The Green Kubo Relations

Transcription

1 Chaper 4. The Green Kubo Relaions 4.1 The Langevin Equaion 4.2 Mori-Zwanzig Theory 4.3 Shear Viscosiy 4.4 Green-Kubo Relaions for Navier-Sokes Transpor Coefficiens

2 Chaper The Langevin Equaion In 1828 he boanis Rober Brown (1828a,b) observed he moion of pollen grains suspended in a fluid. Alhough he sysem was allowed o come o equilibrium, he observed ha he grains seemed o undergo a kind of unending irregular moion. This moion is now known as Brownian moion. The moion of large pollen grains suspended in a fluid composed of much ligher paricles can be modelled by dividing he acceleraing force ino wo componens: a slowly varying drag force, and a rapidly varying random force due o he hermal flucuaions in he velociies of he solven molecules. The Langevin equaion as i is known, is convenionally wrien in he form, dv = - ζv + F R (4.1.1) Using he Navier-Sokes equaions o model he flow around a sphere wih sick boundary condiions, i is known ha he fricion coefficien ζ= 3πηd/m, where η is he shear viscosiy of he fluid, d is he diameer of he sphere and m is is mass. The random force per uni mass F R, is used o model he force on he sphere due o he bombardmen of solven molecules. This force is called random because i is assumed ha <v().f R ()> =,. A more deailed invesigaion of he drag on a sphere which is forced o oscillae in a fluid shows ha a non-markovian generalisaion (see 2.4), of he Langevin equaion (Langevin, 198) is required o describe he ime dependen drag on a rapidly oscillaing sphere, dv() = - ' ζ(-' ) v(' ) + F R () (4.1.2) In his case he viscous drag on he sphere is no simply linearly proporional o he insananeous velociy of he sphere as in (4.1.1). Insead i is linearly proporional o he velociy a all previous imes in he pas. As we will see here are many ranspor processes which can be described by an equaion of his form. We will refer o he equaion d A() = - ' K(-' ) A(' ) + F() (4.1.3) as he generalised Langevin equaion for he phase variable A(Γ). K() is he ime dependen ranspor coefficien ha we seek o evaluae. We assume ha he equilibrium canonical ensemble average of he random force and he phase variable A, vanishes for all imes. < A() F() > = < A( ) F( +) > =, and. (4.1.4)

3 Chaper 4-3 The ime displacemen by is allowed because he equilibrium ime correlaion funcion is independen of he ime origin. Muliplying boh sides of (4.1.3) by he complex conjugae of A() and aking a canonical average we see ha, d C() = - ' K(-' ) C(' ) (4.1.5) where C() is defined o be he equilibrium auocorrelaion funcion, C() < A() A * () >. (4.1.6) Anoher funcion we will find useful is he flux auocorrelaion funcion φ() φ() = < Ȧ() Ȧ* () >. (4.1.7) Taking a Laplace ransform of (4.1.5) we see ha here is a inimae relaionship beween he ranspor memory kernel K() and he equilibrium flucuaions in A. The lef-hand side of (4.1.5) becomes e -s dc() = e-s C() - (-se -s ) C() = sc(s) - C(), and as he righ-hand side is a Laplace ransform convoluion, sc ~ (s) - C() = - K ~ (s) C ~ (s) (4.1.8) So ha C ~ (s) = C() s + K ~ (s) (4.1.9) One can conver he A auocorrelaion funcion ino a flux auocorrelaion funcion by realising ha, d 2 C() 2 = d < da() A * () > = d < [ila()] A * () > = d < A() [-ila * ()] > = - < [ila()] [-ila * ()] > = - φ().

