Green-Kubo formula with ymmetrized correlation function for quantum ytem in teady tate: the hear vicoity of a fluid in a teady hear flow Hirohi Matuoa Department of Phyic, Illinoi State Univerity, Normal, Illinoi, 679-456, USA Phone: 39 438-336 FAX: 39-438-543 hmb@phyiltuedu ASTRACT For a quantum ytem in a teady tate with a contant current of heat or particle driven by a temperature or chemical potential difference between two reervoir attached to the ytem, the fluctuation theorem for the current wa previouly hown to lead to the Green-Kubo formula for the linear repone coefficient for the current expreed in term of the ymmetrized correlation function of the current denity operator In thi article, we how that for a quantum ytem in a teady tate with a contant rate of wor done on the ytem, the fluctuation theorem for a quantity induced in the ytem alo lead to the Green-Kubo formula expreed in term of the ymmetrized correlation function of the induced quantity A an example, we conider a fluid in a teady hear flow driven by a contant velocity of a olid plate moving above the fluid Keyword: Green-Kubo formula, quantum ytem in teady tate, fluctuation theorem, hear vicoity
INTRODUCTION AND A SUMMARY OF OUR RESULTS In the tandard linear repone theory [] for a quantum ytem ubject to a time-dependent external field, the Green-Kubo formula for the linear repone coefficient for a quantity induced by the external field i expreed in term of the canonical correlation function involving the induced quantity For example, for a ytem ubject to an external electric field ocillating along one direction with an angular frequency, we can obtain the following Green-Kubo formula for the complex electrical conductivity " for the ytem: " = V T # $ dt e %i"t j q ; j q t, where i the oltzmann contant, T and V are the initial temperature and the volume of the ytem, and j q ; j q t i the canonical correlation function for the electric current denity operator j q in the interaction picture: " j q ; j q t " $ du j q #iu j q t, eq where " T and the ubcript eq indicate that the tatitical average i taen when no electric field i applied to the ytem o that the ytem i in an equilibrium tate at T For a quantum ytem in a teady tate with a contant current of heat or particle driven by a temperature or chemical potential difference between two
3 reervoir attached to the ytem, the fluctuation theorem for the current wa previouly hown [, 3, 4] to lead to the Green-Kubo formula for the linear repone coefficient for the current expreed in term of the ymmetrized correlation function of the current denity operator For example, for a quantum ytem that i ubject to a contant magnetic field and i driven to a teady heat conduction tate by a temperature difference between two heat reervoir attached to the ytem, we can derive the Green- Kubo formula for it thermal conductivity in term of the ymmetrized correlation function of it heat current denity operator Q j,f picture: in the Heienberg = V " t T lim dt " # $ " % dt % { j Q t,f j Q t, F + j Q t, F j Q t,f }, eq 3 where V i the volume of the ytem and the ubcript eq indicate that the tatitical average i taen when the temperature difference between the reervoir i ept at zero o that the ytem and the reervoir are all in equilibrium at the ame temperature T In thi article, we will how that for quantum ytem in teady tate with contant rate of wor done on the ytem, we can alo derive the fluctuation theorem for a quantity induced in uch a ytem to obtain the Green-Kubo formula expreed in term of the ymmetrized correlation function of the induced quantity The fluctuation theorem for the induced quantity follow from a Earlier, the fluctuation theorem for the current for a claical ytem had been hown to lead to a Green-Kubo formula expreed in term of the time correlation function j q j q t eq of the current denity in the ytem [5, 6, 7]
