PHYSICAL REVIEW E 68, 52 23 Diffusion tie-scale invariance, randoization processes, and eory effects in Lennard-Jones liquids Renat M. Yuletyev* and Anatolii V. Mokshin Departent of Physics, Kazan State Pedagogical University, 422 Kazan, Russia Peter Hänggi Departent of Physics, University of Augsburg, D-8635 Augsburg, Gerany Received 9 Noveber 22; revised anuscript received 3 June 23; published 5 Noveber 23 We report the results of calculation of diffusion coefficients for Lennard-Jones liquids, based on the idea of tie-scale invariance of relaxation processes in liquids. The results were copared with the olecular dynaics data for the Lennard-Jones syste and a good agreeent of our theory with these data over a wide range of densities and teperatures was obtained. By calculations of the non-markovity paraeter we have nuerically estiated statistical eory effects of diffusion in detail. DOI:.3/PhysRevE.68.52 PACS nubers: 66..Cb, 6.2.p The otion of atos and olecules in liquids is a typical exaple of a any-body proble. It is well known, however, that in contrast to gases and solids the physical echaniss of transport processes in liquid atter is not well established. In particular, for diffusive systes, the role of eory effects induced by the disorder of the background ediu is not investigated enough 2,3. There are different theoretical approaches and approxiations for calculating the transport coefficients,4, in soe of the they resort to tie correlation functions see, for exaple, Refs.,3 6. The general approach of tie correlation functions leads to calculation of exact expressions of transport coefficients for hydrodynaic equations. Siilar expressions are known as the Green-Kubo relations. In the present work we suggest an approach for calculating transport properties based on the realization of idea of tie-scale invariance of relaxation processes in liquids 7 by well-known Zwanzig-Mori s eory function foralis 8,9. Recently, this idea has ade it possible to explain the experiental data on slow neutron scattering in liquid cesiu and sodiu 7. Here we have tested this approach on Lennard-Jones LJ liquids. As known, aong a large set of dense fluids these systes are one of the ost popular base for different investigations. So, diffusion coefficients of LJ liquids were calculated for a wide range of densities and teperatures, and then quantitative analysis of the eory effects in diffusion phenoena was executed. The obtained results are copared with the predictions of other theories 6 and the olecular dynaics data. As known, the diffusion coefficient can be expressed in ters of the velocity autocorrelation function VACF at v v t v 2 by the relation D k BT atdt. The latter is one of the Green-Kubo equations relating the integrals of the correlation function to the transport coefficients. Here k B, T, and are the Boltzann constant, teperature, and atoic ass, respectively. On the other hand, the generalized Langevin equation for VACF can be written in the following way: dat (2) t M dt atd, where M (t) is the first-order noralized eory function M (t), and (2) is the second frequency oent of VACF. This equation is obtained by the use of Zwanzig- Mori s projection operators foralis 8. However, the projection operators technique allows us to derive the whole chain of interconnected equations siilar to the above entioned equation. These equations contain the eory functions M i (t) and the even frequency oents (2i) of higher orders, i.e., i,2,3,... This chain can be represented in ters of Laplace transfors as a continued fraction, ãs dte st ats (2) M s /s (2) /s (4) / (2) (2) M 2s /s (2) /s (4) / (2) (2) /s, and Eq. 2 yields 2 3 4 *Electronic address: ry@dtp.ksu.ras.ru Electronic address: av@dtp.ksu.ras.ru D k BT ãs. 5 63-65X/23/685/525/$2. 68 52-23 The Aerican Physical Society
YULMETYEV, MOKSHIN, AND HÄNGGI So, the proble of calculation of transport properties can be reduced to calculation of the certain eory functions M i (t). Although there are icroscopic expressions and soe foral prescriptions for eory functions calculations, it is prohibitively difficult for odels like LJ liquid. A rich variety of odel functions was proposed for this purpose, ost of which have little physical justification. For exaple, hyperbolic secant eory was used to calculate velocity, transverse stress, energy current density correlation functions and the corresponding transport coefficients for LJ liquid 4. The validity and justification of this approach was closely exained in work of Ref.. There is another ore effective and powerful way to calculate transport coefficients of the diffusion constant type. This ethod was first suggested in work of Ref. 2 to calculate self-diffusion in liquid argon, and was used later in