Diffusion in tilted periodic potentials : Enhancement , universality , and scaling

P. Reimann, 1 C. Van den Broeck, 2 H. Linke, P. Hänggi, J. M. Rubi, and A. Pe ́rez-Madrid Institut für Physik, Universita ̈t Augsburg, Universita ̈tsstrasse 1, D-86135 Augsburg, Germany Limburgs Universitair Centrum, B-3590 Diepenbeek, Belgium Department of Physics, University of Oregon, Eugene, Oregon 97403-1274 Department de Fisica Fonamental, Facultat de Fisica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain ~Received 4 September 2001; published 12 February 2002 !


I. INTRODUCTION
Within the realm of thermal equilibrium, the overdamped force-free thermal diffusion of a single Brownian particle as considered, e.g., by Einstein in Ref. ͓1͔, is always reduced when an additional periodic potential is switched on ͓2͔, and it is therefore tempting to conjecture a qualitatively similar behavior at least for time-independent nonequilibrium systems.A first main conclusion of our present explorations is that the opposite is the case: The effective diffusion coefficient of a Brownian particle in a periodic potential that is driven away from equilibrium by a static ''tilting force'' can become arbitrarily much larger than in the presence of thermal noise alone.A striking consequence of our finding is the possibility to observe thermal diffusion of macroscopic particles on macroscopic time and length scales at roomtemperature in appropriate tilted periodic structures.
A second main result of our present paper are scaling relations for the diffusion coefficient that become asymptotically exact in the limit of weak thermal fluctuations and small deviations from the critical tilt ͑i.e., the threshold at which deterministically running solutions set in͒.Furthermore, the asymptotic behavior of the diffusion leads to a classification into different universality classes with scaling exponent and scaling function depending on the characteristics of the potential at the critical tilt.These conceptsscaling and universality-are a recurrent theme in many branches of statistical physics, such as, e.g., critical phenomena, hydrodynamics, or low-dimensional nonlinear dynamics ͓3͔.Most closely related to our present findings are the scaling and universality phenomena as observed in the context of so-called deterministic diffusion ͓4 -8͔, especially in the presence of noise ͓9-11͔, and of noisy systems at a saddlenode bifurcation ͑e.g., relaxation oscillations͒ ͓12,13͔.
At the basis of all our above mentioned findings is an exact analytical expression for the diffusion coefficient, applicable to arbitrary periodic potentials, arbitrary tilts, and arbitrary strengths of the thermal noise ͓see Eq. ͑22͒ below͔.
Besides describing a real Brownian particle, thermal diffusion in a tilted periodic potential, as we will consider it here, is of relevance in numerous other contexts ͑see also Chap.11 in Ref. ͓14͔͒, such as Josephson junctions ͓15͔, the motion of fluxons in superconductors ͓16͔, rotating dipoles in external fields ͓17͔, the rotation of molecules in solids ͓18͔, superionic conductors ͓19͔, charge density waves ͓20͔, synchronization phenomena ͓21͔ in electrical circuits as described by the Adler equation or in phase locked loops ͓22͔, mode locking in laser gyroscopes ͓23͔, plasma accelerators ͓24͔, diffusion of atoms and molecules on crystal surfaces ͓25͔, particle separation by electrophoresis ͓26͔, biophysical processes such as neural activity ͓27͔ and intracellular transport ͓28͔, and possibly also for the explanation of the matterantimatter asymmetry of the universe ͓29͔.Also worth noting is the fact that the Brownian motion in a ''traveling periodic potential'' ͑pump͒ of the form V 0 (xϪvt) can be readily mapped onto a static tilted periodic potential ͓30͔.Moreover, our results near criticality ͑marginal stability͒ are universal for general dynamical systems close to a saddle-node bifurcation ͓12,13͔, such as for instance relaxation oscillations.
With the present work we continue and explain in more detail our brief account in Ref. ͓31͔.The organization of the paper is as follows: In Sec.II we introduce the model and the basic quantities of interest, namely, the average particle current and the effective diffusion coefficient.In Sec.III we derive as our first main result the relation ͑15͒ between the diffusion coefficient and the first two moments of the first passage time distribution.As a consequence, the closed analytical expression ͑22͒ can be inferred.Section IV is devoted to the exploration of universality and scaling properties of the diffusion coefficient near the threshold of deterministically running solutions ͑critically tilted periodic potentials͒, predicting a giant enhancement of the free thermal diffusion under suitable conditions.The latter result is exemplified in Sec.V for the special case of a mechanical Brownian particle that moves in a critically tilted geometrical profile.The summary and discussion of our findings is presented in Sec.VI.
x ˙͑t ͒ϭϪVЈ"x͑ t ͒…ϩ͑ t ͒, ͑1͒ V͑x ͒ªV 0 ͑ x ͒ϪxF, ͑2͒ where is the viscous friction coefficient ͑static mobility͒ of the particle and where the total potential V(x) consists of a periodic part V 0 (x) with period L, and a homogeneous, static ''tilting force'' F. Further, thermal fluctuations are modeled as usual ͓32͔ by Gaussian white noise with zero average and correlation ͗͑t͒͑s͒͘ϭ2kT␦͑tϪs͒, where T is the temperature, k is the Boltzmann constant, and ͗¯͘ indicates the ͑nonequilibrium͒ average over a statistical ensemble of realizations in Eq. ͑1͒.Finally, as compared to a full fledged Newtonian equation of motion, an inertia term mx ¨(t) is missing on the left-hand side of Eq. ͑1͒.In other words, this inertia term is assumed to have a negligibly small effect in comparison with the other forces appearing in Eq. ͑1͒, hence the name overdamped motion ͓14,32͔.
A first basic quantity of interest is the average particle current in the long-time limit ͑i.e., after transients due to initial conditions have died out͒ The analytical solution for this current goes back to Stratonovich ͓33͔ and has subsequently been rederived many times ͓see, e.g., Chap.11 in Ref. ͓14͔; the explicit formula will be given in Eq. ͑18͒ below͔.The fact that such an exact closed solution can be given without any further restrictions in the model ͑1͒ is rather exceptional and has given this model the status of a ''hydrogen atom'' in the context of Brownian motion theory.
In our present study, the quantity of central interest will be the effective diffusion coefficient . ͑6͒ Exact analytical results are known in two special cases.First, in the absence of the periodic potential V 0 (x) in Eq. ͑1͒ a straightforward calculation yields the so-called Einstein relation ͑7͒ for arbitrary values of the static tilt F ͓34͔.Second, in the absence of a tilt F, the following analytic prediction for the diffusion coefficient is due to ͓2,35͔ It has been furthermore demonstrated in Ref. ͓2͔ that D рD 0 , basically by applying the Cauchy-Schwartz inequality to the denominator in Eq. ͑8͒.
The evaluation of the diffusion coefficient ͑6͒ in the presence of both an arbitrary tilt F and an arbitrary periodic potential V 0 (x) is not obvious.One of the main objectives of the present work is the derivation of such a general, exact diffusion formula, analogous to Stratonovich's result for the current.

