Brownian motion in a periodic potential under an applied bias : the transition from hopping to free conduction

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Brownian motion in a periodic potential under an applied bias : the transition from hopping to free

conduction

P. Nozières, G. Iche

To cite this version:

P. Nozières, G. Iche. Brownian motion in a periodic potential under an applied bias : the transition from hopping to free conduction. Journal de Physique, 1979, 40 (3), pp.225-232.

�10.1051/jphys:01979004003022500�. �jpa-00208902�

Brownian motion in a periodic potential ^{under an} applied ^{bias :}

the transition from hopping ^to free conduction

P. Nozières and G. Iche

Institut Laue-Langevin, ^{156X, 38042 Grenoble} Cedex, France

(Reçu ^{le 13} septembre 1978, accepté le 14 novembre 1978)

Résumé.

Des particules placées ^{dans un} potentiel périodique ^sont soumises à un champ directeur ; le frottement est assez faible _pour que l’oscillation dans ^un puits ^de potentiel soit sous-amortie. Nous développons ^{une for-}

mulation stochastique ^du problème, ^{fondée sur une} hypothèse Markovienne d’un puits à l’autre. Le paramètre important ^est l’énergie moyenne 03B4 dissipée ^en traversant un puits, comparée ^{soit à T,} soit à la chute du potentiel appliqué, y. Nous montrons que le régime change ^{très vite} lorsque y ^{= 03B4.} Lorsque 03B4 ~ y, T, ^{nous retrouvons} les résultats classiques de Frenkel. Dans la limite opposée 03B4 ~ T, la mobilité augmente; ^nous donnons dans

ce cas une solution détaillée pour y quelconque. ^Nous généralisons ainsi les résultats obtenus _par Ambegaokar

et Halperin [2] ^{dans le cas suramorti.}

Abstract.

We consider particles ^{in an} arbitrary large bias, subject ^{to a} periodic potential ; the friction is such that the oscillation in the potential ^{wells is} underdamped. ^We develop ^a stochastic formulation based on the assump- tion that the friction is Markovian from one well to the next. The relevant parameter ^{is the mean} energy 03B4 lost in crossing ^{one well,} compared ^{to either T or to the} drop ⁱⁿ bias energy y. We show that a rapid change ^of regime

occurs at y ⁼ 03B4. When 03B4 ~ y, T, ^{we recover} the old results of Frenkel. When 03B4 ~ T, the mobility is increased :

we give ^a detailed solution in this limit for arbitrary bias. We thus extend the results obtained by Ambegaokar

and Halperin [2] ^{in the} overdamped ^case.

Classification

Physics ^Abstracts

05.40

1. Introduction.

This note is concemed with the Brownian motion of a _gas of independent [1] particles,

immersed in a periodic potential Uo(x), ^and subject

to an applied ^{force F. Our} problem ^{is one} dimensional,

controlled by ^{the usual} Langevin equation

fi is the friction coefficient (with the dimension of a

frequency). ^{U =} Uo - Fx is the total potential energy,

and R(t) the random thermal force responsible ^for

fluctuations. We wish to find the _average velocity

v of the particles, expressed ^{in terms of a} mobility

Il ⁼ TIF.

The problem ^is of interest in itself, bearing ^{as it}

does on the motion of ions in crystal lattices, ^or along

the surface of a crystal. ^{It also occurs in} completely

different situations

for instance the phase slippage

in a Josephson junction. ^{As shown} by Ambegaokar

and Halperin [2], ^the voltage V ^{across the} junction

and the phase difference 0 of the order parameter obey

the equations

R and C are the resistance and capacity ^{of the} junc- tion,1 ^the constant current fed in by ^the generator, and Io ^{the maximum} Josephson ^{current. L is the} fluctuating ^{noise current.} (2) is similar to (1). ^{The mean}

drift of the phase, 0, ^is analogous ^{to our} velocity v

-

and it yields the average d.c. voltage ^{across the} junction. ^Our mobility J1 ^{is thus} tantamount to the observed d.c. resistance.

We return to our mechanical problem, keeping ⁱⁿ

mind that transposition ^{to the} Josephson junction ^is straightforward. ^{As shown} by ^Kramers [3] ^{in his}

classic _paper on Brownian motion over a single poten- tial well, ^different regimes may apply depending ^on

the value of q .

