10
DIFFUSION THROUGH SHARP POTENTIAL BARRIERS. 2. TIME CHARACTERISTICS OF DIFFUSION A. N. Malakhov UDC 539.219.3:621.382 The time characteristics of one-dimensional diffusion through sharp potential bar- riers are studied. A method is proposed for finding the characteristic diffusion time from its Laplace transform. The diffusion time is found as a function of the width and height of the potential barriers for a number of examples. A phys- ical interpretation is given for the results. i. FORMULATION OF THE PROBLEM AND METHODS OF SOLUTION i. This paper is a direct continuation of [i]. Knowing the exact solutions of the dif- fusion equation, we formulate the problem of finding the characteristic times of diffusion through potential barriers, the characteristic escape times of diffusing particles from poten- tial wells, etc. First we explain what we mean by the escape time. We consider, e.g., diffusion through a potential barrier of finite width (Fig. la). As shown in [I], as t ~ ~ no particles re- main at the barrier itself; instead all the particles are distributed equiprobably on either side of the barrier so that the probabilities 1 -I§ m Po(t) ~ W (X,t)~LT, J1 ' Pz(t)= = | Pi(t) = (x,t)dx, f Wa(x,t)dx , - I+L having initial values P0(0) : i, Pl(0) = P2(0) = 0, assume the final values P0(~) = P2(=) = i/2, PI(~) = 0. It is entirely natural to assume that particles leave the barrier in a time t = 8, at which P0(8) = (I/2)P2(=) = 1/4, i.e., the time in which the probability of par- ticles being beyond the barrier reaches half the final value. Similarly, we can introduce the escape time from other potential profiles as well, as long as the corresponding prob- abilities P(t) vary monotonically as t increases. 2. If now we take the probability density W2(x, t) given in [i], then 8 should be deter- mined from the condition P2 (e) = ~ ~+ i " ch ~+i " erfc . n=O ~/ 2D8 = ~/4. (1) where W r(x.t) - 1 .exp - 2Dr " q 27TDt The deviation of 8 = 8(8) from condition (i) is very problematical. We thus select a different approach: we determine 8 not from P(t) but from its Laplace transform P(p), al- though time does not appear in it. The idea of this approach is that instead of determining the escape time 8 from P(6) =(I/2)P(~) we first determine the value P0 of the operational variable of the Laplace transform, having the dimension of reciprocal time, so that pop(po ) ~ }~ pP(p) = (I/2)P(| (see the limiting relations (13) [I]), and we then replace P0 by i/a8, where a is an appro- priately chosen coefficient. In other words, we propose to find the time taken by diffuslng Nizhegorod State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 34, No. 6, June, 1991. Original article submitted July ii, 1990. 0033-8443/91/3406-0571512.50 1992 Plenum Publishing Corporation 571

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Page 1: Diffusion through sharp potential barriers. 2. Time characteristics of diffusion

DIFFUSION THROUGH SHARP POTENTIAL BARRIERS.

2. TIME CHARACTERISTICS OF DIFFUSION

A. N. Malakhov UDC 539.219.3:621.382

The time characteristics of one-dimensional diffusion through sharp potential bar- riers are studied. A method is proposed for finding the characteristic diffusion time from its Laplace transform. The diffusion time is found as a function of the width and height of the potential barriers for a number of examples. A phys- ical interpretation is given for the results.

i. FORMULATION OF THE PROBLEM AND METHODS OF SOLUTION

i. This paper is a direct continuation of [i]. Knowing the exact solutions of the dif- fusion equation, we formulate the problem of finding the characteristic times of diffusion through potential barriers, the characteristic escape times of diffusing particles from poten- tial wells, etc.

First we explain what we mean by the escape time. We consider, e.g., diffusion through a potential barrier of finite width (Fig. la). As shown in [I], as t ~ ~ no particles re- main at the barrier itself; instead all the particles are distributed equiprobably on either side of the barrier so that the probabilities

1 -I§ m Po(t) ~ W (X,t)~LT, J1 ' Pz(t)= = | Pi(t) = (x,t)dx, f Wa(x,t)dx ,

- I+L

having initial values P0(0) : i, Pl(0) = P2(0) = 0, assume the final values P0(~) = P2(=) = i/2, PI(~) = 0. It is entirely natural to assume that particles leave the barrier in a time t = 8, at which P0(8) = (I/2)P2(=) = 1/4, i.e., the time in which the probability of par- ticles being beyond the barrier reaches half the final value. Similarly, we can introduce the escape time from other potential profiles as well, as long as the corresponding prob- abilities P(t) vary monotonically as t increases.