III. EVALUATION OF THE DIFFUSION COEFFICIENT
In order to evaluate the diffusion coefficient ͑6͒ we will take advantage of another quantity that is analytically known for the model ͑1͒, namely, the moments of the first passage time.To define these quantities, we consider the stochastic process ͑1͒ with an arbitrary but fixed seed x(0)ϭx 0 and we denote by t(x 0 →b) the time until an arbitrary but fixed point b is reached for the first time.Then the nth moment of the first passage time is the statistical average In what follows, we will temporarily restrict ourselves to the case FϾ0 and bϾx 0 , since otherwise the averages in Eq. ͑9͒ may diverge.Then, for the one-dimensional dynamics ͑1͒, these moments of the first passage time are given by the well-known closed analytical recursion ͑see, e.g., Sec.7 in Ref. ͓32͔ and further references therein͒ for nϭ1,2 . . .and with T 0 (y→b)ϵ1.Note that the convergence of the integrals in Eq. ͑10͒ is guaranteed by our assumption that FϾ0 and bϾx 0 .In principle, it is quite plausible that all properties of the stochastic process ͑1͒ should be expressible in terms of the moments ͑10͒ and, in this sense, available in closed analytical form.In practice, the explicit connection between a given quantity of interest and the moments is, however, not at all obvious.
We now come to the first main point of our paper, namely, the derivation of an exact expression for the diffusion coefficient D in terms of the mean first passage time T 1 (x 0 →b) and the so-called first passage time dispersion To this end, we denote by a an arbitrary point between x 0 and b.Then the time t(x 0 →b) that the stochastic process in Eq. ͑1͒ needs to travel from x 0 to b can be decomposed into the time to travel from x 0 to a, plus the time to travel from a to b.For a white noise driven process ͑1͒, the latter two times are statistically independent of each other ͓36͔.Further, since the process ͑1͒ is homogeneous in time, all statistical properties of t(x 0 →b) are exactly the same as those of t(x 0 →a)ϩt(a→b) with t(x 0 →a) and t(a→b) being sta-tistically independent of each other.It then follows readily from the definitions in Eqs.͑9͒ and ͑11͒ that the mean first passage time T 1 (x 0 →b) and the dispersion ⌬T 2 (x 0 →b) are additive quantities, i.e., As a second consequence it follows that t(x 0 →x 0 ϩlL) is statistically equivalent to a sum of l independent, random variables t(x 0 →x 0 ϩL),...,t"x 0 ϩ(lϪ1)L→x 0 ϩlL…, and due to the periodicity ͑3͒, they are identically distributed.
Invoking the central limit theorem, the distribution of the first passage times t(x 0 →x 0 ϩlL) thus approaches ͑for large l͒ a Gaussian distribution with mean value lT 1 (x 0 →x 0 ϩL) and variance l⌬T 2 (x 0 →x 0 ϩL).
Next, we introduce a discrete set of ''coarse-grained states'' along the x axis ͕x m ªx 0 ϩmlL͖ mϭϪϱ ϱ with mutual distance lL, where l is a large but fixed integer ͓37͔.The process x(t) is said to be in a certain ''state'' from the instant of time it hits the associated point x m until the moment it hits one of the adjacent neighboring points x mϮ1 .It follows that both the current ͗x ˙͘ and the diffusion coefficient D are iden- tical for the original process x(t) and its coarse-grained counterpart due to the long-time limits in the respective definitions ͑5͒ and ͑6͒.Next, we note that ''backward transitions'' x m ‫ۋ‬x mϪ1 require climbing up an ''energy ramp'' of height lLF by thermal activation and are thus suppressed by a Boltzmann factor ͑barometric formula͒ exp͕ϪlLF/kT͖ compared to x m ‫ۋ‬x mϩ1 , i.e., ''sliding down the ramp.''For sufficiently large l we, therefore, can safely neglect transitions x m ‫ۋ‬x mϪ1 .The remaining ''forward transitions'' between neighboring ''states'' x m and x mϩ1 are identically distributed random events ͓38͔ with a probability distribution that is identical to the first passage time distribution for the original process x(t).In particular, the moments of the first passage time T n (x m →x mϩ1 ) are thus identical for the original process x(t) and its coarse-grained counterpart.On the other hand, we have seen above that for sufficiently large l, all these moments and hence the entire coarse-grained process is completely fixed by the mean first passage time T 1 (x 0 →x 0 ϩL) and the dispersion ⌬T 2 (x 0 →x 0 ϩL).As our main conclusion we thus find that if two processes (1) yield the same values of T 1 (x 0 →x 0 ϩL) and ⌬T 2 (x 0 →x 0 ϩL) then ͗x ˙͘ and D will also be the same in the two cases.
With the above construction at our disposal, we may conclude ͓31͔ that The proof of these relations follows from the consideration of the special case with a potential V 0 (x)ϵ0 in Eq. ͑1͒, implying ͗x ˙͘ϭF/ and Eq.͑7͒.The evaluation of T 1 (x 0 →x 0 ϩL) and ⌬T 2 (x 0 →x 0 ϩL) according to Eqs. ͑10͒ and ͑11͒ is straightforward and one sees that the relations ͑14͒, ͑15͒ are indeed fulfilled.But from our conclusion at the end of the preceding paragraph, it now follows that Eqs.͑14͒ and ͑15͒ are also satisfied for a process ͑1͒ with an arbitrary periodic potential V 0 (x).We remark that here and in the following, the reference point x 0 is arbitrary.By introducing Eqs.͑10͒ and ͑11͒ into Eq.͑15͒ an analytical formula for D is recovered within our so far used restriction that FϾ0 ͑otherwise several terms in this formula would diverge͒.To remove this restriction we first rewrite the y integral in Eq. ͑10͒ with nϭ1 as where we have introduced dy e ϪV͑ y ͒/kT , ͑19͒ and where ''I Ϯ '' indicates that the index may be chosen to be either ''ϩ'' or ''Ϫ''.The equivalence of these indices in Eq. ͑18͒ follows by interchanging in the denominator the order of the two integrations in combination with some additional steps ͑see also Appendix A͒.We remark that the wellestablished formula ͑18͒ can also be used, by reversing steps, as a derivation of Eq. ͑14͒.For later use, we also note that By similar manipulations as used in the derivation of Eq. ͑18͒ ͑the details of which are given in Appendix A͒, we obtain for the diffusion coefficient ͑15͒ a central result of this paper, namely ͓31͔, with arbitrary reference points x 0 and arbitrary indices in I Ϯ both in the numerator and the denominator.So far, we have restricted ourselves tacitly to FϾ0.It is not difficult to see that our results above for both the current and the diffusion coefficient remain valid also for FϽ0.Moreover, for both of them, the limit F→0 does not give rise to any problems: In Eq. ͑18͒ one recovers the obvious result ͗x ˙͘ϭ0 ͓28͔ and by observing that in Eqs.͑19͒ and ͑20͒ the y integral is x independent one readily recovers Eq. ͑8͒ from Eq. ͑22͒.
In other words, the above closed expressions for both the current ͑18͒ and the diffusion ͑22͒ are exact analytical results for arbitrary periodic potentials V 0 (x) and arbitrary forces F. While the current formula has been known for more than 30 years, the corresponding compact expression for the diffusion coefficient has, to the best of our knowledge, been derived here for the first time ͑see also the discussion in Sec.VI͒.
For V 0 (x)ϵ0 one recovers from Eq. ͑22͒ by means of a straightforward calculation the Einstein relation ͑7͒.For nontrivial potentials V 0 (x), the above general analytical expression is still rather complicated.One may evaluate it numerically for arbitrary potentials V 0 (x), and one may try to simplify it analytically for some special limits.A particularly interesting such limit will be considered in the next section.We remark that an analytical discussion of other limits in Eq. ͑22͒ is also possible, but will not be pursued further here ͑see also the remark below on the weak noise limit͒.
As far as the numerical evaluation of our formula ͑22͒ is concerned, a representative example for the sine potential is depicted in Fig. 1.The purpose of this figure is threefold.First, it confirms within the numerical accuracy of the simulations that our analytical prediction ͑22͒ is indeed exact.Second, it contains a comparison with a formula for the diffusion coefficient recently proposed by Constantini and Marchesoni ͓39͔ of the form

