Approximate analytical solution of the jump rate problem in a symmetric well with spatially varying friction

Physica A 196 (1993) 83-92 North-Holland SDI: 0378-4371(93)E0009-4

Approximate analytical solution of the jump rate problem in a symmetric well with spatially varying friction R. Ferrando, R. Spadacini and G.E. Tommei Dipartimento di Fisica dell'UniversitY, Centro di Fisica Superfici e delle Basse Temperature/ CNR and Unit~ INFM, via Dodecaneso 33, 16146 Genova, Italy

Received 10 August 1992 The jump rate problem for a particle in a periodic potential is studied by an analytical solution of the Klein-Kramers equation. The very general case of a position-dependent friction is considered. The low-friction solution is obtained extending to the symmetric well the Wiener-Hopf method developed by Mel'nikov and Meshkov in the study of the escape rate from a metastable well with position-independent friction. An analytical expression for the jump rate, valid in the whole damping range, is obtained by a multiplicative bridging formula. Explicit results are presented for a cosine potential and a cosine friction and the low and high damping limits are discussed.

1. Introduction Since the fundamental p a p e r of Kramers [1], the escape-rate p r o b l e m of a classical particle from a metastable well has been widely studied both analitycally and numerically [2-4]. On the contrary, much less is known about the escape-rate p r o b l e m from a periodic well [2,5] and no analytical solution has b e e n found in the case of a position-dependent friction. In this p a p e r we p r o p o s e an analytical solution for the jump rate of a Brownian particle in a periodic well in the general case of position-dependent friction. If m e m o r y effects are disregarded, the motion of the particle can be studied by the K l e i n - K r a m e r s equation ( K K E ) , a F o k k e r - P l a n c k equation with a periodic potential [6], which describes the time evolution of the probability density in phase space. If the thermal energy k B T of the particle is smaller than the amplitude of the potential, the particle performs a j u m p diffusion [7], spending a long time near the b o t t o m of some well before hopping suddenly to another well. In this case one of the quantities of m a j o r physical interest is the total j u m p rate rj (the total j u m p rate, in the one-dimensional case, is the double of 0378-4371/93/$06.00 © 1993- Elsevier Science Publishers B.V. All rights reserved

84

R. Ferrando et al. / Approximate analytical solution of the jump rate problem

the directional jump rate, i.e. of the jump rate to the left or to the right). Even in the case of a position-independent friction the relationship between the jump rate in a periodic potential, rj, and the escape rate from a metastable well, r, is not trivial, because in the former case the particle has two escape paths and in the latter only one. At high friction the diffusion is spatially controlled and we expect rj = 2 r [2], at low friction the diffusion is energy controlled and therefore rj = r. At intermediate friction we expect that r < rj < 2r [51. In section 2 we calculate analytically rj for a periodic potential and inhomogeneous friction. The dependence on position of the friction may be important in the diffusion of adsorbates at crystal surfaces [8] and in the discussion of the relationship between the Kramers-Klein equation and the transition state theory [9]. The solution is obtained extending to the symmetric well the W i e n e r - H o p f method developed by Mel'nikov and Meshkov [3] for the metastable problem with position-independent friction. The solution given by the W i e n e r - H o p f method is valid if the dissipation integral is smaller than k B T and therefore the friction has to be small at the bottom of the well. Moreover, recrossing events are neglected and this is correct if the friction is small also at the top of the barriers. Results are plotted for a cosine potential with amplitude A = 5k BT and different values of the parameter which describes the spatial variation of the friction. In section 3 we propose a solution valid in the whole damping range obtained by a bridging procedure [2]. In section 4 the conclusions are presented.

2. The Wiener-Hopf method in the periodic problem with position-dependent friction We consider the K K E [1] with a potential U ( x ) of periodicity a and amplitude A (see fig. 1),

oi_ Ot

ol V ox

F(x)ol m

o(

Ov + rl(x) -~v v f +

m

-~v / '

(1)

where F ( x ) = - U ' ( x ) and m, ~(x) and f ( x , v, t) are the mass of the particle, the position-dependent friction and the phase-space probability distribution respectively. We look for a stationary solution of eq. (1) with the boundary conditions of equilibrium distribution at the bottom of the well (near x = 0 in fig. 1) and of two perfectly absorbing walls at the top of the barriers (in x = +-a/2). The latter condition implies that the recrossings are neglected, an approximation which is valid if the friction is not high in the barrier region.

R. Ferrando et al. / Approximate analytical solution o f the jump rate problem

85

U 0 E

I n n n n

I I I I I

I I I

I + +

l I I I I i

i + t i + I

2A -0/2

0

+ 0/2

+

Fig. 1. The periodic potential considered in the text.