2. If now we take the probability density W2(x, t) given in [i], then 8 should be deter- mined from the condition

P2 (e) = ~ ~+ i " ch ~+i " erfc . n=O ~/ 2D8

= ~/4. (1)

where

W r ( x . t ) - 1 . e x p - 2 D r " q 27TDt

The deviation of 8 = 8(8) from condition (i) is very problematical. We thus select a different approach: we determine 8 not from P(t) but from its Laplace transform P(p), al- though time does not appear in it. The idea of this approach is that instead of determining the escape time 8 from P(6) =(I/2)P(~) we first determine the value P0 of the operational variable of the Laplace transform, having the dimension of reciprocal time, so that

pop(po ) ~ }~ pP(p) = (I/2)P(|

(see the limiting relations (13) [I]), and we then replace P0 by i/a8, where a is an appro- priately chosen coefficient. In other words, we propose to find the time taken by diffuslng

Nizhegorod State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 34, No. 6, June, 1991. Original article submitted July ii, 1990.

0033-8443/91/3406-0571512.50 �9 1992 Plenum Publishing Corporation 571

Page 2: Diffusion through sharp potential barriers. 2. Time characteristics of diffusion

particles to escape beyond the barrier not from the evolution of the probability density (as t ~ ~) but from the evolution of the probability-density transform (as p ~ 0).

3. We now choose a so that the escape times, determined from P(t) and from pP(p), would coincide for free diffusion. This is due to the fact that as was shown in [i], probability densities for the potential profile under consideration are expressed in terms of the "base" probability density WG(X, t), which exactly describes free diffusion.

For free diffusion (no potential field) it follows from [i] that for all x we have

r(x) = (B127) e -u V(x,t) = Wr(x,t).

We note that y 2 = Bp, B = 2/D.

We distinguish the point X = s on the X axis and find the escape time of free diffusing particles beyond point s On the basis of the above O can be obtained from

Since obviously P(~) = 1/2. The free-diffusion escape time, found in terms of P thus is

OCB - - t I" 2,19, (2)

where t I E s

Next we determine the escape time in terms of the Laplace transform of the probability. Since

PP(p)- p ~ Y(x)dx- 1 e-~l 1 T 7 " ~ ) 2 '

we determine 7o = ~p0/D from the condition e-~0 s = I/2. The corresponding value of ~0 is 0.693. Substituting 7o = /2/Da@fr, we find @fr = tl'4"163/a" If this value is to coincide with (2), we must take a= 1.90. Since the definition of the escape time is rather arbitrary, however, for simplicity we assume that 8fr = 2t I and replace the operational variable p by 1/28 in the calculation of the escape time, i.e., we take 7s =

2. DIFFUSION TIME THROUGH A BARRIER OF FINITE WIDTH

I. We turn now to diffusion through a potential barrier of finite width (Fig. la) and as an illustration we use the above method to calculate the time taken by diffusing particle to escape beyond this barrier.

Integrating the function Y2(x), given by Eq. (16) in [I], from x = s + L to x = = and multiplying the result by p = ~2/B, we obtain

p~2(p) = i (1-~ 2) e -c*'x'a 2 1 - R 2 e - 2 A ~ '

where 6 = 7s I =̂ L/s and R = R(S) = (i -- e-~)/(l + e-G). When p (or 6) varies from ~ to 0 the function PP2(P), and hence P2(t) vary from 0 to 1/2. To find the escape time e, there- fore, we must find 6 from

2(i - R z) e -(*+k}~ = 1 - R 2 e -2k~,

and then set 6 = r

The simplest case to solve is i = i, for which

e - 4t i in'211 + 4e-B/(1 + 2e "B + e'~48)]. (3)