͑24͒
with ͗x ˙͘ given by Eq. ͑18͒.This prediction is expected to capture the correct qualitative behavior of D under rather general conditions and can be shown to become asymptotically exact in any of the three limits F→0, F→ϱ, V 0 (x)/kT→0.As far as the quantitative behavior of D under general conditions is concerned, our findings in Fig. 1 show that the formula from Eq. ͑24͒ is at most a rough approximation-see also the discussion at the end of the next section.Third, the most interesting feature in Fig. 1 is the resonancelike behavior of the diffusion coefficient around that value of the tilt F for which the potential in Eq. ͑2͒ ceases to exhibit local extrema, which apparently gets more and more pronounced as the thermal noise strength kT and thus the bare ͑force-free͒ diffusion coefficient D 0 ϭkT/ in Eq. ͑7͒ decreases.In the next section we will consider in more analytical detail this special limit of weak noise in combination with a tilt F close to its critical value.
We finally remark that, for both numerical and analytical purposes, the reformulation ͑22͒ of the original analytical result in Eqs.͑15͒ and ͑10͒ simplifies matters a lot in the weak noise limit, since only a very small ͑and usually quite evident͒ region of z values then contributes significantly to the integrals in Eq. ͑21͒.The most involved case is the nearcritical regime treated in detail below.In any other case, the weak noise limit can be handled rather straightforwardly by standard steepest descent-type methods.

IV. UNIVERSALITY AND SCALING NEAR THE CRITICAL TILT
We now turn to the case of a critically tilted periodic potential, i.e., we choose FϭF c such that V(x) in Eq. ͑2͒ exhibits a strictly monotonic behavior with the exception of exactly one inflection point within each period L. In other words, the tilted potential just ceases to display any local maxima and minima ͑saddle-node bifurcation ͓13͔͒, corresponding to the threshold beyond which deterministically running solutions set in.When looking upon x as a phaselike variable, we may also speak of relaxation oscillations in this context.
Without loss of generality we assume that F c Ͼ0 ͑potential ''tilted to the right''͒ and that the inflection point is at xϭ0 ͑modulo L͒, i.e., V(x) is strictly monotonically decreasing, VЈ(x)Ͻ0, unless x is a multiple of L. Next we assume that for FϭF c ϩ⑀ with F c Ͼ0 and ⑀ small, ͑25͒ the potential V(x) can be written in the general form V͑x ͒ϭϪ sgn͑x ͉͒x͉ q Ϫ⑀xϭϪx͉x͉ qϪ1 Ϫ⑀x, ͑26͒ in the vicinity of the critical point xϭ0.We furthermore restrict ourselves to the cases Ͼ0, qϾ1, ͑27͒ implying differentiability with VЈ͑x ͒ϭϪq͉x͉ qϪ1 Ϫ⑀. ͑28͒ In the remainder of the interval ͓ϪL/2,L/2͔, the potential V(x) may be an arbitrary strictly monotonically decreasing smooth function, while outside ͓ϪL/2,L/2͔ it is fixed by Eqs.͑2͒,͑3͒.An example with qϽ1 is given at the end of this section.
We remark that in the generic case we have For instance, this is so whenever V 0 (x) in Eq. ͑2͒ is an analytic function of x.Nevertheless, more general q values are also worth studying, as our results below will demonstrate.They can be readily realized experimentally by tailoring the form of V 0 (x) accordingly.We finally note ͓using Eqs.͑2͒,͑3͔͒ that V(L/2)ϪV(ϪL/2)ϭϪFL.Assuming that Eq. ͑26͒ is a rough approximation for V(x) in the entire interval ͓ϪL/2,L/2͔ it follows that Ϫ2(L/2) q Ϫ⑀L ϷϪFL and hence Our second main assumption throughout this section is that the thermal energy kT is small, in the sense that kTӶLF c .͑31͒ Our main goal in what follows is to determine the behavior of the diffusion coefficient D for asymptotically small ⑀ and kT.
It is instructive to extend for a moment the approximation ͑26͒ to the entire x axis and to consider the corresponding dynamics ͑1͒ for ⑀ϭ0 and in the zero temperature limit (t)ϵ0, i.e., x ˙͑t ͒ϭ q ͉x͑t ͉͒ qϪ1 .͑32͒ A straightforward calculation then shows that in order to reach the inflection point xϭ0 from a seed x(0)Ͻ0, an infinite amount of time is needed if qу2, while a finite time is sufficient when qϽ2 ͑but still qϾ1͒.On the other hand, for x(0)→Ϫϱ a finite time suffices to reach a small neighborhood of xϭ0 for qϾ2, while this traveling time diverges for qр2.Analogous results are recovered for the traveling times in the region xϾ0.As a consequence, fundamentally differ-ent kinds of behavior of the corresponding mean first passage times for qϾ2 and qϽ2 are expected also for asymptotically small but finite kT ͑but still ⑀ϭ0͒ ͓40͔: For qϾ2 the motion of the particle ͑1͒ is dominated by the passage through the vicinity of xϭ0, where Eq. ͑31͒ is valid, whereas for qϽ2 one expects that the passage through this region is negligible, and outside this region the influence of the noise can be ignored.A similar behavior is still expected for finite ⑀-values, provided they become sufficiently small as kT approaches zero, while otherwise again fundamentally different realms may be encountered.In the following subsections, a more rigorous, quantitative version of these heuristic arguments will be elaborated.
Regarding the case qϽ1 we restrict ourselves to the specific example of a critically tilted piecewise linear potential.More precisely, we consider the diffusion of particles in a step-type potential, i.e., V(x) decreases proportional to the integer part of Ϫx/L, with kT much smaller than the potential step.This problem is a priori not simple at all, and was in fact the question that motivated the entire present investigation.However, with the general framework developed above now at hand, one readily finds from Eqs. ͑22͒ and ͑15͒ the result Dϭ2D 0 /3 for small kT, i.e., the free diffusion coefficient is reduced by the factor 2/3.

