Flooding dynamics of diffusive dispersion in a random potential
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Flooding dynamics of diffusive dispersion in a randompotential
Michael Wilkinson, Marc Pradas and GerhardKlingAbstract
We discuss the combined effects of overdamped motion in a quenchedrandom potential and diffusion, in one dimension, in the limit where the diffusioncoefficient is small. Our analysis considers the statistics of the mean first-passagetime T ( x ) to reach position x , arising from different realisations of the randompotential: specifically, we contrast the median ¯ T ( x ), which is an informative de-scription of the typical course of the dispersion, with the expectation value h T ( x ) i ,which is dominated by rare events where there is an exceptionally high barrier todiffusion. We show that at relatively short times the median ¯ T ( x ) is explained bya ‘flooding’ model, where T ( x ) is predominantly determined by the highest bar-riers which is encountered before reaching position x . These highest barriers arequantified using methods of extreme value statistics. Keywords
Diffusion, Ornstein-Uhlenbeck process
Michael WilkinsonChan Zuckerberg Biohub,499 Illinois Street,San Francisco,CA93107, USAE-mail: [email protected] PradasSchool of Mathematics and Statistics,The Open University,Walton Hall,Milton Keynes, MK7 6AA,EnglandE-mail: [email protected] KlingUniversity of Aberdeen,Business School, King’s College,Aberdeen AB24 3FX,ScotlandE-mail: [email protected] Michael Wilkinson, Marc Pradas and Gerhard Kling
There are many situations where particles move under the combined influence ofthermal diffusion and a static (or quenched) random potential [1]. The particlesmight be electrons, holes or excitons diffusing in a disordered metallic or semicon-ductor sample [2], or molecules diffusing in a complex environment such as thecytoplasm of a eukaryotic cell [3]. The state of knowledge of this problem is sur-prisingly under-developed, and in this work we present new results on the simplestversion of this problem, in one dimension, where the equation of motion is˙ x = − d V d x + √ D η ( t ) . (1)Here V ( x ) is a random potential, D is the diffusion coefficient, and η ( t ) is a whitenoise signal with statistics defined by h η ( t ) i = 0 , h η ( t ) η ( t ′ ) i = δ ( t − t ′ ) (2)( h·i denotes expectation value throughout). We assume that V ( x ) is a smoothrandom function, defined by its statistical properties, which are stationary in x ,and independent of the temporal white noise η ( t ). The one and two-point statisticsof this potential are h V ( x ) i = 0 , h V ( x ) V ( x ′ ) i = C ( x − x ′ ) . (3)The correlation function C ( ∆x ) is assumed to decay rapidly as | ∆x | → ∞ . We alsoassume that the tails of the distribution of V are characterised by a large-deviation‘rate’ (or ‘entropy’) function J ( V ), so that when | V | is large, the probability densityfunction of V is approximated by P V ∼ exp[ − J ( V )] (4)where throughout we shall use P X to denote the probability density function(PDF) of a random variable X . If P V is a Gaussian distribution, then the en-tropy function is quadratic, J ( V ) ∼ V / C (0).It has been proposed that the behaviour of this system is diffusive, with an ef-fective diffusion coefficient which vanishes very rapidly as D →
0: Zwanzig [4] gavean elegant argument which implies that, when V ( x ) has a Gaussian distribution,the effective diffusion coefficient is D eff ∼ D exp (cid:20) − C (0) D (cid:21) . (5)An earlier work by De Gennes [5] proposes a similar expression. We discuss the ori-gin of this result, and present a generalisation of it to non-Gaussian distributions,in section 2. When D is small, this estimate for the diffusion coefficient dependsupon rare events where the potential is unusually large, and it is very difficultto verify equation (5) numerically. In addition, numerical experiments show thatthe model exhibits sub-diffusive behaviour and it has been suggested that thereis anomalous diffusion, in the sense that h x i ∼ t α , with 0 < α < looding dynamics of diffusive dispersion in a random