This procedure is not different from that employed in the metastable case [3] and it reduces the Kramers problem in a periodic potential to an escape problem from a symmetric well. The motion of the left- and right-going particles in the well can be separated putting [3,6]

= + x / E [ E - V(x)l,

x) =y(v+-(E, x), x),

(2)

where plus and minus refer to right- and left-going particles respectively. The condition of total absorption at the barrier top implies that f+(E, -a/2) =f-(E,

a/2) = O,

(3)

i.e. there is no right-going flux at the left barrier and no left-going flux at the right barrier. Now we define the dissipation variable s by means of the following relationship: ds

d----x= +--rl(x) ~ - 2rnU(x).

(4)

The dissipation variable s differs from the action variable defined in ref. [3] by the factor ~?(x). Substituting eqs. (2), (4) into the stationary form of eq. (1) and noticing that at low friction the most part of the escaping particles comes from a narrow energy range below E = 0 (see fig. 1) [3], we obtain

86

R. Ferrandoet al. / Approximateanalyticalsolution of the jump rateproblem Of+-(E,s) 0 ( ,- E Of+-(E,s) ) Os = O---E f - ( ' s) + kBT OE "

(5)

Eq. (5) is a diffusive equation with uniform drift in the energy space, in which the dissipation variable s plays the role of a time [3]; therefore a relationship between the function f at different dissipations can be obtained if the Green function of eq. (5), F, is known

f'-(E, s) = f

F(E - E', s - s') f+-(E', s') d E ' .

(6)

The Green function F has a simple Gaussian form,

F ( E - E',s)=

1

/~

exp -

( E - E' + s) 2)\ 4-k'BBits

"

(7)

We define the functions ~-+(E),

E<0, crP+-(E)=f±(E, +-a/2),

E>0,

(8a) (8b)

where x+(E) is the inversion point of a trajectory of negative energy E on the right side of the well and x-(E) is the corresponding point on the left side (see fig. 1). The boundary condition of thermal equilibrium at the well bottom (E ~ lOBT) implies that ~_+(E)_

mtOosc exp(

2~kB T

E+2A] ka T / ,

(9)

where COos¢ is the small oscillation frequency. In our problem the well is symmetric, therefore the relationship between ~ + and ~ - is • +(E) = @ - ( E ) .

(10)

In the limit Eb k BT

2A k BT

--

>> 1

(11)

the dissipation of the particles in an energy range of k BT below the barrier top can be approximated [3] by the dissipation S 1 at the barrier top, defined by

R. Ferrando et al. / Approximate analytical solution of the jump rate problem

87

a/2

(12)

*?(x) ~ / - 2 m U ( x ) d x .

S,= -a12

According to eq. (6) qb- propagates in • ÷ with a dissipation corresponding to a single travel from the left to the right side of the well. In the case of a metastable well with only one escape path, the dissipation corresponding to a round trip has to be considered [3]. Taking into account eqs. (3), (6), (12) we obtain 0

qb+(E) =

f r(E-E',S1)q~-(E')dE'

(13)

and by the condition (10) we get an integral equation of the Wiener-Hopf type [10] with Gaussian kernel 0

dP+(E)= f r(E - E', S,) dP+(E') dE'.

(14)

Eq. (14) with the condition (9) is solved by standard methods [3,10] and the escape rate from the right side (which is one half of the total escape rate) is obtained as r + = -i-f m

~+(E) dE.

(15)

0

Finally the result for the total jump rate in the underdamped regime is

rllf)

°)°so exp (2k-BT)- exp ( l f l ° g ( 1 - e x p [ - A ( u22++ l1) ] ) d U ) u fit

--

(16)

o

with A-

S'

kB T .

(17)

Formula (16) is valid at low dissipation and high barriers, i.e. when both the following conditions are satisfied: A < 1,

(18a)

2 A >>k B T .

(18b)

88

R. Ferrando et al. / Approximate analytical solution of the jump rate problem

In fig. 2a results are reported for Q _00 in the case o f a cosine potential

U(x) =

-A

/2-,-,x~]

1 + cos~ ---~-- J]

(19)

and of a cosine friction

r/(x) = %[1 + A c o s ( ~ - - ~ ) ] ,

(20)

for a fixed potential amplitude A = 5 k B T (corresponding to an activation barrier o f 1 0 k a T ) and for 3 different values of the parameter A: A = 0, 0.5, - 0 . 5 . Positive A correspond to a friction stronger at the well b o t t o m than at the barrier and this situation s e e m s to be of interest in surface diffusion of

rj (If) 024

(°)

