Thermoelectric efficiency in momentum-conserving systems
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Thermoelectric efficiency in momentum-conservingsystems
Giuliano Benenti , , Giulio Casati , and CarlosMej´ıa-Monasterio , CNISM and Center for Nonlinear and Complex Systems, Universit`a dell’Insubria,via Vallegio 11, 22100 Como, Italy Istituto Nazionale de Fisica Nucleare, Sezione di Milano, via Celora 16, 20133Milano, Italy Laboratory of Physical Properties TAGRALIA, Technical University of Madrid,Av. Complutense s/n, 28040 Madrid, Spain Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68FIN-00014 Helsinki, FinlandE-mail: [email protected] , [email protected] , [email protected] Abstract.
We show that for a two-dimensional gas of elastically interacting particlesthe thermoelectric efficiency reaches the Carnot efficiency in the thermodynamic limit.Numerical simulations, by means of the multi-particle collision dynamics method, showthat this result is robust under perturbations. That is, the thermoelectric figure ofmerit remains large when momentum conservation is broken by weak noise.
Keywords : thermodynamic efficiency, thermoelectricity, conservation laws, anomaloustransport
Submitted to:
NJP
1. Introduction
Understanding and controlling the behaviour of out-of-equilibrium systems is one of themajor challenges of modern statistical mechanics. From a fundamental point of view, thechallenge is to understand the origin of macroscopic transport phenomenological laws,such as diffusion equations, in terms of the properties of microscopic dynamics, typicallynonlinear and chaotic [1, 2]. The problem is extremely complex for coupled flows, sofar barely studied from the viewpoint of statistical mechanics and dynamical systems[3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. In particular, it is of primary importance for thermoelectrictransport [13, 14, 15, 16] to gain a deeper understanding of the microscopic mechanisms hermoelectric efficiency
G Benenti et al leading to a large thermoelectric efficiency; see Ref. [17] for a review on the fundamentalaspects of heat to work conversion.Within linear response, and for time-reversal symmetric systems ‡ both themaximum thermoelectric efficiency and the efficiency at the maximum output power [22,23, 24, 25, 26] are monotonous growing functions of the so-called figure of merit ZT = ( σS /κ ) T , which is a dimensionless combination of the main transport coefficientsof a material, that is, the electric conductivity σ , the thermal conductivity κ and thethermopower (Seebeck coefficient) S , and of the absolute temperature T . The maximumefficiency reads η max = η C √ ZT +1 − √ ZT +1+1 , where η C is the Carnot efficiency, while the efficiencyat maximum output power P max is given by η ( P max ) = η C ZTZT +2 . Thermodynamics onlyimposes ZT ≥ η max → η C , η ( P max ) → η C when ZT → ∞ .Since the different transport coefficients are interdependent, it is very difficultto find microscopic mechanisms which could provide insights to design materialswith large ZT . While for non-interacting models it is well understood that energyfiltering [27, 28, 29] allows us to reach the Carnot efficiency, very little is known forinteracting systems [30]. It has been recently shown [31] that the thermoelectric figureof merit ZT diverges in the thermodynamic limit for systems with a single relevantconserved quantity, an important example being that of momentum-conserving systems,with total momentum being