potential 3 We should mention that there are also exact results [9, 10, 11] on a closely re-lated model (motion in a quenched velocity field, which is not the derivative of a po-tential with a well-defined probability distribution) showing that h x i ∼ (ln t ) / .This ‘Sinai diffusion’ process is fundamentally different, because the particle be-comes trapped in successively deeper minima of the potential, from which it takesever increasing time intervals to escape.We will argue that, while equation (5) and its generalisation to non-Gaussiandistributions describes the long-time asymptote of the dispersion of particles, thediffusive behaviour only emerges at very long times. At intermediate times, thedynamics of typical realisations is not diffusive. We show that it is determinedby the time taken to diffuse across the largest potential barrier which must betraversed to reach position x . The diffusion process is able to traverse a barrier ofheight ∆V after a characteristic time T ∼ exp( ∆V /D ) [12], and as time increasesthe height of the barriers which can be breached, leading to ‘flooding’ of theregion beyond, increases. According to this picture, the dispersion distance x isdetermined by a problem in extreme value statistics: how large must x be before wereach a barrier of height ∆V ≈ D ln t ? By considering the solution of this problemin extreme value statistics, we argue that the median ¯ T (with respect to differentrealisations of the potential V ( x )) of the mean-first-passage-time (averaged over η ( t )) satisfies ln (cid:16) x ˜ x (cid:17) ∼ J ( D ln ¯ T /
2) (6)where ˜ x is a lengthscale which characterises the typical distance between extremaof the potential. In the case where the potential has a Gaussian distribution, thisimplies that the dispersion is sub-diffusive, satisfyingln (cid:16) x ˜ x (cid:17) ∼ D (ln ¯ T ) C (0) (7)which is quite distinct from the usual anomalous diffusion behaviour, characterisedby power-laws such as h x i ∼ t α . After a sufficiently long time, the dynamicsbecomes diffusive, with a diffusion coefficient given by (5).Our arguments will depend upon making estimates of sums of the form S N = N X j =1 exp( f j /ǫ ) (8)where f j are independent identically distributed (i.i.d.) random variables, and ǫ is a small parameter, which we identify with the diffusion coefficient D . We termthe S N ‘extreme-weighted sums’, because the largest values of f j make a dominantcontribution to S N as ǫ →
0. In section 2 we show how the mean-first-passage timeis related to sums like (8), and in section 3 we analyse some of their statistics, whichare used in section 4 to justify our principle result, equation (6). Section 5 describesour numerical investigations, and section 6 is a summary.
Our discussion of the dynamics of (1) will focus on the mean first passage problem:what is the mean time T ( x ) at which a particle released from the origin reaches Michael Wilkinson, Marc Pradas and Gerhard Kling position x . First passage problems are discussed comprehensively in the book byRedner [13]. The result that we require can be found in multiple sources: [14] is theearliest reference that we are aware of and the key formula, equation (9) below,was already applied to equation (1) by Zwanzig [4].In this section we first quote the general formula for the mean first passagetime T ( x ), as a functional of the potential V ( x ). If particles are released at x = 0,the mean first passage time to reach position x is given by T ( x ) = 1 D Z x d y exp[ V ( y ) /D ] Z y d z exp[ − V ( z ) /D ] (9)where the averaging is with respect to realisations of the noise η ( t ) in the equationof motion (1), with V ( x ) frozen, so that T ( x ) is a functional of V ( x ).We then (subsection 2.1) discuss the result obtained by Zwanzig [4] for theexpectation value h T ( x ) i (averaged with respect to realisations of V ( x )). Zwanziggave the result for a potential with Gaussian fluctuations, which we extend tothe case of a general form for the large-deviation entropy function (as definedby equation (4)). The result obtained by Zwanzig suggests that the dispersion isdiffusive, with a diffusion coefficient D eff which vanishes in a highly singular fashionas D →
