Title. Author(s)Izumida, Yuki; Okuda, Koji. CitationPhysical review E, 80(2): Issue Date Doc URL. Rights. Type.

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1 Title Onsager coefficients of a finite-tie Carnot cycle Author(s)Izuida, Yuki; Okuda, Koji CitationPhysical review E, 80(2): Issue Date Doc URL Rights 2009 The Aerican Physical Society Type article File Inforation PRE80-2_ pdf Instructions for use Hokkaido University Collection of Scholarly and Aca

2 Onsager coefficients of a finite-tie Carnot cycle Yuki Izuida and Koji Okuda Division of Physics, Hokkaido University, Sapporo , Japan Received 6 June 2009; published 24 August 2009 We study a finite-tie Carnot cycle of a weakly interacting gas which we can regard as a nearly ideal gas in the liit of T h T c 0 where T h and T c are the teperatures of the hot and cold heat reservoirs, respectively. In this liit, we can assue that the cycle is working in the linear-response regie and can calculate the Onsager coefficients of this cycle analytically using the eleentary olecular kinetic theory. We reveal that these Onsager coefficients satisfy the so-called tight-coupling condition and this fact explains why the efficiency at the axial power ax of this cycle can attain the Curzon-Ahlborn efficiency fro the viewpoint of the linear-response theory. DOI: /PhysRevE PACS nuber s : Ln I. INTRODUCTION Iproving efficiency of heat engines has been a long challenge since the Industrial Revolution. Substantial progress of our knowledge cae fro Carnot s insight: he conceived a atheatical odel of an idealized heat engine, now called the Carnot cycle, and showed that there is an upper liit of the efficiency of all the existing heat engines which can be attained only when the engines are working infinitely slowly quasistatic liit to vanish the irreversibility. In the fundaental physics, properties of heat engines working at the axial power have also been studied since the study by Curzon and Ahlborn 1,2 see also 3. Although the quasistatic Carnot cycle has the highest efficiency, it outputs zero power because it takes infinite tie to output a finite aount of work. y contrast, Curzon and Ahlborn considered a finite-tie Carnot cycle which exchanges heat at a finite rate with the reservoirs according to the linear tie-independent Fourier law. Under the assuption of endoreversibility that irreversible processes are occurred only through these heat exchanges, they derived a rearkable result: the efficiency at the axial power ax is given by ax =1 T c CA, 1 T h where T h and T c are the teperatures of the hot and cold heat reservoirs, respectively, and the above CA is usually called the Curzon-Ahlborn CA efficiency. Previously we studied a finite-tie Carnot cycle of a weakly interacting gas which we can regard as a nearly ideal gas to confir the validity of the CA efficiency fro a ore icroscopic point of view 4,5. We perfored extensive olecular dynaics MD coputer siulations of the finite-tie Carnot cycle of the two-diensional low dense hard-disk gas and easured the efficiency and the power for the first tie. Our siulations revealed that our ax agrees with CA only in the liit of T 0 where T T h T c, but exceeds CA at soewhat large T. We also confired this behavior of ax analytically using the eleentary olecular kinetic theory. Therefore the phenoenological prediction of Eq. 1 sees to be valid only in the liit of the sall teperature difference for our finite-tie Carnot cycle. Recently, Van den roeck 6 considered the heat engine described as the Onsager relations J 1 = L 11 X 1 + L 12 X 2, J 2 = L 21 X 1 + L 22 X 2, and showed that CA is the upper liit of ax in this heat engine see also Sec. V. The recent studies 7 11 on the various theoretical heat engine odels also support the results in 6. These results see to eet our previous result of the finite-tie Carnot cycle, though it is unclear why our syste realized the upper liit of ax in T 0. Moreover it is also unclear whether the finite-tie Carnot cycle can be understood in the fraework of the Onsager relations because the explicit calculations of the Onsager coefficients L ij for that cycle do not exist to our knowledge. In this paper, we apply the fraework of the Onsager