4 Chaper 4-4 Then we ake he Laplace ransform of a second derivaive o find, - φ ~ (s) = e -s d 2 C() 2 = s e-s C() = e -s dc() + s e -s dc() + s 2 e -s C() = s 2 C(s) - s C(). (4.1.1) Here we have used he resul ha dc()/ =. Eliminaing C(s) beween equaions (4.1.9) and (4.1.1) gives K ~ (s) = φ ~ (s) C() - φ (s) s (4.1.11) Raher han ry o give a general inerpreaion of his equaion i may prove more useful o apply i o he Brownian moion problem. C() is he ime zero value of an equilibrium ime correlaion funcion and can be easily evaluaed as k B T/m, and dv/ = F/m where F is he oal force on he Brownian paricle. ζ(s) = C F (s) C F (s) mk B T - s (4.1.12) where C ~ F 1 (s) = < F() F ~ (s) > (4.1.13) 3 is he Laplace ransform of he oal force auocorrelaion funcion. In wriing (4.1.13) we have used he fac ha he equilibrium ensemble average denoed <.. >, mus be isoropic. The average of any second rank ensor, say < F() F() >, mus herefore be a scalar muliple of he second rank ideniy ensor. Tha scalar mus of course be 1/ 3 r{< F() F() >}= 1/3 < F() F() >. In he so-called Brownian limi where he raio of he Brownian paricle mass o he mean square of he force becomes infinie,

5 Chaper 4-5 β ζ(s) = 3m e -s < F() F() > (4.1.14) For any finie value of he Brownian raio, equaion (4.1.12) shows ha he inegral of he force auocorrelaion funcion is zero. This is seen mos easily by solving equaion (4.1.12) for CF and aking he limi as s. Equaion (4.1.9), which gives he relaionship beween he memory kernel and he force auocorrelaion funcion, implies ha he velociy auocorrelaion funcion Z() = 1/ 3 <v().v()> is relaed o he fricion coefficien by he equaion, Z(s) = k B T/m s + ζ(s) (4.1.15) This equaion is valid ouside he Brownian limi. The inegral of he velociy auocorrelaion funcion, is relaed o he growh of he mean square displacemen giving ye anoher expression for he fricion coefficien, Z() = lim ' 3 1 < v() v(' ) > = lim ' 3 1 < v() v(' ) > = lim 1 < v() r() > = lim d 2 < r() >. (4.1.16) Here he displacemen vecor r() is defined by r() = r() - r() = ' v(' ). (4.1.17) Assuming ha he mean square displacemen is linear in ime, in he long ime limi, i follows from (4.1.15) ha he fricion coefficien can be calculaed from k B T m ζ() D = 1 lim d < r() 2 > = 1 lim 6 6 < r() 2 >. (4.1.18) This is he Einsein (195) relaion for he diffusion coefficien D. The relaionship beween he diffusion coefficien and he inegral of he velociy auocorrelaion funcion (4.1.16), is an example of a Green-Kubo relaion (Green, 1954 and Kubo, 1957).

6 Chaper 4-6 I should be poined ou ha he ranspor properies we have jus evaluaed are properies of sysems a equilibrium. The Langevin equaion describes he irregular Brownian moion of paricles in an equilibrium sysem. Similarly he self diffusion coefficien characerises he random walk execued by a paricle in an equilibrium sysem. The idenificaion of he zero frequency fricion coefficien 6πηd/m, wih he viscous drag on a sphere which is forced o move wih consan velociy hrough a fluid, implies ha equilibrium flucuaions can be modelled by nonequilibrium ranspor coefficiens, in his case he shear viscosiy of he fluid. This hypohesis is known as he Onsager regression hypohesis (Onsager, 1931). The hypohesis can be invered: one can calculae ranspor coefficiens from a knowledge of he equilibrium flucuaions. We will now discuss hese relaions in more deail.