For a low barrier (B << I, R = (1/2)8 << i) whence we have @ = 2tl+ L = 2ti(l + I) 2 = efr , i.e., the escape time coincides with the free-diffusion time (to the far boundary of the barrier). For a high barrier (e-B ~ p << i, R = i -- 2p) Eq.(3) leads to 8 = (i/4)tle2~. If I # I, then for the high and low barriers, respectively, we easily obtain (for the high barrier we take the auxiliary condition I >> 0)

1 t ,A2e2B 1 A 2 - e2~- (4) @ " @fr @ T " @fr 8 (I + A) 2

With rising barrier height, therefore, the time taken to escape beyond the barrier in- creases and is proportional to e2S for a high barrier: diffusing particles overcome a poten- tial barrier more slowly when it is higher.

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Page 3: Diffusion through sharp potential barriers. 2. Time characteristics of diffusion

2. Before going on to other examples, for the most general case (see Fig. L in [i]) we give expressions for the Laplace nransforms of the probabilities of diffusing particle being in the respective intervals of the x axis, multiplied by the operational variable.

For positive x

~t 1 [dl(e 711 - i) + d~ - e-~l*)J, " o "r x < 11, PP~ cP" = 2 A

p ~ (p) = 1 dt [ - -T~2)] 1 x l e, 2 ~ n a~,Ce ~'~ - e ~'' ') + ~ r - e , ~< ,

[ ] 1 dt r[ IT d l (e ~13 - e ~1"12) + < ( e 712 -~1"13) PP2 ( p ) - 2 A * 2 2 - e ,

(5a)

12 < ,1[ < 13,

" i ~t i11 i I [din ~'lN 7'.-I) PP"-: (P) - 2 ;% 2"" " " "I/I.-t -I (e - e +

+ <_l(e -TIN-I -- e-71N)], Is_ 1 < X < IN,

1 dt ppw(p) m 2 ~ II g2-....lie -wi" x > i.

For negative x

^ [ - _ ] PPo (p) " i ' T d i ( e ~'l ' - 1) + do(1 - e -~' ' ' ) , - I < x s O,

^ 1 dt - - - - -T[2)], p'9 (p) - -~ -K-- ]T [ d:(e ''2 e"*) + <(e "'1 - e

- - [2 ( X < I [ , ,

^ 1 d - - " _ - ~n,~, - ~ -~ ~, ~ [ <J'~ _ ~ + <,;", ;"~ ~].

-I3r x �9 -If . . . . . - . . . . ~ �9 �9 . . . . . . , . .

PP,-,IP) = 2 -~-~,~2"'" "~- , ,-, -

,,-".-, ;,..,] - , . , , ,_, . . , . + -1 -- '

/

In the interval - - s

A pPw (p ) =

< x < Ii

I d, . W , e T~, 2 -r- % =, "''" , *,-I~

( 5 b )

" + ( e ~'11 - i IdiOt (e~'ll -~l'~ 1 ) (e~f[1 -~fll ] P P o ( P ) = 1 - + e ~'l i ) + ~ - e + d l d t - e ) . (5c)

The notation used here accords completely with that introduced in [i]. The normaliza- tion condition has the form

573

Page 4: Diffusion through sharp potential barriers. 2. Time characteristics of diffusion

H N

Z p Pk (p) + Z p P~(p) = I. k = O k=O

x ~ O x ~ O

3. Let us now consider the diffusion of particles from the central region z bounded by two barriers of finite width (Fig. Ib). For this symmetric case (N = 2, s = s = s s = s = s + L, ~ = 6• = ~, 6~ = 6~ = --8) on the basis of Eq. (5) and [I]

A (I/2)(I - R 2 ) e -* 'c~§

1 -- R2e "21fL - Re -271 + Re "2~(I§

This expression varies from 0 to I/2 as 7s varies from ~ to 0. The escape time of diffusing particles beyond the potential barriers is thus determlned by

2(i - R2)e -a~ = I - R2e "2x~ - Re -2~ + Re -2~I§ (6)

Ofr �9

For low barries (6 << i), as should be the case, it follows from (6) that 8 = 2ts L =

For high barriers (e-6 = p << i) with the auxiliary condition A >> p we can find

If we require that ~0 << i, i.e., if in the final account the relative width of the high barriers is limited by e-6 << ~ << e6, then 6 = ~p7~-and

e c x e ~ x e ~. (7) = = OcB 2(I+A)2

If we consider high thin barriers, when I = p = e-6 (the higher the barriers, the thin- ner they are), then according to analysis of Eq. (6) the escape time 8 = 3.7"t I is no more than double the free-diffusion time. We can thus speak of some "tunneling" of diffusion through thin but high barriers.

