The realization of idea of time-scale invariance for relaxation processes in
liquids has been performed by the memory functions formalism. The best
agreement with experimental data for the dynamic structure factor $S(k,\omega)$
of liquid cesium near melting point in the range of wave vectors ($0.4
\ang^{-1} \leq k \leq 2.55 \ang^{-1}$) is found with the assumption of
concurrence of relaxation scales for memory functions of third and fourth
orders. Spatial dispersion of the four first points in spectrum of statistical
parameter of non-Markovity $ \epsilon_{i}(k,\omega)$ at $i=1,2,3,4$ has allowed
to reveal the non-Markov nature of collective excitations in liquid cesium,
connected with long-range memory effect.Comment: REVTEX +3 ps figure

Hänggi, P.

Mokshin, A. V.

Shurygin, V. Yu.

Yulmetyev, R. M.

English

arXiv

Yulmetyev, Renat M.

Mokshin, Anatolii V.

Hänggi, Peter

OPUS Augsburg

PHYSICAL REVIEW E, VOLUME 64, 057101Time-scale invariance of relaxation processes of density fluctuation in slow neutron scatteringin liquid cesiumR. M. Yulmetyev,1,* A. V. Mokshin,1,† P. Hänggi,2 and V. Yu. Shurygin31Department of Physics, Kazan State Pedagogical University, 420021 Kazan, Mezhlauk 1, Russia2Department of Physics, University of Augsburg, Universitätsstrasse 1, D-86135 Augsburg, Germany3Department of Physics, Yelabuga State Pedagogical Institute, 423630 Elabuga, Kazanskay 89, Russia~Received 2 April 2001; published 18 October 2001!The realization of the idea of time-scale invariance for relaxation processes in liquids has been performed bythe memory functions formalism. The best agreement with experimental data for the dynamic structure factorS(k ,v) of liquid cesium near melting point in the range of wave vectors (0.4 Å21<k<2.55 Å21) is foundwith the assumption of concurrence of relaxation scales for memory functions of third and fourth orders.Spatial dispersion of the first four points in the spectrum of the statistical parameter of non-Markovity e i(k ,v)at i51,2,3,4 has allowed us to reveal the non-Markov nature of collective excitations in liquid cesium,connected with long-range memory effect.DOI: 10.1103/PhysRevE.64.057101 PACS number~s!: 05.40.2a, 05.20.2y, 05.90.1m, 47.10.1gThe dynamic structure factor S(k ,v) of liquid metals~lithium, sodium, rubidium, lead, cesium, aluminum, potas-sium! represents unique information on the collective excita-tions in these systems ~see Ref. @1#!. Most of existing theo-ries for S(k ,v) are based on models of the linearized andgeneralized hydrodynamics @2#. Despite much effort andconsiderable recent progress, understanding of experimentaldata for liquid metals remains an interesting challenge. Hereadequate understanding and explanation of high-frequencycollective excitations does not exist. The reason is that thenature of such excitations in liquid metals is not described byany of hydrodynamic models. On account of specific short-range and oscillatory behavior of ion-ion potential in liquidmetals, the hydrodynamics here is not applicable.In the present work, for an explanation of a specific natureof high-frequency collective excitations in liquid cesium @3#we use one of most fundamental ideas of modern physics—the idea of invariance. In particular, we suggest the idea oftime-scale invariance of relaxation processes in liquids. Ex-perimental data on S(k ,v) give direct information about re-laxation processes of density fluctuation in liquids. Under theassumption, for any experimentally observable relaxationprocess, there corresponds a multilevel hierarchy of intercon-nected relaxation processes. Actually, in experiment the‘‘top’’ of its relaxation ‘‘iceberg’’ is observed only. On acertain relaxation level, an invariance ~equiscaling! of twonearest interconnected relaxation