relations to our previous study of the finite-tie Carnot cycle and analytically calculate the Onsager coefficients for it. Although there are a few analytic calculations of the Onsager coefficients for the steady state of heat engines such as rownian otors 7,12, we believe that the present study is the first exaple of the calculation for the cyclic heat engine odel where the two heat reservoirs do not contact with the working substance siultaneously. We will show that these Onsager coefficients satisfy the so-called tight-coupling condition and therefore we can give an explicit explanation why ax of this cycle attains the CA efficiency in the liit of T 0, as observed in 4,5, fro the viewpoint of the linear-response theory. We also perfor the MD coputer siulations to check the validity of our analytic calculations of the Onsager coefficients. The organization of this paper is as follows. First we introduce our finite-tie Carnot cycle odel in Sec. II and describe the olecular kinetic theory in Sec. III. The ain result of this paper, the analytic calculations of the Onsager coefficients for our odel, accopanied by the results of the MD siulations for checking the validity of our analytic calculations, are shown in Sec. IV. In Sec. V, we introduce the general fraework of the heat engine used in 6 and discuss the efficiency at the axial power of our finite-tie /2009/80 2 / The Aerican Physical Society

3 YUKI IZUMIDA AND KOJI OKUDA (a) FIG. 1. a Scheatic illustration of a two-diensional finitetie Carnot cycle odel. Hard-disk particles as a weakly interacting nearly ideal gas are confined into the cylinder. The piston oves at a finite constant speed u and the theralizing wall with the length S is set on the left botto of the cylinder during the isotheral processes. b Teperature-volue T-V diagra of a quasistatic Carnot cycle for a two-diensional ideal gas. Carnot cycle using those Onsager coefficients according to that fraework. We suarize this study in Sec. VI. II. MODEL We first introduce a theoretical odel for a finite-tie Carnot cycle of a two-diensional weakly interacting gas which we can regard as a nearly ideal gas 4,5. To iic the weakly interacting nearly ideal gas, we confine a low dense N hard-disk particles with diaeter d and ass into a cylinder with rectangular geoetry and let the collide with each other Fig. 1 a. The head of the cylinder is a piston and it oves back and forth at a constant speed u. The usual quasistatic Carnot cycle u 0 consists of four processes Fig. 1 b : A isotheral expansion process V 2 in contact with the hot reservoir at the teperature T h, adiabatic expansion process V 2 V 3, C isotheral copression process V 3 V 4 in contact with the cold reservoir at T c, D adiabatic copression process V 4, where V k s k=1,...,4 are the volues of the cylinder at which we switch each of the four processes. V k s satisfy the relations V 3 = T h /T c V 2 and V 4 = T h /T c in the case of the twodiensional ideal gas. Since we regard our syste of harddisk particles as a nearly ideal gas, we apply these relations to our syste. In the case of a finite-tie cycle, we also switch each process at the sae volue V k as in the quasistatic case. Defining x,y coordinates as in Fig. 1 a, we let the piston ove along the x axis at a finite constant speed u. Here, we express the x length and the y length of the cylinder as l and L, respectively and the volue of the cylinder as V=Ll. Then, the x length l k at the switching volue V k can be defined as l k V k /L. When a particle with the velocity v= v x,v y collides with the piston oving at the x velocity u, its velocity changes to v = v x 2u,v y, assuing perfectly elastic collision. Then the colliding particle gives the icroscopic work v 2 v 2 /2=2 uv x u 2 against the piston. In the isotheral processes A and C, we set the theralizing wall with the length S at the position as in Fig. 1 a. When a (b) particle with the velocity v= v x,v y collides with the theralizing wall, its velocity stochastically changes to the value governed by the distribution function f v,t i = 1 2 k T i 3/2v y exp v2 2k T i v x +,0 v y +, T i i=h in A, cin C, where k is the oltzann constant 13. The icroscopic heat flowing fro the