A random walk model in a one dimensional disordered medium with an
oscillatory input current is presented as a generic model of boundary
perturbation methods to investigate properties of a transport process in a
disordered medium. It is rigorously shown that an admittance which is equal to
the Fourier-Laplace transform of the first-passage time distribution is
non-self-averaging when the disorder is strong. The low frequency behavior of
the disorder-averaged admittance, $ -1 \sim \omega^{\mu}$ where $\mu <
1$, does not coincide with the low frequency behavior of the admittance for any
sample, $\chi - 1 \sim \omega$. It implies that the Cole-Cole plot of $$
appears at a different position from the Cole-Cole plots of $\chi$ of any
sample. These results are confirmed by Monte-Carlo simulations.Comment: 7 pages, 2 figures, published in Phys. Rev. 

G. Franco

G. Pfister

G.H. Weiss

J. Klafter

J.M. López

J.P. Bouchaud

J.W. Haus

K.P.N. Murthy

K.W. Kehr

Klaus W. Kehr

Mitsuhiro Kawasaki

P.E. de Jongh

S. Alexander

S.H. Noskowicz

T. Odagaki

Takashi Odagaki

Y. Yasuda

English

arXiv

A random walk model in a one-dimensional disordered medium with an oscillatory input current is presented as a generic model of boundary perturbation methods to investigate properties of a transport process in a disordered medium. It is rigorously shown that an admittance which is equal to the Fourier-Laplace transform of the first-passage time distribution is non-self-averaging when the disorder is strong. The low-frequency behavior of the disorder-averaged admittance, -1 similar to omega(mu), where mu appears at a different position from the Cole-Cole plots of chi of any sample. These results are confirmed by Monte Carlo simulations

Kawasaki, M.

Odagaki, T.

Kehr, K. W.