Comparing the diffusion time through a single barrier [Eq. (4), Fig. la] and through two barriers [Eq. (7), Fig. ib], we see that the diffusion corresponding to Fig. la is substan- tially slower, namely slower by a factor of %eB >> i. This is due to the existence of a free space to the left of the barrier in Fig. la, to which the diffusing particles move rather than overcome the high barrier. And since in the final account (as t ~ ~) half of the par- ticles still "should get through" the barrier, this process becomes more prolonged when the barrier is higher.

4. To verify this statement, we consider diffusion from a left-bounded space through the

same barrier (Fig. 2c). In this example N = 2, s = I, s = i, s = s + L, s = ~, 61 = 6, 62 = --6, 61 = =, and ~2 = 0. From (5a) and [i] it follows that

p~2(p) = ( 1 / 2 ) (1 - a 2 ) ( 1 + e -25 ) e - c 1 §

1 - R2e -2~6 - Re -46 +Re -~4§ 6 ~ 0 ) I .

The equation for the escape time e has the form

(i - R 2) (I + e -2~) e -(I+A)~ = 1 - R~e -2A~ - Re -4~ + Re -(4+2A)~.

For a low barrier we obtain e -(3+~)6 + e -(I+~)6 -- i = 0. If we take i = i, then we

arrive at @ = 8fr'2.16, where @fr = 2t2s For a high barrier under the same limitations on the relative barrier width, e -8 << ~ << e6, we can find

e = 2t! = eCB (i + A) 2 "

As in the previous case, here too we have also obtained a proportionality law @ ~ eS.

At the same time, upon comparing the diffusion times corresponding to Fig. Ic and to,the preceding example we readily see that the diffusion times almost doubled both for free diffu-

574

Page 5: Diffusion through sharp potential barriers. 2. Time characteristics of diffusion

sion and for the case of a high potential barrier [compare Eq. (8) with Eq. (7)]. This can be interpreted as follows. In the example under consideration (Fig. ic), in contrast to the previous example (Fig. ib), before reaching the barrier, half of the diffusing particles first have to travel to a reflecting boundary and then move back toward the barrier. In other words, for half of the diffusing particles the "path to the barrier" grows from s to 3s The escape time increases accordingly.

5. Let us now consider the next example, in which an absorbing boundary lies beyond the barrier; particles traveling past the barrier no longer turn back. An infinite potential well at the point x = s + L (Fig. Id) corresponds to this case, where N = 2, ~i = i, s =

+ L, s = s s = ~, ~I = 6, 62 = --m, 61 = +m, and 62 = 0. At such values of s and 6k it follows from (5a) and [i] that

^ ( 1 - R ) ( 1 + e -2a) e - ( * § PP2(P) = - e -2A6 Re -46 + e 6 ) 0 1 R - -(r ) 1 .

The equation for determining the escape time becomes

2 ( 1 - R ) ( I + e -m6) e -(~+A)6 = 1 - R e -2~16 - Re -4a + e -(4+2A)6

If the barrier is low, then for i = i and ~ = 2 we obtain O = 8fr/1.85 and 8 = 8fr/2.15, re- spectively: in comparison with the previous example the escape time through a low barrier decreased by a factor of roughly four. Such is the influence of an absorbing boundary im- mediately behind the barrier.

As for a high barrier, with the same condition for the relative barrier width, calculat- ing gives an escape time that is given by Eq. (8). This means that an infinitely deep poten- tial well lying beyond the barrier did not change the escape time.

6. To understand why an absorbing boundary beyond a high barrier did not affect the escape time through this barrier we consider one more example, that of diffusion in a so- called two-level system, whose potential profile has two minima (Fig. 2). For such a pro-

file N = 3, s = s ~2 = 3~, ~3 = 5s s = s s = ~3 = ~, 61 = B, 62 = --~, 63 = ~i = ~, and 82 = ~3 = 0.