(i) If 1 ^{is much} larger ^{than the} oscillation frequency

COm in the bottom of the well, ^the motion of the particle

is overdamped. Locally, ^{the current is the sum of a}

diffusion and a conduction current, and the particle density n(x) obeys ^a Smoluchowski equation

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004003022500

226

This case has been fully analysed by Ambegaokar ^and Halperin [2], who found the steady solutions of (3)

for arbitrary values of the force F. The periodic poten- tial is of importance only ^{when its} depth U.. ^is > ^T.

The relevant parameter ^{is the} applied potential drop

y ⁼ Fa over one period a ^of Uo, ^as compared ^to Uom. ^As long as y Uom, ^the mobility ^is small,

controlled by thermally ^activated hopping ^{over the}

barriers. A cross-over occurs when y N Uom, ^{such as}

to wash out the extrema of U : then the mobility

increases rapidly, until it reaches its limit F/mq

obtained when Uo ⁼ 0. Note that everywhere, ^the mobility ^{is -} 1 lil.

(ii) ^{In the} opposite limit 1 « Cùrn, the particles

oscillate freely ^{in the} potential ^wells. When U. - T,

the situation is complicated, because of correlations between successive wells. Here, ^we only consider the

simpler ^case U. » ^{T : when a} particle has fallen by

more than T below the top ^{of a} given well, it is hope- lessly trapped, until much later it succeeds in desorbing again. The conduction is thus a succession of statisti-

cally independent hops. Still, ^a single hop ^can jump

over several wells

hence an added complication

which we shall study ^here.

As shown in the simpler problem ^of desorption [4],

the relevant quantity ^{is the mean} energy 8 dissipated by ^a particle ^while crossing ^a single ^{well of U. The}

relevant parameters will be the ratios y/b ^and 5/T.

In the absence of an applied ^force (y ⁼ 0), ^{we recover}

the two regimes ^described by ^Kramers [3].

(iia) ^à > T : a particle ^{which enters a well of} Uo

will loose so much _energy before it reaches the other side that it is definitely trapped ^{in that} well. Conduc- tion is a succession of single ^well hops ; moreover, the hopping ^{rate is} given by the usual absolute rate

theory ^of Eyring (which follows from perfect trapping

+ detailed balance [4] : ^{see section} 4). ^{One thus}

recovers the diffusion coefficient and mobility ^found by ^Frenkel [5] many years ago ; both are independent

of _il. That regime ^was considered by Ivachenko and Zilberman [6] ^{in their treatment of the} Josephson

characteristics.

(iib) à ^{T : a} particle which reaches the top ^{of a} well can diffuse _up and down in energy many times before it is trapped again. ^{Both the} hopping ^{rate and}

the length ^of hops ^are affected. The resulting ^{low field} mobility turns out to be ’" T/b. ^Since the energy loss

is of order

it follows that y - il - ’ -

Assume now that the applied ^force (i.e. y) ^increases.

A particle ^{which has} barely ^{crossed the i’ maximum}

reaches the (i ⁺ 1 )lh ^{with an} average energy (y -£5)

above the top, ± fluctuations of amplitude ^{A. When} reaching ^wall (i ⁺ n), ^the energy of this particle (still

measured from the top) ^is roughly (y - 5) n ± ^L1 Jn :

for large ^{values of} n, the average energy transfer will

always ^{take over} fluctuations. As a result, ^{a catastro-} phy ^{occurs, if} v exceeds (5 : then every particle ^will eventually ^run away, with an ever increasing energy.

Actually, ^{such a} divergence ^does not occur, of ^{course :} when a run away particle ^starts gaining energy, the

mean energy loss (4) ^increases

^{and the} run--away stops ^{when 5} reaches y again. ^The mobility ^{is thus} always ^finite

yet ^{with two distinct} regimes.

-

if y 5, ^{all the} conducting particles ^{have an}

energy within T from the top ^{of the} well ; ^a hopping

model is valid ;

-

if y > 5, ^the distribution n(e) spreads ^to very

high energies, such that 5 ’" _y. The mobility ^{is much} larger.