02

0.16 / / / 7

/

/ / / /

0.12

iii / /

0.08

I

004

0

i

i

i

Iii

III

002

i

0.04

iil

I

~1

ir

006

Iii

t

0,08

i

I

01

i

i

it

I

i

i

i

0.12

i

Ii

i

i

i

0.14

0.07

//

(b) 0.06

/ / /

0.05 0.04 / 0,03

/ /

J

002

I

OOl i

o

i I i i I i I I Ilil 0.005

I~1

II

I~ll~lillllll 0.01

IIIIII 0.015

'~! 0.02

")/0

Fig. 2. (a) Low friction jump rate rlm(eq.(16)) as a function of 3'0 (see eqs. (20), (21)) for A = 0, 0.5 and -0.5 (full, dashed and dash-dotted lines respectively). (b) Extremely low fraction range in the case of homogeneous friction A = 0; the full line is the result of eq. (16), the dashed line is the underdamped Kramers result (eq. (24)). In both panels the rate is given in the same arbitrary units.

R. Ferrando et al. I Approximate analytical solution of the jump rate problem

89

_(If) is plotted as a function of the dimensionless adsorbates [8]. The rate ,j quantity

•

a ~k~___~

(21)

Having defined A g-

(22)

2kBT ,

from eqs. (17), (19), (20) it results

a =8

oV

(1+

(23)

As can be seen in fig. 2a the effect of a friction stronger at the well bottom than at the barrier top is to enhance the escape rate: in fact a particle in the well can gain more easily the energy needed to jump away and once it has reached the vicinity of the barrier top suffers a lower dissipation. If the recrossing events were taken into account this effect should result stronger. In the opposite situation (A < 0), the escape rate decreases. In fig. 2b the result at homogeneous friction (h = 0) is compared to the very low friction Kramers result r 0f) [1],

(2A)

r(lf) -- 8rl°A -k~ "rrkBT exp

(24)

The Kramers formula gives accurate results at extremely low friction, % < 10-3; in fact it has been obtained [1,2] assuming that all the particles that reach the energy corresponding to the barrier top really escape form the well and therefore it gives a rather large overestimation of the rate as ~ exceeds 0.1, as can be seen by the asymptotic expansion of eq. (16) for A---)0 [3]:

r(lf) j

--

~

r(k'f)[1 -- ~r( ½ )

V A / ' r r ] ~-- r(k'f)(1

- 0.82V~)

(25)

where ~" is the Riemann zeta function. It should be noticed that eq. (25) is formally identical to the corresponding formula for the metastable problem, but in the present homogeneous friction case A is one half of the metastable one, being calculated on a single travel from the left to the right of the well. This means that at the lowest order in zl (Kramers order [1], eq. (24)) the metastable and the periodic problem give the same result, but with different corrections as the friction increases.

90

R. Ferrando et al. / Approximate analytical solution o f the jump rate problem

3. Jump rate in the whole damping range The expression for the jump rate in eq. (16) is valid only if the condition (18a) is fulfilled. This condition implies that the friction must be small: for example, taking A = 0 and g = 2.5, from (18a) and (23) we obtain approximately % < 0.1. However, an expression approximately correct in the full damping range may be found by a multiplicative bridging formula [2], connecting the low- and the high-friction results. In the case of position dependent friction an expression valid in the overdamped range may be obtained by a generalization of well-known Kramers result [1,2] for homogeneous friction and metastable potential. In our problem the friction at the barrier top 7b has to be considered (in fact, at high and intermediate friction, the resistance against escape is essentially localized there [1,2,11]) and the result has to be multiplied by a factor 2 due to the two escape paths. Therefore we have rlhf)_ °Josc(,/ 7b 7b exp - k ~ 7r \ ~/4wzb + 1 -- 2w b

(26)

where wb describes the curvature at the barrier top. For the potential shape of eq. (19) we have wb = Wosc and for the friction shape of eq. (20) we have 78 =70( 1 - A).

(27)

Also in the overdamped limit the effect of a spatially varying friction is qualitatively the same as in the underdamped case: a friction lower at the barrier top enhances the jump rate. An expression for rj (see fig. 3), approximately valid in the whole damping (j

...............

028 I

~

\

ooB

o

.....

1........ ~ . . . . . 1 2

J ..... 3

u. . . . . . . • . . . . 4 5

J .... 6

I .... 7

I,,~J 8

. . . .

9

T0 Fig. 3. Jump rate rj in the whole friction range (see eq. (28)) for A = 0, 0.5 and - 0 . 5 (full, dashed

and dash-dotted lines respectively). The rate is given in the same arbitrary units as in fig. 2.