4 quantum extenion of a general relation ie, Equation in [8] derived by Croo for claical ytem In Sec II, a an example of uch ytem, we will conider a fluid in a teady hear flow driven by a contant velocity of a olid plate moving above the fluid We will derive the fluctuation theorem for the hear tre on the fluid and obtain the Green-Kubo formula for it hear vicoity in term of the ymmetrized correlation function of it hear tre operator P F in the Heienberg picture: = V " t T lim dt " # $ " % dt % { P t P t + P t P t }, 4 F F F F eq where V i the volume of the fluid and the ubcript eq indicate that the tatitical average i taen when the plate remain at ret o that the fluid, the plate, and a heat reervoir attached to the fluid are all in equilibrium at the ame temperature T A teady tate in quantum ytem are accompanied by either contant rate of wor done on the ytem or contant current of heat or particle through the ytem, the fluctuation theorem for a quantity induced in uch a ytem therefore alway lead to the Green-Kubo formula expreed in term of the ymmetrized correlation function of the induced quantity In Appendix, for the ae of completene, we will derive the fluctuation theorem for the heat current through a quantum ytem in a teady heat conduction tate with a contant heat current driven by a temperature difference between two heat reervoir attached to the ytem to obtain the Green-Kubo formula 3 for it thermal conductivity Our derivation of the fluctuation theorem for the heat current i imilar to that of the fluctuation theorem for the
5 hear tre in Sec IIE but different from the exiting one [3, 4, 5] a it employ the quantum extenion of the relation by Croo mentioned above GREEN-KUO FORMULA FOR SHEAR VISCOSITY Shear vicoity of a fluid In thi ection, a an example of quantum ytem in teady tate with contant rate of wor done on the ytem, we conider a fluid driven to a teady hear flow The fluid i placed between two plate both perpendicular to the y- axi and each with a urface area A and the depth of the fluid along the y-axi i h The fluid i alo attached to a heat reervoir at an invere temperature efore an initial time t = and after a final time t =, the both plate remain at ret During the time interval [, ], the plate under the fluid remain at ret while the plate above the fluid i moving along the x-direction to induce the hear flow in the fluid We aume that the plate above the fluid move at a contant velocity v during a time interval [", " # "], where " i a fixed poitive contant atifying " < < " On the macrocopic level, the average wor done on the fluid by the moving plate during the time interval [, ] i approximately W AP yx v", where P yx i the average hear tre exerted on the fluid by the plate For mall v, we define the hear vicoity of the fluid by the following linear repone relation:
6 P yx " v h Total Hamiltonian by The total Hamiltonian for the fluid, the plate, and the heat reervoir i given H t,v = H + H int { r a + X t x ˆ } a, 3 where H int i the potential energy due to the interaction between the particle in the fluid and thoe in the moving plate and H i the ret of the total Hamiltonian X t i the x coordinate of the center of ma of the moving plate and r a i the poition vector, meaured from the center of ma, for the a-th atom in the moving plate For the fluid and the plate, we impoe periodic boundary condition at the boundarie perpendicular to the x axi We then aume that X t atifie X t = X " t and X = X = o that H t,v = H " t,v and the moving plate return to it initial poition at the final time t = and H,v = H, v = H + H int { r a } a = H, 4 We alo aume that the velocity X t of the center of ma of the moving plate atifie X = X = and that during the time interval i the fixed contant atifying " < < ", X t [", " # "], where " remain contant at v: X t = v