In _summary, in the underdamped ^case (ii), ^{the rele-}

vant parameters ^are 8/T (which fixes the nature of

hopping), ^and y/5. ^The purpose of this _paper is to

apply ^{to such a} problem the stochastic approach ^of

reference [4], ^which provides explicit ^{answers both in}

cases (Ha) ^and (iib), ^for arbitrary y. In a somewhat different language, ^{we thus} generalize ^{the results of}

Kramers [3]. ^The stochastic description is formulated in section 2. Section 3 deals with the case 5 T, leading ^to the notion of effective temperatures. ^The absolute rate regime ⁸ > T is briefly sketched in section 4. Our approach ^deals only with the limit ri co., U. » ^{T. The} intermediate friction _case, il - rom, requires ^a numerical treatment, ^as done by Kurkijârvi ^and Ambegaokar [10] (via ^{a molecular} dynamics simulation of the random force).

Throughout ^the paper, ^we consider only ^{a static}

bias

albeit a large ^{one. The} problem ^{of linear}

response ^{to an a.c.} field has been considered recently by ^{a number} of authors [7], in relation with the optical properties ^of superionic conductors. The correspond- ing mobility depends ^on frequency, ^{and one} expects ^a

resonance effect to occur. It seems that these various papers ^are limited to our cases (i) ^and (Ha), ^since they predict ^{a finite zero} frequency mobility ^{in the low}

friction limit. (As ^{we shall} see, such a conclusion becomes ^erroneous (when 1 --+ 0.) ^Our approach ^is quite different in spirit, ^{and it} complements ^{the above}

papers : ^we exclude shallow wells or finite frequency,

but ^we allow for large ^{biases or} very small friction.

2. Stochastic formulation.

Consider a given

maximum (Xm Un) ^{of the total} potential U(x). ^The

energy ^E is measured from that maximum (the origin

of a is thus n dependent). ^{We focus our attention}

on particles which either cross Xn (e > 0), ^{or which}

are reflected ^near Xn (e 0). ^{In each case, we define}

the flow ^of particles ^{in the} energy range (e, ^{e +} de) :

in this _range, a number f (E) ^de of particles ^{either cross}

Xn ^{or are reflected} per unit time. For each maximum of

U, ^{we thus} introduce four functions

- À(8) ^and Zee) describe the flow toward right

or left when e > 0

-

f"L(e) and , f"R(E) describe reflection ^{on the} left

or on the right ^of Xn ^{when a 0.}

These four functions completely ^describe the par- ticle distribution. ^For instance, ^the total current across the nth maximum is

In practice, ^{we are} interested in steady, ^uniform solutions, in which all the f,, ^are independent ^{of time}

and of ^n. In thermal equilibrium ( y ⁼ 0), ^{all the} fn

reduce to a Boltzmann exponential :

(a is the chemical potential, v ^{is the} velocity). ^{In a}

finite bias _y, the four f ^will depart ^from (6) ^{in a} layer

of width - T near the top of the well.

The density ^of particles is controlled by ^{the bottom}

of the wells. As long as y « Uom, the ordinate of the minimum is E ^{= -} Uom - y if reckoned from the

2

left maximum, e ^{= -} Uo. + if ₂ ^{reckoned from the}

right. ^{Let ç be} the energy measured from that mini-

mum. Since a reflection occurs at every period ² n/co.,

the density p(ç) ^{near the} minimum is given by

Note that (7) imposes ^a relationship between fL

and fR. ^The latter is best expressed by introducing ^the

total number of particles in each well

(We ^{can extend the} integral ^{to oo since} only j ^T

is relevant.) (7) appears ^{as a} boundary ^{condition on} fL ^and fR, valid when 8 - - Uo. « - ^{T : 1}

(8) fixes ^{the scale of fL and f,}

hence that of

f ând 7The ^current I, given by (5), ^is proportional ^to

N : the average velocity v ⁼ I/N is well defined (1).