the only relevant constant of motion. While the mechanismis generic, it has been illustrated in Ref. [31] only for a toy model, i.e., a diatomic chainof hard-point elastically colliding particles.In this paper, we show by means of extensive multi-particle collision dynamicssimulations that the momentum-conservation mechanism leads to the Carnot efficiencyin the thermodynamic limit also in the more realistic case of two-dimensionalelastically colliding particles. Furthermore, we show that this mechanism leads to asignificant enhancement of the thermoelectric figure of merit even when the momentumconservation is not exact due to the existence of an external noise. This robustnessis particularly relevant in experiments for which inelastic or incoherent processes areunavoidable to some extent. In this case, the figure of merit saturates with the size ofthe system to a value higher, the weaker is the noise. Finally, we discuss the validityrange of linear response.The paper is organised as follows. In Sec. 2, in order to make the paper self-contained, we review the theoretical argument of Ref. [31] explaining the divergence ofthe thermoelectric figure of merit ZT in the thermodynamic limit for systems with asingle relevant constant of motion. In Sec. 3 we explain our out-of equilibrium multi-particle collision dynamics simulations. Our numerical results are presented in Sec. 4.We finish with concluding remarks in Sec. 5. ‡ Thermodynamic bounds on efficiency for systems with broken time-reversal symmetry are discussedin [18, 19, 20, 21] hermoelectric efficiency G Benenti et al
2. Theoretical argument
The equations connecting fluxes and thermodynamic forces within linear irreversiblethermodynamics read as follows [32, 33]: J ρ J u ! = L ρρ L ρu L uρ L uu ! −∇ ( βµ ) ∇ β ! , (2.1)where J ρ and J u are the particle and energy currents, µ the chemical potential and β = 1 /T the inverse temperature (we set the Boltzmann constant k B = 1). The kineticcoefficients L ij ( i, j = { ρ, u } ), are related to the familiar transport coefficients as σ = L ρρ T , κ = 1 T det L L ρρ , S = 1 T (cid:18) L ρu L ρρ − µ (cid:19) , (2.2)where L denotes the (Onsager) matrix of kinetic coefficients and we have set the electriccharge of each particle e = 1. Thermodynamics imposes det L ≥ L ρρ ≥ L uu ≥ L uρ = L ρu follows from the Onsager reciprocity relations. The thermoelectric figure ofmerit reads ZT = ( L uρ − µL ρρ ) det L = σS κ T. (2.3)Furthermore, the Green-Kubo formula expresses the kinetic coefficients interms of correlation functions of the corresponding current operators, calculated atthermodynamic equilibrium [34, 35]: L ij = lim ω → Re L ij ( ω ) , (2.4)where L ij ( ω ) ≡ lim ǫ → Z ∞ dte − i ( ω − iǫ ) t lim Ω →∞ Z β dτ h J i J j ( t + iτ ) i , (2.5)where h · i = (cid:8) tr (cid:2) ( · ) exp − βH (cid:3)(cid:9) / tr [exp( − βH )] denotes the equilibrium expectationvalue at temperature T and Ω is the system’s volume. Within the framework of Kubo’slinear response approach, the real part of L ij ( ω ) can be decomposed into a singularcontribution at zero frequency and a regular part L reg ij ( ω ) asRe L ij ( ω ) = 2 π D ij δ ( ω ) + L reg ij ( ω ) . (2.6)The coefficient of the singular part defines the generalized Drude weights D ij § , whichcan be expressed as kD ij = lim t →∞ lim l →∞ l ) t Z t dt ′ h J i ( t ′ ) J j (0) i , (2.7)where in the volume Ω( l ) we have explicitly written the dependence on the system size l along the direction of the thermodynamic