Random walks in disordered lattice, CTRW, memory and dipole transport Текст научной статьи по специальности «Физика»

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aänetic Resonance in Solids

Electronic Journal

Volume 19, Issue 2 Paper No 17201, 1-11 pages

2017

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Short cite this: Magn. Reson. Solids 19, 17201 (2017)

Random walks in disordered lattice, CTRW, memory and dipole transport

F.S. Dzheparov

National Research Center "Kurchatov Institute", ITEP, Moscow 117258, Russia National Research Nuclear University "MEPhl", Moscow 115409, Russia E-mail: dzheparov@itep.ru

(Received October 30, 2017; revised November 11, 2017; accepted November 14, 2017)

Application of CTRW (continuous time random walks) to dipole hopping transport is reviewed. Conditions of applicability of basic kinetic equations to spin systems are indicated. Correct versions of derivation of the CTRW-equations are presented. Existence of different forms of memory kernels is demonstrated. Correction of Scher-Lax memory kernel within geometrical memory approach is fulfilled in accordance with leading terms of concentration expansion. Approximate solution for autocorrelation function is considered. Modern state of numerical simulation and experimental measurements of autocorrelation function in nuclear polarization delocalization are described. It is shown, that application of the CTRW was more successful in description of dipole transport than for hopping conductivity.

PACS: 45.10.Hj, 76.60.-k, 82.56.-b, 76.30.-v, 05.40.-a

Keywords: random walks, random media, disordered lattice, survival probability, dipole transport, hopping transport, master equation, memory kernel, projection operator, spin diffusion

1. Introduction

Random walks in disordered lattice are described by the equations

Q

QtP'0 = -ZJ (wj'Pi0 - wvPj0^ Pi0(t = 0) = 8i0, (1)

where pi0(t) is the "probability" to find an "excitation" at position ri, if it started at r0 = 0 at t = 0 and 8ij is the Kronecker's symbol. Here wJi is the transition rate for transfer from ri to rJ. The problem of disordered sites will be considered below, when positions rJ are statically and randomly distributed on sites of regular lattice, wJi depends on rJJ = ri - rJ, and observables are directly related to solution pi0(t), averaged over all possible positions {rj.}. Occupation number representation allows to rewrite Eqs.(1) as [1]

Q

^0 =-Zz nxnz (zxPx0 - WxzPz0 ) = -(^(0)p5)x0 , (2)

or

rs

Q-p. =-Zz(nzWzxPx0 -nxWxP) = -((1)P)x0, (3)

where the propagator p (t) gives the probability to find the excitation at lattice site x, when initially it was at the site 0, and wxz = w. (ri = x, r. = z). Here nr is occupation number of the site r by a donor (nr = 1(0) if the site r is (not) occupied by the donor), while the donor is an impurity, which can carry the excitation. Eqs. (3) are equivalent to Eqs. (2) because nx p = Px0, while equivalence of Eqs. (1) and (2) is evident, if we omit in (2) all empty sites for which nx = 0 and, consequently, Px0 = 0. The problem consists in calculation of the observable propagator

Px0 (t) = (P> (0). (4)

Random walks in disordered lattice, CTRW, memory and dipole transport Here <• • •) is averaging over all possible donors positions in infinite sample.

Solutions to Eqs. (2) and (3) are of the form Px0 = (exp (-At)) n0 / c, where the operator A = A(0)

x0

or A = A(1) and c = (nx) is the probability of occupation (concentration). The initial condition

Po (t = 0) = nAo / c (5)

ensures, that the excitation can be placed at site 0 if it is occupied by a donor, and produces evident normalization

I x Pxo (t) = 1.

Occupations of different sites are assumed as independent and having no dependence on the x (with small c ■ 1 as a rule):

<nr) = c, <nrnx) = cArx + c2(1 - A), <n;r=]nr ) = nm=1<nr ) = cm.

J J J J

All rj are different in the last relation. Coincidence in indexes can be treated using the identity nr2 = nr.