In order to complete ^our formulation we need a set of dynamical equations governing ^the behaviour of

1, 7: fL, fR ^{near a = 0. Let us} follow the motion of a

given particle, ^which may be viewed as a succession of well crossings in either direction. Our only approxi-

mation is to assume that these successive steps ^are statistically independent : energy transfers to the heat bath follow a Markovian _process on the _gross scale of individual well crossings (no correlation between successive crossings). ^{Such an Ansatz is not} very restrictive (we ^{make no} assumption ^{on what} happens

inside each well). Under such conditions, ^the physical input ^{of the} theory ^{is the} probability W(e, e’) ^{that a} particle entering ^{a well with} energy ^e ends _up with energy e’ on the other side. W(e, a’) ^{is defined in the}

absence of a force : it is the same whether the particle goe§ toward left or right.

From our Markovian Ansatz, it follows that in a

steady state, the various f obey ^the following ^sto-

chastic equations

(For clarity, ^we have restored the well index n. There-

after, ^we drop ^{it since we} only ^consider homogeneous states.) ^The difference in origin ^{of e on different wells}

has been absorbed in the argument ^{of W. The} integral equations (9) contain all the physics ^{of our} pi;oblem;

together ^{with the} boundary ^condition (8), they ^deter-

mine 1 ,f, ^{fR and} fL unambiguously-hence ^{the mobi-} lity ^Jl.

In such a Markovian model, ^the only parameter ^is W(E, e’) (and ^{of course} y). ^This energy transfer proba- bility ^{for a} single crossing ^{is normalized}

Moreover, detailed balance in thermal equilibrium implies ^that

The mean energy loss b defined earlier is just ^{the first}

moment of W

^"

(’) ^{Let us stress} again ^that (8) only ^{holds if} Uom > T, y, in such a

way that the particles ^{in a} range - T ^near the bottom of the wells

are in thermal equilibrium despite ^the applied bias. If this condition is not met, one cannot separate top ^from bottom, ^{and the} density ^N

is strongly ^modified.

228

We may equally well define a fluctuation amplitude ^L1

via the second moment

à and L1 must be such as to obey (11). ^{When à T} (case iib), ^{we can} expand exp f3(e - e’), ^and (11) yields

at once

(14) ^is nothing ^but Einstein’s relation in _{energy space.}

In the opposite ^{limit à » T, there is no} simple expres- sion for A ; ^from ( 11 ), ^{we can} only ^{conclude that} W(a, E’) ^is negligible ^{if e’}

^{8 » T :} particles mostly

loose _energy to the heat bath. -

Our problem ^{is now clearly laid out. We} only

need to solve the integral equations (9).

3. The weak friction limit, ô « ^T.

^{In view} of (14),

the various energies ^of interest are such that _y, à « L1 T. The kernels W in (9) ^{extend over a} range e’ ^- e ± L1, much smaller than the scale of variation

of f, which is of order T. We may thus expand 1(8’) ⁱⁿ

powers of (e’

e), thereby transforming (9) ^{into a set}

of differential equations. ^The coefficients of these

equations ^are obtained with the help ^of (11).

Consider for instance the first equation (9), ^which

we write more explicitly

If 8 » d, the relevant e’ are always positive.

(15) ^involves 1 only, ^{and the} integration ^{in the cor-} responding ^term may be extended to F’ ^{= -} oo . In

leading order, ^the resulting differential equations

read as

(We ^used (10)-(13).) ^The only effect of the bias is to shift the mean energy loss by ± y, depending ^{on the}

direction of flow. If ô is constant, integration ^of (16) yields ^{at once}

~ ~ C and C are unknown constants. The effective ^tem- peratures ^are given by

The distributions are Maxwellian, ^{as a result of a} random walk in energy for particles going ^{either to the}

right ^{or to the left. As} long as y b, T remains > 0, and the distribution f is ^{localized near the top of the}

well (the assumption ô ^{= const. is then} justified). ^As predicted, ^a catastrophy ^occurs when y > b. Then the

particles going ^{to the} right ^{have a} negative tempera- ture, correspondin ^{to the run away phenomenon}

described earlier. extends ^to large energies, ^and we

must allow for an a-dependence ^{of ô}

^{i.e. of} fia).

The general solution of (16) ^is

It has a maximum at that _energy for which ô _{= y ;} thereafter, it decreases rapidly. The solution is well

defined, ^however large y. The change ^of regime ^at

y ⁼ b is smooth - yet quite rapid.