A. Evaluation of the diffusion coefficient
In this subsection we derive our central results ͑53͒-͑55͒ for the scaling behavior of the effective diffusion coefficient.
We first focus on the evaluation of the integral ͐ x 0 x 0 ϩL dx I Ϯ (x) appearing in Eqs.͑18͒ and ͑22͒.To keep things simple, we temporarily focus on the index ''ϩ'' and make the specific choice x 0 ϭϪL/2, but it is clear that the final result will be valid for both indices and any x 0 .Because of Eq. ͑31͒ one can replace the lower integration limit xϪL in Eq. ͑19͒ by Ϫϱ to a very good approximation ͓see also Eq. ͑A1͒ in Appendix A͔.Next we evaluate a part of the integral ͐ ϪL/2 L/2 dx I ϩ (x), namely ͓cf.Eq. ͑21͔͒,

͑33͒
where we have introduced Here A is a dimensionless number, and in the second relation in Eq. ͑34͒ we have exploited Eq. ͑30͒.In the following, we will always assume that A is very large, while a is so small that Eq. ͑26͒ can be applied in the region ͉x͉рa ͓such a choice of A and a is guaranteed to be possible due to Eq. ͑31͔͒.Observing that for small kT only z values very close to zero contribute notably in the second integral in Eq. ͑33͒, a Taylor expansion of the integrand about zϭ0 yields after a straightforward calculation the result

͑35͒
In the last identity we have exploited the fact that a in Eq. ͑34͒ belongs to the region where Eq. ͑26͒ applies if kT becomes sufficiently small, and that within this region ͉VЈ(x)͉ will be much smaller than it is outside it.In the same way one finds that In the remaining integral

Ϫϱ
x dy e y͉͑y͉ qϪ1 ϩ␥ ͒ .͑40͒ Note that K(x,␥) is a dimensionless function and that both its arguments ͑x and ␥͒ are dimensionless as well.We also remark that if we had worked with I Ϫ (x) instead of I ϩ (x) then the results ͑35͒ and ͑37͒ would have been recovered without any modification, while on the right-hand side of Eq. ͑38͒ the integrand K(Ϫx,␥) would have appeared, which of course gives the same result as K(x,␥) after integration over x.Upon adding up the three contributions ͑35͒, ͑36͒, ͑38͒ one sees that for qϾ2 the last one dominates for any small but fixed choice of a in Eq. ͑34͒ as kT becomes small.In particular, the latter contribution ͑38͒ converges for kT→0, which is basically a consequence of the fact that the mean first passage time to infinity is finite for qϾ2.Recalling that the final result does not depend on the specific choice x 0 ϭϪL/2 and the index ''ϩ'', we can conclude that ͓31͔ up to a relative error that approaches zero as kT decreases and provided that ⑀ also decreases such that the dimensionless number ␥ in Eq. ͑39͒ remains constant.Note that Eq. ͑41͒ has the form of a scaling law with a completely universal, dimensionless scaling function G 1 for any given value of the ''critical exponent'' q.The equivalence of both signs in Eq. ͑42͒ is obvious, but we have kept both of them in order to indicate the effect of the two possible signs in Eq. ͑41͒.
In the opposite case qϽ2 things are more complicated since later we will be interested not only in asymptotically small kT and ⑀ with ␥ from Eq. ͑39͒ kept fixed, but also in the case of negative ⑀ values such that the corresponding negative ␥ values diverge logarithmically as kT approaches zero.To this end, we henceforth set Aª͑L q /kT͒ 2/q .͑43͒ In this way, Aӷ͉␥͉ as kT→0 and a form ͑34͒ tends to zero ͓other choices than in Eq. ͑43͒ with the same properties would also be possible͔.It follows that the contributions of order A Ϫq in Eqs.͑35͒ and ͑36͒ can be neglected for asymptotically small kT.By closer inspection one further can deduce that in the remaining integrals ͑35͒ and ͑36͒ a ␥ value that diverges at most logarithmically with kT has an asymptotically negligible effect as well, i.e., we can formally set ⑀ϭ0 in those integrals.The remaining integrals in Eqs.͑35͒ and ͑36͒ converge if one formally lets a tend to zero, implying the asymptotically exact approximation

͑44͒
where Note the implicit kT dependence of G ˆ1 via Eq.͑43͒.In spite of this dependence, one finds that the first term on the righthand side of Eq. ͑44͒ is negligible in comparison with the second for asymptotically small kT with the possible exception of very large, negative ␥ values.In the latter case, we may evaluate Eq. ͑45͒ by means of a saddle point approximation with the result G ˆ1(␥)ӍS(␥), where Ϫ1 ͉͒␥/q͉ q/͑qϪ1 ͒ ͖.

͑46͒
The Heaviside step function ⍜͑Ϫ␥͒ has been introduced in order to make S(␥) well defined for arbitrary ␥.Closer in-spection shows that the difference G ˆ1(␥)ϪS(␥) is for all values of ␥ negligible in comparison with the second term on the right-hand side of Eq. ͑44͒.In other words, we obtain the result

͑47͒
As already noticed, for positive or moderately negative ␥ values the second term on the right-hand side dominates.Since the quantity ͑47͒ is basically equivalent to the mean first passage time T 1 (x 0 →x 0 ϩL) ͓cf.Eq. ͑A3͔͒ this approximate kT independence in Eq. ͑47͒ is in agreement with our heuristic discussion above ͓see also below Eq. ͑31͔͒.On the other hand, for sufficiently large negative ␥ values, the first term on the right-hand side in Eq. ͑47͒ takes over, reproducing the expected Arrhenius-type behavior ͑46͒ for the escape time over a potential barrier ͓32͔.Finally, we note that also in the case qϾ2 a saddle point approximation for large negative ␥ values in Eq. ͑42͒ leads to the very same result S(␥) as in Eq. ͑46͒.
The above results imply for the current ͗x ˙͘ in Eq. ͑18͒ that The ''crossover'' case qϭ2 requires a separate treatment that is relegated to the Appendix B.
For the evaluation of the numerator in Eq. ͑22͒ one proceeds in exactly the same way as for the denominator, and we only report here the final results, As in Eq. ͑42͒, we have kept in Eq. ͑51͒ both signs, though their equivalence is obvious.Moreover, for qϾ4/3 a saddle point approximation for large negative ␥ values in Eq. ͑51͒ again leads to the same expression ͓S(␥)͔ 2 /2 as appearing in Eq. ͑52͒.For the ''crossover'' case qϭ4/3, see Appendix B.
The implications of the above results for the diffusion in Eq. ͑22͒ are for 1ϽqϽ4/3.͑55͒ We recall that G 1 (␥), S(␥), and G 3 (␥) from Eqs. ͑42͒, ͑46͒, and ͑51͒, respectively, are dimensionless scaling functions of their dimensionless argument ͑39͒ that are completely universal for any given q value.Similarly, the fraction L q /kT as well as all the integrals appearing in Eqs.͑49͒-͑55͒ are dimensionless numbers.Under the approximative assumption that Eq. ͑26͒ is valid in the entire interval ͓ϪL/2,L/2͔ one obtains for those integrals the result The special ''crossover'' values qϭ2 and qϭ4/3 are addressed in Appendix B. Basically, these q-values continuously ͑but not smoothly͒ match together the results ͑53͒-͑55͒, involving certain logarithmic corrections similarly as it is the case for crossover exponents in the context of critical phenomena.

