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Heat flow in chains driven by noise

Heat flow in chains driven by noise. Hans Fogedby Aarhus University and Niels Bohr Institute ( collaboration with Alberto Imparato , Aarhus). Outline. Equilibrium Non equilibrium Fluctuation theorem Driven bound particle Driven harmonic chain General fluctuation theorem

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Heat flow in chains driven by noise

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  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 • Fluctuationtheorem • Driven boundparticle • Driven harmonicchain • General fluctuationtheorem • Summary Heat flow in chains

  3. Equilibrium Single degree of freedom - particle in potential U(x) at temperature T Staticdescription Thermostat Temperature T Q(t) Particle Dynamic description U(x) k Substrate x Heat flow in chains

  4. Heat distribution Fluctuating heat transferred in time t Heat distribution function Characteristicfunction at long times Heat distribution – General result • Fogedby-Imparato ‘09 Heat flow in chains

  5. Harmonic oscillator Harmonic potential Partition function Heat distribution function Properties of P(Q) • Distribution normalizable • Distribution even in Q • Mean <Q> = 0 • Log divergence for small Q • Exponentialtails for large Q • Independent of k Plot of P(Q) vs Q Heat flow in chains

  6. Non equilibrium • Gibbs/Boltzmannschemedoes not exist • Phasespace distribution unknown • No freeenergy • Dynamic description: Hydrodynamics Transport equation Master equation Langevin/Fokker Planck equations • Close to equilibrium (wellunderstood): Linear response Fluctuation-dissiptiontheorem Kuboformula Transport coefficients • Far from equilibrium (open issues): Low d model studies Fourier’slaw Fluctuationtheorems Large deviation functions Heat flow in chains

  7. Fluctuationtheorem Example: System driven by two heat reservoirs T2 T1 System Q1 Q2 • Heat reservoirs drive system • Non equilibriumsteadystate set up • Transport of heat • Heat is fluctuating Heat flow in chains

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

  9. Gallavotti-Cohen fluctuationtheorem (FT) • Evans et al. ‘93 • Gallavottiet al. ‘95 • FT holds far from equilibrium • FT yields fundamental symmetry for large deviation function • FT demonstrated under general conditions • FT generalizesordinary FD theoremclose to equilibrium Large deviation function F(q) q Heat flow in chains

  10. FT for generatingfunction Distribution and characteristicfunction Cumulantgeneratingfunction Cumulantgeneratingfunctionm(l) m(l) l l+ l- Legendretransform (steepestdescent) Fluctuation– dissipationtheorem Fluctuationtheorem (FT) Heat flow in chains

  11. Driven boundparticle Equations of motion Heat exchange Characteristicfunction m(l) is the cumulantgeneratingfunction • Derrida-Brunet ‘05, Visco ’06, Fogedby-Imparato ’11, Sabhapandit ‘11 Heat flow in chains

  12. Cumulantgeneratingfunction (CGF) m(l) Cumulantgeneratingfunction Branch points m(l) Properties l l- l+ Heat flow in chains

  13. Numerical simulations Heat flow in chains

  14. Driven harmonicchain Hamiltonian Equations of motion Heat exchange Characteristicfunction Cumulantgeneratingfunction • Saito-Dhar ‘11, Kundu et al. ‘11, Fogedby-Imparato ‘12 Heat flow in chains

  15. Mathematical details Solution Dispersion law w 2k1/2 acoustic acoustic p p 0 Heat exchange Noise distribution Cumulantgeneratingfunction Identities (Gaussianpath integral) Heat flow in chains

  16. Cumulantgeneratingfunction Cumulantgeneratingfunction m(l) l- l+ l T1=10, T2 =12, G=2, k=1, N=10 Oscillating amplitude Large deviation function T1=10, T2 =12, G=2, k=1, N=10 Heat flow in chains T1=10, T2 =12, G=2, k=1, N=10

  17. Exponentialtails Cumulantgeneratingfunction Ln[..] singular for |B(w)|2 f(l) =-1/4G2 yieldsbranch points l+=1/T1and l-=1/T2 Linear tails in F(q) Exponentialtails in P(q) Large deviation function T1=10, T2 =12, G=2, k=1, N=10 Heat flow in chains

  18. Large N approximation Cumulantgeneratingfunction Oscillating amplitude Large N approximation Heat flow in chains

  19. General fluctuationtheorem Heat flow in chains

  20. Summary • Analysis of cumulantgeneratingfunction (CGF) for single particle model and harmonicchain • Gallavotti- Cohen fluctuationtheorem (FT) shownnumerically (Evans et al. ’93) and theoretically under general assumptions (Gallavotti et al. ’95) • FT holds for boundparticle model and for harmonicchain • Large N approximation for harmonicchain • General fluctuationtheorem Heat flow in chains

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