Green-Kubo relation for thermal radiation in non-reciprocal systems
eepl draft
Green-Kubo relation for thermal radiation in non-reciprocal sys-tems
F. Herz and
S.-A. Biehs
Institut f¨ur Physik, Carl von Ossietzky Universit¨at, D-26111 Oldenburg, Germany
PACS – Thermal Radiation
PACS – Nonequilibrium and irreversible thermodynamics
PACS – Fluctuation phenomena, random processes, noise, and Brownian motion
PACS – Quantum electrodynamics
Abstract –We rederive the Green-Kubo relation establishing a connection between the near- andfar-field heat transfer between two objects out of equilibrium to the equilibrium fluctuations ofthese objects in an arbitrary environment. Employing the scattering approach in combination withthe fluctuation-dissipation theorem, we generalize the previously derived Green-Kubo expressionto the case of non-reciprocal objects and non-reciprocal environments.
Introduction. –
In the last decade the theory offluctuational electrodynamics has been succesfully ap-plied to describe nanoscale heat fluxes in many-body sys-tems [1–11]. It could be shown that the heat flux betweentwo objects can be enhanced by the presence of a thirdobject placed in between the considered objects [1, 12, 13].Furthermore, the heat flux can be actively tuned by addingan intermediate object [14–20], and the heat flux can alsobe enhanced by coupling to the surface modes of an in-termediate object or a nearby environment [21–24]. Innon-reciprocal many-body systems new interesting effectshave been highlighted like a persistent heat flux [4,11,25],a giant magneto-resistance [26, 27], a Hall effect for ther-mal radiation [28, 29], and non-trivial angular and spinmomentum of thermal light [30]. It could also be shownthat the coupling to non-reciprocal surface modes evenallows for an efficient heat flux rectification [31].In such many body systems, it is still possible to es-tablish a link between the heat flux between two objectssurrounded by an arbitrary environment which can consistof many other objects and the equilibrium fluctuations inthe many-body system as shown by Golyk et al. [32] withinthe scattering approach. This link is given by the Kuboor Green-Kubo relation [32]. To be more precise, let usconsider two compact objects of arbitrary material andshape placed within an arbitrary environment as sketchedin Fig. 1. When H α/β is the total power absorbed by thetwo objects labeled as α and β having in general two dif-ferent temperatures T α/β placed in an environment which has another temperature T b , then the mean power (cid:104)(cid:104) H α/β (cid:105)(cid:105) received by object α or β is in general finite. In globalequilibrium, i.e. for T α = T β = T b ≡ T , (cid:104)(cid:104) H α/β (cid:105)(cid:105) eq mustvanish. But the fluctuations do not vanish in global equi-librium in general. As shown by Golyk et al. [32] assumingthat the thermal radiation fields have the Gauss property,the fluctuations in global equilibrium can be linked to theheat flux out of equilibrium. This link is established bythe Green-Kubo relation − d (cid:104)(cid:104) H β (cid:105)(cid:105) d T α (cid:12)(cid:12)(cid:12)(cid:12) T α = T β = T = 12 k B T (cid:90) + ∞−∞ d t (cid:10)(cid:10) H α ( t ) H β (0) (cid:11)(cid:11) eq , (1)where k B is the Boltzmann constant. Note, that this re-lation has been derived under the assumption of “micro-reversibility” of the objects and the environment whichmeans that both objects and environment are reciprocal.In this case one can use the Kubo-symmetry relation [33] (cid:10)(cid:10) H α ( t ) H β (0) (cid:11)(cid:11) eq = (cid:10)(cid:10) H α ( − t ) H β (0) (cid:11)(cid:11) eq to further simplifythe Green-Kubo relation to − d (cid:104)(cid:104) H β (cid:105)(cid:105) d T α (cid:12)(cid:12)(cid:12)(cid:12) T α = T β = T = 1 k B T (cid:90) + ∞ d t (cid:10)(cid:10) H α ( t ) H β (0) (cid:11)(cid:11) eq . (2)This is exactly the form given in Ref. [32].In this letter, we will generalize this Green-Kubo re-lation to include the non-reciprocal case by starting thederivation without making any assumption on the “micro-reversibility” or reciprocity of the objects and environ-ment. Therefore, our generalized Green-Kubo relationp-1 a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p . Herz S.