Ref. 3 to investigate viscosity effects in liquid argon. Here we use this approach to analyze diffusion tie scales and to calculate the diffusion coefficient in LJ liquid over a wide density and teperature ranges. As known, the eory functions M i (t) have characteristic tie scales, which can generally be defined at fixed i by i M i(s) dtm i (t), where i is the relaxation tie of a certain correlation function M i (t). These tie scales characterize the corresponding relaxation processes and can have different nuerical values. Nonetheless, on a certain relaxation level for exaple, on the ith level the scale invariance of the nearest interconnected relaxation processes can exist. We shall consider below the following approxiation i i. Physically it eans the occurrence of tiescale invariance on the ith relaxation level. In case of i, we have an approxiate equality of VACF relaxation tie and the relaxation tie of the firstorder eory function. Then, fro Eq. 5 and the second equality of Eq. 4 at one can obtain D k BT (2) /2, (2) 4n 3 drgrr 2 3 r Ur r r r Ur rr, 6 where n is the density, g(r) is the radial distribution function, and U(r) is the interparticle potential. At i we obtain the equality of the relaxation ties of first- and second-order eory functions 4. In this case, 2, and fro Eqs. 4 and 5 we have D k BT /2 (4) (2) (2) (2), (4) 8n drgr 3 3 dur dr Ur r 82 n 2 3 3 Ur rr r r r rr r 2 r r Ur PHYSICAL REVIEW E 68, 52 23 2 rr r Ur rr drdr r 2 r 2 Ur r Ur r r Ur r r Ur r r r r Ur r dg3 r,r r Ur rr 2, r Ur rr where g 3 (r,r ) is the triplet correlation function, and is the cosine of the angle between r and r. Following the sae procedure for third- and second-order relaxation ties 3 2 and i2), we coe to the following expression: D k BT (4) (2) 2 3/2 (2) 3/2 (6) (2) (4) 2 /2. In a ore general case with i i, i,2,..., we can obtain the expression of diffusion coefficient by the first even frequency oents: (2), (4),..., (2i). The second, the fourth, and the sixth frequency oents contained in Eqs. 6 8 were approxiately obtained for LJ liquid by the authors of Ref. 6. Using these values, the diffusion coefficients were calculated with the help of Eqs. 6 8. The coparison shows that the values of D obtained fro Eq. 7 have the best agreeent with the olecular dynaics data for the whole studied range of densities and teperatures. The results for D*D(/ 2 ) /2 fro Eq. 6 are presented in Fig. by triangles, fro Eq. 7 are shown by solid curves and fro Eq. 8 are presented by circles. We also represented here the results of the authors of Ref. 6 for coparison with our ones. Naely, the diffusion coefficient obtained fro Ref. 6 fro the approxiation of the second-order eory function by the hyperbolic secant are shown by dotted curves, whereas those obtained in Ref. 6 fro the prescription of Joslin and Gray 5 are shown by broken and chain curves. Fro Fig. one can easily see that the results of Eq. 7 have a better agreeent with the olecular dynaics data,6 presented by full circles in ore cases than data found fro other theories. This result is of great interest for the study of diffusion phenoena. It confirs the possibility of equality of relaxation ties of first- and second-order eory functions, i.e., and 2, respectively. As can be seen fro Fig., the approxiation with hyperbolic secant eory shows a sufficiently satisfactory agreeent for soe cases. Joslin and Gray odels yield an understated result in coparison with the olecular dynaics data. The distinction between these theories and olecular dynaics data grows with the decreasing of density. How- 7 8 52-2
DIFFUSION TIME-SCALE INVARIANCE,... PHYSICAL REVIEW E 68, 52 23.6 T * =3.46.4.2 D *.8.6.4.2.5 T * =2.5.5.8.6.4.2 T * =.8 T * =.23.5.5 FIG.. The reduced density n*n 3 dependence of diffusion coefficient at the reduced teperature T*k B T/3.46, 2.5,.8, and.23. The triangles show results of Eq. 6, the solid line presents our results fro Eq. 7, the circles show results of Eq. 8, the dotted line presents the results of approxiation with hyperbolic secant eory of Ref. 6, the broken and the chain line correspond to results of two odels for diffusion coefficient obtained on the basis of Joslin and Gray study in ters of the first two and three Mori s coefficients, correspondingly 6,5. The full circles represent olecular dynaics data of Ref. and correspond to diffusion values in points well within the fluid region of the LJ phase diagra 6. Presented data bring out clearly that the results of our theory are in excellent agreeent with olecular dynaics data without any deviations. ever, all exained odels, including even Eqs. 6 and 8, begin to reproduce qualitatively olecular dynaics data as the triple point of LJ syste with the reduced paraeters n*.849 when T*.773 is