processes can exist. Suchinvariance can be easily taken into account by the memoryfunctions formalism, which most adequately describes non-equilibrium statistical processes in condensed matter physics.Later we consider a normalized time correlation function~TCF! a(t)5^dA*(0)dA(t)&/^udA(0)u2& of particle-densityfluctuation in liquid metal dA(t)5A(t)2^A(t)&, wheredA(t)5N21/2( j51N exp(ikrj), k is wave vector, and r j definesthe position of the j th particle of the liquid. With the help of*Email address: rmy@dtp.ksu.ras.ru†Email address: mav@dtp.ksu.ras.ru1063-651X/2001/64~5!/057101~4!/$20.00 64 0571Zwanzig and Mori’s projection operators method @4,5# it ispossible to construct the infinite chain of connected non-Markov kinetic equations as follows:dM i~ t !dt 52V i112 E0tdtM i11~ t2t !M i~t !, i50,1, . . . ,~1!where M 0(t)5a(t). Here M i(t) is a memory function of ithorder and V i2 are general relaxation parameters with the di-mension of the square of frequency. These parameters areconnected with even frequency moments of spectral densityof TCF a(t) by the following relationship @6# V125I2 , V225I4I2212I2 , V325(I6I22I42)/(I4I22I23).Every relaxation process can be described with the help ofa characteristic time scale usually named as relaxation time.So, for example, the relaxation time for initial TCF a(t) canbe determined as follows ta5Re*0`dta(t). Similarly, relax-ation time on the second relaxation level @at i51 in Eqs. ~1!#for memory function M 1(t) would be expressible as t15Re*0`dtM 1(t), where the symbol Re means the real part. Itis convenient to describe long-range memory effects in theunderlying system with the help of time scales ta and tM1 .For example, the presentation of the dimensionless non-Markovity parameter was introduced earlier in Ref. @6#, e15t0 /t1, as an criterion of describing of non-Markovity forany relaxation processes. As pointed out in Ref. @6# values ofe1 allow to obtain a quantitative and qualitative estimate ofnon-Markovity effects and statistical memory in relaxationprocesses. Parameter e1 allows to divide all relaxation pro-cesses into three important cases. Markovian processes cor-respond to e1→` , while quasi-Markovian processes are ap-propriate in situations with e1@1 and e1.1. The limitingcase e1;1 describes non-Markovian processes. In this casethe time scale of memory processes and correlations ~or jun-ior and senior memory functions! coincide with each other.Thus, the introduced parameter e1 characterizes non-Markovity and memory effects for any relaxation processes.©2001 The American Physical Society01-1BRIEF REPORTS PHYSICAL REVIEW E 64 057101The infinite set of values of parameters e i , where i51,2,3, . . . , was also entered on the basis of the simpleformula e i5t i21 /t i , where t i is a relaxation time of thememory function of ith order @7#. The whole set of values ofnon-Markovity parameter e i forms the statistical spectrum,which is connected with collective and statistical propertiesof the system and allows also to estimate in detail the non-Markovian properties of the underlying relaxation process.In Ref. @8# the conception of non-Markovity parameter forthe frequency-dependent case was generalized . This param-eter is determined by the following expression e i(v)5$m i21(v)/m i(v)%1/2 at the fixed wave vector k. Here i51,2,3 . . . and m i(v) is the power frequency spectrum ofthe ith relaxation level, which is defined in the followingway m j(v)5@Re*0`dte ivtM j(t)2.The appropriate values of e i(v) and e i will correspond toevery j th equation of the chain ~1!. As the chain of the Eq.