7 Chaper Mori-Zwanzig Theory We will show ha for an arbirary phase variable A(Γ), evolving under equaions of moion which preserve he equilibrium disribuion funcion, one can always wrie down a Langevin equaion. Such an equaion is an exac consequence of he equaions of moion. We will use he symbol il, o denoe he Liouvillean associaed wih hese equaions of moion. These equilibrium equaions of moion could be field-free Newonian equaions of moion or hey could be field-free hermosaed equaions of moion such as Gaussian isokineic or Nosé-Hoover equaions. The equilibrium disribuion could be microcanonical, canonical or even isohermalisobaric provided ha if he laer is he case, suiable disribuion preserving dynamics are employed. For simpliciy we will compue equilibrium ime correlaion funcions over he canonical disribuion funcion, f c, f c ( Γ ) = e -βh (Γ ) dγ e-βh (Γ ) (4.2.1) We saw in he previous secion ha a key elemen of he derivaion was ha he correlaion of he random force, F R () wih he Langevin variable A, vanished for all ime. We will now use he noaion firs developed in 3.5, which reas phase variables, A(Γ), B(Γ), as vecors in 6Ndimensional phase space wih a scalar produc defined by dγ f (Γ)B(Γ)A*(Γ), and denoed as (B,A*). We will define a projecion operaor which will ransform any phase variable B, ino a vecor which has no correlaion wih he Langevin variable, A. The componen of B parallel o A is jus, P B(Γ,) = (B(Γ,),A * (Γ)) A(Γ). (4.2.2) (A(Γ),A * (Γ)) This equaion defines he projecion operaor P. The operaor Q=1-P, is he complemen of P and compues he componen of B orhogonal o A. (QB(),A * (B(),A * ) ) = (B() - A, A * ) (A,A * ) = (B(),A * ) - (B(),A * ) (A,A * ) = (4.2.3) (A,A * ) In more physical erms he projecion operaor Q compues ha par of any phase variable which is random wih respec o a Langevin variable, A.

8 Chaper 4-8 B QB = (1-P)B = B - (B,A * )(A,A * ) -1 A A * PB = (B,A * )(A,A * ) -1 A Figure 4.1. The projecion operaor P, operaing on B produces a vecor which is he componen of B parallel o A. Oher properies of he projecion operaors are ha, PP = P, QQ = Q, QP = PQ =, (4.2.4) Secondly, P and Q are Hermiian operaors (like he Liouville operaor iself). To prove his we noe ha, (PB,C * ) * = ((B,A * )A,C * ) * (A,A * = ) * (B,A * ) * (A,C * ) * (A,A * ) = (B *,A) (A *,C) (A,A * ) = (A,B * ) (C,A * ) (A,A * ) = ((C,A * )A,B * ) (A,A * ) = (PC,B * ). (4.2.5) Furhermore, since Q=1-P where 1 is he ideniy operaor, and since boh he ideniy operaor and P are Hermiian, so is Q. We will wish o compue he random and direc componens of he propagaor eil. The random and direc pars of he Liouvillean il are iql and ipl respecively. These Liouvilleans define he corresponding random and direc propagaors, eiql and eipl. We can use he Dyson equaion o relae hese wo propagaors. If we ake eiql as he reference propagaor in (3.6.1) and eil as he es propagaor hen,

9 Chaper 4-9 e il = e iql + dτ e il(-τ) ipl e iqlτ. (4.2.6) The rae of change of A(), he Langevin variable a ime is, d A() = e il ila = e il i(q + P) LA. (4.2.7) Bu, e il ipla = e il (ila,a * ) (ila,a * ) A = e il A iω A(). (4.2.8) (A,A * ) (A,A * ) This defines he frequency iω which is an equilibrium propery of he sysem. I only involves equal ime averages. Subsiuing his equaion ino (4.2.7) gives, da() = iω A() + e il iqla. (4.2.9) Using he Dyson decomposiion of he propagaor given in equaion (4.2.6), his leads o, da() = iω A() + dτ e il(-τ) ipl e iqlτ iqla + e iql iqla. (4.2.1) We idenify eiql iqla as he random force F() because, (F(),A * ) = (e iql iqla,a * ) = (QF(),A * ) =, (4.2.11) where we have used (4.2.4). I is very imporan o remember ha he propagaor which generaes F() from F() is no he propagaor eil, raher i is he random propagaor eiql. The inegral in (4.2.1) involves he erm, ipl e iql iqla = iplf() = iplqf() = (ilqf(),a * ) A (A,A * ) = - (QF(),(iLA) * ) A (A,A * ) as L is Hermiian and i is ani-hermiian, (il)*=(d/)*=(dγ/ d/dγ) =d/=il, (since he