Juelich Shared Electronic Resources

PHYSICAL REVIEW B 1 MARCH 2000-IVOLUME 61, NUMBER 9Absence of self-averaging in the complex admittance for transport through random mediaMitsuhiro Kawasaki and Takashi OdagakiDepartment of Physics, Kyushu University, Fukuoka 812-8581, JapanKlaus W. KehrInstitut fu¨r Festko¨rperforschung, Forschungszentrum Ju¨lich GmbH, D-52425 Ju¨lich, Germany~Received 3 June 1999; revised manuscript received 15 November 1999!A random walk model in a one-dimensional disordered medium with an oscillatory input current is pre-sented as a generic model of boundary perturbation methods to investigate properties of a transport process ina disordered medium. It is rigorously shown that an admittance which is equal to the Fourier-Laplace transformof the first-passage time distribution is non-self-averaging when the disorder is strong. The low-frequencybehavior of the disorder-averaged admittance, ^x&21;vm, where m,1, does not coincide with the low-frequency behavior of the admittance for any sample, x21;v . It implies that the Cole-Cole plot of ^x&appears at a different position from the Cole-Cole plots of x of any sample. These results are confirmed byMonte Carlo simulations.Response of a system to an external field is a standardinformation to be utilized in the study of condensed matter.Recently, the frequency response method1 and the intensitymodulated photocurrent spectroscopy2 have been introduced,where an oscillatory perturbation is applied at one end of asystem and a response from the other end of the system ismeasured. Thus, these experimental techniques can be con-sidered to belong to a generic method which can be calledthe boundary perturbation method ~BPM!. In the presence ofa periodically forced boundary condition on one end of asystem, the output from the other end of the system is inproportion to the perturbation in the linear regime and theproportionality constant is called the admittance. The fre-quency dependence of the admittance contains various infor-mation of the dynamics of the system. The BPM is expectedto provide useful information on transport properties of dis-ordered media.The analysis of transport of particles in a random mediumattracts great interest since the transport mechanism is basicfor the understanding of many physical phenomena, fromelectrical conductivity to thermal properties.3 Usually it isassumed that a sample used in experiments is sufficientlylarge that it can be considered to be composed of a largenumber of subsystems. If each of the subsystems is itselfmacroscopic, the boundary effect is negligible and each sub-system may be considered to be a realization of the systemwith a particular choice for the disorder. Thus, a measure-ment of any observable in such a system corresponds to anaverage over all the subsystems, i.e., an average over theensemble of all realizations of the disorder. Systems forwhich this assumption is valid are said to be self-averaging~SA!. Since the assumption implies no sample dependence, itguarantees reproducibility of experimental results in anysample.However, it is known that the disorder-averaged quantityfollows a dynamics that is different from the dynamics of thequantity of each sample. If the dynamics of particles in adisordered sample is assumed to be stochastic and Markov-ian, i.e., to follow a master equation, the disorder-averagedPRB 610163-1829/2000/61~9!/5839~4!/$15.00master equation has a memory effect.4 Thus, the dynamics ofan averaged quantity may be non-Markovian even thoughthe process for each sample is Markovian. There are severalreports on absence of SA in transport phenomena in disor-dered media: for mean square displacement,5 for mean first-passage time of the Sinai model.6In this report, we study a random walk model in a one-dimensional disordered medium with an oscillatory inputcurrent as a generic model of the BPM. It is rigorouslyshown that the admittance, which is equal to the Fourier-Laplace transform of the first-passage time distribution~FPTD!, is non-self-averaging ~non-SA! when the disorder isstrong.We consider a general random walk in a one-dimensionallattice segment of N11 sites. The lattice sites are denoted byintegers, n50,1, . . . ,N . The dynamics of a random walkingparticle can be described by a master equation for the prob-ability Pn(t) that the particle is at site n at time t>0. Themaster equation for the model is written asP˙ n~ t !5wn n21Pn21~ t !2~wn21 n1wn11 n!Pn~ t !1wn n11Pn11~ t !, ~1!where wm n denotes the random jump rate of a particle fromsite n to site m. The probability distribution of jump ratescharacterizes the random medium. We introduce a perturba-tion at the left end, so that the equation for P0(t) is given byP˙ 0~ t !52w1 0P0~ t !1w0 1P1~ t !1J~ t !. ~2!J(t) is the oscillatory current perturbation at site 0. The rightend of the system is assumed to be a sink and the equationfor PN21 is given byP˙ N21~ t !5wN21 N22PN22~ t !2~wN22 N211wN N21!PN21~ t !. ~3!Since the current perturbation J(t) oscillates in time arounda positive average with the amplitude DJ , the response of the5839 ©2000 The American Physical Society5840 PRB 61BRIEF REPORTSoutput current wN N21PN21(t) oscillates around its station-ary state with the amplitude wN N21DPN21 at the same fre-quency with a phase-shift. The admittance is defined as theratio of these two amplitudesx~v![wN N21DPN21DJ . ~4!We first note that the admittance can be related to the firstpassage time distribution FN 0(t) ~FPTD!, which is the prob-ability density that the particle which starts at site 0 at time 0arrives for the first time at site N at time t. Since site N is asink, the FPTD is given by the output current from site Nwhen there is no input current and the particle starts from site0 at time 0. The Fourier-Laplace transform of a system ofmaster equations for the case is the same as a system ofequations for DPn /DJ derived from the master equationsEqs. ~1!