~1! is infinite, the sets of e i and e i(v) are also infinite. Nowlet us use the approximation M i11(t)'M i(t). It means theapproximate equality of relaxation time scales of memoryfunctions of ith and (i11)th orders, i.e., t i11't i . Then thenon-Markovity parameter for the whole frequency range isapproximately equal to unity, e i11(v);1 ~and also e i11;1), and chain ~1! becomes closed. Applying Laplace trans-form to the ith equation of the chain ~1! we get M̃ i(s)5@2s1(s214V i112 )1/2#/2V i112 .The results of experiment for liquid cesium were pub-lished in Ref. @3#. The dynamic structure factor S(k ,v) hasbeen measured by inelastic slow neutron scattering ~INS!near the melting point at T5380 K. The wave vector kchanges in the range from 0.2 Å21 up to 2.55 Å21. Thenumerical results for S(k) were obtained from experimentaldata @3# too.It is well known that S(k ,v) is connected with TCF ofdensity fluctuation in the following way: S(k ,v)5@S(k)/p#lime→10Re@ ã(k ,iv1e)# , where Laplace trans-form ã(k ,s)5*0`dte2sta(k ,t) was found by us as follows.We have taken advantage of the correlation approximationfor the fourth order memory function @9#M 4~ t !'M 3~ t !. ~2!From the physical point of view it means the concurrenceof time scales of TCF’s M 3(t) and M 4(t). In this case thechain of the connected kinetic equations ~1! becomes a sys-tem consisting of four equations. Using Laplace transform,we receive the following equation for the dynamic structurefactor S(k ,v):S~k ,v !5S~k !2p V12V22V32~4V422v2!1/2$V14V341v23~22V12V341V14V422V14V3212V12V22V422V12V22V321V24V42!1v4~V3422V12V4212V12V3222V22V421V22V32!1v6~V422V32!%21. ~3!05710Relaxation frequency parameters V12 and V22 are definedas follows @11#:V125KBTk2@mS~k !#21; v125V12 ; V225v222v12 ;v2253v12S~k !1v l2 ;v l25N/mVE drg~r !@12cos~kr !#,z2u~r !. ~4!In Eqs. ~4! the following designations are introduced:KBT is thermal energy, g(r) is a radial distribution functionof particles, u(r) is a pair interparticle potential of interac-tion, and axis z is chosen in the direction of a wave vector k.Calculating the frequency parameter V22 we use the well-known approximation @10# V2253V12S(k)1vE2 $12@3 sin(kR0)/kR0#2@6 cos(kR0)/(kR0)2#1@6 sin(kR0)/(kR0)3#%2V12, where vE is the Einstein frequency. In our case vE hasthe following value vE54.1231012 s21 @3#. This frequencyparameters can also be calculated through I i (i52,4).Theoretical formulas for calculation of relaxation fre-quency parameters V32 and V42 are also known @11#. How-ever the final result of these calculations contains some er-rors. For example, in paper @11# it is shown that results ofcalculated frequency moments in papers @12–14#, vary from10% to 50%, and the distinction reaches up to 30 times forseparate values of the wave vector. Therefore it is more con-venient to obtain these parameters by comparing results oftheory and experiment. Relaxation frequency parameter V42is easy to find from comparison of developed theory withexperiment on zero frequency. Namely, it follows V425p2V14V34@S(k ,0)#2„@S(k)#2V24…21 from Eq. ~3! at v→0.The spectrum of S(k ,v), Eq. ~3!, allows investigations ofcollective excitations in liquid cesium to be made in detail.Our analysis show that the position of collective excitationpeak vc(k) in spectrum S(k ,v) depends on the combinationof frequencies V12 , V22 , V32, and V42. Our numerical calcu-lations demonstrate that vc(k) value is most sensitive to re-laxation frequency V22. On the other hand, collective effectsturn out to be connected with non-Markov processes in liq-uids.In Fig. 1 the comparison of our theory ~solid line! andexperimental data ~circles! @3# for S(k ,v) for liquid cesiumat T5380 K is shown. From Fig. 1 it is evident that ourtheory absolutely agrees with the experiment in the wholerange of values of wave vector k. It is possible to see thegood qualitative coincidence between the experiment and ourtheory from Fig. 2. Here dispersion of the frequency of col-lective