Most important experimental realizations of the process (1) are hopping conductivity, for which the CTRW theory of Scher and Lax [2] was developed, Förster electronic energy transfer [3] and spin-polarization transfer [1]. The simplest transition rates are of the form

Wz*x = w0exp(-|z - x| / rl wxx = 0 (6)

for conductivity problem, and

w„x = wo r6/(z - x )6, wxx = 0 (7)

for dipole transitions in electronic energy and spin-polarization transfers. Here w0 corresponds to the transition rate at distance r0 between donors. More advanced representations for transition rates can be found in special literature, see, for example, [4] for spin polarization transfer. It should be noted that in the problem of spin-polarization transfers the propagator Px0 (t) represents polarization (instead of probability) of a spin at the site x, when initially polarization was localized at the site 0.

Polarization transport in disordered media is very important in spin kinetics and magnetic resonance. Its manifestations are related, first of all, with spin-lattice and spin-spin relaxation [5-7] as well as with establishing and retaining of (quasi)equilibrium [8]. The discovery of the model system 8Li-6Li is of great importance, because it gives a unique possibility for direct measurement of the survival probability P00 (t) (see [4] and reference therein) contrary to other systems, where the process (1) can not be directly observable.

The conditions of applicability of Eqs. (1-3) to description of spin polarization transport in the subsystem of impurity spins require that the speed of the process should be small a) relative to the speed of phase relaxation of impurity spins and b) in comparison with the speed of fluctuations of local fields on these spins [1, 4]. The model system 8Li-6Li in the LiF single crystal represents excellent example for experimental study of the problem. It consists of one beta-active nucleus 8Li with high initial polarization and of 6Li nuclei with negligible thermal initial polarization. The nuclei 6Li have small controllable concentration, and the difference of gyromagnetic ratios of 6Li and 8Li is rather small: (g8 - g6) / g6 = 0.0057. The matrix LiF consists of nuclei 7Li and 19F with higher gyromagnetic ratios that ensures with high quality the conditions of applicability, mentioned above, in comfortable external magnetic field in the range 200G < H0 < 3kG [4]. The evolution of the polarization P00(t) of the nuclei 8Li is directly observable via measurement of the beta-decay asymmetry (ß-NMR).

We use here the simplest transition rates (7) for the problem of spin polarization transport in order to concentrate our attention on specifics of random walks in disordered media. Quantitative comparison with experimental results requires to take into account that transition rate wzx depends as well on the orientation of x - z relative to external static magnetic field, on correlation of local fields on donor spins and on difference in spin values of 8Li and 6Li [4].

The problems of calculation of the propagator (4) or memory kernels Nxz (t) in corresponding master equation

rs

-Px0 (t) = -j/t 'X z [Nzx (t ')Px0 (t - t') - Nxz (t ')Pz0 (t - t')], Px0 (t = 0) = ¿x„ (8)

belong to the most complex problems of modern statistical mechanics and they have not adequate analytical solution up to now. Nonseparable version of CTRW theory, developed by Scher and Lax in two articles [2], produced an integral representation for approximate solution of the problem and extended previously developed separable CTRW of Montroll and Weiss [9] (which operated with kernels of the form: Nzx(t) = Xzx • Yx(t)).

CTRW is the abbreviation of continuous time random walks. The meaning of the expression consists in following. If the excitation is placed at site x , then its escape, according to Eqs. (1)-(3), is

described by the law Qx (t) = exp (-t / rx), with one characteristic hopping time rx = (Xz nzwzx) , while the Eq. (8) corresponds to the escape law with dQx (t )/dt =-j0 dt 'X z Nzx (t ')Qx (t -1') and its result can be written as Qx(t) = jdrexp(-t/ r)Wx(r) with continuous distribution of times Wx(r).