7 3 Initial eigentate efore the initial time t =, we aume that the total ytem coniting of the fluid, the plate, and the reervoir i in it equilibrium tate characterized by the invere temperature Jut before t =, through a meaurement of the energy in the total ytem, we find the total ytem to be in an eigentate i of H, with a correponding energy eigenvalue E i : H, i = E i i 5 We aume that the initial eigentate i elected by the following canonical enemble ditribution: eq i " Z # exp [ $#E i ], 6 where Z i the partition function defined by Z $ " exp [#E i ] i 4 Final eigentate Jut after the final time t =, through a meaurement of the energy in the total ytem, we find the total ytem to be in an eigentate f of H, with a correponding energy eigenvalue E f : H, f = E f f 7 Note that the et of all the initial eigentate i the ame a the et of all the final eigentate: { i } = { f }
8 5 Time-revered proce and the principle of microreveribility In thi ubection, to mae thi article to be elf-contained, we will derive the principle of microreveribility 6 whoe direct conequence 7 a well a,, and 3 will be ued in Sec IIE, where we how the fluctuation theorem for the hear tre The reader who are familiar with thee equation may wih to ip thi ubection 5 Principle of microreveribility During the time interval [, ], the tate t of the total ytem evolve according to the Schrödinger equation with H t,v o that it final tate i related to it initial tate by " = U v, 8 where U v i the time evolution operator at t = Since the time reveral operator atifie i = "i and = = I, the time-revered tate defined by r t " # $ % t 9 evolve according to the following Schrödinger equation: i t " t = i r t # " $ % t & = # i $ % t " $ % t * +, = #H $ % t,v " $ % t = #H $ % t,v# # " $ % t = # H $ % t,v " r t,
9 where the time-revered Hamiltonian H " # t, v i defined by H " # t,v $ H " # t, v We aume our total Hamiltonian atifie H " # t, v = H t, #v o that H, i invariant with repect to time-reveral: H, = H ", = H, The final tate of the time-revered bacward proce i related to it initial tate by r " = # U v r, 3 where U v i the time evolution operator at the end of the bacward proce, which i controlled by H " # t, v = H t, #v o that U v = U " v 4 For any, we then find " = " r # = U v " r = U v "# = U v U v " 5 o that U v = " U v = "U v ", 6
which i called the principle of microreveribility Uing thi equation, we can how that the tranition probability for the forward proce from an initial eigentate i to a final eigentate f i equal to the tranition probability for the bacward time-revered proce from f " f to i " i : f U v i = i U v f = i U "v f, 7 which follow from 4 and i U v f = i, U v f = U v f, i = f U v i, 8 where i anti-unitary o that for any pair of tate, and ", atifie #", # = #, #", 9 where, " i the inner product between and " 5 Ueful property of the time reveral operator The following property of will be alo ueful when we how the fluctuation theorem for the hear tre in Sec IIE If n i an eigentate of an obervable A with a real eigenvalue a n o that A n = a n n, then eigentate of A " A with the eigenvalue a n: n " n i an A n = A n = A n = a n n = a n n
More pecifically, i i an eigentate of H, = H, with the energy eigenvalue E i : H, i = H, i = E i i and f i an eigentate of H, = H, with the energy eigenvalue E f : H, f = H, f = E f f, which alo implie eq " f = Z # exp [ $#E f ] = eq f, 3 where [ ] $ exp "E f = $ exp "E f = $ exp ["E i ] = Z " # f f [ ] the time reveral operator provide a one-to-one map from { f } = { i } i becaue f to f and 6 Cumulant generating function for the hear tre For a forward proce from an initial eigentate i to a final eigentate f, we define the wor W i, j done on the fluid by W i, f E f " E i 4 and the correponding hear tre by
