Green-Kubo formula for heat conduction in open systems

Transcription

1 Green-Kubo formula for heat conduction in open systems Abhishek Dhar Raman Research Institute, Bangalore, India. Collaborators: Anupam Kundu(Raman Research Institute) Onuttom Narayan(UC Santa Cruz) Green-Kubo formula for heat conduction in open systems p.1/22

2 Response to small temperature difference T + Τ I T I = G T G is the conductance of the system. For macroscopic systemsitisusefultodefinetheconductivity κ = GL/A. (L =length of system, A =cross-sectional area.) κexpectedtobeanintrinsicpropertyofthesystem and is given by linear response theory. Green-Kubo formula for heat conduction in open systems p.2/22

3 Green-Kubo formula for thermal conductivity κ = 1 lim lim τ L k B T 2 L d τ dt J x (t)j x (), J x (t) = j x (x,t)dx,where j x (x,t)istheheatflux densityin x-direction. Listhelinearsizeofthesystem (assumed to be isotropic). No rigorous derivation. Requires assumption of local thermal equilibrium. Luttinger s mechanical derivation also assumes relation between responses to applied field and temperature gradient. Limit of infinite system size necessary. Order of limits important. Green-Kubo formula for heat conduction in open systems p.3/22

4 Heatconductioninfinitesystems Hence the usual Green-Kubo formula cannot be directly used for small systems and systems with anomalous transport. Conductance of finite systems: earlier results. Include infinite reservoirs connected to finite systems. Get results identical to Landauer and nonequilibrium Green s function approach. Proved for non-interacting(e.g. harmonic) systems. Steady state fluctuation theorem implies the Green-Kubo formula. Valid for finite OPEN systems. Proved for anharmonic lattices connected to Markovian stochastic baths. Green-Kubo formula for heat conduction in open systems p.4/22

5 Steadystatefluctuationtheorem Cohen-GallavottiSSFT:Let β = 1/T R 1/T L.Consider rateofentropyproductionovertime τ. Q = τ dtj(t), s = ( β)q/τ P(s) P( s) = exp[sτ] τ Let Z(λ) = exp[ λq] exp[g(λ)τ].ssftthenimplies the symmetry relation g(λ) = g( β λ). Expanding both sides and comparing coefficients gives: (Gallavotti, Lebowitz-Spohn, Andrieux-Gaspard): G = lim T j T T = 1 k B T 2 j()j(t) T. Green-Kubo formula for heat conduction in open systems p.5/22

6 Outlineofourderivation Write equation for phase space distribution P(x,v,t).Find O( T)correctionto P = e βh /Z. Express J T intermsof J(t)J fp (). J fp isa specified current operator. Use continuity equations to relate J()J(t) and J()J b (t). J b isaninstantaneouscurrentoperator depending on heat flux from baths. Relate J()J b (t) to J()J fp (t) andthen,using time-reversalinvariance,to J(t)J fp () (Detailed balance). Green-Kubo formula for heat conduction in open systems p.6/22

7 OnedimensionallatticeHamiltonian:Langevinreservoirs j 1, L m i j N, R T L T R j i + 1, i H = N l=1 [ ml v 2 l 2 + V (x l ) ] + N 1 l=1 U(x l x l+1 ) m 1 v 1 = f 1 γ L v 1 + η L m l v l = f l l = 2, 3,...N 1 m N v N = f N γ R v N + η R. Green-Kubo formula for heat conduction in open systems p.7/22

8 Definitions f l = H/ x l η L (t)η L (t ) = 2γ L k B T L δ(t t ). η R (t)η R (t ) = 2γ R k B T R δ(t t ). Localenergy: ǫ l (t) Continuity equation: ǫ l (t)/ t = j l,l 1 j l+1,l + δ l,1 j 1,L + δ l,n j N,R This defines local current operator. Totalcurrent: J = l=1,n 1 j l+1,l Green-Kubo formula for heat conduction in open systems p.8/22

9 Fokker-Planckequation P(x,v,t) t where ˆLH = ˆL B = γl m 1 v 1 = ˆL H P(x,v,t) + ˆL B P(x,v,t), N ( + v l + f l x l v l m l l=1 ( v 1 + k ) BT L + γr m 1 v 1 m N ) v N ( v N + k BT R m N v N ) Let T = 1 2 (T L + T R ) and T = (T L T R ). Green-Kubo formula for heat conduction in open systems p.9/22

10 Fokker-Planckequation P(x,v,t) = t ˆLP(x,v,t) + ˆL T P(x,v,t), ( ˆL = ˆL H + γl v 1 + k ) BT + γr m 1 v 1 m 1 v 1 m N v N ˆL T (v) = k ( ) B T γ L 2 γr 2. 2 m 2 1 v1 2 m 2 N vn 2 ( v N + k BT m N v N Attime t = systemisinequilibrium: P = P = exp( βh)/z. Solve Fokker-Planck equation perturbatively upto order T. Let P(x,v,t) = P (x,v) + p(x,v,t). Green-Kubo formula for heat conduction in open systems p.1/22

11 SolutionofFokker-Planckequation p(x,v,t) = = β t t with J fp (v) = γl 2 dt eˆl(t t ) ˆL T (v) P (x,v) dt eˆl(t t ) J fp (v) P (x,v), [ v 2 1 k BT m 1 ] + γr 2 [ v 2 N k BT m N ]. Note that J fp = ( β P ) 1ˆL T P = [( βp) 1 P/ t] P=P Green-Kubo formula for heat conduction in open systems p.11/22