flows. Non-zero Drude weights, D ij = 0, are § For i = j = ρ , we have the conventional Drude weight D ρρ . k See Ref. [36] for a detailed discussion and derivation of Eq. (2.7). hermoelectric efficiency G Benenti et al a signature of ballistic transport [37, 38, 39, 40], namely in the thermodynamic limitthe kinetic coefficients L ij scale linearly with the system size l . As a consequence, thethermopower S does not scale with l . We now discuss the influence of conserved quantities on the figure of merit ZT . Makinguse of Suzuki’s formula [41] for the currents J ρ and J u , one can generalize Mazur’sinequality [42] by stating that, for a system of finite size l (along the direction of theflows), C ij ( l ) ≡ lim t →∞ C ij ( t ) = lim t →∞ t Z t dt ′ h J i ( t ′ ) J j (0) i = M X n =1 h J i Q n ih J j Q n ih Q n i , (2.8)where for readability, in the right hand side of the equation we have omitted thedependence on l . The summation in Eq. (2.8) extends over all the M constants ofmotion Q n , which are orthogonal, h Q n Q m i = h Q n i δ n,m , and relevant for the consideredflows. That is, h J ρ Q n i 6 = 0 and h J u Q n i 6 = 0.From Eq. (2.8) one can define the finite-size generalized Drude weights as D ij ( l ) ≡ l ) C ij ( l ) . (2.9)Therefore, the presence of relevant conservation laws directly implies that the finite-sizegeneralized Drude weights are different from zero. If the thermodynamic limit l → ∞ can be taken after the long-time limit t → ∞ , so that the generalized Drude coefficientscan be written as D ij = lim l →∞ D ij ( l ) , (2.10)and moreover D ij = 0, then we can conclude that the presence of relevant conservationlaws yield non-zero generalized Drude weights, which in turn imply that transportis ballistic. We point out that, in contrast to Eq. (2.10), one should take thethermodynamic limit l → ∞ before the long-time limit t → ∞ . While it remainsan interesting open problem for which classes of models the two limits commute ¶ ,numerical evidence suggests that it is possible to commute the limits for the modelsconsidered in Ref. [31] and in the present paper.Let us first consider the case in which there is a single relevant constant of motion, M = 1. We can see from Suzuki’s formula, Eq. (2.8), that the ballistic contributionto det L vanishes, since it is proportional to D ρρ D uu − D ρu , which is zero from (2.8)and (2.10). Hence, det L grows only due to the contributions involving the regular partin Eq. (2.6), i.e., slower than l , which in turn imply that the thermal conductivity κ ∼ det L /L ρρ grows sub-ballistically. Furthermore, since σ ∼ L ρρ is ballistic and S ∼ l , we can conclude that ZT = σS Tκ ∝ lk . (2.11) ¶ See Ref. [36] for a proof of the commutation of the two limits for a class of quantum spin chains. hermoelectric efficiency G Benenti et al
Thus ZT diverges in the thermodynamic limit l → ∞ .The situation is drastically different if M >
1, as it would be the case for integrablesystems, where typically the number of orthogonal relevant constants of motion equalsthe number of degrees of freedom. In that case, due to the Schwartz inequality, D ρρ D uu − D ρu = || x ρ || || x u || − h x ρ , x u i ≥ , (2.12)where x i = ( x i , ..., x iM ) = 1 p l ) h J i Q i p h Q i , ..., h J i Q M i p h Q M i ! , (2.13)and h x ρ , x u i = P Mk =1 x ρk x uk . The equality arises only in the exceptional case when thevectors x ρ and x u are parallel. Hence, for M > L ∝ l , sothat heat transport is ballistic and ZT ∼ l .