-A. Biehs Fig. 1: Sketch of the considered configuration. Two compactscatterers or objects are coupled to independent heat bathswhich keep the objects in local thermal equilibrium at temper-tures T α/β . Both objects are immersed in an arbitrary envi-ronment in local equilibrium with temperature T b . Here theenvironment is sketched as a cavity so that the cavity wallsbring the environmental field into local equilibrium with tem-perature T b . applies to all possible cases including the case of non-reciprocal objects and reciprocal environment, reciprocalobjects and non-reciprocal environment etc. We show thatour relation reproduces Eq. (1) only if the two objects arereciprocal and if further the environment is reciprocal. Scattering approach. –
In the following we willmake use of the scattering approach together with a basis-independent representation of the electromagnetic fieldsand currents as used in Refs. [32, 34–36]. To make ourmanuscript self-consistent we will give first a brief intro-duction into this approach. The scattering approach startswith the Helmholtz equation [35] (cid:18) H − V − ω c (cid:19) E ( r ) = (3)where H = ∇ × ∇ × and the potential of a scattererwith the permittivity tensor (cid:15) and permeability tensor µ occupying a volume V is defined by V = ω c (cid:0) (cid:15) ( r ) − (cid:1) + ∇ × (cid:18) − µ − (cid:19) ∇ × . (4)Here we consider a homogeneous medium for the scat-terer so that (cid:15) ( r ) , µ ( r ) are constant within the volume ofthe scatterer and equal to outside the scatterer so that V is only nonzero within the scatterer. The solution ofthe Helmholtz equation is determined by the Lippmann-Schwinger equation E ( r ) = E ( r ) + (cid:90) V d r (cid:48) G ( r , r (cid:48) ) V E ( r (cid:48) ) . (5) Here E is the solution of the free Helmholtz equationand G is the Green tensor defined as solution of the freeHelmholtz equation for a point source (cid:18) H − ω c (cid:19) G ( r , r (cid:48) ) = δ ( r − r (cid:48) ) . (6)Defining now the position basis by the ket vectors | r (cid:105) with the usual completeness (cid:90) d r | r (cid:105)(cid:104) r | = , (7)and Dirac orthonormality (cid:104) r | r (cid:48) (cid:105) = δ ( r − r (cid:48) ) (8)and introducing the notations for vectors and tensors E ( r ) = (cid:104) r | E (cid:105) , (9) G ( r , r (cid:48) ) = (cid:104) r | G | r (cid:48) (cid:105) (10)we can bring the Helmholtz equation in a basis-independent form (cid:18) H − V − ω c (cid:19) | E (cid:105) = | (cid:105) (11)where | (cid:105) is the zero vector. The solution given by theLippmann-Schwinger equation is then | E (cid:105) = | E (cid:105) + G V | E (cid:105) . (12)and can be expressed as | E (cid:105) = | E (cid:105) + G T | E (cid:105) . (13)introducing the T-operator of the scatterer T := V ( − G V ) − . (14)In the remaining part of the manuscript we will con-sider two compact scatterers within an arbitrary environ-ment. In this case the scattering potential consists of asum V α + V β of the two scatterers labeled by α and β .Note that the T-operator is not just the sum of the T-operators T α/β of the two scatterers. Since in the follow-ing the environment is assumed to be arbitrary we willreplace G and | E (cid:105) be the Green tensor G and the back-ground field | E b (cid:105) which are the solutions of the Helmholtzequations for the considered system without the two scat-terers. For example if we would consider two scatterersclose to a halfspace then G and | E b (cid:105) would be the Greentensor and the field of the halfspace geometry, only. Total electric field. –