R. Ferrando et al. / Approximate analytical solution of the jump rate problem

91

range, may be obtained [2] as rj =

(2A)

"rr .(Lf)_(hf)exp k ~ t'Oosc tj tj

(28)

This multiplicative form of bridging is commonly used [3] and seems to give the best results; in fact in ref. [5] it has been shown that it furnishes a rather good approximation in the homogeneous case. However, if the friction varies by orders of magnitude in the lattice cell, eq. (28) may give inaccurate results.

4. Conclusions

In conclusion we summarize the differences between the results obtained in the case of a metastable potential with only one escape path and homogeneous friction [3] and the results in the case of a periodic potential with position dependent friction. The existence of two escape paths implies that the periodic rate at high friction is the double of the metastable rate; in fact at high friction the particle can escape only if it is in the vicinity of an escape point. If a particle gains an energy higher than the activation barrier at a position far from the escape points, it cannot leave the well because it dissipates its energy before reaching the barrier. At extremely low friction the situation is different: the metastable rate and the periodic rate tend to coincide. In fact at very low friction the diffusion is energy limited [1,2] and a particle can escape if it gains sufficient energy, irrespective of its position and of the direction of its velocity (in the metastable case, if the particle has a velocity directed towards the reflecting wall, it escapes after the reflection). However the corrections to the extremely low friction limit are different in the two cases and in general the periodic rate is higher than the metastable rate. An inhomogeneous friction, which may be interesting in atomic diffusion at surfaces [8], has relevant effects both at low and high damping. At low friction the dissipation integral (12) has to be considered instead of the action integral [3]. If the friction at the barrier top is lower than at the well bottom, the escape rate increases, while it decreases in the opposite case. In the overdamped regime the effective value of the friction is that at the barrier top [11] and the escape rate is inversely proportional to this effective friction. Therefore the rate is enhanced decreasing the damping in the barrier region. The results of the Wiener-Hopf method in the periodic case with positionindependent friction have been tested against numerically exact results [5] with a good agreement, especially at high potential barriers (g > 2). Also in the

92

R. Ferrando et al. / Approximate analytical solution of the jump rate problem

i n h o m o g e n e o u s p r o b l e m this analytical solution should be rather accurate, e v e n if the bridging p r o c e d u r e might be a slightly worse a p p r o x i m a t i o n , in particular if the friction varies rapidly near the barrier top. T h e m e t h o d p r o p o s e d here for periodic systems can be applied also to m e t a s t a b l e potentials with p o s i t i o n - d e p e n d e n t friction. In conclusion, we r e m a r k that the spatial variation of the friction is a p r o b l e m of growing interest in noise-activated rate processes (ref. [9] and references therein); we expect a spatially varying d a m p i n g to be even m o r e relevant in diffusion in i n h o m o g e n e o u s (e.g. periodic) systems. T h e exact solution o f the rate p r o b l e m in periodic potentials [5] will be generalized to i n h o m o g e n e o u s friction in a forth-coming paper [12]. T h e exact m e t h o d of solution can also be applied to systems where the friction varies by orders of m a g n i t u d e within the lattice cell, a case in which the analytical solution p r o p o s e d here b e c o m e s rather inaccurate.

References [1] H.A. Kramers, Physica 7 (1940) 284. [2] P. H~inggi, P. Talkner and M. Borkovec, Rev. Mod. Phys. 62 (1990) 251, and references therein. [3] V.I. Mel'nikov and S.V. Meshkov, J. Chem. Phys. 85 (1986) 1018. [4] M. Biittiker, E.P. Harris and R. Landauer, Phys. Rev. B 28 (1983) 1268; B. Carmeli and A. Nitzan, Phys. Rev. Lett. 51 (1983) 233. [5] R. Ferrando, R. Spadacini and G.E. Tommei, Phys. Rev. A 46 (1992) R699. [6] H. Risken, The Fokker- Planck Equation (Springer, Berlin, 1989), and references therein. [7] C.T. Chudley and R.J. Elliott, Proc. Phys. Soc. London 77 (1961) 353. [8] G. Wahnstr6m, Surf. Sci. 159 (1985) 311; Phys. Rev. B 33 (1986) 1020. [9] R. Krishnan, Surjit Singh and G.W. Robinson, Phys. Rev. A 45 (1992) 5408. [10] P.M. Morse and H. Feshbach, Methods of Mathematical Physics, vol. I (McGraw-Hill, New York, 1953). [11] R. Ferrando, R. Spadacini and G.E. Tommei, Surf. Sci. 265 (1992) 273. [12] R. Ferrando, R. Spadacini and G.E. Tommei, to be published.