P yx i, f W i, f Av" # E i = E f 5 Av" We then define the cumulant generating function for the hear tre by G P P,v " #lim $ % & [ ] v $ ln exp #$ PP yx i, f, 6 where the tranient proce average i defined by C i, f f U i " v eq i C i, f # 7 i, f In the following, we will aume that the limit in 6 exit Uing the cumulant generating function, we can then obtain the average hear tre by P yx = lim P yx i, f " # v = $G P $% P % P = 8 and the hear vicoity by = h dp yx = h " G P 9 dv v = "v"# P # P =v = lim "# P yx i, f v can alo be written a lim "# P yx i, f v - & = lim Tr/ $ eq " # / % dt P F t * +,, 3
3 where we have defined the denity matrix eq correponding to the initial canonical enemble ditribution eq i 6 by eq " exp [ $#H, Z # ] 3 We have alo defined the hear tre operator P F in the Heienberg picture by P F t A int "H F t,v #, 3 a "x a int where H F t,v U v t H int t,v U v t with U v t being the time evolution operator for the total ytem at t and x a i the x coordinate of the poition vector for the a-th atom in the moving plate Uing f U v i = i U v f f U v i, f f f =, and eq i = eq i can how 3 a follow i, we
4 lim "# P yx i, f % E f = lim "#,& Av i, f v $ E i * f U v i + eq i i U = lim v H, f f U v i $ i U v f f U v H, i "#, Av i, f + eq i = lim, i "# i % U v H, U v $ H, & Av * + eq i - % H = lim Tr + F,v $ H, eq & "# / Av * - = lim Tr/ + eq "# / : - % = lim Tr/ < + eq & "# / < dt A, a 4 6 5 - + lim "# Tr / / : dt P F 3H F int 3x a : + eq dt < t * <, $ ; 7 9 8 +lim - ; "# Tr / X + eq dt t $ v : / Av X t $ v Av, a 4 6 5 3H F int 3x a 7 9 8, a 4 6 5 3H F int 3x a 7 9 8 where H F t,v U v t H t,vu v t and we have alo ued 33 H F,v " H, Av = dt dh t,v F Av # = dt = # dt dt % int $H # F A + & $x * + a a ",, + # ", X t dt Av + a % & X t Av, # $H F int $x a + a X t dt Av = dt % int $H F # A+ & $x * +, X dt t " v a a # Av + X t " v % int $H dt F # Av + & $x * a a ", * % & $H F int + a $x a % & + a * $H F int $x a % & * $H F int $x a * 34
5 7 Fluctuation theorem for the hear tre Uing 7,,, and 3, we can how the fluctuation theorem for the hear tre, where G P Av " # P,v = G P # P, "v, 35 G P P,"v = " lim # $% = "lim # $ % The fluctuation theorem follow from [ ] " v # ln exp "# PP & yx f, & i # ln + exp "# PP & yx f, & i * + & i, & f [ ] & i U & " v f eq & f 36, - [ ] v exp " A#v $ P P yx i, f = exp ["$ P P % yx f, % i ], v 37 which we can how a follow [ ] v [ ] f U v i % eq i exp " A#v $ P P yx i, f = exp ["$ P P yx i, f ]exp #Av"P yx i, f & i, f & [ ] % eq f = exp "$ P P yx f, i i, f & = exp "$ P P yx f, i i, f % eq i [ ] i U v f = exp ["$ P P yx f, i ], v i U v f %eq f %eq i
6 38 where we have ued 7, f U v i = i U "v f, and P yx E i i, f = E f Av" = E i E f Av" = P # yx f, # i, 39 where and are ued We have alo ued [ ] = exp ["{ E f E i }] = $ eq f exp "Av#P yx i, f $ eq i = $ eq % f $ eq i, 4 where eq " f = eq f 3 i ued In 38, we have alo ued the fact that the time reveral operator provide a one-to-one map from the et of all the eigentate of H, to itelf and that i { } and f { } { } = i { } = f For a general quantum ytem in a teady tate with a contant rate of wor, we can generalize 37 a $ exp &" W i, f %& # * eq i * + eq f # $ = exp," W + f, + i & % & # - # #, 4 where eq " f eq i [ ] and the wor W i, f = exp #$W i, f done on the ytem during a forward proce from an initial eigentate i to a final eigentate f i proportional to ", where i a quantity, lie Av, that drive the teady tate We alo aume that the total Hamiltonian atifie H" # t,$ = H t, % $ $, where " i or In Appendix, we will find a imilar relation hold for heat
7 current ee A4 We alo note that 4 i a quantum extenion of a general relation ie, Equation in [8] derived by Croo for claical ytem 8 Green-Kubo relation Uing the fluctuation theorem for the hear tre 35, we find G P Av, v = G P, "v = 4 Uing thi equation, we can derive the Green-Kubo relation, = V T lim " # $ " P yx i, f v v =, 43 where V = Ah and T =, a follow = G P A"v, v v v = = A" G P # P,v # P # P = v= + G P # P, v v # P =v = = $ A" % P yx i, f + A" & v v = h, where we have ued + A" G P # P,v v# P # P = v = 44 G P " P,v v " P = v = = G P, v = 45 v v =