Let’s assume that we have twocompact objects α and β with scattering potentials V α/β ,T-operators T α/β occupying a volume V α/β . The two scat-terers are placed in an arbitrary environment. Then thetotal electric field | E (cid:105) is determined by the backgroundfield | E b (cid:105) and the field radiated and scattered by bothp-2reen-Kubo relation in non-reciprocal systemsobjects. Hence the total electrical field can be expressedby | E (cid:105) = | E b (cid:105) + i ωµ G (cid:0) | J α (cid:105) + | J β (cid:105) (cid:1) (15)where G is the Green’s tensor of the environment withoutthe two objects α and β and µ is the permeability ofvacuum. The currents | J α/β (cid:105) are due to direct thermalradiation and the scattered radiation and can thereforebe expressed by a fluctuational current and an inducedcurrent | J α/β (cid:105) = | J fl α/β (cid:105) + | J ind α/β (cid:105) . (16)Within the scattering approach the induced current canbe expressed by the scattered field coming from the envi-ronment and the other object, i.e. for object α we have forexample | J ind α (cid:105) = 1i ωµ T α | E β (cid:105) (17)where T α is the T-operator of object α describing thescattering and | E β (cid:105) is the field of the environment andobject β so that | E β (cid:105) = | E b (cid:105) + i ωµ G | J β (cid:105) . (18)This means, that at this step we ignore any self-polarisation of the object which might be included at alater stage by replacing polarisabilities by dressed polar-isabilities. Now, since | J α (cid:105) and | J β (cid:105) are related by themultiple scattering between the objects, we can expressboth by means of | J fl α (cid:105) , | J fl β (cid:105) , and | E b (cid:105) . We obtain | J α (cid:105) = G − D αβ G (cid:20) | J fl α (cid:105) + T α G | J fl β (cid:105) + 1i ωµ T α (1 + GT β ) | E b (cid:105) (cid:21) (19)and the corresponding expression for | J β (cid:105) . Hence, thetotal electric field can also be expressed by these quantitiesonly and takes the form | E (cid:105) = A αβ | E b (cid:105) + i ωµ (cid:18) O α G | J fl α (cid:105) + O β G | J fl β (cid:105) (cid:19) (20)where we have introduced the tensors O β = (cid:0) + GT α (cid:1) D βα , (21) A αβ = O β (cid:0) + GT β (cid:1) , (22)and the Fabry-Perot-like “denominator” D αβ = (cid:0) − GT α GT β (cid:1) − . (23)The expresssions for O α and D βα can be obtained from O β and D αβ by interchanging α ↔ β . Note, that D αβ GT α = GT α D βα . Equilibrium correlation functions. –
Since wehave now the expression for the total electric field of thetwo objects in presence of an environment we can deter-mine the correlation functions of the total fields. To thisend, we assume that the background field is in local ther-mal equilibrium at temperature T b so that by using thefluctuation-dissipation theorem, the correlation functionreads [37, 38] (cid:10)(cid:10) E b ( r ) ⊗ E † b ( r (cid:48) ) (cid:11)(cid:11) leq = a ( T b ) G ( r , r (cid:48) ) − G † ( r (cid:48) , r )2i (24)where we have introduced a ( T ) := 2 µ ¯ hω ( n ( T ) + 1) (25)with n ( T ) = 1e ¯ hω/k B T − . (26)Note that we treat the fields fully quantum mechanicallyas operators and therefore the correlation function de-pends on the ordering. The above given anti-normallyordered correlation function is related to the normally or-dered by the expression [37, 38] (cid:10)(cid:10) E † b ( r ) ⊗ E b ( r (cid:48) ) (cid:11)(cid:11) leq = nn + 1 (cid:10)(cid:10) E b ( r ) ⊗ E † b ( r (cid:48) ) (cid:11)(cid:11) ∗ leq . (27)Therefore, we can express all relations in terms of the anti-normally ordered correlation functions only. Eq. (24) canbe brought in the basis-independent representation andreads (cid:10)(cid:10) | E b (cid:105)(cid:104) E b | (cid:11)(cid:11) leq = a ( T b ) G − G † . (28)If we further assume that the fluctuating currents withinthe objects are in