12 Finding J T The expectation value of total current is given by: J T = dxdv J p(x,v) = β dt dxdv J eˆlt J fp P = β dt J(t)J fp (). J fp doesnothaveanyobviousphysical interpretation. We need an expression in terms of J(t)J(). Green-Kubo formula for heat conduction in open systems p.12/22

13 Relating JJ fp to JJ b Definition: J b (t) = 1 2 (j 1,L j N,R ) = 1 2 [ γl v1(t) 2 + η L (t)v 1 (t) ] 1 2 [ γr vn(t) 2 + η R (t)v N (t)]. We prove: J()J b (t) = J()J fp (t) = J(t)J fp (). Recall: J fp (v) = γl 2 [ v k BT m 1 ] γr 2 [ v 2 N + k BT m N ]. Green-Kubo formula for heat conduction in open systems p.13/22

14 Relating JJ fp to JJ b Proof: J()η L (t)v 1 (t) = J()η R (t)v N (t) =. This is proved using Novikov s theorem. Also J() T =. Thus J()J b (t) = J()J fp (t). Butweneed J(t)J fp (). Use time-reversal symmetry. Green-Kubo formula for heat conduction in open systems p.14/22

15 Relating JJ fp to JJ b Let W(q,t q, )denotethetransitionprobabilityfrom q = (x,v )to q = (x,v)intime t. J fp (t)j() = dq dq J fp (v)j(x,v )P (x,v )W(x,v,t x,v, ) = dq dq J fp (v)j(x,v )P (x, v)w(x, v,t x, v, ) = dq dq J fp ( v)j(x, v )P (x,v)w(x,v,t x,v, ) Used: = J(t)J fp (). (i) detailed balance principle (ii) Jisoddinthevelocities, J fp iseven. Green-Kubo formula for heat conduction in open systems p.15/22

16 Relating JJ b to JJ Final step of proof: dt J(t)J() = (N 1) dt J()J b (t). Define D k (t) = k l=1 ǫ l N l=k+1 ǫ lfor k = 1, 2,...N 1. Fromcontinuityequation D k = 2j k+1,k (t) + 2J b (t). Define A(t) = N 1 l=1 D l.then: 2J(t) 2(N 1)J b (t) = da(t)/dt. Multiplying by J() taking... and integrating from t = to givesaboveresult.use: A()J() =, A( )J() =. Green-Kubo formula for heat conduction in open systems p.16/22

17 Finalresult J T = β dt J(t)J fp () J(t)J fp () = J()J b (t) = Definecurrent: J = J/(N 1). 1 N 1 dt J(t)J(). G = lim T J T T = 1 k B T 2 dt J(t) J(). Green-Kubo formula for heat conduction in open systems p.17/22

18 Othersystems Easy to generalize above proof to lattice models in arbitrary dimensions. Result valid for fluid system coupled to Maxwell baths.withcurrentdefinedas J = J,weget: L G = lim T J T T = 1 K B T 2 dt J(t) J() T. Also proved above result for Nose-Hoover baths and for an exponentially correlated stochastic bath. Green-Kubo formula for heat conduction in open systems p.18/22

19 Discussion Exact Green-Kubo like expression for the linear response conductance in a system connected to heat baths. Results valid in arbitrary dimensions and sizes. Derived both for lattice and fluid models Various bath models have been considered. Markovian, non-markovin and deterministic. Differences with the usual Green-Kubo formula: (i)noneedtofirsttakelimitofinfinitesystemsize. Result valid for finite systems. (ii)correlation function has to be evaluated not with Hamiltonian dynamics, but for an open system evolving with heat bath Green-Kubodynamics. formula for heat conduction in open systems p.19/22

20 Discussion (iii) Assumption of local thermal equilibrium is not necessary. Equilibration in absence of bias necessary. Likely that for systems with normal transport our formula will reduce to usual formula. Proof? For systems with anomalous transport(low dimensions), the present formula has to be used. Form of correlation functions very different. Boundary conditions important. Extension of steady state fluctuation theorem. Quantum systems Green-Kubo formula for heat conduction in open systems p.2/22

21 Linearresponsetheory Start with equilibrium phase space distribution correspondingtohamiltonian H : ρ(x 1,x 2...x N,p 1,...p N ) J =. Addperturbationto H : H = H + V e st V = e φ(x i ) Solveeqn.ofmotion ρ t = {ρ,h}tolinearorderin V tofind Calculate J. Gives ρ neq = ρ eq + δρ σ = () J()J(t) eq dt Green-Kubo formula for heat conduction in open systems p.21/22

22 Green-Kubo formula for heat transport Energy density:ǫ(x, t), current density: j(x, t). Continuity equation ǫ/ t +. j =. Fourier s law: j = D ǫgives: ǫ t D 2 ǫ = Consider fluctuations in the energy density in equilibrium: S(x, t) = ˆǫ(x, t)ˆǫ(, ) ˆǫ(x, t) ˆǫ(, ). Assume that fluctuations decay by transfer of energy inthesamewayastheprocessofnonequilibrium energy transfer. Thus: S(x, t) t D 2 S(x,t) = t > Green-Kubo formula for heat conduction in open systems p.22/22