A similar analysis holds when 8 « - 4. Then (9)

involves only fR ^and fL, ^{and the} corresponding ^diffe-

rential equations ^are

The characteristic equation ^{of (20) is}

Up ^{to errors -} y2, ^{its roots are s =} 0, : ⁼

Well inside the well (8 « - L1), ^{the last root domi-}

nates : the distribution has the usual Boltzmann form

exp[ - 8/7$ ^{with an} unperturbed temperature (energy

loss and gain ^{due to the bias} balancing ^{to 0 in a round} trip). ^{With an} accuracy - y, the amplitudes of fL ^and f, ^{are found to be}

(21) ^is consistent with our general ^statement (8)

-

which incidentally fixes the integration ^constant

C ⁼ [Nwm/2 nT] exp(- PUOm)’

In order to complete ^our solution, ^{we must relate} the integration ^constants well above the well

(C and ë) ^to that well below (C). ^That implies ^a

solution of (9) ^{in the} intermediate range 1 e 1 - ^{d .}

In that _energy, reduction to a differential system ^{is not} of much use, since the integration boundary ^{at e = 0}

makes all the coefficients _very complicated. ^{It is} simpler ^{to solve the} integral equation directly, using

a standard Wiener Hopf approach. ^{Let us assume}

that W(e, e’) depends only ^{on the} difference (e

e’),

a condition which is certainly ^{met in the narrow}

transition _range of width L1 near the top of the well.

The integrals ^{in (9) are then} convolutions, ^{and our} equations ^{are a set of four} coupled ^Wiener Hopf equations ^{for the} quantities f, f, fR, fL. ^{We show in}

the appendix that these equations ^{can be solved} explicitly. In the limit 5, y T, ^a direct solution is

quite easy, and it yields ^{the result}

(22) provides ^the required link between e > 0 and 8 0.

The current ^{density (5) is dominated} by energies e 1"’01 T, T > A, ^a range in which the asymptotic

forms (17) ^are valid. Below the run away threshold,

y b, ^{we thus find}

(We may also note that according ^to (5), I ⁼ f (s ⁼ 0) : (23) then follows from (A.11).) ^{In the low field limit,}

y - 0, ^the corresponding mobility ^is

This is to be compared ^to the result of the standard Frenkel hopping ^model [5]

We see that the mobility is increased by ^{a factor 2} Tlb.

By Einstein’s relationship, ^{the same} correcting ^factor

occurs in the diffusion coefficient D. The physical origin of this correction is a compromise ^between

two conflicting tendencies : a reduction - b/T ^{of the} desorption ^{rate at which a} particle ^{leaves the well in}

which it is trapped, together ^{with an} increase - Tlô

of the corresponding ^{mean free} path ^{before it is} trapp- ed again. ^{We defer a} discussion of that point ^{to sec-}

tion 4. Here, ^{we note that the} mobility M ^{is N} 1 lil,

as in the overdamped ^case.

For larger y, (23) departs ^from linearity. ^{If 5 were}

constant, ^I would be oo at y ^{= ô. Such a} singularity

is cured by ^the increase of b with a. In that region, ^the

current is completely ^dominated by ^{the run} away distribution 1 ^{From (19), it} follows that

Departures ^from (23) become sizeable when 1

ylô - TIU..,. Thereafter, ^{the current increases} rapidly, the detailed I - F characteristics depending

on the form of b(e). When y is several times b, ^the distribution 7(e) ^{extends to} energies larger ^than Uom ;

the drift velocity ^{is much} larger ^{than any thermal} velocity, ^{and the} periodic potential ^is unimportant :

in that fictitious limit,

The corresponding characteristic is sketched on

figure ^1.

Fig. ^1.

^{The current as a} function of the applied bias y ^{= Fa.}

At y ⁼ ô, there is a continuous change ^of regime ^from hopping ^to

free conduction.

4. The intermediate friction case T « à « Uom.

The oscillation is still underdamped, ^and (9) ^remains

valid. Its solution is however drastically changed.