excitations vc(k), obtained from the position of lat-eral peaks ~the points present experimental data; the circlesare our theoretical values!, is presented. We calculated thefirst four points in a statistical spectrum of the frequency-dependent non-Markovity parameter e i(k ,v) for the wholerange of values of the wave vector k. The results of suchcalculations are shown in Fig. 3. In our opinion, the param-eter e1(k ,v) represents special interest. From Fig. 3 it isapparent that this parameter has maxima on frequencies thatcoincide with collective excitations in S(k ,v). The spectrum1-2BRIEF REPORTS PHYSICAL REVIEW E 64 057101FIG. 1. Theoretical ~solid line! and experimental @3# (s s s s) values of dynamical structure factor S(k ,v) for liquid cesium atT5380 K at various wave vectors k5uku.of S(k ,v) falls down smoothly at large values of k, and thefrequency-dependent non-Markovity parameter also fallsdown smoothly to zero. Moreover, a ‘‘burst’’ of valuese1(k ,v) is observed on common frequencies with S(k ,v).As indicated by Fig. 3, the non-Markovity parameter e i(v)for this range of frequencies v must satisfy the conditione i(k ,v).1. The non-Markovity parameter e1(k ,v) has thedistinct expressed maximum on frequencies, relevant to col-lective excitations. Proceeding from the received results, it isreasonable to speak about the non-Markov nature of collec-tive excitations in liquid cesium. This important conclusionmust be taken into account while constructing the appropri-ate theories. The behavior of e i(k ,v) for levels i52,3,4 isvery interesting here. Further calculation shows that the re-lationship e3(k ,v)'1 takes place for the high k-value range1.15 Å21<k<2.55 Å21 and for all frequencies v . Becauseof this, the more simple approximation M 3(t)'M 2(t) canbe applied for this case. Moreover, there are cases in whiche2(k ,v)'1 for all v . It is possible to assume that goodagreement with the experiment gives correlation approxima-tion for the memory function of the second order M 2(t)'M 1(t) in these cases.Thus, the frequency-dependent non-Markovity parametere i(k ,v) introduced in Ref. @8# allows us to reveal two im-portant features. First, its behavior allows us to judge theproperties of non-Markovity in the whole range of the fre-quency spectrum and at various values of the wave vector k.Then, with its help it is possible to judge the applicability ofcorrelation approximation of the junior order, for which lessdifficult calculations are required.The results of this Brief Report can be summarized asfollows.~i! S(k ,v) for liquid cesium is found on the basis of thehypothesis of time-scale invariance of relaxation processes in05710liquids. We have assumed that relaxation times of memoryfunctions M 4(t) and M 3(t) are approximately equal in thiscase and relaxation time scales are invariant. Therefore, wehave used correlation approximation for the memory func-tion of fourth order, M 4(t), for closing the chain of the con-nected kinetic equations ~1!. As a result we have receivedgood agreement with the experiment for all values of wavevector k (0.4 Å21<k<2.55 Å21).~ii! For the estimation of the received results thefrequency-dependent parameter of non-Markovity e i(v) (i51, 2, 3, and 4) was calculated for all the values of thewave vector k. It turned out that the first point in the spec-trum of non-Markovity parameter e1(v) has frequency de-pendence similar to the behavior of the dynamic structurefactor for liquid cesium. The maximum of e1(k ,v) andS(k ,v) for values of the wave vector 0.4 Å21<k<2.55 Å21 appear at the same frequencies.~iii! On the basis of our calculations we have establishedFIG. 