3. Momentum-conserving gas of interacting particles
In this section we analyse the consequences of our analytical results in a two-dimensionalgas of interacting particles. We consider a gas of point-wise particles in a rectangulartwo-dimensional box of length l and width w . The gas container is placed in contactwith two particle reservoirs at x = 0 and x = l , through openings of the same size asthe width w of the box. In the transversal direction the particles are subject to periodicboundary conditions.The dynamics of the particles in the system are solved by the method of Multi-particle Collision Dynamics (MPC) [43], introduced as a stochastic model to studysolvent dynamics. The MPC simplifies the numerical simulation of interacting particlesby coarse graining the time and space at which interactions occurs. MPC correctlycaptures the hydrodynamic equations [44, 45]. It has been successfully applied to modelsteady shear flow situations in colloids [46], polymers [47], vesicles in shear flow [48],colloidal rods [49], and more recently to study the steady-state of a gas of particles in atemperature gradient [50].Under MPC dynamics the system evolves in discrete time steps, consisting on freepropagation during a time τ , followed by collision events. During propagation, thecoordinates ~r i of each particle are updated as ~r i → ~r i + ~v i τ , (3.1)where ~v i is the particle’s velocity. For the collisions the system’s volume is partitionedin identical cells of linear size a . Then, the velocities of the N particles found inthe same cell are rotated with respect to the center of mass velocity by a randomangle. In two dimensions, rotations by an angle + α or − α with equal probability p (+ α ) = p ( − α ) = 1 / ~v i → ~V CM + ˆ R ± α (cid:16) ~v i − ~V CM (cid:17) , (3.2)5 hermoelectric efficiency G Benenti et al where ~V CM = N P N i =1 ~v i is the center of mass velocity and ˆ R θ is the two-dimensionalrotation operator of angle θ . Furthermore, to guarantee Galilean invariance, the collisiongrid is shifted randomly before each collision step. It has been shown that for thesedynamics, the equation of state of the gas of particles corresponds to that of an idealgas [43]. Moreover, the time interval between successive collisions τ and the collisionangle α tune the strength of the interactions and consequently affects the transportcoefficients of the gas of particles. When α is a multiple of 2 π , the particles do notinteract, propagating ballistically from one reservoir to the other as they cross thesystem. For any other value of α , the particles interact, exchanging momentum duringthe collision events. The value α = π/ k ( k = L, R for the left and the right reservoir), particles ofmass m enter the system at rate γ k obtained by integration of the appropriate canonicaldistribution to give γ k = w (2 πm ) / ρ k T / k , (3.3)where ρ k and T k are the particle density temperature. Assuming that the particles inthe reservoirs behave as ideal gas, the particle injection rate is related to the value ofthe chemical potential µ k of the reservoir k as µ k = T k ln γ k T / k ! + µ , (3.4)with µ an arbitrary constant whose value does not qualitatively modify the resultsdiscussed in this paper; hereafter we set µ in such a way that µ = 0 + . Whenever aparticle from the system crosses the boundary which separates the system from reservoir k , it is removed (absorbed in the reservoir), i.e., it has no further effects on the evolutionof the system.
4. Discussion of numerical results
We have numerically studied the nonequilibrium transport of the model defined in Sec. 3,coupled to two ideal particle reservoirs. The nonequilibrium state is imposed by settingthe values of T and µ/T in the reservoirs to different values, meaning that from eachof the reservoirs, the particles are injected into the system at different rates and witha different distribution of their velocities. Out of equilibrium the kinetic coefficients L ij can be computed, in the linear response regime, by direct measurement of theparticle and energy currents in the system. Using (2.1), it is enough to perform twononequilibrium numerical simulations: one with T L = T R and µ L /T L = µ R /T R , and onewith T L = T R and µ L /T L = µ R /T R . In the first simulation the reservoirs’ temperatures + This arbitrariness is intrinsic in classical mechanics and can only be removed by means of semiclassicalarguments, see Ref. [10]. hermoelectric efficiency G Benenti et al are set to T L = T − ∆ T / T R = T + ∆ T /
2, so that the temperature gradient isgiven by ∆
T /l , while µ L /T L = µ R /T R . Conversely, in the second simulation we set T L = T R = T and using (3.4), we set the particle injection rates γ L and γ R so that∆ ( µ/T ) = µ L /T L − µ R /T R = ( µ L − µ R ) /T .In all simulations the mean particle density and mean temperature in the reservoirswas set to n = N/lw = 22 .
75 ( N is the mean number of particles) and T = 1,respectively. We parametrize the gradients in terms of a single parameter by setting,∆ T = ∆ ( µ/T ) ≡ ∆ (in units where k B = e = 1). The rotation angle for the collisionsin the MPC scheme was set to α = π/
2, unless otherwise specified. The length of thecollision cells in the MPC scheme was set to a = 0 . τ = 0 .