0. We shall argue that this result is a consequence of the expectation valueof T ( x ) being dominated by very rare large excursions of the potential V ( x ), andthat for typical realisations of the potential the dispersion is much more rapidthan the value of h T ( x ) i suggests. This requires a more delicate analysis of thestructure of the integrals in the expression for T ( x ), equation (9). In subsection2.2, we discuss how these integrals may be approximated by sums, analogous to(8), in the limit as D → h T ( x ) i = 1 D Z x d y Z y d z (cid:28) exp (cid:20) V ( y ) − V ( z ) D (cid:21) (cid:29) ∼ x D h exp( − V /D ) ih exp( V /D ) i (10)where in the second line we consider the leading order behaviour as x → ∞ . If themotion were simple diffusion, with V = 0, equation (9) would evaluate immediatelyto h T i = x / D , so that it is reasonable to identify x / h T i as the effective diffusioncoefficient. Hence, assuming that the PDF of V ( x ) is symmetric between V and − V , we have D eff = D [ h exp( V /D ) i ] . (11)When D is small, h exp( V /D ) i is dominated by the tail of the PDF of V , so that h exp( V /D ) i = Z ∞−∞ d V P V exp( V /D ) looding dynamics of diffusive dispersion in a random potential 5 ∼ Z ∞−∞ d V exp[ V /D − J ( V )] ∼ r πJ ′′ ( V ∗ ) exp[ V ∗ /D − J ( V ∗ )] (12)where V ∗ is the stationary point of the exponent, satisfying DJ ′ ( V ∗ ) = 1 . (13)From this we obtain D eff ∼ DJ ′′ ( V ∗ )2 π exp (cid:20) J ( V ∗ ) − V ∗ D (cid:21) . (14)In the Gaussian case, where J = V C (0) + 12 ln(2 πC (0)) (15)equation (14) agrees with (5).2.2 Summation approximationsIn order to understand the implications of equation (9), we should consider thebehaviour of the integral S ( x ) = Z x d y exp[ − V ( y ) /D ] (16)in the limit as D →
0. When D is small this quantity may be estimated from theminima of the potential: S ( x ) ∼ N X j =1 s πDV ′′− j exp( − V − j /D ) ≡ N X j − exp[ − ˜ V − j /D ] (17)where V − j are the values of the N minima between 0 and x , occurring at positions x − j , and where we have defined˜ V − j = V − j − D πD | V ′′− j | ! (18)Note that T ( x ) = 1 D Z x d y exp[ V ( y ) /D ] S ( y ) (19)and consider how to estimate T ( x ) in the limit as D →
0. Note that S ( y ) isdetermined by the values of the minima of V ( y ) in the interval [0 , y ], jumpingby an amount exp[ − ˜ V − j /D ] at x − j . Similarly, if V + j are local maxima of V ( x ), Michael Wilkinson, Marc Pradas and Gerhard Kling occurring at positions x + j , then T ( x ) jumps at local maxima. The evolution of S ( x ) and T ( x ) are therefore determined by a pair of coupled maps: S → S ′ = S + exp[ − ˜ V − j /D ] (at minima x − j ) T → T ′ = T + D exp[ ˜ V + j /D ] S (at succeeding maximum x + j ) (20)where we have defined again ˜ V + j = V + j + D ln (cid:16) πD | V ′′ + | (cid:17) . These equations are difficultto analyse in the general case, but in the next section we discuss an approach whichcan be used to treat the limit where D is small. We have seen that when D is small the integrals defining the mean first passagetime are approximated by sums over extrema of the potential, as described byequation (17). Accordingly, we study properties of random sums of the form (8)where ǫ is a small parameter and where the f j are drawn from a distribution forwhich the probability for f j being greater than f is Q ( f ). In the case where f hasa Gaussian distribution, (8) is a sum of log-normal distributed random variables.There is some earlier literature on these sums which shows very little overlapwith our results, see [15] and references therein, also [16], which discusses a phasetransition which arises in a limiting case. We also consider sums of the form T N = N X n =1 exp( g n /ǫ ) S n (21)where g j are drawn from the same i.i.d. distribution as the f j . This is a modelfor the summation which approximates the integral T ( x ) defined by equation (19).When ǫ is sufficiently small, these sums are determined by the largest values of f j and g j , and for this reason we shall refer to S N and T