2. Comparison of the values of frequency of collectiveexcitations in liquid cesium vc5vc(k) at T5380 K with calcu-lated values from experimental data @3# (• • • •) and our theory(s s s s).1-3BRIEF REPORTS PHYSICAL REVIEW E 64 057101FIG. 3. Frequency dependence of the first four points in statistical spectrum of non-Markovity parameter e i5e i(v); (• • • •)correspond to i51, (1 1 1 1) relate to i52, (s s s s) present i53, solid line reflects values at i54. In a wave vector range0.4 Å21<k<1.1 Å21 collective excitations exist.the non-Markov nature of collective excitations at the micro-scopic level in liquid cesium. It should be pointed out thatthe developed approach is true, especially for the solution ofnonperturbative problems. Nevertheless, we believe that asystematic and general theoretic approach, such as we haveexpounded, should play a useful role in analyzing and clas-sifying experimental data simulations and more elaboratemodels. It is especially true for the construction of theoriesdescribing high-frequency and short-range relaxation pro-cesses. The calculated frequency parameter e1(v) for everyconcrete value of wave vector k allows to describe the effectsof amplification or attenuation of non-Markovity within the05710whole frequency interval. From the preceding it is clear thatthe idea of time-scale invariance has large prospects for thedescription of high- and short-wave stochastic and relaxationprocesses in liquids.The authors acknowledge Dr. T. Scopigno, Professor A.G. Novikov, and Dr. R. K. Sharma for useful discussion ofINS and inelastic x-ray scattering data for liquid metals andDr. L. O. Svirina for technical assistance. This work waspartially supported by the Russian Humanitarian ScienceFund ~Grant No. 00-06-00005a! and the NIOKR RT Foun-dation @Grant No. 14-78/2000~f!#.@1# T. Scopigno, T. Balucani, G. Ruocco, and F. Sette, Phys. Rev.E 63, 011210 ~2001!.@2# P. A. Egelstaff, Adv. Phys. 11, 203 ~1962!.@3# T. Bodensteiner, Chr. Morkel, W. Gläser, and B. Dorner, Phys.Rev. A 45, 5709 ~1992!.@4# H. Mori, Prog. Theor. Phys. 33, 423 ~1965!; 34, 765 ~1965!.@5# R. Zwanzig, Phys. Rev. 124, 1338 ~1961!.@6# V. Yu. Shurygin, R. M. Yulmetyev, and V. V. Vorobjev, Phys.Lett. A 148, 199 ~1990!.@7# V. Yu. Shurygin and R. M. Yulmetyev, Zh. Eksp. Teor. Fiz.102, 852 ~1992! @Sov. Phys. JETP 75, 466 ~1992!#.@8# R. Yulmetyev, P. Hänggi, and F. Gafarov, Phys. Rev. E 62,6178 ~2000!.@9# The introduction of approximation ~2! is related to a specificfeature of liquid metals. In particular, it is connected with theshort range of ion-ion potential. It can be shown that timecorrelation of Fourier components of a local momentum den-sity, local energy density, and local current density occur inmemory functions M 1(t), M 2(t), and M 3(t), respectively.Space and momentum functions arise in appropriate dynamicvariables, depending on position and momentum of moleculesof liquids. The reapproachment of time-relaxation scale on thethird and fourth relaxation levels occurs through the entangle-ment of these functions and the short range of ion-ion interac-tions.@10# J. Hubbard and J. L. Beeby, J. Phys. C 2, 556 ~1969!.@11# V. Yu. Shurygin and R. M. Yulmetyev, Metallofizika ~Kiev!12, 55 ~1990!.@12# E. Michler, H. Hahn, and P. Schofield, J. Phys. F: Met. Phys.7, 869 ~1977!.@13# G. S. Dubey, D. K. Chaturvedi, and R. Bansal, J. Phys. C 12,1997 ~1979!.@14# I. Ebbsjo, T. Kinell, and I. Waller, J. Phys. C 13, 1865 ~1980!.1-4

Time-scale invariance of relaxation processes of density fluctuation in slow neutron scattering in liquid cesium

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Time-scale invariance of relaxation processes of density fluctuation in
  slow neutron scattering in liquid cesium

Time-scale invariance of relaxation processes of density fluctuation in slow neutron scattering in liquid cesium

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