theralizing wall can be calculated by the difference between the kinetic energies before and after the collision. We su up the above icroscopic work and heat during one cycle. When Q h denotes the heat flown fro the hot reservoir during A and Q c denotes the heat flown fro the cold reservoir during C, the total work W during one cycle is expressed as W Q h +Q c. Then the power P and the efficiency can be defined as P Ẇ and W/Q h P/Q h. Here, the dot denotes the value divided by one cycle period or the value per unit tie throughout the paper. In our syste, one cycle period is 2 l 3 l 1 /u. III. MOLECULAR KINETIC THEORY In this section, we review the results of the olecular kinetic theory of the finite-tie Carnot cycle obtained in our previous work. The details of the derivation of the equations in this section are described in 4. If we assue that, even in a finite-tie cycle, the gas relaxes to a unifor equilibriu state with a well-defined teperature T sufficiently fast and the particle velocity v is governed by Maxwell-oltzann distribution at T, we can easily derive the tie-evolution equation of T using the eleentary olecular kinetic theory. Such an assuption of the fast relaxation to a unifor equilibriu state at a welldefined teperature T is valid when the energy equilibration in the cylinder due to the interparticle collisions is uch faster than the speed of the energy transfers through the theralizing wall and the piston. This situation is surely realized when u is sall and the interaction length S between the gas and the reservoir is sufficiently sall. If we assue that the gas is sufficiently close to a two-diensional ideal gas, the internal energy of the gas can be approxiated as Nk T. Then, we can derive the tie-evolution equation of the gas teperature T for each of the four processes A D as A :Nk dt = q h w e, C :Nk dt = q c w c, :Nk dt = w e, D :Nk dt = w c, where q i =q i t,t i=h in A, cin C is the heat flowing into the syste per unit tie in the isotheral processes and w j =w j t,t j=e in the expansion processes A and, c in the copression processes C and D is the work against the piston per unit tie. y counting the nuber of the particles colliding with the theralizing wall and the piston, we can derive the specific fors of q i and w e as

4 ONSAGER COEFFICIENTS OF A FINITE-TIME CARNOT T u=10-3 u=10-4 quasistatic V as the solutions of /dt=0 by expanding as T st i =T i +a 1 i u +a 2 i u 2 +O u 3, assuing that u is sall. These expansion coefficients a 1 i,a 2 i, etc. can be deterined order by order. If we assue that the relaxational process to T st h is sufficiently fast, st the heat Q h flowing into the syste during the steady part T t =T st h can be calculated up to O u as Q h st = 0 l 2 l 1 /u q h t,t h st dt = Q h qs FIG. 2. Teperature-volue T-V diagra for the steady cycle obtained by nuerically solving Eqs. 5 for u=10 4 dotted line and u=10 3 dashed line. The solid line is a theoretical quasistatic Carnot cycle for a two-diensional ideal gas. Since the dotted line is very close to the solid line, the case of u=10 4 ay be regarded as the quasistatic cycle. The paraeters used are N=100, T h =1, T c =0.7, S=0.05, =1, k =1, l 1 =1, l 2 =1.5, and L=1. q i t,t = 3SNk T i T 2 k T 4 V t, 6 A w e t,t = 2uNL A 2 T V t 4 T u + u u/a T dv x A Tvx u 2e v x, where A 2k /. w c is also obtained by changing u u in Eq. 7. We can nuerically solve Eqs. 5 for the entire cycle at various piston speeds. y using the final teperature of each process as the initial teperature of the next process repeatedly, we can obtain a steady cycle at a given u. In Fig. 2, we have shown the teperature-volue T-V diagra for the steady cycle at u=10 4 dotted line and u=10 3 dashed line, respectively. The solid line is the theoretical quasistatic line of a two-diensional ideal gas. Fro this figure, we can see that as u becoes larger, the cycle deviates fro the theoretical quasistatic line and the teperatures during the isotheral processes A and C relax to the steady teperatures T h st T h and T c st T c, respectively. If it is considered that the relaxation to the steady teperature is very fast, the isotheral process ay approxiately be divided into the relaxational part and the steady