B. Discussion and examples
In this subsection we discuss Eqs.͑53͒-͑55͒.As far as the result ͑53͒ for qϾ2 is concerned, the most remarkable feature is the divergence of D/D 0 when kT tends to zero for any fixed ␥ value.In other words, we recover a giant enhancement of thermal diffusion ͑cf.Fig. 1͒.Specifically, for qϭ3, i.e., the most important case in practice ͓cf.Eq. ͑29͔͒, the scaling function G(␥)ªG 3 (␥)/G 1 3 (␥) appear-ing in ͑53͒ is depicted in Fig. 2. From this plot it follows that the scaling form ͑53͒ is rather well satisfied already for moderately small kT values and that the enhancement of diffusion is most pronounced for ␥ values in Eq. ͑39͒ of the order unity.A similar behavior is recovered for any other qϾ2.In other words, for qϾ2 the diffusion coefficient as a function of F exhibits a pronounced peak at FϭF c with a height of the order D 0 (L q /kT) 2/q and a width ⌬Fϭ⌬⑀ of the order 1/q ͓kT͔ 1Ϫ1/q ͓cf.͑39͔͒.Exploiting that D 0 ϰT ͓cf.͑7͔͒ we have, in particular, that D͑␥ max ͒ϳD 0 1Ϫ2/q ϰT 1Ϫ2/q , ͑58͒ ␥ max Ӎ0. ͑59͒ Further, we note that for large negative ␥ values one can exploit in Eq. ͑53͒ the saddle point approximation ͑46͒ for G 1 (␥) and the corresponding approximation ͓S(␥)͔ 2 /2 for G 3 (␥) as discussed below ͑52͒.
Returning to the height of the peak in the special case q ϭ3 the numerics from Fig. 2 in combination with Eq. ͑53͒ implies that for qϭ3, FϭF c .͑60͒ We remark that in the present special case qϭ3, the integrations in Eq. ͑42͒ can be exchanged and the integral over x performed, with the result Similarly, the integral over x can also be performed in Eq. ͑51͒.Further analytical simplifications are only possible for ␥ϭ0.In this case, Eq. ͑61͒ can be expressed as

͑62͒
where ⌫(z)ª͐ 0 ϱ dt t zϪ1 e Ϫt is the Gamma function with ⌫(1/6)Ӎ5.57.Due to a remarkable mathematical identity by Sigeti and Horsthemke ͓12͔ one further finds for the scaling function in Eq. ͑51͒ the result By means of this result one analytically recovers Eq. ͑60͒.
Next, we turn to the discussion of the result ͑54͒ for 4/3 ϽqϽ2.The salient difference in comparison with Eq. ͑53͒ is a competition between the two terms in the denominator on the right-hand side of Eq. ͑54͒.For any fixed ␥ value, the first term is negligible when kT becomes sufficiently small.Thus D/D 0 increases proportional to ͓kT͔ 4/qϪ3 , i.e., we find again a giant enhancement of thermal diffusion.More subtle is the behavior of Eq. ͑54͒ as a function of ␥ for a small but fixed kT value.For arbitrary positive as well as for moderately negative ␥ values it is still the second term in the denominator that dominates and thus the ␥ dependence of D is governed by G 3 (␥).These predictions are confirmed by comparison with a direct evaluation of the exact formula ͑22͒, see Fig. 3 for an example.In contrast to the case q ϭ3 ͑cf.Fig. 2͒ the asymptotic scaling form ͑54͒ with S(␥) ªG 3 (␥)/G 1 3 (␥) in Eq. ͑53͒ for qϭ3 versus its dimensionless argument ␥ from Eq. ͑39͒ by numerically evaluating Eqs.͑42͒,͑51͒.Full and dashed-dotted lines: Same as the respective lines in Fig. 1, but now plotted in the form (D/D 0 )(kT/L q ) 2/q ͓cf.Eq. ͑53͔͒ versus ␥ ͓cf.Eqs.͑25͒,͑39͔͒ with qϭ3 ͓cf.Eqs.͑23͒,͑29͔͒.
→0 is approached only for rather small kT values in Fig. 3.
On the other hand, for large negative ␥ values we can make use of the saddle point approximation ͓S(␥)͔ 2 /2 for G 3 (␥) as discussed below Eq. ͑52͒.Since S(␥) from Eq. ͑46͒ increases very fast with decreasing ␥, the right-hand side of Eq. ͑54͒ increases very fast as long as G 3 (␥) governs the ␥ dependence.However, again due to this fast increase, the first summand in the denominator starts to compete with the second summand and ultimately takes over, leading to a decrease of D proportional to 1/S(␥).Thus a peak appears at a ͑negative͒ ␥ value for which both terms in the denominator are of the same order of magnitude.The detailed quantitative calculation is straightforward and leads to the result where ␥ max is defined via the transcendental equation For small kT one thus obtains with Eq. ͑46͒ the leading order solution Ͻ0. ͑66͒ The width ⌬␥ of the peak is found to be of the order First of all, we note the logarithmic kT dependence in Eq. ͑66͒.In other words, the peak region is self-consistently described by our calculations, see above Eq.͑43͒.Second, taking into account Eq. ͑7͒ we see that the maximal effective diffusion coefficient in Eq. ͑64͒ is in fact independent of kT.
In other words, the maximal enhancement of diffusion is even stronger than for qϾ2, see Eq. ͑58͒.Under the approximative assumption that Eq. ͑26͒ is valid in the entire interval ͓ϪL/2,L/2͔ one obtains with Eq. ͑56͒ the explicit result ͓31͔ independent of kT.These asymptotic predictions for kT→0 are nicely confirmed already for moderately small kT values by comparison with a direct numerical evaluation of the exact formula ͑22͒ in Fig. 4. Note that while the maximizing ␥ value in Eq. ͑66͒ tends to Ϫϱ as kT→0, the corresponding tilt ⑀ϭFϪF c in Eq. ͑39͒ tends to zero.While kT may become arbitrarily small, the case kTϭ0 is not included in our above calculations, basically since this limit is singular in Eq. ͑10͒ and thus in Eq. ͑22͒.The basic physical reason for this singularity is the fact that the passage time through the interval ͓ϪL/2,L/2͔ remains finite for any finite kT but becomes infinite for kTϭ0.
Finally, we turn to the discussion of the result ͑55͒ for 1 ϽqϽ4/3.For positive and moderately negative ␥ values the two integrals on the right-hand side of Eq. ͑55͒ dominate and thus the diffusion coefficient is essentially constant.Consequently, the diffusion D is proportional to the bare value D 0 and in the limit V 0 Ј(x)→F c ϩ⑀, corresponding to q→1, the correct behavior ͑7͒ is also recovered.On the other hand, with increasingly negative ␥ values, the first term in the numerator in Eq. ͑55͒ takes over, while in the denominator the integral is still dominating.In other words, we essentially recover the same behavior as in Eq. ͑54͒.Especially, a peak of the form ͑64͒-͑68͒ arises for any 1ϽqϽ2, cf.Fig. 4.
In the case qϾ2 the approximation by Constantini and Marchesoni in Eq. ͑24͒ can be evaluated by means of Eqs.͑18͒,͑31͒,͑39͒, and ͑41͒, leading to Comparison with the exact asymptotics ͑53͒ shows that this approximation does not capture the correct scaling functions and exponents.A similar disagreement is obtained for 1Ͻq Ͻ2.On the other hand, one readily finds that the main qualitative features are correctly reproduced in all cases.