10 Chaper 4-1 equaions of moion are real). Since Q is Hermiian, ipl e iql iqla = - (F(),(QiLA) * ) A (A,A * ) = - (F(),F() * ) A - K() A, (4.2.12) (A,A * ) where we have defined a memory kernel K(). I is basically he auocorrelaion funcion of he random force. Subsiuing his definiion ino (4.2.1) gives da() = iω A() - dτ e il(-τ) K(τ) A + F() = iω A() - dτ K(τ) A(-τ) + F(). (4.2.13) This shows ha he Generalised Langevin Equaion is an exac consequence of he equaions of moion for he sysem (Mori, 1965a, b; Zwanzig, 1961). Since he random force is random wih respec o A, muliplying boh sides of (4.2.13) by A*() and aking a canonical average gives he memory funcion equaion, dc() = iω C() - dτ K(τ) C(-τ). (4.2.14) This is essenially he same as equaion (4.1.5). As we menioned in he inroducion o his secion he generalised Langevin equaion and he memory funcion equaion are exac consequences of any dynamics which preserves he equilibrium disribuion funcion. As such he equaions herefore describe equilibrium flucuaions in he phase variable A, and he equilibrium auocorrelaion funcion for A, namely C(). However he generalised Langevin equaion bears a sriking resemblance o a nonequilibrium consiuive relaion. The memory kernel K() plays he role of a ranspor coefficien. Onsager's regression hypohesis (1931) saes ha he equilibrium flucuaions in a phase variable are governed by he same ranspor coefficiens as is he relaxaion of ha same phase variable o equilibrium. This hypohesis implies ha he generalised Langevin equaion can be inerpreed as a linear, nonequilibrium consiuive relaion wih he memory funcion K(), given by he equilibrium auocorrelaion funcion of he random force.

11 Chaper 4-11 Onsager's hypohesis can be jusified by he fac ha in observing an equilibrium sysem for a ime which is of he order of he relaxaion ime for he memory kernel, i is impossible o ell wheher he sysem is a equilibrium or no. We could be observing he final sages of a relaxaion owards equilibrium or, we could be simply observing he small ime dependen flucuaions in an equilibrium sysem. On a shor ime scale here is simply no way of elling he difference beween hese wo possibiliies. When we inerpre he generalised Langevin equaion as a nonequilibrium consiuive relaion, i is clear ha i can only be expeced o be valid close o equilibrium. This is because i is a linear consiuive equaion.

12 Chaper Shear Viscosiy I is relaively sraighforward o apply he Mori-Zwanzig formalism o he calculaion of flucuaion expressions for linear ranspor coefficiens. Our firs applicaion of he mehod will be he calculaion of shear viscosiy. Before we do his we will say a lile more abou abou consiuive relaions for shear viscosiy. The Mori-Zwanzig formalism leads naurally o a non- Markovian expression for he viscosiy. Equaion (4.2.13) refers o a memory funcion raher han a simple Markovian ranspor coefficien such as he Newonian shear viscosiy. We will hus be lead o a discussion of viscoelasiciy (see 2.4). We choose our es variable A, o be he x-componen of he wavevecor dependen ransverse momenum curren J (k,). J(k,) k J (k,) J (k,) J(k,)= J (k,) + J (k,) Figure 4.2. We can resolve he wavevecor dependen momenum densiy ino componens which are parallel and orhogonal o he wavevecor, k. For simpliciy, we define he coordinae sysem so ha k is in he y direcion and J is in he x direcion. J x (k y,) = mv xi () exp(ik y y i ()) (4.3.1) In 3.8 we saw ha. J = ik P yx (k,) (4.3.2) where for simpliciy we have dropped he Caresian indices for J and k. We noe ha a zero wavevecor he ransverse momenum curren is a consan of he moion, dj/=. The quaniies we need in order o apply he Mori-Zwanzig formalism are easily compued.

13 Chaper 4-13 The frequency marix iω, defined in (4.2.8), is idenically zero. This is always so in he single variable case as -<A*dA/> =, for any phase variable A. The norm of he ransverse curren is calculaed N (J(k),J * (k)) = < i=1 N p xi e iky i p xj e -iky j > j=1 2 = N<p x1 > + N(N-1) <p x1 p x2 e ik(y 1-y 2 )> = Nmk B T (4.3.3) A equilibrium p xi is independen of p x2 and (y 1 -y 2 ) so he correlaion funcion facors ino he produc of hree equilibrium averages. The values of <p x1 > and <p x2 > are idenically zero. The random force, F, can also easily be calculaed since P P yx (k) = (P yx (k),j(-k)) J =, (4.3.4) < J(k) 2 > we can wrie, F() = iqlj = (1-P) ik P yx (k) = ik P yx (k). (4.3.5) The ime dependen random force (see (4.2.11)), is F() = e iql ik P yx (k) (4.3.6) A Dyson decomposiion of eqil in erms of eil shows ha, e il = e QiL + ds e il(-s) PiL e QiLs (4.3.7) Now for any phase variable B, PiLB = < J * J ilb > NmkB T = -< B(iLJ) * J > NmkB T J = - ik<bp yx (-k)> NmkB T (4.3.8)