approached 7. The next step of our study is the estiation of statistical eory effects in diffusion processes of LJ liquid. It can be successfully done within the fraework of eory function and diensionless non-markovity paraeter 8 foralis. The last one is the ratio of the relaxation tie of initial correlation function and the relaxation tie of eory function /. Using Eq. 5 for the relaxation tie ã(s), we obtain D k B T. 9 Fro the second equality in Eq. 4 at s and Fro Eqs. 9 and we coe to the final expression (2) D k B T 2. On the basis of the definition 9 it is possible to define the spectru of non-markovity paraeter i i / i see Ref. 8 for ore detail. Then the general for of this paraeter for i,2,3,... can be expressed in ters of Mori s coefficients static correlation functions as i 2 4 2 44 i 4 4 2 if i is odd 34 i 4 2 34 i 4 2 4 2 if i is even. 44 i 2 Obviously, the paraeter i obtained by Eq. 2 estiates the eory effects of the ith level relaxation process, which is described by the eory function M i (t). To investigate the eory effects of VACF itself, it is necessary to study the zero point in spectra of the non- Markovity paraeter calculated by Eq.. Therefore, the paraeter has been calculated for a wide range of densities and teperatures for LJ liquid. The results of our calculations are presented in Fig. 2. Fro this figure one can ε 3 25 2 5 5 =.2 =.3 =.4 =.5 =.6 =.7 2 3 4 5 T * FIG. 2. Variation of the non-markovity paraeter for VACF of LJ liquid with teperature at different densities. Correspondence between the curves and the densities is presented in the inset. The behavior of the paraeter indicates a quantitative change of ratio between VACF tie scale and eory relaxation tie. The crossover fro Markovian ( ) to a quasi-markovian ( ) character of the diffusion process at low teperature and high densities is noteworthy. Referring to the obtained data, the diffusion process is in fact quasi-markovian in a wide teperature region at high densities. 52-3
YULMETYEV, MOKSHIN, AND HÄNGGI see that the non-markovity paraeter always satisfies the condition. This is the evidence of weak statistical eory effects in diffusion processes, which is usually observed in Markovian and quasi-markovian processes. In this case is uch saller than the VACF relaxation tie. The received result helps us to understand the reason for aazing efficiency of Markovian approxiation in the analysis of diffusion in LJ liquids see, for exaple, one of the recent investigations in Ref. 9. It also enables us to estiate the source of unsatisfactory agreeent of Eq. 6 with the olecular dynaics data. The point is that Eq. 6 is based on the assuption of and/or, which is true for non-markovian processes only, while Eqs. 7 and 8 are applied both to Markovian and non-markovian phenoena. Furtherore, in Fig. 2 we can see that the paraeter soothly increases with the increase of the reduced teperature T*k B T/, at the sae tie it decreases with the increase of density. So, non-markovian effects falloff and/or Markovity is enhanced when the teperature of the syste increases while the value of the density n*n 3 is fixed, and it reflects the aplification of randoness during diffusion. However, a saturation is attained at a certain value of teperature, and the values of the paraeter alost cease to change in this case. As can be seen fro Fig. 2, such saturation is observed at lower teperatures for a ore dense ediu. The density n* dependence of the paraeter is shown in Fig. 3 for the four isothers. The behavior of these curves has a nonlinear character. In this figure we can observe aplification of eory effects, that is, aplification of regularity and robustness in the syste which appears with the increase of density. For exaple, for the isother T*.8 the paraeter decreases ore than five ties within density range.2.7. Non-Markovian effects are enhanced and begin to doinate in the vicinity of the triple point of LJ liquid, where a well-known negative correlation in the behavior of VACF is observed. The ratio between VACF tie scale and relaxation tie of eory in this phase region is 2.5 5. This is a quantitative evidence of considerable eory effects incipient diffusion process of LJ syste near the triple point, where the density variation is short ranged and of the order of a couple of olecular diaeters 2. The results of this work can be suarized as follows. i We have presented an approach for calculation of transport coefficients. The approach is based on the tiescale invariance idea, developed within fraework of Zwanzig-Mori s foralis. Our theory allows one to calculate transport properties in ters of frequency oents of corresponding correlation functions without any adjustable paraeters. ii Three different expressions were obtained