N as extreme-weighted sums.We write the distribution function for f in the form Q ( f ) = exp[ −J ( f )] (22)where J ( f ) is a large deviation rate function. We are interested in the asymptoticbehaviour of statistics of the sums S N and T N for small ǫ and large N . Thesums vary wildly in magnitude and the mean is dominated by the tail of thedistribution of f . Unless N is sufficiently large, values of f j which determine themean are unlikely to be sampled. This suggests that it will be useful to characterisethe distribution of the S N by the median, rather than the mean. We denote themedian of X by ¯ X and its expectation by h X i .3.1 Estimate of median of S N The sum S N may be well approximated by its largest term, which is s N = exp( ˆ f/ǫ ) (23) looding dynamics of diffusive dispersion in a random potential 7 where ˆ f is the largest of the N realisations, f j , with index j = ˆ j . We write S N ≡ exp[ ˆ f/ǫ ] F ≡ s N F (24)where F = 1 + N X j =1 j =ˆ j exp[ − ( ˆ f − f j ) /ǫ ] . (25)If F is close to unity, we can estimate ¯ S N by ¯ s N . Let us first estimate ¯ s N andreturn to consider F later. Note that¯ s N = exp(¯ˆ f/ǫ ) (26)where ¯ˆ f is the median of the largest value of N samples from the distribution of f . This is determined by setting the probability for N samples to be less than f to be equal to one half: h − Q (¯ˆ f ) i N = 12 . (27)When N ≫
1, this is determined by the tails of the distribution, where Q ( f ) isapproximated using (22): exp h − N exp[ −J (¯ˆ f )] i = 12 (28)so that ¯ˆ f satisfies ln N − ln ln 2 = J (¯ˆ f ) . (29)An important special case is where the f have a Gaussian distribution, so that inthe case where h f i = 0 and h f i = 1, Q ( f ) = 1 √ π Z ∞ f d x exp( − x / ∼ √ πf exp( − f /
2) (30)implying that J ( f ) = f f + ln (2 π )2 (31)so that ¯ˆ f f = ln N − ln ln 2 − ln (2 π )2 . (32)In the limit where N is extremely large, we can approximate ¯ˆ f by¯ˆ f ∼ √ N (33)and consequently the median of ¯ s N is approximated by¯ s N ∼ exp (cid:18) √ Nǫ (cid:19) . (34)Next consider how to estimate the quantity F in equation (24), when ǫ ≪ N ≫
1, either F is close to unity or else it is the sum of a large numberof terms which make a comparable contribution. The value of F depends upon ˆ f . Michael Wilkinson, Marc Pradas and Gerhard Kling
The f j which contribute to F are i.i.d. random variables, each with a PDF whichis the same as that of the general f j , except that there is an upper cutoff at ˆ f : theadjustment of the normalisation due to the loss of the tail, f > ˆ f , can be neglected.If the PDF of f is P f = exp[ − J ( f )] (35)then the expectation value of F is obtained as followsexp( ˆ f/ǫ )[ h F i −
1] = ( N − Z ˆ f −∞ d f P f exp( f/ǫ ) ∼ N Z ˆ f −∞ d f exp[ f/ǫ − J ( f )] ∼ N exp[ f ∗ /ǫ − J ( f ∗ )] p J ′′ ( f ∗ ) / Z √ J ′′ ( f ∗ ) /
2( ˆ f − f ∗ ) −∞ d y exp( − y ) (36)where f ∗ satisfies ǫJ ′ ( f ∗ ) = 1 . (37)Noting that h S N i = N h exp( f/ǫ ) i ∼ √ π N exp[ f ∗ /ǫ − J ( f ∗ )] p J ′′ ( f ∗ ) (38)we have exp( ˆ f/ǫ )[ h F i − ∼ h S N i " r J ′′ ( f ∗ )2 ( ˆ f − f ∗ ) ! . (39)Hence, we obtain a rather simple approximation for S N , depending upon the ex-treme value ˆ f of the sample of N realisations of the f j : S N ∼ exp( ˆ f/ǫ ) + h S N i " r J ′′ ( f ∗ )2 ( ˆ f − f ∗ ) ! . (40)The median of S N is therefore approximated by¯ S N ∼ exp(¯ˆ f/ǫ ) + h S N i " r J ′′ ( f ∗ )2 (¯ˆ f − f ∗ ) ! (41)where ¯ˆ f is the solution of equation (32). looding dynamics of diffusive dispersion in a random potential 9 T N We shall see that equation (41) gives a quite precise approximation for the median¯ S N , but it is not immediately clear when either of the two terms is dominant. Inorder to clarify the structure of equation (41), we consider an approximate formof the equation determining ¯ˆ f , and transform to logarithmic variables. As well asleading to a transparent understanding of equation (41), this facilitates making anestimate for ¯ T N in the limit where ǫ ≪ N ≫