25. Forthese values and small ∆ the system exhibits reasonably linear temperature and chemicalpotential profiles in the bulk, with some nonlinear boundary layer near the contacts,arising from the fact that the mean free path of the particles near the boundaries isdifferent than in the bulk ∗ , yielding a contact resistance [50]. We performed numericalsimulations with up to Ω = 10 ( l = 500 and w = 2), so that systems with mean numberof particles up to N = 4 . × were considered. Using the Suzuki’s formula (2.8), the current-current correlation functions C ij ( t ) canbe obtained analytically. The particle current is J ρ = P Ni =1 v x,i and the energy current J u = m P Ni =1 (cid:0) v x,i + v y,i (cid:1) v x,i where the coordinate x corresponds to the direction ofthe thermodynamic gradients, thus the direction of the flows.Furthermore, for the MPC model there exists a single relevant constant of motion,namely the x -component of the total momentum Q = p x = m P Ni =1 v x,i . The otherconstants of motion, i.e. momentum in the transverse direction, energy and number ofparticles, are irrelevant since they are orthogonal to the considered flows. Therefore, inthis case M = 1.Applying Eq. (2.8) and integrating over the equilibrium state at temperature T andfixed number of particles N , we obtain that the finite-size correlators are C ρρ ( l ) = N Tm , C ρu ( l ) = 2 N T m , and C uu ( l ) = 4 N T m . (4.1)To verify Eq. (2.8) we have numerically computed the equilibrium current-currenttime correlation functions for the isolated system, averaged over an equilibrium ensembleof initial conditions with N = 1000 particles of mass m = 1 and temperature T = 1.A square container of size l = 2 and periodic boundary conditions in both directionswas considered. The results, shown in Fig. 1, verify our the analytical expressions.Note that the initial values C ρρ (0) and C ρu (0) of the time-averaged correlation functions C ρρ ( t ) and C ρu ( t ) are equal to their asymptotic values C ρρ ( l ) = lim t →∞ C ρρ ( t ) and C ρu ( l ) = lim t →∞ C ρu ( t ). On the other hand, it is easy to compute analytically ∗ The MPC collisions at the boundaries are implemented without taking into account the particles inthe reservoirs hermoelectric efficiency G Benenti et al t C i j ( t ) / N Figure 1.
Equilibrium current-current correlation functions. From bottom to top: C ρρ ( t ) (in red), C ρu ( t ) (in green) and C uu ( t ) (in blue), averaged over the ensemble ofrealisations. The dashed horizontal lines indicate their corresponding analytical valuesfrom Eq. (4.1). C uu (0) = 6 N T /m , and numerical data show that C uu ( t ) converges algebraically toits asymptotic value C uu ( l ) = lim t →∞ C uu ( t ) = 4 N T /m . This asymptotic behaviourmay be due to the slow decay of the energy hydrodynamic modes.Equation (4.1) also shows that the dependence of the correlations on the size l comes exclusively through the number of particles N . The thermodynamic limit requireskeeping the density of particles fixed, so that the number of particles has to scale linearlywith the volume of the system: N ∝ Ω( l ) = lw . Therefore, Eqs. (4.1) imply that thefinite-size generalized Drude weights of Eq. (2.9) do not scale with l . Using Eqs. (2.8)-(2.10) we obtain, for fixed w , the generalized Drude weights D ρρ = nT m , D ρu = D uρ = nT m , and D uu = 2 nT m . (4.2)As a consequence of the finiteness of the Drude weights, the transport is ballistic,meaning that all kinetic coefficients L ij scale linearly with the size of the system: L ij ∼ l .This prediction is confirmed by the numerical results shown in panel a of Fig. 2.More importantly, as discussed in Sec. 2.2, due to the conservation of totalmomentum, the ballistic contribution to the determinant of the Onsager matrix is zero.Indeed, it can be readily seen from Eq. (4.2) that D ρρ D uu − D ρu = 0. Hence a scalingdet( L ) slower than l is expected. From the nonequilibrium numerical simulations thescaling of the determinant with l is consistent with det( L ) ≈ l . (dotted-dashed curvein Fig. 2- b ). It is worthwhile recalling that different analytical methods such as modecoupling theory and hydrodynamics predict, for momentum conserving systems in twodimensions, a logarithmic divergence of the thermal conductivity with the size of thesystem [1, 2]. Therefore, one should expect that det( L ) ∼ l log( l ). We show in Fig. 2- b hermoelectric efficiency G Benenti et al l L ij l de t L a b Figure 2.