The kernel W(8’, e) corresponds mostly ^{to an} energy

loss, (E - a’) ^{- - à} (the positive ^tail being exponen-

tially small). ^{Let us} first consider particles going

toward the left near the top ^{of the} nth maximum

(e - T). ^After crossing ^{the left} well, they hit maximum

(n - 1) ^{with an} energy e’ - - ô - y « - T : they

are hopelessly trapped in the left well. Conversely, by ^detailed balance, the distributions À(c) ^and fnL(F)

should originate mostly ^from energies

they ^are in thermal equilibrium ^{with the well on the}

left. (They ignore the existence of other wells.) ^We may thus write

(Put ^{another way, the} asymptotic ^{form (8) extends}

throughout (e T).) +:-

By ^{the same} token, ^the distributions £(a) ^{and fnR(F)}

involve particles ^{that come from well} (n ⁺ 1), ^where

most of them had energies ^e’ f"Oo.I E - b _{+ y.} For a

small bias, y - ^T 5, ^these energies ^are again ^far

below - T, and the above argument ^holds. fn ^and fnR

are in thermal equilibrium ^{with the well on the} right

of the nth maximum, ^{and we} may write

(again ^a continuation of the asymptotic ^form (8)). ^On

the other hand, ^{for a} large bias, y - 5, (28) ^{is not}

230

correct. There is a fair probability ^{that the} original

e’ was > 0. Then the particles ^are coming ^from

maximum (n ⁺ 2)

^{or even further}

^no simple

thermal equilibrium argument ^holds.

We _may verify ^these statements by solving (9) ⁱⁿ

iterative fashion. We consider (27) ^and (28) ^{as zeroth}

order solutions. When inserted on the right ^hand

side of (9), they yield ^the following improved expres- sions (we ^use (11)) :

We known that the kernel W(e, e’) ^is important only

when E - e’ ^’" b. In the small bias limit, fly % 1, the correction terms in (29) ^are clearly small : the iteration converges, and our Ansatz (27) ^and (28) ^are

thus correct. For larger y, ^we must treat (29a) ^and (29b) separately. In the latter the integral ^becomes large when y - ô : the iteration does not converge, and as surmised, (28) ^no longer holds. The situation is more subtle for the first equation (29a) (in ^which efly _may overcome the smallness of the integral).

By using (11), ^we may rewrite its bracket as

The correction ^to f ^looks small, ^{as we} guessed

earlier. Unfortunately, it extends over a much broader energy range, - y, and its contribution to the current I is similar to that of (27). (Physically, that correction

originates ^from particles ^{that have} spilled ^{over maxi-}

mum (n - 1), ^{and lost} only ^an energy y - ^ulti- mately, ^{it is} responsible ^for the runaway phenomenon when y ⁼ £5.)

In conclusion, the solution (27) ^and (28) ^holds only if y ^£5 (but ^for any value of fly). ^{We do not know}

the solution when y - £5; ^{it is non} universal, depend- ing ^{on the} shape ^of W(8’, e). ^The only ^{fact we know}

for sure is that run away ^occurs when y ^{= £5. In the}

limit y £5, ^{the current I is} easily found from (5) :

When y - ^0, we recover the Frenkel mobility (25).

It is instructive to consider these results as a random walk problem. By Einstein’s relation, ^{to the} mobility

Il is associated ^a diffusion coefficient (’)

Such a diffusion _may be viewed as a succession of

statistically independent hops, between which the

particle ^is trapped ^{in some well. Let u be the} desorption

rate for that well, ^{i.e. the} probability per unit time that the particle leaves the well for ^the first ^{time since}

it has been trapped. ^{u is a sort} of desorption frequency.

Let also 1 = na be the mean free path ^before the par- ticle is trapped again somewhere else. The diffusion coefficient is [8]

In the Frenkel regime, ^D follows from (25) ^and (31 )

In this _case, the mean free path ^is simply ^{1 = a : a} particle ^which spills ^{over the} top ^{of one} well, ^{with an}

energy ^{8 -} T, ^is automatically trapped ^{in the next one.}

Moreover, by ^{detailed balance, such} perfect trapping implies ^{that the} desorption ^{rate is} given by ^{the abso-}

lute formula of Eyring [4]

(33) then follows immediately ^from (32).