part in the case of a finitetie cycle, where T t instantaneously changes fro the initial teperature of the isotheral process to the steady teperature in the relaxational part and T t keeps the steady teperature in the steady part. These relaxational parts in the isotheral processes A and C are surely issed in the original odel by Curzon and Ahlborn 1,4,14. Moreover we note that the heat flow per unit tie q i as shown in Eq. 6 is tie dependent even during T=T i st through the volue of the cylinder after the relaxational processes in A and C. Therefore the assuption of the tie-independent heat flows in the original phenoenological odel in 1 does not see to be applied to our odel. Although it ay be ipossible to find the exact solutions of Eqs. 5, we can find the analytic expressions of the steady teperature T i st during the isotheral processes A and C 7 2NA Th 1 + 3S u L ln V 2, 8 where the quasistatic heat for an ideal gas in the isotheral process A is defined as Q qs st h =Nk T h ln V 2 /. Q c during T t =T st c can also be obtained by changing u u, T h T c, and V 2 / V 4 /V 3 in Eq. 8. If we neglect the contributions of the heat transfers through the theralizing wall during the relaxational parts in A and C, the steady part of the work during one cycle W st Q st h +Q st c becoes W st = Nk T h T c ln V 2 2NA 1 3S + L Th + Tc u ln V 2. If we assue that the adiabatic processes and D even at a finite u satisfy the quasistatic adiabatic relation TV= const of a two-diensional ideal gas, the additional heat transfers Q h add and Q c add during the relaxational part in A and C, respectively, can be estiated as Q h add = Nk 4L Th 3SA 1+ T h T c u 9 10 up to O u and Q c add can also be obtained by changing u u and T h T c in Eq. 10. Therefore the total Q h, Q c, and W becoe Q h =Q h st +Q h add, Q c =Q c st +Q c add, and W=W st +Q h add +Q c add, respectively. IV. CALCULATION OF ONSAGER COEFFICIENTS After the preliinaries in the previous sections, we can now calculate the Onsager coefficients for our finite-tie Carnot cycle in the linear-response regie T 0 as follows. The first step is an appropriate choice of the therodynaic forces and fluxes for this syste. A typical way to choose these therodynaic forces and fluxes is to introduce the rate of the total entropy production during one cycle, Q Q h c. 11 T h T c This is just the entropy increase of the two reservoirs during one cycle divided by the cycle period 2 l 3 l 1 /u because the entropy of the working substance does not change after one cycle. Using the relation Q c =W Q h and considering the linear-response regie T 0, it can be rewritten as

5 YUKI IZUMIDA AND KOJI OKUDA u W = 2 l 3 l 1 T + T Q T 2 h, 12 where T T h +T c /2 and we have neglected T 3 Q h and uw T ters, the reason of which we will clarify later. According to the linear-response theory, can be expressed as the su of the product of the therodynaic force and its conjugate therodynaic flux 15,16, = J 1 X 1 + J 2 X 2, where we define the therodynaic forces as X 1 W 2 l 3 l 1 T, X 2 T T 2 and their conjugate fluxes as J 1 u, J 2 Q h Moreover the linear-response theory assues the Onsager relations between the fluxes and forces 15,16, W J 1 = u = L 11 2 l 3 l 1 T + L T 12 T 2, 16 W J 2 = Q h = L 21 2 l 3 l 1 T + L T 22 T 2, 17 where L ij s are the Onsager coefficients and the nondiagonal eleents should satisfy the syetry relation L 12 =L 21. Fro these relations between the therodynaic fluxes and forces, we understand that =J 1 X 1 +J 2 X 2 is the quantity of the second order of the therodynaic forces, which explain why we neglected the higher order ters like T 3 Q h and uw T in Eq. 12. Moreover, although we considered the contributions of the additional heat transfers Q add h and Q add c to the total heat and the work in Sec. III, we can easily show that their effects do not contribute to in the liit of T 0. Therefore we can indeed neglect the in the linearresponse regie and use Q h =Q st h and W=W st in the calculations of the Onsager coefficients below. We are now in a position to calculate the Onsager coefficients explicitly. First we deterine L 11 and L 21 as follows. To calculate L 11, we consider the relation between u and X 1 in the case of