8 9 Green-Kubo formula for the hear vicoity To derive the Green-Kubo formula for the hear vicoity 4 from the Green-Kubo relation 43, we firt how lim P yx i, f = lim "# v - & Tr $ eq dtp * / "# % F t + /,, 46 Uing f U v i = i U v f f U v i, f f =, f eq i = eq i i, and [, H, ] =, we find eq P yx i, f v = = * i, f * i, f " E i # E f & $ % Av f U v i eq i # + i U v H, f f U v i " i U v H, f f U v H, i $ % + Av + i U v f f U v H, i & + Av + i eq # + U = i v H, U v " U v H, U v H, + H, & + * $ % + Av + i eq i, # = Tr H F, v " H, & /, # eq $ + - % Av Tr & / eq dtp + $ F t 3 4 - % + +, where we have alo ued 34 We then obtain 47
9 lim P i, f yx "# v v = - & = lim Tr $ eq dtp * / "# % F t + /, v = - = lim dt "# % dt Tr $ eq P F t / t % t = lim dt "# % dt C % P t,t { P t + P t P t } F F F v = 48 where we have defined the ymmetrized correlation function of the hear tre operator C P by Tr% C P t,t # " eq P t P t + P t P t & $ { } F F F F { P t P t + P t P t } F F F F eq v = 49 The ubcript eq indicate that the tatitical average i taen when both of the plate remain at ret o that the fluid, the plate, and the heat reervoir are all in equilibrium at the ame temperature T Uing thi equation in the Green-Kubo relation 43, we finally obtain the Green-Kubo formula for the hear vicoity : = V T lim " P i, f yx = V " t " # $ v v = T lim dt " # $ " % dt % C P t,t 5
3 CONCLUSIONS In thi article, we have hown that for quantum ytem in teady tate with contant rate of wor done on the ytem, the fluctuation theorem for a quantity induced in uch a ytem lead to the Green-Kubo formula for it linear repone coefficient expreed in term of the ymmetrized correlation function of the induced quantity A an example of uch ytem, we have conidered a fluid driven to a teady hear flow and derived the fluctuation theorem for the hear tre on the fluid to obtain the Green-Kubo formula for it hear vicoity expreed in term of the ymmetrized correlation function of it hear tre operator For a quantum ytem in a teady tate with a contant current of heat or particle driven by a temperature or chemical potential difference between two reervoir attached to the ytem, the fluctuation theorem for the current wa alo previouly hown to lead to the Green-Kubo formula for the linear repone coefficient for the current expreed in term of the ymmetrized correlation function of the current denity operator A teady tate in quantum ytem are accompanied by either contant rate of wor done on the ytem or contant current of heat or particle through the ytem, the fluctuation theorem for a quantity induced in uch a ytem therefore alway lead to the Green-Kubo formula expreed in term of the ymmetrized correlation function of the induced quantity ACKNOWLEDGMENTS
I wih to than Michele oc for contant upport and encouragement and Richard F Martin, Jr and other member of the phyic department at Illinoi State Univerity for creating a upportive academic environment APPENDIX: GREEN-KUO FORMULA FOR THERMAL CONDUCTIVITY A Thermal conductivity Conider a ytem driven to a teady heat conduction tate by a temperature difference between two heat reervoir, A and We aume that their temperature, T A and T, atify T A > T The invere temperature aociated with T A and T are defined by A " T A and " T, repectively We aume that the ytem and the reervoir are ubject to a contant magnetic field We define the thermal conductivity of the ytem by the following linear repone relation for the average heat current J Q : J Q = A T A " T L # A T L $%, A where A and L are the cro-ectional area and the length of the ytem " i defined by " # " $ " A = T $ T A % T A $ T T, A where T i defined by