When we write PP2(P) in accordance with Eq. (5a) and [i], after a number of transforma- tions we find

^

pP2(p) = P2(m) (e-3a+ e -s6) 2A

( i + e -Ba) ( , '] + I ) + 2& 4a ( a - l )

where A = e-~ + e-6 + e-(~+6). The escape time, therefore, is determined by

(e -3a + e -sa) 4A -- ( i + e - S a ) ( A + i ) + 2 e - 4 a ( A - i ) . (9)

We point out a very noteworthy circumstance. Since the barrier height 6 and potential- well depth B enter the expression for A symmetrically, Eq. (9) remains the same after the interchange B ~ ~. The escape time 86+ u of particles diffusing from point x = 0 through a barrier of height B into a potential Well of depth ~ coincides with the escape time %D+~ of particles from point x = 4s through a barrier of height D into a potential well of depth 6, i.e., with the time of back diffusion. This is due to the steady-state probability distribu- tion in the system. For the two-level system (Fig. 2) under consideration we easily obtain

1 e "~ e "~+~

Po(- ) - l+e.~+e_B§ . ' PI(-) = l+e_~+e_B+ ~ ' Pz(| = l+e_~+e_B+ ~ '

and if, e.g., ~ < B, then it is clear that the (particle) probability flux corresponding to 8~+~ is smaller than for 8u+ 8. On the other hand, the final probability P2(~) is also smaller than P0(~). The difference between the final values completely compensates for the differ- ence between the fluxes, thus causing the escape times to be the same.

For a high barrier (e6 >> I) and a deep well (e~ >> I) it follows from (9) that the escape time

575

Page 6: Diffusion through sharp potential barriers. 2. Time characteristics of diffusion

w(m3

eat) e~(,+) P~ ~ a

% %#~ z

m 2-

i .~(,t) d

C

I

Fig. 1

IQ<e) Fig. 2

e B �9 e ~ 2 e B �9 e ~

is determined by the ratio e~/e u. If the barrier is relatively high (e~/e~ >> i), then the escape time @ = 4tle~ no longer depends on the barrier height but is determined only by the well depth. If the well depth beyond the barrier is rather large (e~/eU << i), then 8 = 4tle~. This value of the escape time, first, agrees completely with (8) if we note that i = 2 in the case under consideration and, second, no longer depends on the depth of the poten- tial well, which can be arbitrarily large, as in the preceding example.

If the barrier remains high and the potential well is assumed to have a small depth (e~ m i), then from (9) we find that 8 = ti"8.35 = @fr/2.15, which once again is at variance with the preceding example of a low barrier, if we note that 88~ ~ = 8~+~.

7. The escape time of particles diffusing past a high ptoential barrier (e~ >> I) de- pends essentially on the behavior of the potential profile in front of and beyond the bar- rier and can vary over broad limits, from values of the order of the free-diffusion time 8fr to values of the order of @fre2~.

For cases when particles diffuse from bounded space through a high potential barrier in a rather deep potential well, such that the probability that diffusing particles turn back into the initial region is very small, the escape time is proportional to e ~ and therefore, the diffusion rate ~e-~.

We note that this factor e-8 for the diffusion rate through a high potential barrier, whose profile is shown in the upper right-hand corner of Fig. 2, was obtained by Kramers [2] using an approximate method which assumes the existence of a Gaussian quasi-steady distribu- tion in the region of the potential minimum A and a parabolic profile in the region of B.

8. To end this section we turn to the mean time of arrival at the boundary for the first time, which differs in meaning from the escape time. We find, e.g., the mean arrival time of a Brownian particle (starting from the point x = 0) at the point x = 4s for the profile shown in Fig. 2. As we know (see, e.g., [3]), in this case an absorbing boundary, which cor- responds to an infinitely deep potential well, must be placed at the point x = x b = 4s In this way we obtain the situation depicted by a dashed line in Fig. 2.

576

Page 7: Diffusion through sharp potential barriers. 2. Time characteristics of diffusion

The mean time of the first arrival is [3]

<r> = ~ P(t)dt.

w

where P(t) = 1 - Q(t), O(t) = ~rpW(X,t)dx. Turning to the Laplace transform, we obtain

<r> = lim (l-pQ(p))/p, Q(p) = ~ Y(x)dx. p~O 41

Calculating pQ(p) by means of (5a), we arrive at

<r>o~4i = 4t,(1 + e ~) (2 + e'~).