T=0. Expanding Eq. 9 by T and putting T=0, we can obtain the relation W = 4NA T 1 + L 3S ln V 2 u. Coparing Eq. 18 with Eq. 16, L 11 is deterined as L 11 = 18 l 3 l 1 T 2N 1/2 1 3S + L 2 k. 19 lnv 2 Likewise Q h with T=0 can be evaluated up to the linear order in W using Eqs. 8 and 18 as Q h = k T 4 3/2 W 1 3S + L 2 k 2 l 3 l 1 T. Coparing Eq. 20 with Eq. 17, L 21 is deterined as L 21 = 20 k T 4 3/2 1 3S + L 2 k. 21 Next we deterine L 12 and L 22 as follows. To calculate L 12, we consider the relation between u and X 2 in the case of W=0. In this case, regardless of a finite teperature difference, useful work cannot be obtained because the engine runs so fast that it cannot output positive work. Therefore, this case is called the work-consuing state. The speed of the piston at the work-consuing state can be obtained as a solution of W=0 in Eq. 9. Considering in the linearresponse regie Th Tc T/ 2 T, it becoes u = k T 4 3/2 T 1 3S + L 2 k T 2. Fro Eqs. 16 and 22, we can obtain L 12 = 22 k T 4 3/2 1 3S + L 2 k. 23 Fro Eqs. 21 and 23, we can confir the syetry relation L 12 =L 21 as expected. To deterine the last coefficient L 22, we consider the heat flow Q h at the work-consuing state using Eqs. 8 and 22, Q h = Nk 2 ln V S + L 2 T 5/2 T l T 2. 3 l 1 24 Fro Eqs. 17 and 24, the last coefficient L 22 turns out to be L 22 = Nk 2 ln V 2 T 5/ S + L l 3 l 1 Note that the positivity of the rate of the total entropy production, =J 1 X 1 +J 2 X 2 0, should restrict the values of the Onsager coefficients to L 11 0, L 22 0, and L 11 L 22 L 12 L We can confir that L ij of our finite-tie Carnot cycle surely satisfy these relations. These analytic expressions of the Onsager coefficients L ij are the ain result of this paper. To confir the validity of the above analytic calculations of the Onsager coefficients, especially T dependence of the, we perfored the event-driven MD coputer siulations 4,5,17 of our two-diensional finite-tie Carnot cycle, following the procedure described in Sec. II

6 ONSAGER COEFFICIENTS OF A FINITE-TIME CARNOT Onsager coefficient MD L11 MD L21 MD L12 MD L FIG. 3. T dependence of the Onsager coefficients. The dashed line, the solid line, and the dotted line indicate the theoretical Onsager coefficient L 22 Eq. 25, L 12 Eq. 23 =L 21 Eq. 21 and L 11 Eq. 19, respectively. The paraeters used in the MD siulations are T= and d=0.01. The other paraeters are the sae as in Fig. 2. The MD data were obtained by averaging cycles after transient 5 cycles in the siulations. To calculate L 12 and L 22 at given T, we fix the teperature difference T to a sufficiently sall value and find the piston speed where the work becoes 0. Then we can deterine L 12 and L 22 nuerically as L 12 = J 1 X 2 = ut2 T, 26 L 22 = J Q 2 ht 2 = 27 X 2 T fro Eqs. 16 and 17. Next, to calculate L 11 and L 21 at given T, weset T=0. Fixing u to a sufficiently sall value , we can deterine L 11 and L 21 nuerically as L 11 = J 1 = 2 l 3 l 1 Tu, 28 X 1 W L 21 = J 2 = 2 l 3 l 1 TQ h 29 X 1 W fro using Eqs. 16 and 17. Figure 3 shows T dependence of these Onsager coefficients deterined by the MD siulations as well as the analytic results. We can see fairly good agreeent between the MD data and the analytic lines in the range we studied. These data clearly support the validity of our analytic expressions of the Onsager coefficients which has been obtained under soe theoretical assuptions. V. EFFICIENCY AT THE MAXIMAL POWER To see how the Onsager coefficients derived in Sec. IV indeed govern the behavior of the finite-tie Carnot cycle, we briefly introduce the general fraework of the heat engine obeying the Onsager relations 6. The basic setup is as follows. We consider a general steady or cyclic process in which the work is extracted fro the heat flow between the T FIG. 4. Scheatic illustration of the heat engine governed by the Onsager relations Eqs. 30 and 31. sall teperature difference. see Fig. 4 The work W done against the external force F is W= Fx where x is the therodynaically conjugate variable of F. We define a therodynaic force as X 1 =F/T and the corresponding therodynaic flux as J 1 =ẋ. We also choose X 2 =1/T c 1/T h as another therodynaic force and J 2 =Q h as the corresponding