Dependence of the kinetic coefficients on the length of the system l , obtainedfrom nonequilibrium simulations with ∆ = 0 . a we show the kineticcoefficients L ρρ (circles), L uρ (crosses), L ρu (pluses) and L uu (squares). The dashedline stands for the linear scaling ∼ l . In panel b we plot the determinant of the Onsagermatrix L (symbols), as a function of the length of the system. The different curvescorrespond to the scalings ∼ l (solid), ∼ l log( l ) (dashed) and ∼ l . (dotted-dashed).Parameter values: m = 1, T = 1, n = 22 . α = π/ w = 2, a = 0 . τ = 0 . (dashed curve), that such scaling is also consistent with our numerical results, thoughdeviations are larger than for the algebraic behaviour at small system sizes. Since wehave no reason to expect an algebraic sub-ballistic behaviour of the heat conductivity,we will assume in what follows that its behaviour is logarithmic. ZT From Eqs. (2.2) and Eqs. (4.2) we obtain that the electric conductivity also scales linearlywith the size of the system: σ = Anm l , (4.3)with A constant. The dependence on l of the Seebeck coefficient cancels out to give,asymptotically in l , S = 1 T (cid:18) D ρu D ρρ − µ (cid:19) = 2 . (4.4)Since the ballistic contribution to det( L ) vanishes, i.e. D ρρ D uu − D ρu = 0, we cannotderive an explicit expression for the heat conductivity κ . However, as discussed in theprevious section, for momentum conserving two-dimensional systems it is predicted that κ diverges logarithmically with respect to the size of the system: κ ∼ log( l ).Fig. 3 shows the dependence of the transport coefficients on the size of the system,for different values of the thermodynamic forces. The electric conductivity verifiesEq. (4.3) independently of the value of the thermodynamic force ∆, with the constant A = π/
4. Instead, the Seebeck coefficient shows a clear dependence on ∆, verifying9 hermoelectric efficiency
G Benenti et al l σ l S l κ Figure 3.
Transport coefficients as a function o the length of the system l , for differentthermodynamic gradients, ∆ = 0 . . .
025 (diamonds),0 . . a the dashed line corresponds to Eq. (4.3) with A = π/ b to S = 2. In panel c the dashed line stands for linear scaling ∼ l , while thesolid line corresponds to log( l ). Eq. (4.4) (asymptotically in l ) only in the limit of small forces (in Fig. 3, S is shown for µ = 0). We have found that S converges to the value S = 2 predicted by (4.4) as 1 / ∆.The heat conductivity κ exhibits the logarithmic behaviour up to a size l = l ⋆ dependent on the strength ∆ of the thermodynamic forces. For any value of ∆, the heatconductivity grows as κ ∼ log( l ), for l > l ⋆ . The smaller the ∆ the larger the range ofvalidity of the logarithmic κ is. We have obtained numerically that the characteristiclength l ⋆ grows linearly with 1 / ∆.Through Eq. (2.3), this characteristic length l ⋆ does also determine the behaviourof the figure of merit ZT . In Fig. 4 we show ZT as a function of l , for different valuesof ∆. We observe that for any value of ∆, ZT is in reasonable agreement with an initialgrow l/ log( l ) for l < l ⋆ and for larger sizes saturates to a maximum value ( ZT ) max .Our results show that as a consequence of the existence of a single relevant conservedquantity, the values of ZT are greatly enhanced when the system under consideration islarge enough. Moreover, ZT does not grow unboundedly, but reaches a maximum valuethat grows with ≈ (1 / ∆) . (see the inset of Fig. 4). The deviations at short sizes areprobably due to the slow convergence of the Seebeck coefficient to its asymptotic value2. In the above discussion on the behaviour of the transport coefficients and the figureof merit ZT as a function of ∆, we should keep in mind that such coefficients andconsequently also ZT are defined in the linear response regime, i.e. in the limit ofsmall thermodynamic forces, formally for ∆ →