for the diffusion coefficient of LJ liquid and tested together with other ε =τ /τ 25 2 5 5 PHYSICAL REVIEW E 68, 52 23 T * =.23 T * =.8 T * =2.5 T * =4.5..2.3.4.5.6.7.8 FIG. 3. Variation of the non-markovity paraeter with the reduced density at four different teperatures T*.23,.8, 2.5, and 4.5 reveals a nonlinear aplification of eory effects at atter densifying. The horizontal dash line in the botto of the figure corresponds to the quantitative value a full non-markovian relaxation scenario. The attenuation of eory effects occurs with the decrease of density and the increase of teperature; it is accopanied by the increase of the paraeter and partial disordering of LJ syste. The increase of density results in ordering and aplification of eory effects of diffusion process with sharp reduction of values. theories and the olecular dynaics data over a wide range of densities and teperatures. The coparison showed that the equation obtained fro the condition of approxiate equality of relaxation ties of the first- and second-order eory functions has the best agreeent with the olecular dynaic data. This equation includes only the second and the fourth frequency paraeters, which were calculated with a high degree of precision. iii The non-markovity paraeter calculated for the diffusion phenoena in LJ liquid reveals a Markovian character of theral otions of particles. Finally, the values of this paraeter deonstrate the cross over fro Markovian to quasi-markovian relaxation scenario at low teperatures and high densities and allow one to estiate quantitatively this cross over. The authors acknowledge Professor M. Howard Lee for helpful discussions and Dr. L.O. Svirina for technical assistance. This work was partially supported by the Russian Foundation for Basic Research Grant No. 2-2-646, Russian Ministry of Education Grant No.E-2-3.-538, and RFHR Grant No. 3-6-28a. U. Balucani and M. Zoppi, Dynaics of the Liquid State Clarendon, Oxford, 994. 2 P. Hänggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 25 99. 3 R. Morgado, F.A. Oliveira, G.G. Batrouni, and A. Hansen, Phys. Rev. Lett. 89, 6 22. 4 J.P. Boon and S. Yip, Molecular Hydrodynaics McGraw- Hill, New York, 98. 52-4
DIFFUSION TIME-SCALE INVARIANCE,... PHYSICAL REVIEW E 68, 52 23 5 J.P. Hansen and I.R. McDonald, Theory of Siple Liquids Acadeic Press, London, 986. 6 K. Tankeshwar, K.N. Pathak, and S. Ranganathan, J. Phys. C 2, 5749 987; 2, 367 988; J. Phys.: Condens. Matter, 68 989;, 693 989; 2, 589 99. 7 R.M. Yuletyev, A.V. Mokshin, P. Hänggi, and V.Yu. Shurygin, Phys. Rev. E 64, 57 2; R.M. Yuletyev, A.V. Mokshin, P. Hänggi, and V.Yu. Shurygin, JETP Lett. 76, 47 22; R.M. Yuletyev, A.V. Mokshin, T. Scopigno, and P. Hänggi, J. Phys.: Condens. Matter 5, 2235 23. 8 R. Zwanzig, Phys. Rev. 24, 338 96; H. Mori, Prog. Theor. Phys. 33, 423 965. 9 As an exaple of recent successful researches of the diffusion phenoena executed with the use of technique of projection operators, we can offer the results of M. Fuchs and K. Kroy, e-print cond-at/25255. M.D. Heyes, J. Che. Soc., Faraday Trans. 79, 74 983; Phys. Rev. B 37, 5677 988. M. Howard Lee, J. Phys.: Condens. Matter 8, 3755 996. 2 R.M. Yuletyev, Phys. Lett. A 56A, 387 976. 3 R.I. Galeev, V.Yu. Shurygin, and R.M. Yuletyev, Ukr. Fiz. Zh. 3, 396 99. 4 It is iportant to note that the initial tie correlation function ay be presented as linear cobination of exponential functions a(t) i b i e t/ i at generalization of closure to tiedependence at this relaxation level, M 2 (t)m (t) detailed discussion of this finding will be published soon. 5 C.G. Joslin and C.G. Gray, Mol. Phys. 58, 789 986. 6 As an experiental data we use results of Heyes, which are well known and applied often for different theories testing Refs. 6,7. The VACF in these olecular dynaics siulations is obtained by averaging over all atos in the syste and over several tie origins; the integral over all tie of the VACF yields the diffusion coefficient 5. For a ore detailed description of the olecular dynaics siulations, the reader is referred to Ref.. 7 U. Balucani et al., J. Phys. C 8, 333 985; K. Rah and B.Ch. Eu, Phys. Rev. E 6, 45 999. 8 V.Yu. Shurygin, R.M. Yuletyev, and V.V. Vorobjev, Phys. Lett. A 48, 99 99. 9 A.M. Berezhkovskii and G. Sutann, Phys. Rev. E 65, 62R 22. 2 Fro the physical point of view the crossover fro a Markovian to a quasi-markovian character of the diffusive process at low teperatures and high densities is not a suprise. For a ore quantitave description of this crossover it should be recognized that the range of negative correlation is observed in the vicinity of the triple point of LJ liquid. As known, this correlation region is analogous but ore considerably to such a range for the hard-sphere syste. 52-5