local thermal equilibrium at temper-atures T α/β we obtain for the correlation functions ofthe fluctuating currents by using again the fluctuation-dissipation theorem (cid:10)(cid:10) | J fl α/β (cid:105)(cid:104) J fl α/β | (cid:11)(cid:11) leq = 1( ωµ ) a ( T α/β ) χ α/β (29)where we have introduced the generalized susceptibility ofparticle α and βχ α/β = T α/β − T † α/β − T α/β G − G † T † α/β . (30)Since | E b (cid:105) , | J fl α (cid:105) , and | J fl β (cid:105) are uncorrelated, the corre-lation function of the total field | E (cid:105) is just the sum ofthe correlation functions of these three quantities. It iseasy to show that in global thermal equilibrium when T b = T α = T β ≡ T , the sum of the three correlationfunctions gives for the correlation function of the totalfield (cid:10)(cid:10) | E (cid:105)(cid:104) E | (cid:11)(cid:11) eq = a ( T ) G full − G † full
2i (31)where G full = A αβ G (32)p-3. Herz S.-A. Biehsis the full Green’s function of the two particles placed inthe environment described by G . That means our localequilibrium ansatz in the sense of Rytov’s fluctuationalelectrodynamics reproduces the correct global equilibriumresult as it should be. Hence, as already pointed out inRefs. [36] the scattering approach naturally reproducesRytov’s theory, when the correct correlation functions forthe isolated objects α and β are identified. Radiative heat transfer. –
We can now determinethe total power received or emitted by object α , for in-stance, when assuming a local thermal equilibrium of thebackground field and the fluctuational currents withinboth objects. It is given by the symmetrically orderedexpression for the dissipated power in object α (cid:104)(cid:104) H α (cid:105)(cid:105) = (cid:88) i (cid:90) V α d r (cid:68)(cid:68)(cid:8) E i ( r , t ) , J i,α ( r , t ) (cid:9) S (cid:69)(cid:69) (33)where E i ( r , t ) and J i,α ( r , t ) are the i -th component of thetotal field and current operators in the Heisenberg pic-ture inside object α . Here, { A, B } S = ( AB + BA ) / (cid:104)(cid:104) H α (cid:105)(cid:105) = 2Re (cid:88) i (cid:90) ∞ d ω π (cid:90) V α d r (cid:68)(cid:68)(cid:8) E i ( r ) , J † i,α ( r ) (cid:9) S (cid:69)(cid:69) (34)where E i ( r ) and J i,α ( r ) are now the i -th component ofthe Fourier components of the total field and current op-erators. As before we suppress the frequency argumentfor convenience. The brackets (cid:10)(cid:10) . . . (cid:11)(cid:11) denoting the non-equilibrium average are to be taken as ensemble averagesfor the background field and the fluctuational currents be-ing separately in local thermal equilibrium. Using rela-tion (27) we can reexpress the above formula in the basis-independent notation as (cid:104)(cid:104) H α (cid:105)(cid:105) = Re (cid:90) ∞ d ω π Tr (cid:2)(cid:10)(cid:10) | E (cid:105)(cid:104) J α | (cid:11)(cid:11)(cid:3) n + 1 n + 1 . (35)Here the factor (2 n + 1) / ( n + 1) is in some sense sym-bolic, because the correct temperature in the function n can only by inserted later when the non-equilibrium av-erage is replaced by the local equilibrium averages. Sincethere is no correlation between the background field andthe fluctuational currents this does not introduce any am-biguity, but allows us to start with a compact expressionfor the dissipated power in object α expressed by the op-erator trace. To this end, we have extended the integralover whole space which is possible because (cid:104) r | J α (cid:105) vanishesfor r / ∈ V α . This allows us then to introduced the operatortrace Tr (cid:2) A (cid:3) = (cid:88) i (cid:90) d r (cid:104) r | A ii | r (cid:105) (36) for an operator A as defined in Ref. [35]. Inserting theexpressions for the total field | E (cid:105) and the currents | J α (cid:105) and using the fact that the background field and the fluc-tuating currents are uncorrelated together with the abovederived correlation functions in Eqs. (28) and (29) we ob-tain (cid:104)(cid:104)| E (cid:105)(cid:104) J α |(cid:105)(cid:105) = i ωµ O α G (cid:104)(cid:104)| J fl α (cid:105)(cid:104) J fl α |(cid:105)(cid:105) leq G † D † αβ G − † +i ωµ O β G (cid:104)(cid:104)| J fl β (cid:105)(cid:104) J fl β |(cid:105)(cid:105) leq G † T † α G † D † αβ G − † + A αβ (cid:104)(cid:104)| E b (cid:105)(cid:104) E b |(cid:105)(cid:105) leq − i ωµ (1 + T † β G † ) T † α G † D † αβ G − † = i ωµ (cid:20) a ( T α ) O α G χ α G † D † αβ G − † + a ( T β ) O β G χ β G † T † α G † D † αβ G − † + a ( T b ) A αβ G − G †