The Scher-Lax theory was directed on calculation of frequency dependent diffusion (or conductivity), which are defined by

(x 2(t)) = X xX 2 Pxo (t),

and it produced important progress in the field at that time. But the theory was ineffective in description of other important quantity, survival probability (or autocorrelation function) P00 (t), which is directly measurable in fine optical [10, 11] and beta-NMR [4, 12] studies. Other important property - conceptual foundation of the relations between "microscopic" equations (1)-(3) and master equations (8) is still unrecognized by absolute majority of workers, that is clearly seen from legend about the derivation of the master equation which exists 37 years and passed in erroneous form throgh many reviews, see for example [13, ch.5] and [14].

The aims of this article consists in explaining of nonseparable version of CTRW, constructed by me and my colleagues for qualitatively correct description of the autocorrelator P00 (t) in problems of dipole hopping transport, in improvement of some details of the theory, in short description of corresponding modern numerical and experimental results and in concentrated clarifying of those connections of the master equations (8) with primary equations (1)-(3), which are fundamentally important for correct calculation of the P00 (t) and for the problem of random walks on disordered sites as a whole. We resume our experience of construction and application of the CTRW theory in the Section 6 with some differences relative to "incredible possibilities of the CTRWs" indicated in Ref. [14].

2. Reformulation and generalization of the CTRW theory

The method of approximate construction of the propagator Px0 (t), developed in Ref. [2], was reformulated in [15, 16] (see also [17, 18] for different approach) basing on the following assumptions:

a) the first term in square brackets in (8) is the rate of polarization outflow from x to z , while the second term is the rate of inflow from z to x;

b) inflow rate from z to x does not depend on how the polarization reached z. The processes of inflow to the site and outflow from it are now separated, and to determine the kernels Nzx (t) we can treat the simple case in which an exact averaging can be carried out. For this purpose, we choose the process of the outflow of polarization from an arbitrary site in exactly solvable problem

8 5

—F = — > nw F —F = nw F (9)

xx ¿j/ z zx xx' ^t z#x zzx xx' \ >

Fzx (t = to) = nz^zx / C F„(t < t0) = 0. According to the assumptions above, the average Fzx (t) = ^ Fzx (t must satisfy the equations

|f„ (t) = —z Nzx (t') Fxx (t — t'), (t) = joidt'Nzx (t') Fxx (t — t'), (10)

Fzx(t = to) = A, Fzx(t < to) = 0. As a result, the equation for the kernel Nzx (t) obtains the form

jVNzx (t')(exp ( —X q nq Wqx (t — t') = ( nz Wzx eXp (—X q nq Wqxt )) (11)

with

(eXp ( — X q nqWqx^ eXp (Xq h (1 — C (1 — ^^ ))) , (12)

(nz Wzx exp (—X q nq Wqxt^ = cWzx e—Wzxt exp (X q#z ln (1 — c (1 — e—Wqxt))). (13)

The Eqs. (8) and (11) give qualitatively satisfactory description of those properties of the hopping conductivity and delocalization of excitations, which are related with (x2(t= Xxx2Px0 (t), but, as it

is demonstrated in the next section, they are erroneous in the description of the autocorrelator P00 (t), which is directly measurable in the optical [3, 10, 11] and beta-NMR [4, 12] experiments.

3. Main weakness of the Scher-Lax CTRW theory

In order to sharpen the problem we can consider the continuum media approximation, when impurity concentration c ^ 0 and lattice prime cell volume Q^- 0 at a fixed value of impurity density n = c / Q. In the continuum media approximation the Eqs. (8) obtain the form

fp(x,110) = —j0tdt'jd3z[[(t')P(x,t — t' 10) — Nxz(t')P(z,t — t'|0)] =

= —j0tdt'jd3z Zxz (t')P(z, t — t '|0), P(z, t = 0 10) = £(x), (14)

where c>(x) is Dirac's delta-function, the probability density P(x, 110) = Px0(t)/ Q has normalization j d3xP(x, 110) = 1 and the kernels can be written as

Nzx (1) = J™ dte ^Xx (t) = nWzxQ (1 + Wzx)/ Q (1), (15)

Q(1) = j0™ dte—AtQ(t) = (l +j d3 zNzx (1))—\

Q (t) = (exp (-X q «, wvt)) = exp (-n j d 3q (l - eWqxt)) = exp (-(ßt )1/2). (16)

Here and below we apply the same symbols for time dependent functions and for their Laplace transformations distinguishing them by the argument. The Förster constant ß = (l6/9 )in2 w0r06 is defined by the last equality in (16).