–~3!. It implies that F˜ N 0(s5iv) for the case wherethere is no input current and the particle starts from site 0 attime 0 is identical to wN N21DPN21 /DJ . Thus, the admit-tance is equal to the Fourier-Laplace transform of the outputcurrent in the problem described above, i.e., the FPTD.We make the low-frequency expansion of the admittanceto see the behavior near the static limit. Since the admittanceis given by the Fourier-Laplace transform of the FPTD, theadmittance at zero frequency equals one due to normaliza-tion of the FPTD. The mean first-passage time ~MFPT! t¯which is the first moment of the FPTD is given by t¯5i dx(v50)/dv . Thus, the low-frequency expansion of theadmittance is given byx~v!512i t¯ v1O~v2!. ~5!Since the MFPT is given by7t¯5 (k50N21 1wk21 k1 (k50N22 1wk11 k(i5k11N21)j5k11iw j21 jw j11 j, ~6!the MFPT t¯ is larger than (k50N211/wk21 k . By taking theaverage of Eq. ~6! over the distribution for the jump rates,one sees that the average of the MFPT ^ t¯& is larger than thefirst inverse moment of a random jump rate ^1/w&. Thus,when the first inverse moment of a jump rate diverges, whichis called strong disorder, the disorder-averaged MFPT di-verges. It indicates that the disorder-averaged admittance isnon-analytic. Since the admittance is a function of iv , wecan write as^x~v!&21;~ iv!m, ~7!where 0,m,1. Since this behavior of the disorder-averaged admittance is completely different from the behav-ior of the admittance of each sample given by Eq. ~5!, theadmittance is non-SA when the disorder is strong.On the other hand, when the first inverse moment of ajump rate exists, it is easily shown from Eq. ~6!, that theMFPT is finite provided that there is no correlation betweenwk11 k and wi11 i , where i>k11. The condition of inde-pendence of the neighboring jump rates holds for the sitedisordered model called random trap model and the bonddisordered model called a random barrier model. Thus, whenthe first inverse moment of a jump rate exists, the low-frequency behavior of the disorder-averaged admittance isthe same as the one of each sample given by x21;iv .Furthermore, if the second inverse moment of a jump rateexists, the prefactor of v does not depend on a realization ofthe chain when the chain is sufficiently long.In order to test the foregoing observation, we performed acomputer simulation in one-dimensional chain of 100 sites.Figure 1 shows the low-frequency behavior of the imaginarypart of the admittance obtained by the Monte-Carlo simula-tion where the relation wi11 i5wi i11 is assumed and theprobability distribution of jump rates is given byP~w !5H awa21 if 0,w,1,0 otherwise, ~8!where 0,a,1. The power-law probability distribution cor-responds to the waiting-time distribution C(t);t2a21 usedin the study of dispersive transport where a is proportionalto temperature,8 and it has been shown that the power-lawdistribution is common for activation processes with randomactivation energy.9In the literature of experiments of the BPM,1,2 the Cole-Cole plot of the admittance is employed for the analysis. It isa parametric plot of the imaginary part of the admittanceagainst the real part. The shape of the Cole-Cole plot is in-dependent of the length of the chain for a sufficiently longchain due to time-scale invariance of the Cole-Cole plot.Thus, the Cole-Cole plot shows the size-independent charac-teristics of a medium.In order to see the non-SA property of the admittance inthe Cole-Cole plot, we first note that the inverse of the ad-mittance satisfies the following recursive relation:xn21~v!5S ivwn11 n111wn21 nwn11 nDxn2121 ~v!2wn21 nwn11 nxn2221 ~v!, ~9!FIG. 1. Low-frequency behavior of the imaginary part of theadmittance when the exponent of the probability distribution ofjump rates is 1/2 and the length of a chain is 100. Each solid linerepresents the admittance for each of 50 samples. The dashed linerepresents the admittance averaged over 5000 samples. Though theadmittance of each sample is proportional to v , the averaged ad-mittance is proportional to Av .PRB 61 5841BRIEF REPORTSwhere xn21(v) is the inverse of the admittance for a chain oflength n.10 The recursive relation proves inductively that theinverse of the admittance obeys the following four inequali-ties in the frequency region @0,vn# , where vn is the positivesmallest zero of xn218:xn219.0,xn21218.0,~10!xn219.xn21219,xn218,xn21218.Since the second inequality shows vn<vn21, the fourth in-equality implies xn218,xn21218,,x121851 in the region@0,vn# . It is straightforward to show that x218,1 impliesthat the Cole-Cole plot of the admittance of each sample canexist only outside a semicircle whose center is at (1/2,0) andradius is 1/2, which we call the Debye semicircle, when v,vn .On the other hand, we can show in the following way thatthe Cole-Cole plot of the averaged admittance is located in-side the Debye semicircle when the disorder is strong. Weconsider the angle u between the tangent of the Cole-Coleplot at (1,0) and the horizontal axis. For