ‘11, Kundu et al. ‘11, Fogedby-Imparato ‘12 Venice 2012Heat flow in chains14

15. Heat exchange Noise distribution Identities (Gaussian path integral) Cumulant generating function Solution  p Dispersion law acoustic  0   Mathematical details acoustic Venice 2012Heat flow in chains15

16. Cumulant generating function Oscillating amplitude Large deviation function Cumulant generating function T 1 =10, T 2 =12, , , N=10 Venice 2012Heat flow in chains16   

17. Ln[..] singular for |B  2 f( ) =-1/4   yields branch points     and    Linear tails in F(q) Exponential tails in P(q) T 1 =10, T 2 =12, , , N=10 Large deviation function Exponential tails Cumulant generating function Venice 2012Heat flow in chains17

18. Large N approximation Oscillating amplitude Cumulant generating function Large N approximation Venice 2012Heat flow in chains18

19. Venice 2012Heat flow in chains19 General fluctuation theorem

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 2012Heat flow in chains20