The master equation (14) can be rewritten in the Laplace representation as

P(x, 110) = Q(1) (ö(x) + jd3zNxz (1)P(z, 110)) = P(s) (x, 11 0) + P(r)(x, 110), (17)

or, in the time representation

P(x,110) = P(s)(x, 110) + P(r)(x,110), P(s)(x,110) = Q(t)S(x). (18)

The solution (15) indicates that Nxz (1) is a smooth function of |x - z|, therefore Eqs. (17) and (18) separate the propagator on singular P(s)(x, 110) and regular P(r)(x , 110) parts near x = 0. At that, the singular part is defined only by the singular part of the memory kernel Z^ )(t) = c>(x - z) j d 3qNqx (t) of the Eq. (14).

Simple analysis indicates, that for long time

(x2(t)) = jd3x x2P(x, 110) - n-2/3ßt, (19)

that is in agreement with scaling arguments [1, 19] and expected for diffusion long time asymptotics. But the solution P(s)(x,110) = F0(t)S(x) with F0(t) = Q(t) is incorrect [1, 11, 15, 16]. It decays with

time exponentially, while more slow behavior, diffusion like as F0(t) -(ßt) 3/2 should be expected, because F0(t) is the survival probability, and it should be of the order of probability F1(t) to find the excitation on a donor, placed near the origin, that is

^(t - jd3x d(x <n-1/3)Px(0r)(t ^ «>) -1 • 2 1 3/2 ~ (ßt)3/2. (20)

n (x (t))

Here Heaviside's function $( x) is applied.

The relations (17) and (18) indicate, that correct F0(t ^-<x>) can be obtained if the

singular part of the memory kernel )(t) = £(x - z)ZD (t) has correct long time tail

ZD (t ^-<x>) = -F0(t)/ J™dt' F0(t') —(ßt) 3 2 [16]. This conclusion contradicts to main strategy of

application of the memory functions method when reasonable approximation for short-term memory kernels produces satisfactory long time behavior for the solution of master equation (that was fulfilled in the relation (19) for example). Therefore, we should look for justification of the applicability of the memory function method (i.e. Eq. (8)) and for modification of the memory kernels.

4. Correction of the CTRW theory

The justification of the applicability of the memory function method can be based on derivation of the Eq. (8) applying the Nakajima-Zwanzig projection operator technique to Eqs. (2) or (3). Similar attempt was undertaken in Ref. [20] for Eqs. (2) choosing the projection operator it as simple averaging ttA = (A for any A. But initial condition was applied in incorrect form Px0 (t = 0) = Sx0,

therefore all equations, derived in [20], are incorrect [21]. Nevertheless, authors of reviews [13] and [14], as well as more than three hundreds other workers insist that the Ref. [20] gave convincing derivation of the master equation (8) ignoring the criticism of the Ref. [21]. It should be noted

nevertheless that similar derivation can be fulfilled both for Eqs. (2) and (3) with correct initial condition Px0 (t = 0) = nxSx0 / c applying other projection operator

* A = 4). (21)

Unfortunately, the results of these derivations produce no indication on the way to improve the singular part of memory kernels.