C. Basic physical mechanism
The basic physical mechanism responsible for the enhancement of the thermal diffusion may be understood along the following heuristic argument.As discussed below Eq. ͑32͒, for qϾ2, ⑀ϭ0, and small kT, the noisy dynamics ͑1͒ is dominated by the passage through the ''dynamical bottle-FIG.4. Diffusion coefficient ͑22͒ versus the tilt F for a potential V(x) defined via Eqs.͑25͒,͑26͒ with qϭ3/2 for all x ͓ϪL/2,L/2͔ while outside ͓ϪL/2,L/2͔ it is fixed by Eqs.͑2͒,͑3͒.
neck'' located at the inflection point xϭ0, cf.Eq. ͑26͒.Since ⑀ϭ0, a very small perturbation due to thermal noise is already sufficient to kick the particle across the inflection point xϭ0.This small variation in comparison with an unperturbed particle is subsequently greatly enhanced by the further dynamical evolution.The result is a huge dispersion for a statistical ensemble of particles subjected to different realizations of the noise.It is quite suggestive that the same basic mechanism subsists also for nonvanishing ⑀ values at least as long as they are so small that the corresponding ␥ values in Eq. ͑39͒ remain finite when kT approaches zero.Due to the same mechanism, one expects a very strong change of the particle current ͗x ˙͘ upon small changes of the tilt F near the critical value F c .In other words, a relation like in Eq. ͑24͒ is expected to be correct at least qualitatively.However, it is clear that this heuristic argument can be applied only for qϾ2, because only then the ''bottleneck'' near xϭ0 dominates the dynamics ͓cf. the discussion below Eq. ͑32͔͒.Since we have found an enhanced diffusion also for ⑀ϭ0 and 4/3рqр2 we have to conclude that such a simple argument only captures a part of the essential physics and that a relation of the form ͑24͒ cannot be strictly correct ͓cf.Fig. 1 and the discussion below Eq. ͑69͔͒.
Interestingly enough, a similar heuristic explanation in terms of dynamically enhanced thermal fluctuations seems again applicable for 2ϾqϾ1 near the peak in the diffusion coefficient described by Eqs.͑64͒-͑68͒: As discussed above Eq.͑64͒, in this peak region both summands in the denominator of Eq. ͑54͒ are comparable, reflecting the fact that the passage time through a very small neighborhood of xϭ0 is comparable to the traveling time through the remainder of ͓ϪL/2,L/2͔.On the other hand, the large negative ␥ value in Eq. ͑66͒ and the concomitant Arrhenius-type form of Eq. ͑46͒ indicate that the former time scale is governed by a thermally activated escape process across a potential barrier.On this basis, it is quite plausible that the dynamical enhancement of these thermally induced ͑rare͒ escape events will be maximal when both time scales are comparable.
The discussion below Eq. ͑32͒ implies that for qϾ2 the mean first passage time T 1 (x 0 →x 0 ϩL) is dominated by a small neighborhood of the inflection point xϭ0, while for 2ϾqϾ1 the region outside this small neighborhood is no longer negligible.From our results in Eqs.͑50͒,͑52͒ we can conclude that a similar crossover occurs for the first passage time dispersion ⌬T 2 (x 0 →x 0 ϩL) at qϭ4/3.In view of our central relations ͑14͒ and ͑15͒ the need to distinguish between two q regimes for the current ͗x ˙͘ in Eqs.͑48͒,͑49͒ and three q regimes for the diffusion D in Eqs.͑53͒-͑55͒ is then immediately clear.
We finally note that a suitable comparison between the directed and the diffusive transport is provided by the dimensionless number ͓41͔ L͗x͑t ͒͘ ,

͑70͒
where the second equality follows from Eqs. ͑14͒,͑15͒ and the third from Eqs. ͑5͒,͑6͒.For large positive ␥ one finds that Q becomes very small, while for large negative ␥ one finds that Q approaches unity.For qϭ3 and ␥ϭ0Ӎ␥ max ͓cf.Eq. ͑59͔͒ the relation ͑63͒ implies Qϭ1/3, whereas for 2Ͼq Ͼ1 and ␥ϭ␥ max ͓cf.Eq. ͑66͔͒ one finds that Qϭ2/9.In other words, in the most interesting regime close to the maximum of the diffusion coefficient the particle dispersion ͗x 2 (t)͘Ϫ͗x(t)͘ 2 is comparable to L times the mean dis- placement ͗x(t)͘ and similarly the first passage time disper- sion is comparable to the square of the mean first passage time.

V. MACROSCOPIC THERMAL DIFFUSION
In this section we consider the thermally induced diffusion of a real mechanical particle of spherical shape that moves in a liquid under the action of gravitation along the rigid surface of a critically tilted periodic geometrical profile, see Fig. 5.
The position of the particle is described by its coordinate y along some horizontal axis and its vertical position z.Gravitation is pointing in the negative z direction and the motion is constrained by a rigid surface according to z уh(y), where h(y) has the shape of a periodic profile with period L 0 that is critically tilted ''to the right,'' i.e., h͑ yϩL 0 ͒ϭh͑ y ͒Ϫh 0 , ͑71͒ with h 0 Ͼ0, see Fig. 5.The motion along the third spatial direction ͑perpendicular to both y and z͒ decouples from the motion in the y-z plane and can, therefore, be ignored.Without loss of generality we assume that h(y) has an inflection point at yϭ0 and thus satisfies for small y a relation analogous to Eq. ͑26͒ with ⑀ϭ0, i.e., h͑ y ͒ϭϪ 0 y͉y͉ qϪ1 .͑72͒ Physically, it is quite plausible that for asymptotically small kT, the constraint zуh(y) can henceforth be replaced by zϭh(y) without changing the dynamics along the y direction.A more detailed mathematical justification of this step is possible but not further elaborated at this place.At FIG. 5. A spherical Brownian particle in a liquid, rolling down a critically tilted, periodic surface h(y) under the action of gravitation, see also Eq. ͑71͒.
every given position ͓ y,h(y)͔, the thermal noise acting on the particle consists of a perpendicular and a tangential component relative to the geometrical profile h(y).Under the assumption that the thermal fluctuations can be modeled by Gaussian white noise, these two components are statistically independent at any given time and position.Furthermore, the perpendicular component has no effect because of the constraint zϭh(y).In order to achieve an effective onedimensional description of the form ͑1͒ the appropriate approach is thus to work with the path-length x along the geometrical profile as our generalized coordinate.The infinitesimal line element dx is then specified by the obvious relation dx 2 ϭdy 2 ϩdh 2 and the sign convention in the relation dxϭϮ(dy 2 ϩdh 2 ) 1/2 .Without loss of generality we choose the sign convention such that dxϭdy"1ϩ͓hЈ(y)͔ 2 … 1/2 and the x origin such that The potential energy of the particle with density and radius r due to the effects of gravitation and the buoyancy of the surrounding liquid with density l is given in terms of the generalized coordinate x by V͑x ͒ϭF g h"y͑x ͒…, F g ª͑4/3͒r 3 ͑ Ϫ l ͒g, ͑74͒ where y(x) is the inverse of x(y) ͑which obviously exists͒ and where g is the acceleration due to gravity.Observing that hЈ(0)ϭ0 in Eq. ͑73͒ it follows that V(x) is a critically tilted periodic potential that satisfies Eq. ͑26͒ for small x with ⑀ϭ0, ϭF g 0 .͑75͒ With Eqs.͑2͒, ͑3͒, ͑71͒, ͑73͒, and ͑74͒ it follows that We assume that the frictional force acting on the particle under consideration is of the Stokes form x ˙(t) with friction coefficient ϭ6 eff r,