T " " A + ", A3 A Total Hamiltonian The total Hamiltonian for the ytem and the reervoir i given by H H " I A " I + I " H A " I + I " I A " H + H int = H + H int, A4 where H and H are the Hamiltonian for the ytem and the -th reervoir = A, while H int i a wea coupling between the ytem and the reervoir efore the initial time t = and after the final time t =, we et H int = o that the ytem i detached from the reervoir We aume that H atifie H = " H = "H " A5 o that U = " U = "U " A6 A3 Initial eigentate Jut before t =, the ytem i detached from the reervoir and through a meaurement of the energy in the ytem, we find the ytem to be in an eigentate i of the ytem Hamiltonian H with energy eigenvalue E i efore t =, we aume that the ytem i in an equilibrium tate at the invere
3 temperature = A + o that the initial eigentate i i elected by the following canonical enemble ditribution: i = Z " exp #" E [ i ], A7 where Z i the partition function for the initial canonical enemble for the ytem We chooe thi value for the initial invere temperature for the ytem o that we can later how the fluctuation theorem for heat current We aume that after a long time interval, the teady tate for the ytem hould become independent of it initial invere temperature o that we can chooe it value almot freely a long a the value i not o different from A or Jut before t =, through a meaurement of the energy in each reervoir, we find the -th reervoir = A, to be in an eigentate i of the reervoir Hamiltonian H with energy eigenvalue E i efore t =, we aume that the reervoir i in an equilibrium tate at invere temperature o that the initial eigentate i i elected by the following canonical enemble ditribution: i = Z " exp #" E i [ ], A8 where Z i the partition function for the initial canonical enemble for the reervoir The initial eigentate i for the total ytem i then i i " i A " i and the initial canonical enemble ditribution for the total ytem i i = i A A i i A9
4 According to, i eigenvalue E i o that " i i an eigentate of H = " H with the energy H " i = E i " i, A and i o that " i i an eigentate of H = " H with the energy eigenvalue E i H " i = E i " i A A4 Final eigentate Jut after t =, the ytem i detached from the reervoir and through a meaurement of the energy in the ytem and thoe in the reervoir, we find the ytem to be in an eigentate f eigenvalue E of the ytem Hamiltonian H with energy f while we find the -th reervoir to be in an eigentate f the reervoir Hamiltonian H eigentate f for the total ytem i then f f with energy eigenvalue E f The final of " f A " f Note that the et of all the initial eigentate for the total ytem i the ame a the et of all the final eigentate: { i } = { f } According to, energy eigenvalue E f f o that " f i an eigentate of H = " H with the H " f = E f " f, A and f " f i an eigentate of H = " H with the energy eigenvalue E f o that
5 H " f = E f " f A3 A5 Cumulant generating function for heat current y applying the firt-order time-dependent perturbation theory [9], where we aume H int to be wea, we find that f U i i appreciable only when E f + E A A f + E { f } E i + E A A i + E { i } < " # A4 For ufficiently long, we can then aume that E + E A + E =, A5 where E for the ytem i defined by E " E f # E i A,6 and E for the -th reervoir i defined by E " E f # E i A7 For a forward proce for the total ytem from an initial eigentate i to a final eigentate f, we define the heat tranferred from the -th reervoir into the ytem by
6 Q i, f "#E A8 and the average heat current through the ytem by J Q i, f $ & & % Q A = # E { } A i A, f + #Q i, f " " A A f # E A A { i } + E f " # E { i } A9 We then define the cumulant generating function for the heat current by G Q Q, "#; $ % lim & ln & exp [ %& QJ Q i, f ], A where the tranient proce average i defined by C i, f f U i " i C i, f # A i, f In the following, we will aume that the limit in A exit Uing the cumulant generating function, we can then obtain the average heat current by J Q = lim " # J Q = $G Q $% Q %Q = A and the thermal conductivity by " L % = $ # A T & J Q = * * = " L % $ G Q # A T A3 & *+ Q + Q = * =