More constructive approach was realized in Refs. [15, 16, 21], see also [22]. Correct initial condition Px0(t = 0) = nxSx0 / c separates only those solutions of the Eqs. (2) and (3), for which the lattice site r = 0 is occupied by a donor, because the identity nxR(nx) = nxR(1) is valid for any reasonable function R(nx). Therefore,

Px0 (t) = ((exp (—At)) ^ = ((exp (—At))) o, (22)

where (• • •) 0 means averaging over occupation numbers with the condition n0 = 1. Therefore, we can apply the projector n = n0 acting as

n0B = ( B o (23)

for any B . Standard transformations produce the Eqs. (8) and (14) again, but the memory kernel Nzx(t) = Nl(°)(t) depends both on z — x and x = x — 0 , while the propagator Px0 (t) depends on x only. This new type of memory (geometrical, contrary to dynamical one, which depends on z — x only) is much less comfortable for calculations, because the matrix Nl(°)(t) can not be diagonalized by the Fourier transformation. Nevertheless [15, 16], equations for the memory kernels can be constructed following the derivation (9)-(13) with the result:

Jo'dt'Nz0,(t')Qx0 (t —1')= 1 ^((tWo- 1)Qx° «• z * ». (24)

N00'«) = W No'C x Q (t) = (exp (—X „„ WJ)) = n „. (1 + C (e—"zx' — 1)).

x0

Details of the derivation (with taking into account additional exactly solvable model) can be found in Ref. [16].

As a result, the Eqs. (8) with short-term memory (24) produce qualitatively well-formed solution Px0 (t) for all x and t, and the solution is correct up to terms ~ c1 [15, 16]. It should be noted, that last property was not fulfilled in [2].

5. Approximate solution to corrected CTRW equations

Analytical solution of the Eqs. (8) with kernels (24) is absent. Therefore approximate solution for P00 (t) was constructed by matching the short- and long-time asymptotics. Analysis of long-time asymptotics [15, 16] indicates, that for dipole transport

P 1 00

(Pt ^œ) = i-Goo(t) (l + 0 (ptr )), (25)

where Gx0(t) is the solution of Eqs. (8) with kernels (11). It can be obtained using the lattice Fourier transformation. For fit ^ <x>

G°°(0 = (2—) 'keXP("O ") + 0((Pt) )) ' (26)

e 1 I dtNx° (t). The asymptotics

x J 0

G°° (t ^ œ), according to the Laplace's method, is defined by N(k ^ 0). In simple lattice with cubic symmetry and for dipole transition rates (7) we have

N (°) - N (k ) = Dk2 -ok3 + 0(k4). (27)

Simple derivation of this expansion can be found in Ref. [23].

The diffusion coefficient D is model dependent, and its value for Scher-Lax theory in continuum media approximation is

-3/2 2 2/3 _

Dsl = n 4/Vo6 —L- r(l/6)r(5/3) = ~6~ Pn -2/3, (28)

3 6

—n

where ksl = 0.3725 , while o = w0r06 is model independent and it is defined by the dipole long ranging exclusively. For example, if we will assume in Eqs. (15) and (16) that Qa (t) = exp (-(/1)a) with arbitrary a > 0 and y> 0 instead of Q(t) = exp(-(Pt)12), prescribed by the relation (16), then we will

—2n

obtain other value for the diffusion coefficient but the same o = w0r06. As a result, in leading terms,

Poo (ßt = ■

( A \ , ( 1 4a

1/2

1" *

(29)

i(4—Dt)3/2 ^ D3/2 (—ty2 J (Pt) ^ (P)

where p = 0.7801 and p = 1.923.

Short time asymptotics was obtained in Refs. [1, 3, 21, 24]:

P°° (Pt < 1) = 1 -(Pt / 2 )1/2 + O (Pt ), (30)

and the next term 0(Pt ) = dPt + O ((Pt )3/2 ) was defined in [21] by calculation of the coefficient d

basing on exact expansion of the propagator in powers cm of the concentration. These results allow to construct the approximation

Poo (t ) = Q(t) 1 1 Q (t )

1-' *

(uß(t + t)) i (uß(t + t))

1/2

/

(31)

which reproduces the relation (30) up to (ßt) and both terms of the asymptotics (29) with Hßr = 3.613 [11, 16].