therodynaic flux. If we consider the linear-response regie T 0, X 2 can be written as X 2 T/T 2. Moreover the linear-response theory assues the Onsager relations between the therodynaic forces and fluxes 15,16, J 1 = L 11 X 1 + L 12 X 2, 30 J 2 = L 21 X 1 + L 22 X 2, 31 where the nondiagonal eleents of the Onsager coefficients should satisfy L 12 =L 21. Then, the power P=Ẇ and the efficiency = P/Q h of the engine can be expressed as P = J 1 X 1 T, 32 = J 1X 1 T. 33 J 2 Now the efficiency at the axial power ax can be given as follows: we first axiize the power at X 1 = L 12 X 2 / 2L 11 which is deterined as the solution of P/ X 1 =0, then, ax becoes where ax = T 2T q 2 2 q 2, 34 q L L11 L 22 is called the coupling strength paraeter. Note that it takes 1 q +1 due to the positivity of. When q satisfies the tight-coupling condition q =1, ax takes the axial value T/ 2T, which is equal to the CA efficiency up to the lowest order in T. In Refs. 4,5, we studied ax of this finite-tie Carnot cycle extensively by perforing the MD coputer siulations as a nuerical experient to verify the validity of the CA efficiency. We found there that ax agrees with the CA efficiency in the liit of T 0. Our olecular kinetic

7 YUKI IZUMIDA AND KOJI OKUDA theory also confired this property 4 : using Q i =Q st add i +Q i in Sec. III, we can calculate the speed u=u ax at which the power P axiizes as u ax = k T h T c 4A L 3S + 1 Th + Tc ln V 2 + 8Lk 3SA Th T h T c 1+ T h T c 1 ln V 2, and can confir that the efficiency at u=u ax shows 36 W u ax ax u ax = Q st h u ax + Q add h u ax T T 0. 2T 37 Now we can clarify the underlying physics of this behavior of ax. In the liit of T 0, our finite-tie Carnot cycle can be described by the Onsager relations as shown in Sec IV. Then we can confir that q=1 in our finite-tie Carnot cycle fro Eqs. 19, 23, 25, and 35. This condition gives a proof that our finite-tie Carnot cycle shows the CA efficiency in the liit of T 0 as suggested in 4,5 fro the viewpoint of the linear-response theory. VI. SUMMARY In this paper, we have studied a finite-tie Carnot cycle of a two-diensional weakly interacting nearly ideal gas working in the linear-response regie and have explicitly calculated the Onsager coefficients of this syste for the first tie. Molecular dynaics coputer siulations of this cycle have supported the theoretical calculations in spite of soe assuptions in the analysis. We have revealed that the Onsager coefficients of this syste satisfy the tight-coupling condition q=l 12 / L11 L 22 =1 and therefore can understand why our finite-tie Carnot cycle attains the Curzon-Ahlborn efficiency in the linear-response regie T 0 as suggested in Refs. 4,5. It would be an interesting proble to construct and study the collective behavior of the Carnot cycles coupled with each other by using the property of the Onsager coefficients derived in this paper 6, ACKNOWLEDGMENTS The authors thank M. Hoshina for helpful discussions. This work was supported by the 21st Century Center of Excellence COE progra entitled Topological Science and Technology, Hokkaido University. 1 F. Curzon and. Ahlborn, A. J. Phys. 43, H. Callen, Therodynaics and an Introduction to Therostatistics, 2nd ed. Wiley, New York, 1985, Chap I. I. Novikov, J. Nucl. Energy 7, Y. Izuida and K. Okuda, Europhys. Lett. 83, Y. Izuida and K. Okuda, Prog. Theor. Phys. Suppl. 178, C. Van den roeck, Phys. Rev. Lett. 95, A. Goez-Marin and J. M. Sancho, Phys. Rev. E 74, T. Schiedl and U. Seifert, Europhys. Lett. 81, Z. C. Tu, J. Phys. A 41, M. Esposito, K. Lindenberg, and C. Van den roeck, Europhys. Lett. 85, M. Esposito, K. Lindenberg, and C. Van den roeck, Phys. Rev. Lett. 102, R. enjain and R. Kawai, Phys. Rev. E 77, R. Tehver, F. Toigo, J. Koplik, and J. R. anavar, Phys. Rev. E 57, R J. irjukov, T. Jahnke, and G. Mahler, Eur. Phys. J. 64, L. Onsager, Phys. Rev. 37, S. R. de Groot and P. Mazur, Non-Equilibriu Therodynaics Dover, New York, J. Alder and T. E. Wainwright, J. Che. Phys. 31, M. J. Ondrechen,. Andresen, M. Mozurkewich, and R. S. erry, A. J. Phys. 49, C. Van den roeck, Adv. Che. Phys. 135, Jiénez de Cisneros and A. C. Hernández, Phys. Rev. Lett. 98, Jiénez de Cisneros and A. C. Hernández, Phys. Rev. E 77,