14 Chaper 4-14 Subsiuing his observaion ino (4.3.7) shows ha he difference beween he propagaors eqil and eil is of order k, and can herefore be ignored in he zero wavevecor limi. From equaion (4.2.12) he memory kernel K() is <F()F*()>/ <AA*>. Using equaion (4.3.6), he small wavevecor form for K() becomes, K() = k 2 < P yx (k,) P yx (-k,)> Nmk B T (4.3.9) The generalised Langevin equaion (he analogue of equaion ) is lim k dj x (k y,) -k 2 = NmkB T ds < P yx (k y,s) P yx (-k y,) > J x (k y,-s) + ik y P yx (k y,) (4.3.1) where we have aken explici noe of he Caresian componens of he relevan funcions. Now we know ha he rae of change of he ransverse curren is ik P yx (k,). This means ha he lef hand side of (4.3.1) is relaed o equilibrium flucuaions in he shear sress. We also know ha J(k) = dk' ρ(k'-k) u(k'), so, close o equilibrium, he ransverse momenum curren (our Langevin variable A), is closely relaed o he wavevecor dependen srain rae γ(k). In fac he wavevecor dependen srain rae γ(k) is -ikj(k)/ρ(k=). Puing hese wo observaions ogeher we see ha he generalised Langevin equaion for he ransverse momenum curren is essenially a relaion beween flucuaions in he shear sress and he srain rae - a consiuive relaion. Ignoring he random force (consiuive relaions are deerminisic), we find ha equaion (4.3.1) can be wrien in he form of he consiuive relaion (2.4.12), lim P yx () = - ds η(k=,-s) γ(k=,s) k (4.3.11) If we use he fac ha, P yx V = lim(k ) P yx (k), η() is easily seen o be η() = βv < P xy () P xy () > (4.3.12) Equaion (4.3.11) is idenical o he viscoelasic generalisaion of Newon's law of viscosiy equaion (2.4.12). The Mori-Zwanzig procedure has derived a viscoelasic consiuive relaion. No menion has been made of he shearing boundary condiions required for shear flow. Neiher is

15 Chaper 4-15 here any menion of viscous heaing or possible nonlineariies in he viscosiy coefficien. Equaion (4.3.1) is a descripion of equilibrium flucuaions. However unlike he case for he Brownian fricion coefficien or he self diffusion coefficien, he viscosiy coefficien refers o nonequilibrium raher han equilibrium sysems. The zero wavevecor limi is suble. We can imagine longer and longer wavelengh flucuaions in he srain rae γ(k). For an equilibrium sysem however γ(k=) and < γ(k=) γ (k=) >. There are no equilibrium flucuaions in he srain rae a k=. The zero wavevecor srain rae is compleely specified by he boundary condiions. If we invoke Onsager's regression hypohesis we can obviously idenify he memory kernel η() as he memory funcion for planar (ie. k=) Couee flow. We migh observe ha here is no fundamenal way of knowing wheher we are waching small equilibrium flucuaions a small bu non-zero wavevecor, or he las sages of relaxaion oward equilibrium of a finie k, nonequilibrium disurbance. Provided he nonequilibrium sysem is sufficienly close o equilibrium, he Langevin memory funcion will be he nonequilibrium memory kernel. However he Onsager regression hypohesis is addiional o, and no par of, he Mori-Zwanzig heory. In 6.3 we prove ha he nonequilibrium linear viscosiy coefficien is given exacly by he infinie ime inegral of he sress flucuaions. In 6.3 we will no use he Onsager regression hypohesis. A his sage one migh legiimaely ask he quesion: wha happens o hese equaions if we do no ake he zero wavevecor limi? Afer all we have already defined a wavevecor dependen shear viscosiy in (2.4.13). I is no a simple maer o apply he Mori-Zwanzig formalism o he finie wavevecor case. We will insead use a mehod which makes a direc appeal o he Onsager regression hypohesis. Provided he ime and spaially dependen srain rae is of sufficienly small ampliude, he generalised viscosiy can be defined as (2.4.13), P yx (k,) = - ds η(k,-s) γ(k,s) (4.3.13) Using he fac ha γ(k,) = -iku x (k,) = -ikj(k,)/ρ, and ha dj(k,)/ = ikp yx (k,), we can rewrie (4.3.13) as, J(k,) = - k 2 ds η(k,-s) J(k,s) ρ (4.3.14) If we Fourier-Laplace ransform boh sides of his equaion in ime, and using Onsager's