The coefficients a and * prescribe specific evolution of P00 (t) relative to its long time asymptote Pa (t) = (¡ußt) 3 2. At the beginning P00(t) < Pa (t), but with increasing of time we have opposite relation and at ßt ^<x> they coincide. This property (reoscillation) was applied in the Ref. [11] to clarify, that, in agreement with Eq. (31), the onset of the diffusive asymptotic behavior in the kinetics of the electodipole delocalization of excitations in a disordered system of donors takes place at P00 (t ) < 0.03.

Consequent numerical [25, 26] and experimental (beta-NMR) [27, 28] studies revealed, that

a) for small concentrations (i.e. for strongest disorder) the diffusion coefficient is D0 =fin

/?™-2/3

6

with /r0=0.296, this is not very far from tcSL = 0.3725 in the relation (28), and b) for c < 0.1 and for all t autocorrelator P00(t) is of the form

Poo(t) = F00(t ) (1 + z(t) ), (32)

where F00(t) is defined by the relation (31) with more correct diffusion tensor (or coefficient, for

isotropic transfer) and variation ^(t) is relatively small ( -0.1 <%(t) < 1). The variation increases the

reoscillation. It should not contain the short- and long-time asymptotic terms, included in the relation (31).

According to the relation (29), for recalculation to new value of the diffusion coefficient, the relation (31) can be written as

Fcc(t ) = Q (t ) +, 1 - Q (t)./2

( \

1

dsl <p

((t + r))3'2 [ D ((t + r))

1/2

/

(33)

in order to apply the same value p. The multiplayer DSL / D was forgotten in preceding studies, but it did not produced errors in description of results of numerical studies, because 1) for approximations of the numerical results the relation (32) was applied as a whole with the fitting function %(t), and corresponding errors was compensated by %(t), and 2) for studied values of D with corresponding uPt the relative inaccuracy in calculation of F00(t) never exceeded 0.05. Nevertheless we should expect that the correction will become important with increasing of accuracy of theoretical and experimental studies, because it allows to exclude the term ~ (pt) 1/2 from the fitting function ^(t).

6. Conclusions

As a whole we see, that the version of CTRW, invented in Ref. [2] and improved in Refs. [15, 16], produced important part of the basis for consequent quantitative understanding and experimental investigations of the problem of random walks in disordered media with dipole transitions. It should be noted, that this approach allowed for the first time to obtain analytically correct diffusion

long-time asymptotics for autocorrelator JP00 (t) ~ (pt) 3/2, while other methods (see, for example, Refs. [3, 24, 29]) produced reasonable behavior at pt ~ 1, but exponential long time tail. Diffusion long time tail can be obtained for coarse-grained propagator, as in the Ref. [30], this is evident from the relation (20), but it produces no direct information for comparison with precise optical and beta-NMR studies.

We can state, that the CTRW theory was more successful in the description of dipole processes, than for hopping conductivity, where the percolation theory is more applicable. Indeed, CTRW and the relation (28) produce correct dependence of the diffusion coefficient D(c) ~ c4/3 ~ n4/3 for dipole transport in continuum media approximation, but, according to modern knowledge [31], in the

4n 3 —nr 3

is produced by the percolation theory, while CTRW gives other parametric dependence D = Dsl ~ exp(-^SL / slJ2) with icSL ~ 1 [2, 17] (see the Appendix).

It should be noted, that all numeric parameters, indicated in this article for dipole transport, are valid for simplified transition rates (7) only. Results for more realistic description can be found in Refs. [4, 28].

Eqs. (1) and (7) are popular in explaining of dipole transport even in the systems, where the conditions of their applicability, indicated after the Eq. (7) are nor fulfilled. It is natural, that, for these systems, results of the CTRW can be applicable for rough qualitative estimations only.

problem of hopping conductivity D ~ exp(-^p / s), where sc = —^nr03 1 and the constant Kp ~ 1

Appendix. Diffusion coefficients in CTRW theory

Short derivation of the diffusion coefficient for the conductivity problem in continuum media approximation is, probably, absent in existing literature. Therefore, we will give it below.