0. On the other hand, we numericallycomputed the kinetic coefficients, for any given ∆, via the fluxes as discussed atthe beginning of Sec. 4. That is to say, there is no saturation of ZT within linearresponse. On the other hand, the numerically observed saturation (as well as theballistic behaviour of κ for l > l ⋆ ) signals that the range of linear response shrinkswith the system size when computing κ and ZT . At first sight, this failure of linear10 hermoelectric efficiency G Benenti et al l ZT -3 -2 -1 ∆ ( ZT ) m a x Figure 4.
Thermoelectric figure-of-merit ZT as a function of the length of the system l for different thermodynamic gradients, ∆ = 0 . . . . . ∼ l/ log( l ) In the inset, we show themaximum (saturation) value of ZT as a function of ∆. The dashed line is a power-lawfit, ( ZT ) max = ∆ α , with α ≈ − . response for a given ∆ and large l appears counterintuitive, since for fixed ∆ larger l means smaller thermodynamic forces, and it is in the limit of small forces that linearresponse is expected to be valid. There is actually no such problem when computingthe kinetic coefficients L ij . As shown in Fig. 3 for the charge conductivity σ = L ρρ /T ,data at different ∆ collapse on a single curve, showing that for all values of ∆ in thatfigure we are within linear response. The problem arises when considering non-trivialcombinations of the kinetic coefficients, as in κ ∝ det( L ) and consequently in ZT .Our theory predicts the divergence of ZT in the thermodynamic limit and ZT diverges(thus leading to Carnot efficiency) if and only if the Onsager matrix L becomes ill-conditioned, namely the condition number [Tr( L )] / det( L ) diverges (in our model as l/ log( l )) and therefore the system (2.1) becomes singular. That is, the charge and energycurrents become proportional, a condition commonly referred to as strong coupling , i.e. J ρ = cJ u , the proportionality factor c being independent of the applied thermodynamicforces. The Carnot efficiency is obtained in such singular limit and it is in attainingsuch limit that the validity range of linear response shrinks. Therefore, as expected ongeneral grounds, the Carnot efficiency is obtained only in the limit of zero forces andzero currents, corresponding to reversible transport (zero entropy production) and zerooutput power. 11 hermoelectric efficiency G Benenti et al l ZT π /4 π /2 Figure 5.
Figure-of-merit ZT as a function of the length of the system l , for differentvalues of the collision parameter α , for ∆ = 0 .
1, and from bottom to top: α = 0, 1 / / π/ π/
2. The other parameter values are as in Fig. 2.
It is worthwhile noticing that for our model in the non-interacting limit themomentum of each particle is conserved, meaning that the system is integrable and thenumber of conserved observables M ∝ l , thus diverging in the thermodynamic limit.As we have discussed at the end of section 2.2, one expects that in such integrablesituation, ZT does not scale with the system size. To corroborate this expectation weshow in Fig. 5 the dependence of ZT on l for different values of the collisional parameter α . We recall that at the collisions, α = π/ α = 0 corresponds to no interaction. As expected, forthe non-interacting gas, namely for an infinite number of conserved quantities, ZT doesnot scale with l , attaining the value 3 / ZT is observed for any value of α >
0, as then only the totalmomentum is preserved and M = 1. Our data also suggest a rather weak dependenceof ( ZT ) max on α . The results discussed above show the enhancement of ZT , and thus of the thermoelectricefficiency, in systems with conserved total momentum. In real systems, however, totalmomentum is never conserved due to the phonon field, the presence of impurities or ingeneral to inelastic scattering events.In this section we want to explore to what extent the break down of total momentconservation modifies the results obtained above. To address this question numerically,12 hermoelectric efficiency G Benenti et al l σ l S l κ Figure 6.