͑78͒
where eff is the effective viscosity of the surrounding liquid.For a spherical particle that does not rotate and that is surrounded by an unbounded reservoir of liquid, eff is given by the bare viscosity of this liquid.A rigorous quantitative theory describing the thermal motion ͓note that also appears in Eq. ͑4͔͒ of the actual setup we have in mind should include the effects of rotational degrees of freedom and the quite intricate boundary effects, both mechanical and hydrodynamical ͓42͔.Here, we shall adopt the simplifying assumption ͓43͔ that all these effects are approximately captured by an appropriately renormalized viscosity eff .Further, we assume that all kinds of fluctuations within the liquid ͑density, temperature, etc.͒ are approximately captured by the thermal noise (t) in Eq. ͑1͒.Finally, we assume incompressibility of the liquid ͑and the particle͒ such that the densities and l are constant, and we exclude temperature gradients.
Finally, we should justify the assumption that the inertia term mx ¨(t), missing on the left-hand side of Eq. ͑1͒, is indeed negligible, at least in those aspects of the dynamics that are responsible for the asymptotic behavior in Eqs.͑53͒-͑55͒.In practice, this is a very delicate question and a truly satisfactory answer seems only possible by means of a comparison with numerical simulations that fully take into account these small but finite inertia effects.Intuitively, we expect that there is a range of small but still macroscopic particle sizes for which the overdamped description ͑1͒ would be admissible.A closely related question concerns the omission of memory effects both in the noise and the dissipation, which can be justified by closer inspection under the condition that the particle density is much larger than the liquid density l .
In the important case qϭ3, and with gӍ981 cm/s 2 , ͑79͒ ˆª10 Ϫ2 g/cm sӍ water , ͑80͒ ˆª10 g/cm 3 ͓ iron Ӎ7.9 g/cm 3 ͔, ͑81͒ r ˆª0.1 cm, ͑82͒ h ˆ0ª1.5 cm, ͑83͒ T ˆª293°K ͓room temperature͔, ͑84͒ the free diffusion coefficient ͑7͒ can be written, using Eq.͑78͒, as Further, one obtains from Eq. ͑60͒ for the diffusion coefficient at the critical tilt the formula In the actual experimental realization, which is presently under construction in the labs of one of the present authors ͑H.L.͒, each fraction appearing in Eqs.͑85͒ and ͑86͒ is of the order of unity ͓44͔.In particular, for typical shapes h(x) one finds that L q 0 /2 qϪ1 h 0 is a dimensionless number of the order of unity ͓cf.Eqs.͑30͒,͑75͒,͑76͒,͑77͔͒, independently of the actual value of L. It follows that the diffusion coefficient in the critically tilted periodic potential ͑86͒ is enhanced in comparison with the free thermal diffusion coefficient ͑85͒ by about nine orders of magnitude, so that it may well reach macroscopically observable values.The width of the peak of the diffusion coefficient with respect to variations of the po-tential tilt has been estimated above Eq.͑58͒, namely, ⌬F Ӎ 1/q ͓kT͔ 1Ϫ1/q .This width of the peak may be used to estimate the experimentally admissible deviation ⌬h 0 from the exact ''critical'' h 0 value in Eq. ͑71͒.Due to Eq. ͑76͒ we see that ⌬Fϭ⌬h 0 F g /L, yielding the following theoretical estimate for ⌬h 0 : Since the fractions on the right-hand side are again of the order of unity, this theoretically required precision is thus extremely high.
As discussed in the previous section, the largest enhancement of thermal diffusion is expected for 2ϾqϾ1.From Eq. ͑64͒ we then find for the maximal diffusion coefficient the result see also Eqs. ͑56͒ and ͑68͒.Again, all the fractions in the right-hand side are of order unity.In other words, if the fabrication of such a specially tailored profile h(y) with 2Ͼq Ͼ1 is possible with the necessary precision, then the enhancement of thermal diffusion can be further improved by another five orders of magnitude as compared to the case q ϭ3.For the required precision to hit the correct h 0 value in Eq. ͑71͒ one obtains ͓45͔ In comparison with Eq. ͑87͒, this theoretically required precision is considerably lower, especially for q values close to unity.In contrast to this global precision ⌬h 0 , the necessary local precision of the profile h(y) may be a more serious problem in practice.Finally, we remark that the maximally enhanced diffusion in Eq. ͑64͒ and hence in Eq. ͑88͒ is not reached exactly at the critical tilt, cf.Eq. ͑66͒.The corresponding ͑negative͒ deviation from the critical h 0 value in Eq. ͑71͒ is of the same order of magnitude as ⌬h 0 in Eq. ͑89͒, see also Eq. ͑67͒.