Eq. ~7!, u5mp/2,which is less than p/2 since m,1, is obtained. However, theangle u for the Debye semicircle is equal to p/2. It impliesthat the Cole-Cole plot of the averaged admittance appearsinside the Debye semicircle when the disorder is strong.FIG. 2. The Cole-Cole plot of the admittance when the exponentof the probability distribution of jump rates is 1/2 and N5100.Each solid line represents the admittance for each of 30 samples.The dashed line represents the admittance averaged over 5000samples. The circles represents the admittance of each sample atv5531027. Although the circles scatter outside the Debye semi-circle, the diamond, i.e., the averaged admittance is inside of theDebye semicircle.Since the Cole-Cole plot of the admittance of any sampleappears outside the Debye semicircle, the Cole-Cole plotclearly shows non-SA property of the admittance. Figure 2shows the Cole-Cole plot of x and ^x& obtained by MonteCarlo simulation when a51/2 in Eq. ~8!. Since the admit-tance of each sample at the same frequency scatters outsidethe Debye semicircle, the admittance averaged at the samefrequency is inside the Debye semicircle. In the case that thefirst inverse moment of a jump rate exists, the angle u of theCole-Cole plot of the averaged admittance is p/2, which isthe same as the Cole-Cole plot of each sample, because ofthe regular behavior as ^x&21;iv .In conclusion, we have presented a generic stochastic dy-namical model in one-dimensional medium of the BPM andhave shown that the admittance has important informationabout transport in disordered media, e.g., the FPTD. Thus theBPM will be a powerful technique to investigate the dynami-cal process in random media. We have also shown that theadmittance is non-SA when the disorder is strong. Since theabsence of SA is due to nonanalyticity introduced by anaverage over infinite number of samples, the behavior of theaveraged admittance over infinite number of samples givenby a theoretical analysis does not coincide with that of theadmittance observed in experiments.Non-SA behavior, i.e., strong sample to sample depen-dence may be observed in other experiments. For example, itis known that anomalous system-size dependence of the mo-bility, m;N121/a, is observed in the time-of-flight experi-ment of amorphous semiconductors.8 The anomaloussystem-size dependence is due to the fact that the smallestvalue of random jump rates depends on the length of achain.11 Since the smallest value dominates the mobility, themobility is also a random variable. Thus, the anomaloussystem-size dependence implies absence of SA.It is important to note that the system treated in thepresent report is purely one dimensional and it is still anopen problem whether the admittance for higher-dimensionalmedia are SA. Since one can consider a diffusing particle’spath a one-dimensional chain and a higher-dimensional me-dium can be regarded as a bundle of many realizations of aparticle’s path, the total output from the medium is given bythe sum of one-dimensional currents. Thus, the admittancefor such a system may be SA, since the number of paths isvery large when the medium is macroscopic. Thus, in orderto see the non-SA behavior described here one needs to workin one-dimensional system or the case where the transla-tional invariance is violated in only one direction since thereis still the possibility that the SA assumption is valid for two-or three-dimensional systems.This work was supported in part by Grant-in-Aid from theMinistry of Education, Science, Sports and Culture. One ofus ~M.K.! is thankful to the Institut fu¨r Festko¨rperforschung,Forschungszentrum Ju¨lich for its hospitality, where part ofthe present work was done.5842 PRB 61BRIEF REPORTS1 Y. Yasuda, K. Iwai, and K. Takakura, J. Phys. Chem. 99, 17 852~1995!; Y. Yasuda, Heterog. Chem. Rev. 1, 103 ~1994!.2 P.E. de Jongh and D. Vanmaekelbergh, Phys. Rev. Lett. 77, 3427~1996!; G. Franco, J. Gehring, L.M. Peter, E.A. Ponomarev, andI. Uhlendorf, J. Phys. Chem. B 103, 692 ~1999!.3 S. Alexander, J. Bernasconi, W.R. Schneider, and R. Orbach,Rev. Mod. Phys. 53, 175 ~1981!; J.W. Haus and K.W. Kehr,Phys. Rep. 150, 263 ~1987!; G.H. Weiss and R.J. Rubin, Adv.Chem. Phys. 52, 363 ~1983!; J.P. Bouchaud and A. Georges,Phys. Rep. 195, 127 ~1990!.4 J. Klafter and R. Silbey, Phys. Rev. Lett. 44, 55 ~1980!.5 J.M. Lo´pez, M.A. Rodrı´guez, and L. Pesquera, Phys. Rev. E 51,R1637 ~1995!.6 S.H. Noskowicz and I. Goldhirsch, Phys. Rev. Lett. 61, 500~1988!.7 K.P.N. Murthy and K.W. Kehr, Phys. Rev. A 40, 2082 ~1989!.8 G. Pfister and H. Scher, Adv. Phys. 27, 747 ~1978!.9 T. Odagaki, Phys. Rev. Lett. 75, 3701 ~1995!.10 The recursive Eq. ~9! is derived from the following two equationsfor the FPTD obtained by its definition: F˜ i11 i(s)5c˜ i11 i(s)1c˜ i21 i(s)F˜ i11 i21(s) and F˜ i11 i21(s)5F˜ i11 i(s)F˜ i i21(s),where F˜ i j(s) denotes the Laplace transform of the FTPD fromsite j to site i and c˜ i j(s) denotes the Laplace transform of thewaiting time distribution for the jump from site j to site i.11 K.W. Kehr, K.P.N. Murthy, and H. Ambye, Physica A 253, 9~1998!.

Absence of self-averaging in the complex admittance for transport through random media

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Absence of self-averaging in the complex admittance for transport
  through random media

Absence of self-averaging in the complex admittance for transport through random media

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