Making the interchange ~ ~ ~, we find the mean time of the first arrival at the point x = 0 when a Brownian particle starts from x = 4s

<r>4,~o = 4t,(l + e~) (2 + e-~).

The mean times of the first arrival for 0 ~ 4s and 4s + 0, as should be the case, are not the same because $ differs from ~. If ~ = D, then

1 1 2 <r> = <r>o~4l = <r>41~0 = 4tl(3+2e~+e-~ ) = 24ti(i+ ~ + ~ +...).

Hence for the mean frequency of particle transmission through a high potential barrier (e$ >> i, B = E/kT, and E is the activation energy), we obtain the familiar relation

12 = <T> "I = VO e-E/kT

For an arbitrary potential-barrier height

3 e-E/kT i -2E/kT). v = voe-~'kT/(1 + ~ + ~ e

3. DYNAMICS OF THE ESCAPE OF DIFFUSING PARTICLES FROM A POTENTIAL WELL

i. We now turn to the time characteristics of the diffusion of particles inside a poten- tial well and the dynamics of their escape from it. The discussion is confined to the sim- plest example of a symmetric potential well, shown in Fig. le, and the most interesting case of high barriers (e-~ = p << I), i.e., the case of a deep potential well. For the potential profile under consideration with N = I, s = El = s and ~l = $i = $, according to (8a) in [I] and Eq. (5), the Laplace transforms for W0(x, t) and P1(t) have the form

S { e "~ml'x> + e "~m+x) } Yo(X) = --~ e -~Ix[ + R 1- Re -2~I ' Ixl < i, ([1)

-~ (12) PPI(P) = (I-R) e . ~ --,i.

2 (l-Re -z~ )

We determine the escape time of diffusing particles from the potential well. Since PI(0) = 0, Pl(~) = i/2, then according to (12) the equation determining the escape time be- comes

2(1-R)e -~ = 1-Re -z~.

From this we find without difficulty that

e T ecs in-2 v/I-R+R z + 1 - R .

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Page 8: Diffusion through sharp potential barriers. 2. Time characteristics of diffusion

If the barriers are low, then 8 = %fr, as should be the case. from a deep well is

2 1 efre2~ e = c , / p = - ~ - ~ 82~ ~* 8 f r .

The escape time of particles

Once again here we have the factor e2~, indicating a slowing of diffusion, in the given case diffusion of particles from a potential well, which in the end "should all escape" from the well. The explanation for this slowing can be that when the particles encounter the high barrier they "find it easier to slide down" than to move along the plateau of the barriers, i.e., there is a reverse flux of particles from the infinite plateaus to the potential well.

To find the time scale associated with the diffusion of particles inside a deep poten- tial well, we also consider the values of the time t, when t << e~$, i.e., when virtually all the particles are in the potential well and are not "affected" by the high walls. The bar- riers, therefore, can be assumed to be arbitrarily high, by setting R = i. A steady-state distribution W0(x ) = 1/2s is established with time in such a potential well. We find the time in which this distribution is established, taking the values x = _+s for simplicity. From (ii) at R = 1 it follows that

1 26 e -6 i PYo (+1) = ~ - z6 ' " - - " ~ "

i- e ~ ~0

The time required for a steady-state distribution to be established, i.e., the relaxation time 0rel, is determined from

46e -6 = I - e -26, 6 = ~ti/ere I

The root of this equation is 6 = 2.18. If we cosnider the process of establishment at x = 0, then 6 = 1.915. Taking 6 = 2 for simplicity, we arrive at 0re I = (I/8)%fr, which is smaller than the free-diffusion time.

2. A deep potential well, therefore, has two markedly different time scales, the relax- ation time ere I and the escape time 02~ = 46rele2~. Three different regions exist on the time axis (and on the axis 6 = 7s respectively (Fig. 3). Diffusion occurs inside the well at 0 < t ~ 0re I. At ~rel << t << e28 a quasisteady state in which W0(x , t) = 1/2s exists in the well. Particles diffuse from the well at t ~ 025. As before, the probability density W0(x, t) at such t does not depend on x, being a constant in the potential well, and only "melts" as t grows.