2i (1 + T † β G † ) T † α G † D † αβ G − † (cid:21) . Inserting this expression into Eq. (35) and keeping in mindthat the factor (2 n + 1) / ( n + 1) replaces in a ( T ) just n + 1by 2 n +1 at the corresponding local temperature we finallyarrive at (cid:104)(cid:104) H α (cid:105)(cid:105) = − (cid:90) ∞ d ω π ¯ hω (cid:2) ( n ( T α ) − n ( T b )) T α + ( n ( T β ) − n ( T b )) T α (cid:3) (37)where the transmission factors T α and T α describe theheat exchange between object α and the background andobject β . They are defined as T α = 43 ImTr (cid:2) O α G χ α G † D † αβ G †− (cid:3) , (38) T α = 43 ImTr (cid:2) O β G χ β G † D † βα T † α (cid:3) . (39)The corresponding expression for (cid:104) H β (cid:105) can be obtainedby interchanging α ↔ β . Note that by convention theemitted power of object α is negativ, whereas the receivedpower is positiv. We have verified that our results are inagreement with the corresponding expressions in Refs. [32,35]. Obviously, in global thermal equilibrium the totalpower emitted or radiated by object α or β vanishes asexpected for equilibrium. When object β has the sametemperature as object α then both transmission factorscontribute equally. Since in this case there is no net heatflux between the objects α and β , but only from object α to the environment, we can define T α → b = T α + T α as the transmission factor describing the heat flux from α to the background. On the other hand, when object α has the same temperature as the background then thecontribution from T α vanishes. In this situation, thereis only a net heat flux between objects α and β so thatwe can identify T β → α = −T α as the transmission factordescribing the heat flux between both objects.p-4reen-Kubo relation in non-reciprocal systems Green-Kubo relation. –
The derivation of theGreen-Kubo relation hinges now on the evaluation ofquantities like H αβ := (cid:90) + ∞−∞ d t (cid:10)(cid:10) H α ( t ) H β (0) (cid:11)(cid:11) eq = (cid:88) i,j (cid:90) + ∞−∞ d t (cid:90) V α d r (cid:90) V β d r (cid:48) (cid:68)(cid:68)(cid:8) E i ( r , t ) , J i,α ( r , t ) (cid:9) S × (cid:8) E j ( r (cid:48) , , J j,β ( r (cid:48) , (cid:9) S (cid:69)(cid:69) eq (40)for all combinations of α and β for global thermal equilib-rium at temperature T . To evaluate such expressions wehave to evaluate fourth-order correlation functions of thetotal fields and the total currents. By making the assump-tion that the thermal radiation field has the Gauss prop-erty [39], these correlation functions can be expressed interms of the second-order correlation function by virtue ofthe moment theorem [39]. Then by switching to frequencyspace we can express H αβ by [32] H αβ = 2Re (cid:88) i,j (cid:90) ∞ d ω π (cid:90) V α d r (cid:90) V β d r (cid:48) nn + 1 (cid:2)(cid:10)(cid:10) E i ( r ) E † j ( r (cid:48) ) (cid:11)(cid:11) eq (cid:10)(cid:10) J j ( r (cid:48) ) J † i ( r ) (cid:11)(cid:11) eq + (cid:10)(cid:10) E i ( r ) J † j ( r (cid:48) ) (cid:11)(cid:11) eq (cid:10)(cid:10) E j ( r (cid:48) ) J † i ( r ) (cid:11)(cid:11) eq (cid:3) . (41)As before, we introduce the basis-independent notation H αβ = 2Re (cid:90) ∞ d ω π Tr (cid:20)(cid:10)(cid:10) | E (cid:105)(cid:104) E | (cid:11)(cid:11) eq (cid:10)(cid:10) | J (cid:105)(cid:104) J | (cid:11)(cid:11) eq + (cid:2)(cid:10)(cid:10) | E (cid:105)(cid:104) J | (cid:11)(cid:11) eq (cid:10)(cid:10) | E (cid:105)(cid:104) J | (cid:11)(cid:11) eq (cid:21) nn + 1 (42)by extending the volume integrals to all space and usingthe operator trace. In this expression the factor n/ ( n +1) iswell defined and evaluated at the global equilibrium tem-perature T . Then by inserting the expressions for the totalfields and total currents where we can use the expressionsfor | J α/β (cid:105) only, depending on the volume V α/β consideredin the integration. With the equilibrium expressions forthe correlation functions in Eqs. (28) and (29) and againthe assumption that the background field and the fluctu-ating currents are uncorrelated we arrive after a lengthyand tedious calculation at H αβ = 3 (cid:90) ∞ d ω π (¯ hω ) n ( T ) (cid:2) n ( T ) + 1 (cid:3)(cid:0) T α + T β (cid:1) . (43)By comparing this result with the expression for the powerreceived by object α in Eq. (37) we obtain the Green-Kuborelation − (cid:20) d (cid:104)(cid:104) H α (cid:105)(cid:105) d T β + d (cid:104)(cid:104) H β (cid:105)(cid:105) d T α (cid:21) T α = T β = T = 1 k B T H αβ . (44) This relation is not the same as in Eq. (1) derived inRef. [32]. When defining the transport coefficents as inRef. [32] by k αβ = − d (cid:104)(cid:104) H α (cid:105)(cid:105) d T β (cid:12)(cid:12)(cid:12)(cid:12) T (45)so that k αβ ∆ T is the power emitted by object β towardsobject α when heating it by ∆ T with respect to the en-vironment temperature T . With that definition we canexpress the Green-Kubo relation for thermal radiation as k αβ + k βα = 1 k B T H αβ = 1 k B T (cid:90) + ∞−∞ d t (cid:10)(cid:10) H α ( t ) H β (0) (cid:11)(cid:11) eq . (46)Therefore, the equilibrium fluctuations quantified by H αβ are related to both transport coefficients k αβ and k βα . Itis evident from the above expression that H αβ = H βα .Finally, we further find that k αα = 12 k B T H αα = 12 k B T (cid:90) + ∞−∞ d t (cid:10)(cid:10) H α ( t ) H α (0) (cid:11)(cid:11) eq . (47)The corresponding expression for k ββ can be obtained byreplacing α ↔ β . To summarize, we can regard Eq. (46)as the general result for all combinations of α and β . Justby replacing the sought for combination of indicies in thatEq. (46) we find the corresponding result. For example,when replacing β in Gl. (46) by α we obtain (47), etc.Now, by comparing our result in Eq. (46) with that inEq. (1) we see that in general the relation in Eq. (1) con-tains both transport coefficients k αβ and k βα in the Green-Kubo expression when considering H αβ or H βα . Strictly,speaking in this case our relation coincides with that ofEq. (1) only if T α = T β . To understand under whichconditions this equality holds we first write T α/β in thefollowing form T α = −
43 Tr (cid:2) D βα G χ β G † D † βα ˜ χ α (cid:3) , (48) T β = −
43 Tr (cid:2) D αβ G χ α G † D † αβ ˜ χ β (cid:3) , (49)where we have introduced the quantity˜ χ α/β = T α/β − T † α/β − T † α/β G − G † T α/β . (50)We will now show that T α = T β if T tα/β = T α/β and G t ( r α , r β ) = G ( r β , r α ). To this end, we start with Eq. (48)using the fact that the trace is invariant under transposi-tion and the cyclic permutation property. We obtain T α = −
43 Tr (cid:2) ( D βα G χ β G † D † βα ˜ χ α ) t (cid:3) = −
43 Tr (cid:2) ˜ χ tα D ∗ βα G ∗ χ tβ G t D tβα (cid:3) = −