In the opposite ^limit b T, ^the desorption ^{rate is} reduced, ^{as shown} by ^Kramers [3]. Repeating ^the analysis ^{of reference} [4], ^{we find}

(u ^would go ^{to zero} in the absence of friction). ^{On the}

other hand, ^{the mean free} path ^{is much} longer. ^A par- ticle with an energy T above the top ^{will cross a number} of wells - y/(5 before it is reflected and eventually trapped. ^{Hence a mean free} path ^{1 -} aT/b. ^From (32),

it follows that the Frenkel DF is corrected by ^{a factor}

2 ^{2 T}

T T : ^{the exact result 2bT we found} in section 3

( ) ^b

is thus a compromise ^{between a smaller u and a} longer ^1.

The result (30) ^no longer ^{holds near the run} away threshold. There the I-F characteristics should follow from a detailed solution of the integral equations (9),

for which there is no simple ^method.

5. Conclusion.

The main result of this _paper is twofold. First, ^{we show that the} transition from a

hopping regime ^{to a} conduction regime ^{occurs for a} bias y ^{= b - at which a run} away would occur if ô

were constant. Second, ^our stochastic approach

(2) The _particle density ^{is n =} N/a. (31) follows when writing ^that

the total current 1= D grad n - N J1 grad V vanishes in thermal

equilibrium.

displays ^{the two} regimes ^found by ^{Kramers in his} theory of chemical rates. For intermediate frictions (£5 > T), ^{the low bias} mobility is the Frenkel _MF,

resulting ^{from an absolute rate} approach. ^{In the weak}

friction limit (8 T), MF is multiplied by ^{a factor}

2 T/£5; ^the corresponding distributions above the top of the wells are Maxwellian, with effective tempera-

tures T and T given by (18). ^{In the latter} regime, ^{we can}

describe explicitly ^{the run} away region y - ô ; ⁱⁿ

the Frenkel _case, ,on the other hand, ^{we cannot} give

a description.

Acknowledgments.

^{We wish to thank Dr. R. Lan-}

dauber for letting ^us know of his work on Brownian motion of interacting particles ^{in a} periodic potential.

Appendix.

^{We start from the} integral equa- tions (9). ^{In order to eliminate the bottom of the} well,

we first extract from fL and fR their known asymptotic

form (8), by setting

(0 ^{is the} step function). ^{The first} equation (9) ^becomes

With the help ^of (11) ^and (10), ^{the last term of} (A. 2)

may be replaced by

In this _{way, it} is localized in a small layer ^{near e = 0.}

Note that (A. 3) ^{is still exact. A similar} procedure ^is

carried out for the second equation (A. 9).

We next assume that W(8’, e) depends only ^{on the}

difference (E’

e)

^{a valid statement since now all}

the quantities ⁱⁿ (9) ^are restricted to a small layer ^of

width - T Uom ^{near e = 0. The} integrals ⁱⁿ (9) ^are

then convolutions. We therefore _carry a Fourier

transformation, by defining

The set of equations (9) ^{takes the} simple ^form

where m and n are known functions of s

Since land vanish ^{when e, 0, their Fourier trans-}

form is analytic ^{in the} upper half plane ; similarly aL and _aR are analytic in the lower half plane. (A. 5) ^thus

appears ^{as a} standard coupled ^Hilbert problem.

A general ^solution may be obtained ^{for arbitrary}

values of b and _y. Let us define f -.f = f, aR - aL ^{= a.}

Using (A. 5) ^and (A. 6), ^one easily ^{verifies that}

(A. 7) ^{is now a one} dimensional Hilbert problem, ^the

solution of which is given in the book of Muskhe- lishvili [9]. ^{One first writes}

where M + and M - are respectively analytic ^{in the}

upper and lower half s-plane. ^Then

The problem is reduced to a calculation of integrals.

According ^to (5), the difference f ^is enough ^{to obtain}

the current (which in the end is the only thing ^we are really interested in). ^{If we insist on} finding i and 1

separately, ^{we must} solve another Hilbert problem,

the inhomogeneous ^{term in (A. 5)} being ^{now the pre-} viously determined f ^{and a.}

Actually, ^{we shall not use this} general solution, which is too formal. Instead, ^{we consider} directly ^the

limiting case ô « ^{T. We know that} f and f decay

exponentially ^{on a scale - T :} their Fourier trans- forms will have a pole ^{at a} small s - - ifl (the ^tran-

sient region ^{e « A} gives singularities ^at larger

s - 1/,A).