The situation at 9re I << t << 029 can be called a regime of intermediate asymptotic be- havior, when W0(x, t) does not depend on t (see Fig. 3) and, therefore, can be found directly from the transform

W o(x) = pr o(x,p) - 6 2 Yo [ X ' ,32 efr err ) (13a)

which is analogous to the limiting relations (13) in [i]. Only now can we assume that 6 satis- fies the inequality

2 * ~ > p. (13b)

In actual fact for 6 >> 1 and 0 << 1 we can easily find from (ii) that

i ~ (14) PYo (X) = 21 6 + p

Hence Eqs. (13) give W0(x) = I/2s If the correction p/6 in (14) is not discarded but is taken into account, however, then from the transform

[ ] E = I p = ~ P (TI) s

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Page 9: Diffusion through sharp potential barriers. 2. Time characteristics of diffusion

Diffusion Quasisteady ~2~ in well regime

0 0 8re 1 8 f r

0 J , " 2 ~7

Diffusion from well

e~ - t i I m

Fig. 3

we can find the original of the probability density

(i5)

This formula, which holds for ere I << t << e25, clearly shows how the quasisteady distribution density in a deep well begins to "melt" with growing t.

As t + ~ on the basis of the limit theorem wo(x,| ) = lim pY0(x) it follows from (14), 640

as it should, that W0(x , ~) = 0. On our t and 6 scales the conditions t >> 02B and 6 << P correspond to the mathematical conditions t + ~ and 6 + 0. From (14) at 6 << p it follows that

to(x) = EIp-" p

The original corresponding to this transform has the form

,,o(-, ~I : ~ v'-'z'7~ v - ~ . (16)

This formula, which is valid for t >> 825, describes the final decrease in the particle prob- ability density in the deep potential well.

3. Let us analyze the characteristics of diffusion from the potential well, starting from the probability density itself and not its Laplace transform. Writing the Gaussian probability

density as WG(X,t) = (1/21) V 0/~ exp(-x20/412) where % = efr/t, we can easily show on the

basis of [i] that the following exact value of the probability density corresponds to the transform (ii):

"o(-, ~> = ~ ~ ,-'2~"211 + 2 ~ R ~ ~-~ c. ~ ] . n : l

(i7)

This formula holds for any values t > 0, R, Ixl < s and contains one time scale 8fr (% << i), when a quasisteady distribution that is almost independent of x exists in a poten- tial well, as a result of which we set x = 0 for simplicity. In this case

','o(o, ~) : ~ ~ D + 2 ~: R ~ ~-~ ]. n:1

(18)

To move on we use the approximate summation formula

E e - a ' ' ' a ~ : [V"RT~ e J / ' ~ er:Ec (-2v o D : I

( 1 9 ) l

which is more accurate when a and % are smaller. A practically acceptable accuracy is obtained up to ~ = 0.5 and % = I.

Denoting R = e -~ and using (19) instead of (18), we obtain

1 ec~/4,~ c W o(0,t) = ~ erfc (. ). 2~-~-

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Page 10: Diffusion through sharp potential barriers. 2. Time characteristics of diffusion

For a deep well r = 2p, whereby r = p2/82 = t/Sfre2e = t/282B" Consequently, at t >> 8fr

As can be readily checked, for t << 8fr and for t >> Bfr formula (20) goes over into formulas (15) and (16), respectively, which therefore are exact. Moreover, we can easily verifv that (20) is just the original of the transform, given by Eq. (14).

In conclusion, we note that the time scale 62B obtained earlier in the analysis of the Laplace transform has now come from the exact formula (17) for the probability density, thus supporting the legitimacy of the proposed approximate method for finding the time character- istics of a function from its Laplace transform.

LITERATURE CITED

i. A.N. Malakhov, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 34, No. 5 (1991). 2. H.A. Kramers, Physica, ~, No. 4, 284 (1940). 3. R.L. Stratonovich, Selected Problems of the Theory of Fluctuations in Radio Engineering

[in Russian], Sov. Radio, Moscow (1961).

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