The dependence of the transport coefficients on the size l of the system,for ∆ = 0 . ε = 0, 0 .
01 and0 .
1. The other parameter values are as in Fig. 2. we consider the existence of a source of stochastic noise. From a physical point of view,this noise source may model the interactions of the gas with the walls of the container,or the inelastic scattering from impurities in the material. We model the stochasticnoise as follows: after a collision of the particles in a given cell has taken place, withprobability ε the velocities of all the particles in the cell are reflected, namely ~v i → − ~v i .Therefore for any ε > ε is small themomentum conservation is weakly broken and we want to investigate how our resultsdepend on the strength ε of the perturbation.In Fig. 6 we show the dependence of the transport coefficients on l for fixed ∆and different strengths of the noise ε . We observe that for sufficiently strong noiseall transport coefficients appear to become independent of l , as expected in a diffusiveregime in which total momentum is not preserved.More interesting is the behaviour of ZT shown in Fig. 7. We see that at strongernoise, ZT becomes constant, as expected in the diffusive regime. From a mathematicalpoint of view, the absence of conserved quantities ( M = 0) leads to decaying correlationfunctions and zero Drude coefficients (inset of Fig. 7). Thus the transport coefficientsand ZT become size-independent.More importantly, we see that when the convergence toward the diffusive regimeis smooth, meaning that when the conservation of total momentum is only weakly perturbed (small ε ), the enhancement of ZT can be significant. This shows that theeffect described here is robust against perturbations.
5. Conclusions
In summary, we have shown that in two-dimensional interacting systems, with theinteractions modeled by the multi-particle collision dynamics method, the thermoelectricfigure of merit diverges at the thermodynamic limit. In such limit, the Carnot efficiencyis obtained with zero output power. When noise is added to the system, ZT saturatesat large l , to values higher the weaker is the noise strength.13 hermoelectric efficiency G Benenti et al l ZT t -2 -1 C uu ( t ) / N Figure 7.
Figure-of-merit ZT as a function of the length of the system l , for differentnoise intensities ε = 0 (squares), 0 .
01 (triangles) and 0 . . ε = 0, i.e. ZT ∼ l/ log( l ). In the inset: energy current-currentcorrelation for different noise intensities ε , for the same parameter values as in Fig. 6.From top to bottom: ε = 0 , − , − , − , − . The dashed curve stands for ∼ /t . Our findings could be relevant in situations in which the elastic mean free pathis longer than the length scale over which interactions are effective in exchangingmomenta between the particles. Suitable conditions to observe the interaction-inducedenhancement of the thermoelectric figure of merit might be found in high-mobility two-dimensional electron gases at low temperatures. In such systems very large elasticmean free paths have been reported (for instance, up to 28 µ m in Ref. [51]). At lowtemperatures the inelastic mean free path is determined by electron-electron interactionsrather than by phonons. It should be therefore possible to find a temperature windowwhere electron-electron interactions dominate, i.e. are effective on a scale smaller thanthe elastic mean free path and are dominant over phonon effects. It would be, however,highly desirable to test our arguments in such regime, by means of numerical simulationsof quantum systems. Acknowledgments
We acknowledge support by MIUR-PRIN and by Regione Lombardia. CMM is partiallysupported by the European Research Council, the Academy of Finland, and by theMICINN (Spain) grant MTM2012-39101-C02-01.14 hermoelectric efficiency
G Benenti et al
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