According to relations (14) and (15) (or (26) and (27)) the diffusion coefficient in the Sher-Lax theory is

Dsl =

2d

or

I j* M y*

d jddx x2Nx0(À = 0) = —Jddx x2wx0 jo dtexp(-wx0t)Q(t)/ Jq dtQ(t) (A1)

(A2)

Dsl =■

2d- j"™dtQ(t) jddx x2Wx0 exp(-wj) / j™dtQ(t).

Here d is dimension of the space and the continuum media approximation is applied.

For dipole transport and d = 3 we have, according to (16), Q(t) = exp (-(ßt)12), and

Ι i 6 \1/6 0 dtQ(t) = 2 / ß. Then using (A2) and substituting x = y (w0r01) we obtain

Dsl = ß ^ exp (1/y6 )j) ddQ (t ) ( r0't )

0t

that directly produces the relation (28), which is exact in continuum media approximation. Similar result was found in Ref. [18].

Calculations for hopping conductivity with wx0 = w0exp(-x / r0) are much more complex. In order

to obtain the main approximation for w0t» 1 we can substitute [2]

1 -exp( Wx0t) = ^(Wx0t > 1). As a consequence, in continuum media approximation,

Q(w0t » 1) = exp (-n jd3x (1- e Wx0t ))

4^

= exp| nr03ln3 (W0t)

For reasonable approximation at all w0t this relation can be written as

Q (t) = exp (sc ln3 (1 + w0t)) 4n 3

that it is correct in main order for small £ = — nr1. Therefore

j™ dtQ(t) = j0™ dt exp (( ln3 (1 + W0t)) = j0™ -W-exp (s - ^cS3 ) = j

(A3)

(A4)

(A5)

J0 S1/2W0

c0

exp

((y-y3 )/ s!). (A6)

where 5 = ln(1 + w0t). Last integrand has sharp maximum at y = se1/2 =el/2ln (1 + w0t) = 3 1/2, that corresponds to exponentially large w0t, justifies preceding approximations and produces (with exponential accuracy)

rdtQ(t) ~ —exp(2/(33/2e1/2) (A7)

w0 v v "

The preexponent is written here to obtain correct dimensional dependence only. It is evident that

n j d3 xwx0 j™ dtQ(t )exp (-wx 0t) = -j™ dtd-Q (t) = 1. Magnetic Resonance in Solids. Electronic Journal. 2017, Vol. 19, No 2, 17201 (11 pp.)

(A8)

Random walks in disordered lattice, CTRW, memory and dipole transport From other side

njd3xwx0j dtQ(t)exp(-wx0t) =nj dtQ(t)jd3xwx0exp(-wx0t) =

0 0 (A9)

= scW0 j0 dtQ(t) jdy3 exp(-y - w(1te-y ).

Last integrand has sharp maximum at y = y0 (t) = x0 (t) / r0 = ln w0t«ln(1 + w0t), and asymptotic expansion in small parameter sc near this extremum will restore the identity (A8). Calculation of the diffusion coefficient according to the relation (A2) will produce additional factors x2 and y2 in the last integrands of (A9), which can be substituted by x0 (t) and y^ (t) correspondingly. As a result,

n jd3x x2wx0 j0 dtQ(t)exp (-wx0t) = n j0 dtQ(t) jd3x x2wx0exp (-wx0t) =

= nw0 j™dtQ(t) jd3x x2 exp (-x / r0 - w0te-x/r°) « w 0 „ d (A 10)

« r02 j™dtQ(t)ln2 (1 + w0t)n jd3xwx0 exp(-wx0t) = - r02 j™dtln2 (1 + w0t)—Q(t) =

= -r2 j;(de-^c * ^) ln2 (1 + w0t) rfc"*3. Therefore, according to the relations (A2), (A7) and (A10), we obtain that for hopping conductivity

Dsl ~ r0w0exp(-2/(33/2^/2)). (A11)

Similar result can be found in Refs. [2, 17].