16 Chaper 4-16 hypohesis, muliply boh sides by J(-k,) and average wih respec o he equilibrium canonical ensemble we obain, C ~ (k,ω) = C(k,) k 2~ η (k,ω) iω + ρ (4.3.15) where C(k,) is he equilibrium ransverse curren auocorrelaion funcion <J(k,) J(-k,)> and he ilde noaion denoes a Fourier-Laplace ransform in ime, C(ω) = C() e -iω. (4.3.16) We call he auocorrelaion funcion of he wavevecor dependen shear sress, 1 N(k,) VkB T < P yx (k,) P yx (-k,) > (4.3.17) We can use he relaion dj(k,)/ = ikp yx (k,), o ransform from he ransverse curren auocorrelaion funcion C(k,) o he sress auocorrelaion funcion N(k,) since, d 2 < J(k,) J(-k,) > = - < J(k,). J(-k,). > 2 = - k 2 < P yx (k,) P yx (-k,) > (4.3.18) This derivaion closely parallels ha for equaion (4.1.1) and (4.1.11) in 4.1. The reader should refer o ha secion for more deails. Using he fac ha, ρ=nm/v, we see ha, k 2 Vk B T N ~ (k,ω) = ω 2 C ~ (k,ω) + iω C(k,). (4.3.19) The equilibrium average C(k,) is given by equaion (4.3.3). Subsiuing his equaion ino equaion (4.3.15) gives us an equaion for he frequency and wavevecor dependen shear viscosiy in erms of he sress auocorrelaion funcion, ~ η (k,ω) = 1 - N ~ (k,ω) k 2 N ~ (k,ω) iωρ (4.3.2)

17 Chaper 4-17 This equaion is no of he Green-Kubo form. Green-Kubo relaions are excepional being only valid for infiniely slow processes. Momenum relaxaion is only infiniely slow a zero wavevecor. A finie wavevecors momenum relaxaion is a fas process. We can obain he usual Green-Kubo form by aking he zero k limi of equaion (4.3.2 ). In ha case η ~ ~ (,ω) = lim N (k,ω) k (4.3.21) Schemaic Diagram of he frequency and wavevecor dependen viscosiy and sress auocorrelaion funcion. ~ η(k,ω) ~ N(k,ω) k ~ (,) N(k,)= ω Figure 4.3. The relaionship beween he viscosiy, η(k,ω), and he sress auocorrelaion funcion, N(k,ω). A k= boh funcions are idenical. A ω= bu k, he sress auocorrelaion funcion is idenically zero. The sress auocorrelaion funcion is disconinuous a he origin. The viscosiy is coninuous everywhere bu non-analyic a he origin (see Evans, (1981)). Because he are no flucuaions in he zero wavevecor srain rae he funcion N(k,ω) is disconinuous a he origin. For all nonzero values of k, N(k,) =! Over he years many errors have been made as a resul of his fac. Figure 4.3 above illusraes hese poins schemaically. The resuls for shear viscosiy precisely parallel hose for he fricion consan of a Brownian paricle. Only in he Brownian limi is he fricion consan given by he auocorrelaion funcion of he Brownian force. An immediae conclusion from he heory we have oulined is ha all fluids are

18 Chaper 4-18 viscoelasic. Viscoelasiciy is a direc resul of he Generalised Langevin equaion which is in urn an exac consequence of he microscopic equaions of moion.