Similarly, the transient corrections _aR and aL ^are

localized in a small region - ^{A below the} top ^{of the} well : the Fourier transform a(s) ^will consequently

vary only ^{on a scale s -} 1/J ; ^as long ^{as s 5} fi, ^{we can}

232

safely ^{assume that} a(s) ^{is constant.} With these points

in mind, ^{we consider the} integral equation (A. 7)

in the limit of small s. The kernel W(s) may be expand-

ed as

Within leading order in s and fl, (A. 7) takes the form

where e and p are the inverse of T and T defined in ( 18).

The value a(O) ^{must be such as to cancel the} pole ^of f ^{at s = 0.} (Such ^a pole ^{would mean that} f ^remains

finite when 8 --+- + oo.) ^{If we} neglect ^the dispersion of a, replacing a(s) by a(O), ^{we obtain}

(A. 11) ^{holds as} long as s f3 11 Li. Actually, ^{it also} implies ^{y b, since we assumed that} 7(f.) ^{was con-}

vergent ^when 8 - 0. Under such conditions, py ^«-l

and (A.11) ^{reduces to}

From our differential approach, ^{we know that} f and

f ^{must have the form} (17). It follows that

(A. 13) provides ^the required ^{link between the} regions

e » d and e « - J.

- Strictly speaking, ^{the above} proof ^{does not} apply

to the run away case y > tJ. However, ^{we note that} (A. 13) follows from a solution of (9) ⁱⁿ the transition range ( - A, ⁺ d ). ^In this limited ^{region, it does not}

matter much whether f ^{decreases or not when}

8 - T > ^{d .} Consequently, (A. 13) should hold irres-

pective ^{of the ratio} ylô, ^as long ^as 5, y, 4 « ^{T. Mathe-} matically, this contention can be proved by ^an appro-

priate analytic continuation of the above argument.

We did not attempt it, ^{as the} result looks fairly ^obvious.

References [1] ^The problem ^{is much more} complicated ^{when the} particles ^are

coupled. ^The drift is then controlled by ^soliton motion, ^a problem ^{which is} currently investigated by many people : See, among others TRULLINGER, ^{S. E. et} al., Phys. ^Rev.

Lett. 40 (1978) 206, 40 (1978) 1603 ;

GUYER, ^R. A., MILLER, ^M. D., Phys. ^{Rev. A 17} (1978) ^1774.

An alternative approach ^{to the} problem ^{has been} given by LANDAUER, ^R. (private communication).

[2] AMBEGAOKAR, V., HALPERIN, ^B. I., Phys. ^{Rev. Lett. 22} (1969)

1364.

The relationship ^{to a critical} point ^{has been} emphasized by BISHOP, A. R., TRULLINGER, ^S. E., Phys. ^{Rev. B 17} (1978) 2175 ;

SCHNEIDER, T., STOLL, ^E. P., MORF, R., ^{to be} published.

[3] KRAMERS, ^H. A., Physica ⁷ (1940) ^284.

[4] ICHE, G., NOZIÈRES, P., ^J. Physique ³⁷ (1976) ^1313.

[5] FRENKEL, J., ^{See for instance} his book The Kinetic Theory of Liquids (Dover, ^New York) ^1955.

[6] IVACHENKO, Yu, M., ZILBERMAN, L. A., ^{JETP Lett. 8} (1968) ^113.

[7] HUBERMAN, B. A., SEN, ^P. N., Phys. Rev. Lett. 33 (1974) ^1379.

FULDE, P., PIETRONERO, L., SCHNEIDER, ^W. R., STRÄSSLER, S., Phys. Rev. Lett. 35 (1975) 1776;

DIETERICH, W., PESCHEL, I., SCHNEIDER, ^W. R., ^Z. Phys. ²⁷ (1977) ^177.

[8] See for instance the famous review article of CHANDRASEKHAR, S., ^{Rev. Mod.} Phys. 15 (1943) ^2.

[9] MUSHKHELISHVILI, ^N. I., Singular integral equations, ^{P. Noor-} dhoff N. V. (Gröningen, Holland) ^1953.

[10] KURKIJÄRVI, J., AMBEGAOKAR, V., Phys. ^{Lett. 31A} (1970) ^314.