The relation (A11) is written with exponential accuracy only, that is sufficient to clarify its sharp contradiction with the result of the percolation theory.

References

1. Dzheparov F.S., Lundin A.A. Sov. Phys. JETP 48, 514 (1978)

2. Scher H., Lax M. Phys. Rev. B 7, 4491, 4502 (1973)

3. Gochanour C.R., Andersen H.C., Fayer M.D. J. Chem. Phys. 70, 4254 (1979)

4. Abov Yu.G., Gulko A.D., Dzheparov F.S., Stepanov S.V., Trostin S.S. Phys. Part. Nucl. 26, 692 (1995)

5. Abragam A. Principles of Nuclear Magnetism, Oxford University Press, Oxford (1961)

6. Abragam A., Goldman M. Nuclear Magnetism: Order and Disorder, Clarendon Press, Oxford (1982)

7. Salikhov K.M., Semenov A.G., Tsvetkov Yu.D. Elektronnoe spinovoe echo i yego primeneniya, Nauka, Novosibirsk (1976) (in Russian)

8. Atsarkin V.A., Dzheparov F.S. Z. Phys. Chem. 231, 545 (2017)

9. Montroll E.W., Weiss G.H. J. Math. Phys. 6, 167 (1965)

10. Ashurov M.Kh., Basiev T.T., Burshtein A.I., Voron'ko Yu.K., Osiko V.V.. JETP Lett. 40, 841 (1984)

11. Gapontsev V.P., Dzheparov F.S., Platonov N.S., Shestopal V.E. JETP Lett. 41, 561 (1985)

12. Dzheparov F., Gul'ko A., Heitjans P., L'vov D., Schirmer A., Shestopal V., Stepanov S., Trostin S. Physica B 297, 288 (2001)

13. Hughes B.D. Random Walks and Random Environment, Clarendorn Press, Oxford (1995)

14. Kutner R., Masoliver J. Eur. Phys. J. B 90, 50 (2017)

15. Dzheparov F.S. Radiospektroskopiya (Perm) 13, 135 (1980) (in Russian)

16. Dzheparov F.S. Sov. Phys. JETP 72, 546 (1991)

17. Butcher P.N. J. Phys. C 7, 879 (1974)

18. Vugmeister BE. Phys. Stat. Sol. (b) 76, 161 (1976)

19. Haan S.W., Zwanzig R. J. Chem. Phys. 68, 1879 (1978)

20. Klafter J., Silbey R. Phys. Rev. Lett. 44, 55 (1980)

21. Dzheparov F.S., Smelov V.S., Shestopal V.E. JETP Lett. 32, 47 (1980)

22. Bodunov E.N., Malyshev V.A. Sov. Phys. Solid State 26, 1804 (1984)

23. Dzheparov F.S., Shestopal V.E. Theor. Math. Phys. 94, 345 (1993)

24. Huber D.L., Hamilton D.S., Barnett B. Phys. Rev. B 16, 4642 (1977)

25. Dzheparov F.S., Lvov D.V., Shestopal V.E. J. Supercond. NovelMagn. 20, 175 (2007)

26. Dzheparov F.S. JETP Lett. 82, 521 (2005)

27. Dzheparov F.S., Gulko A.D., Ermakov O.N., Lyubarev A.A., Stepanov S.V., Trostin S.S. Appl. Magn. Reson. 35, 411 (2009)

28. Abov Yu.G., Gulko A.D., Dzheparov F.S., Ermakov O.N., Lvov D.V., Lyubarev A.A. Phys. At. Nucl. 77, 682 (2014)

29. Godzik K., Jortner J. J. Chem. Phys. 72, 4471 (1980)

30. Franchi D.S., Loring R.F., Mukamel S. J. Chem. Phys. 86, 6419 (1987)

31. Zvyagin I.P. Transport Phenomena in Disordered Semiconductors, MSU Publ., Moscow (1984) (in Russian)