VI. DISCUSSION
In this paper, we have addressed the problem of overdamped Brownian motion in a tilted periodic potential in the presence of white thermal noise ͓31͔.Our first main result is the compact and exact expression ͑22͒ for the diffusion coefficient, valid for arbitrary tilted periodic potentials and arbitrary strengths of the thermal noise.At the basis of this result lies Eq. ͑15͒, connecting the diffusion coefficient with the mean first passage time and the dispersion of the first passage times.The relations ͑14͒ and ͑15͒ have been previously proposed ͑without proof͒ in the context of random walk theory on discrete lattices ͓46͔, and they are also wellknown asymptotic relationships in the particular case of socalled renewal processes, see, e.g., formulas ͑14͒ and ͑16͒ in Chap. 5 of Ref. ͓47͔.While all these works are concerned with models in discrete space, the use of the above relations for the continuous problem at hand has in fact been advocated ͑without proof͒ in Ref. ͓11͔, and an independent alternative derivation of Eq. ͑15͒, based on the results from Ref. ͓47͔, has recently been presented in Ref. ͓48͔.Note also that a different expression for the diffusion coefficient ͑24͒ was recently proposed in Ref. ͓39͔, which is, in general, only approximatively valid ͑see Fig. 1 and Sec.IV͒.As a side remark we mention that our result has nothing to do with the relation between the diffusion coefficient and the Lyapunov exponent in certain chaotic dynamics ͓49͔.In particular, the concept of Lyapunov exponents is useless in our context since it focuses on infinitesimal deviations between trajectories, while the diffusion coefficient here is governed by deviations ranging over many periods L.
It seem likely that a relation analogous to Eq. ͑15͒ can also be derived under more general conditions, e.g., in higher dimensions or beyond the overdamped limit ͓39,50͔.While in these cases the mean first passage times and the dispersion of the passage times are generally not known in analytical form, such a relation may still be useful for speeding up the numerical determination of the diffusion coefficient.For spatially discrete, periodic, one-dimensional systems, an exact expression for the diffusion coefficient has been derived by Derrida in Ref. ͓51͔, see also Ref. ͓52͔ for various generalizations.It is clear that Derrida's result will become equivalent to Eq. ͑15͒ or Eq.͑22͒ in the continuous space limit.Conversely, by considering periodic potentials V 0 (x) with very high barriers between neighboring local minima ͑''states''͒, an effective spatially discrete model is recovered.Again, it is clear that in this way our result will become equivalent to Derrida's discrete random walk result.In practice, our compact expression ͑22͒ for the diffusion coefficient is, however, considerably simpler than Derrida's.
Turning to our second main result, we recall that quantum mechanics is generally appreciated to be indispensable for the explanation, e.g., of the stability of atoms, molecules, and solids.Yet, verifying and exploiting basic quantum mechanical effects ''more directly'' on a macroscopic scale is currently attracting much attention.A somewhat similar situation arises with respect to the random microscopic fluctuations at the basis of statistical mechanics.As detailed in Sec.V, the present study suggests a very elementary experiment that would make those microscopic fluctuations ''visible'' on a macroscopic scale.In contrast to the quantum mechanical case, here the word ''macroscopic'' literally means ''observable by the naked eye.'' We note that somewhat related phenomena are also known in other far-from-equilibrium systems.For example, the spreading of particles can be enhanced by the critical fluctuations in the vicinity of a nonequilibrium phase transition to a convective state, resulting in a diverging diffusion coefficient Dϳ(TϪT c ) Ϫz , where z is a dynamical critical exponent ͓53͔.Another well-known example is the enhancement of dispersion by turbulence.Diffusion can also be amplified by the coupling to a convective ͑laminar͒ process as exemplified in the well-studied example of Taylor dispersion with a characteristic diffusion coefficient given by DϳD 0 Ϫ1 ϳT Ϫ1 , where D 0 is the usual ͑bare͒ molecular diffusion coefficient: see Ref. ͓54͔ for a review.In contrast to our present case, in those examples strong nonequilibrium fluctuations and/or gradients of the temperature and the velocity of the surrounding medium are a crucial ingredient.Two other examples that come closer to the situation of interest to us are, respectively, the Suzuki scaling law for relaxation from a marginally stable state ͓55͔ and the transient bistability in explosive systems ͓56͔.In these cases, the dispersion is increased by the amplification of the initial thermal spreading in the close vicinity of a metastable point through the subsequent fast dynamical evolution away from this point.From this point of view, our present problem may be considered as the dispersion of particles in a potential displaying a spatially periodic repetition of marginally stable states-recall also our qualitative explanation of the effect below Eq. ͑55͒.Finally, we mention that a somewhat similar, resonancelike enhancement of the free thermal diffusion has also been reported for various systems in the presence of a timedependent external driving force ͓57-61͔.While the behavior of the diffusion coefficient there is reminiscent of the present case, the underlying physical mechanisms are once again quite different.

APPENDIX A
We present in this appendix some details pertinent to the derivation of Eq. ͑22͒.We first introduce the quantities The last identity in Eq. ͑A1͒ follows from Eq. ͑17͒ and similarly for Eq.͑A2͒.Next, we rewrite the first two moments in Eq. ͑10͒ as we can rewrite Eq. ͑A3͒ as x 0 ϩL dx͓ϪKЈ͑x ͒H͑ x ͔͒.

͑A11͒
Using the fact that the product K(x)H(x) is L periodic, a partial integration yields T 1 ͑ x 0 →x 0 ϩL ͒ϭ 1

APPENDIX B
In this appendix we discuss in some detail the special cases qϭ2 and qϭ4/3 that have been omitted in the main text in Sec.IV.
In the case qϭ2 we may choose, similarly as in the discussion below Eq. ͑43͒, a large value of A and then approximately set ⑀ϭ0 and A Ϫq ϭ0 in Eq. ͑35͒.The remaining integral does not converge for a→0 but we may introduce a convergence-inducing term according to .͑B1͒ The first integral converges for a→0 and by actually performing this limit we make an error that vanishes for kT →0, see Eq. ͑34͒.The second integral in Eq. ͑B1͒ can be performed, with the result

͑B2͒
where we have used Eq.͑34͒.Exactly the same result is recovered for Eq.͑36͒.Turning to Eq. ͑38͒, a convergence inducing term for A→ϱ may be introduced according to

͑B4͒
Putting together everything, we finally find that

͑B5͒
where we have introduced As in the case of G 1 (␥) in Eq. ͑42͒, G ˜1(␥) is again a universal, dimensionless scaling function of its dimensionless argument ␥.Further, the saddle point approximation ͑46͒ can be applied for large negative ␥ values.Finally, C is a constant of order unity that depends on the details of the potential V 0 (x).For instance, when the form ͑26͒ is exactly valid on the entire interval ͓ϪL/2,L/2͔ then we have Cϭ0.
A similar calculation yields the result for qϭ4/3, ͑B8͒ where we have restricted ourselves to the leading order term in the weak noise limit kT/F c L→0, i.e., higher order terms analogous to those on the right-hand side of Eq. ͑B5͒ have been omitted.
For the resulting diffusion coefficient ͑22͒ one obtains x 0 ϩL dx L Ϫ2/3 F c ϪV 0 Ј͑x͒ ͬ 3 for qϭ4/3.͑B10͒ The detailed discussion of the result ͑B9͒ can be carried out in complete analogy to the one for 2ϾqϾ4/3 below Eq. ͑55͒.In particular, for sufficiently small kT, the second and third term in the denominator are negligible for positive and moderately negative ␥ values, while for large negative ␥ the saddle point approximation ͑46͒ for G ˜1(␥) and G 3 (␥) ϭ͓S(␥)͔ 2 /2 can be applied.In Eq. ͑B10͒ only the leading order terms for small kT have been kept.Thus the discussion of the dependence for a fixed kT upon variation of ␥ is not possible on the basis of Eq. ͑B10͒ but it is clear that this dependence will be of exactly the same form as the one for 4/3ϾqϾ1 discussed below Eq. ͑68͒.Especially, Eqs.͑64͒-͑68͒ can be taken over for qϭ4/3 without any change.
first integral now converges when A→ϱ, while the second one can be performed.Neglecting all contributions that tend to zero when A→ϱ, we thus obtain ͵