7 J Q can alo be written a lim "# J Q = - & lim Tr / $ " # / % dt Q J,F * t +,, A4 where, the denity matrix correponding to the initial canonical enemble ditribution for the total ytem A9 i defined by " Z % # exp $# H Z A # A exp $# A H A % Z # exp $# H A5 We have alo defined the heat current operator J Q, F in the Heienberg picture by J Q, F t # " dh A,F % $ dt + dh, F dt &, A6 where H, F t U t the total ytem at t and H U t with U t being the time evolution operator for i dh, F dt = [ H,F t, H,F t] A7 o that " dt J Q, F t = # H A,F # H A [ { } + { H,F # H }] A8
8 Recalling Q = E f E { i } A8 and uing f U i = i U f f U i, f f f =, and i = i can how A4: i, we J Q i, f = + i, f E A A { i } + E f E i # E A A % f $ & % ",, A A H = Tr *,F " H - - ", # " % = Tr * $ " dt - &% { } { } + { H,F " H } Q J,F % / t % // % % f U i * i A9 A6 Fluctuation theorem for the heat current We can how the fluctuation theorem for the heat current, where G Q G Q Q, "#; $ " # $ Q,"; = G Q $ Q, ";#, A3 [ ] $ = $lim % & % ln exp $% J Q Q $ f, i + = $lim % & % ln - exp % J Q * Q $ f, i,- i, f The fluctuation theorem follow from [ ] i U $ f $ f / A3
9 [ ] exp "#$ % Q J Q i, f [ & f, & i ], Q = exp "% Q J A3 which we can how a follow [ ] [ ] f U i & i exp "#$ % Q J Q i, f = exp ["% Q J Q i, f ]exp #$"J Q i, f i, f [ ] & f & i f, i Q = exp "% Q J f, i i, f Q = exp "% Q J i, f i U f & i [ ] i U f & f Q = exp ["% Q J f, i ] A33 In thi derivation, we have ued a conequence of the principle of microreveribility, and f U i = i U f = i U " f, A34 J Q = E i, f A A f E A A { i } + E f " E { i } Q = J # f, # i, A35 where A and A3 are ued We have alo ued
3 % exp ["#$J Q i, f ] = exp #$ #E A + #E * & + = exp $ #E + #E A - + #E, #$ #E = exp $ #E $ A #E A $ #E = f i f = i, A + #E / A36 where we have ued A5, E + E A + E =, a well a A = A + " " A = " # A37 and = A + + " A = + # A38 We have alo ued # " f = f, A39 which follow from A and A3 A3 can be written a [ ] # % $ f exp " Q J Q i, f # i [ ] $, Q = exp $" Q J % $ f, % i A4 which i imilar to 4 for the hear tre
3 A7 Green-Kubo formula for the thermal conductivity Uing the fluctuation theorem for the heat current A3, we find G Q ","; = G, "; # = A4 Uing thi equation, we can derive the Green-Kubo relation, = L lim " J Q A T i, f " #$, A4,% & = which follow from = G Q "#,"#; "# "# = = G Q $ Q,"#; $ Q $ Q = "# = + G Q $ Q,"#; "# $ Q = "# = + G Q $ Q,"#; "#$ Q $ Q = "# = = %& J Q i, f + A T,, "# = L where we have ued A and A3 Recalling Q = E H A A [, H ] = H,F, H, F f E { i } A8 and uing [ ] =, we can then how A43 [,H ] = and
3 J Q i, f, " = & & A A $ H = Tr,F % $ H # % & - % = Tr / # %, dt / { } + { H,F % $ H } Q J,F / t + 3 / + * " = + + * + + * + " = A44 Uing thi equation in the Green-Kubo relation A4, we obtain = = = L A T lim " J Q i, f " #$ L lim "Tr + A T * " # $, + " V T lim " # $ " Tr,% & = + *, + " " dt dt j Q,F J Q,F - 3 + t 5 / + 5 4-3 + t 5 / + 5 4 %& = %& =, A45 where V = AL and we have defined the heat current denity operator j Q,F in the Heienberg picture by j Q,F t A Q J, F t A46 Following tep imilar to thoe for the hear vicoity, we then obtain the Green- Kubo formula for, = V " t T lim dt " # $ " % dt % C Q t,t, A47
33 where the ymmetrized correlation function of the heat current denity operator i defined by Tr C Q t,t # " $ % j Q t, F j Q t,f + j Q t,f j Q t & {, F } { j Q t,f j Q t,f + j Q t,f j Q t, F } eq * = A48 The ubcript eq indicate that the tatitical average i taen when the temperature difference between the reervoir i ept at zero o that " = and the ytem and the reervoir are all in equilibrium at the ame temperature T
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