19 Chaper Green-Kubo Relaions for Navier-Sokes Transpor Coefficiens I is relaively sraighforward o derive Green-Kubo relaions for he oher Navier- Sokes ranspor coefficiens, namely bulk viscosiy and hermal conduciviy. In 6.3 when we describe he SLLOD equaions of moion for viscous flow we will find a simpler way of deriving Green-Kubo relaions for boh viscosiy coefficiens. For now we simply sae he Green-Kubo relaion for bulk viscosiy as (Zwanzig, 1965), η = V 1 Vk B T < [ p()v() - <pv> ][ p()v() - <pv> ] > (4.4.1) The Green-Kubo relaion for hermal conduciviy can be derived by similar argumens o hose used in he viscosiy derivaion. Firsly we noe from (2.1.26), ha in he absence of a velociy gradien, he inernal energy per uni volume ρu obeys a coninuiy equaion, ρdu/ = - J Q. Secondly, we noe ha Fourier's definiion of he hermal conduciviy coefficien λ, from equaion (2.3.16a), is J Q = -λ T. Combining hese wo resuls we obain ρ du = λ 2 T. (4.4.2) Unlike he previous examples, boh U and T have nonzero equilibrium values; namely, <U> and <T>. A small change in he lef-hand side of equaion (4.4.2) can be wrien as (ρ+ ρ) d(<u>+ U)/. By definiion d<u>/=, so o firs order in, we have ρd U/. Similarly, he spaial gradien of <T> does no conribue, so we can wrie ρ d U = λ 2 T. (4.4.3) The nex sep is o relae he variaion in emperaure T o he variaion in energy per uni volume (ρu). To do his we use he hermodynamic definiion, 1 E V T = V (ρu) T V = ρc V (4.4.4) where c V is he specific hea per uni mass. We see from he second equaliy, ha a small variaion in he emperaure T is equal o (ρu)/ρc V. Therefore, ρ U. = λ 2 ρ U (4.4.5) ρcv

20 Chaper 4-2 If D T λ/ρc V is he hermal diffusiviy, hen in erms of he wavevecor dependen inernal energy densiy equaion (4.4.5) becomes, ρ. U(k,) = - k 2 D T ρ U(k,) (4.4.6) If C(k,) is he wavevecor dependen inernal energy densiy auocorrelaion funcion, C(k,) < ρ U(k,) ρ U(-k,) > (4.4.7) hen he frequency and wavevecor dependen diffusiviy is he memory funcion of energy densiy auocorrelaion funcion, C(k,ω) = C(k,) iω + k 2 D T (k,ω) (4.4.8) Using exacly he same procedures as in 4.1 we can conver (4.4.8) o an expression for he diffusiviy in erms of a curren correlaion funcion. From (4.1.7 & 1) if φ = - d2c/2 hen, φ(k,) = k 2 < J Qx (k,) J Qx (-k,) > (4.4.9) Using equaion (4.1.1), we obain he analogue of (4.1.11), k 2 D T (k,ω) = C(k,) - iωc(k,ω) C(k,ω) = φ(k,ω). (4.4.1) φ(k,ω) C(k,) - iω If we define he analogue of equaion (4.3.17), ha is φ(k,) = k2 N Q (k,), hen equaion (4.4.1) for he hermal diffusiviy can be wrien in he same form as he wavevecor dependen shear viscosiy equaion (4.3.2). Tha is D T (k,ω) = N Q (k,ω) C(k,) - k 2 NQ (k,ω) iω. (4.4.11) Again we see ha we mus ake he zero wavevecor limi before we ake he zero frequency limi, and using he canonical ensemble flucuaion formula for he specific hea, ρc V C = (, ) 2 Vk T (4.4.12) B

21 Chaper 4-21 we obain he Green-Kubo expression for he hermal conduciviy λ = V k B T 2 < J Qx () J Qx () >. (4.4.13) This complees he derivaion of Green-Kubo formula for hermal ranspor coefficiens. These formulae relae hermal ranspor coefficiens o equilibrium properies. In he nex chaper we will develop nonequilibrium roues o he hermal ranspor coefficiens.