43 Tr (cid:2) G t D tβα ˜ χ tα D ∗ βα G ∗ χ tβ (cid:3) . (51)p-5. Herz and S.-A. BiehsAt this step we assume that both objects and the envi-ronment are reciprocal. Then we have T tα/β = T α/β and G t = G . It follows that in this case χ tα/β = ˜ χ α/β . Itremains to show that D βα G = G t D tαβ to have T α = T β .This can be done by writing this equation D βα G = G t D tαβ as GD − αβt = D − βα G t which is simply G ( − T tβ G t T tα G t ) = ( − GT β GT α ) G t . (52)Obviously both sides of this equation are the same if T tα/β = T α/β and G t = G so that under this condition T α = T β . Hence, only if the objects and the environmentare reciprocal our generalized Kubo-Formula retrieves theexpression in Eq. (1). That means that in general if onlyone part of the system is non-reciprocal then T α (cid:54) = T β and then the generalized Green-Kubo relation applies.Let us consider the following simple example to showthat when certain approximations apply then only the en-vironment is responsible for reciprocity or non-reciprocityof the heat transport. To this end, we consider two identi-cal objects with T α = T β without making any assumptionon the reciprocity of the objects. Then in general the gen-eralized Green-Kubo relation applies, but in this specificcase it can happen that the non-reciprocity is driven bythe environment only. To see this we simplify first theexpressions for T α and T β in Eqs. (48) and (49) by us-ing the single scattering approximation ( D αβ ≈ D βα ≈ T α = −
43 Tr (cid:2) G χ β G † ˜ χ α (cid:3) , (53) T β = −
43 Tr (cid:2) G χ α G † ˜ χ β (cid:3) . (54)It seems that both are the same when χ α = χ β which isapproximately true if the second terms in χ α/β of order( T ) can be neglected or if both objects “see” the sameenvironment. This would mean that the transport coeffi-cients k βα and k αβ are the same as well and therefore thereciprocal Green-Kubo relation would apply. But one hasto keep in mind that the Green’s functions in the expres-sion for T α are those corresponding to G ( r α , r β ) and in T β we have the Green’s function G ( r β , r α ). Hence, onlyif G ( r α , r β ) = G ( r β , r α ) t or G ( r α , r β ) = G ( r β , r α ) the ex-pressions for T α and T β are the same. Therefore if theenvironment is reciprocal then we have k βα = k αβ no mat-ter if the objects themselves are reciprocal or not. Butif the environment is non-reciprocal the transport coeffi-cients are not the same. Hence for identical objects thenon-reciprocity in the transport coefficients is induced bythe non-reciprocal environment only if the above approx-imations apply. Indeed it has recently been shown thatwhen placing two nanoparticles in a non-reciprocal envi-ronment the transport coefficients k βα and k αβ can differ alot allowing for an efficient heat flux rectification [11, 31]. Summary and Conclusions. –
In summary, wehave rederived the general form of the Green-Kubo rela- tion establishing a link between the radiative heat transferbetween two arbitrary compact objects in an arbitrary en-vironment to the equilibrium fluctuations in this system.By this we have generalized a previously derived Green-Kubo relation. In this generalized relation both transportcoefficients describing the heat flux from object α to ob-ject β and from β to α are needed. This is particularlyimportant when heat fluxes between non-reciprocal ob-jects or in non-reciprocal environments are considered likein magneto-optical many-body systems showing persistentheat currents and angular momenta and spins of thermalradiation [4, 11, 25, 30], giant-magneto-resistance [26, 27],Hall effect for thermal photons [28, 29] as well as heat-flux rectification by non-reciprocal surface waves [31]. Asshown in Refs. [4,11,25,30] the non-reciprocity of the con-sidered systems results in a directional global equilibriumheat flux which can result in a directional heat flux in anon-equilibrium situation [28–31]. This is reflected by thegeneralized Green-Kubo relation where the directionalityor non-reciprocity found in the transport coefficients k βα and k αβ is already encoded in the equilibrium fluctuations H αβ . ∗ ∗ ∗ S.-A. B. gratefully acknowledges helpful discussionson the validity of the generalized Green-Kubo relationwith Matthias Kr¨uger and support from Heisenberg Pro-gramme of the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) under the project No.404073166.
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