1. We define η = ln N , σ = ln ¯ S N , τ = ln ¯ T N . (42)Note that η and τ are logarithmic measures of, respectively, distance and time,so that a plot of η versus τ gives information about the dispersion due to thedynamics.Let us consider the limit where the first term in (41), exp[¯ˆ f/ǫ ], is dominant.Note that the condition (32), determining the extreme value of a set of N samples,can be approximated by the requirement that the PDF of f is approximately equalto 1 /N . That is 1 ∼ N exp[ − J (¯ˆ f )]. For the purposes of considering the N → ∞ and ǫ → f by a solution of the equation J (¯ˆ f ) = η . (43)If the second term in equation (41) is negligible, as might be expected when ¯ˆ f − f ∗ ≪
1, equations (42) and (43) then yield a simple implicit equation for σ : η = J ( ǫσ ) . (44)If ¯ˆ f − f ∗ ≫
1, and if the second term in (41) is dominant, then ¯ S N ∼ h S N i = N h exp( f/ǫ ) i , and using the Laplace principle we find σ = η + f ∗ ǫ − J ( f ∗ ) . (45)Note the (45) indicates that d σ d η = 1. Let us compare this with the value of d σ d η obtained from (44), which predicts d η d σ = ǫJ ′ ( f ). The approximation (44) thereforebecomes sub-dominant when 1 = ǫJ ′ ( f ), which is precisely the equation for f ∗ ,equation (37). If we define η ∗ and σ ∗ by writing σ ∗ = f ∗ ǫ , η ∗ = J ( ǫσ ∗ ) (46)then assembling these results and definitions, the relationship between η and σ can be summarised in the following equation η = (cid:26) J ( ǫσ ) 0 < η < η ∗ η = σ − σ ∗ + η ∗ η ≥ η ∗ . (47)Note that η ( σ ) and its first derivative are continuous functions. In the foregoingwe defined ¯ x as the median of x , but it should be noted that our arguments willlead to equations (45) and (46) as N → ∞ if ¯ S N denotes any fixed percentile of S N . Thus far we have considered the behaviour of η as a function of σ rather thanof τ , but it is the function η ( τ ) which describes the dynamics of the dispersion.Consider the form of the sum T N defined in equation (22). When σ < σ ∗ , the valueof S n is almost always determined by ˆ f the largest value of f j , and similarly, onethe factors exp( g j /ǫ ) corresponding to ˆ g , the largest of the g j , will predominateover the others. In one half of realisations, those where ˆ k ( N ) > ˆ j ( N ), the largestvalue of f j contributes to the sum which is multiplied by exp(ˆ g/ǫ ), and we have T N ∼ exp(ˆ g/ǫ ) exp( ˆ f/ǫ ). In cases where ˆ j > ˆ k , T N is expected to be small incomparison to this estimate. Noting that exp( ˆ f/ǫ ) and exp(ˆ g/ǫ ) are independentand both have probability one half to exceed exp(¯ˆ f/ǫ ) and exp(¯ˆ g/ǫ ) respectively,there are one quarter of realisations where exp[( ˆ f + ˆ g ) /ǫ ] exceeds exp[(¯ˆ f + ¯ˆ g ) /ǫ ] andin half of these realisations T N ≪ exp[( ˆ f + ˆ g ) /ǫ ]. If we now use the overbar torepresent the upper octile of the distribution of T N , rather than the median, wehave ¯ T N ∼ exp(¯ˆ g/ǫ ) exp(¯ˆ f/ǫ ) . (48)Using the assumption that the f j and g j have the same PDF, we can concludethat ¯ T N ∼ ¯ S N and hence that τ = 2 σ . The equation describing the dispersion asa function of time is therefore η = J ( ǫτ / , τ < τ ∗ (49)where τ ∗ is determined by the condition that d η/ d τ = when τ = τ ∗ . When τ > τ ∗ , we have ¯ T N ∼ h T i = N h exp( f/ǫ ) i , implying that η = J ( ǫτ ∗ /
2) + τ − τ ∗ , τ > τ ∗ . (50)Equations (49) and (50) are a description of the logarithm of the typical dispersion η as a function of the logarithm of the time, τ . Usually the function J ( V ) has aquadratic behaviour for small values of V , so that the initial dispersion, describedby (49), is sub-diffusive. The factor of one half in (50) indicates that the long-time limit is diffusive. Writing η ∼ τ + ln D eff , we see that the effective diffusioncoefficient is D eff ∼ exp (cid:2) J ( ǫτ ∗ / − τ ∗ (cid:3) (51)which is consistent with (14). In section 2, we showed that the integrals which are used to compute the mean-first-passage time may be approximated by sums when D is small. In section3, we considered the statistics of these sums, S N and T N , defined by equations(8) and (21) respectively. In terms of the calculation discussed in section 2, ourestimate of ¯ T N corresponds, for N < N ∗ , to the value of ¯ T ( x ) being determinedby the difference between the lowest minimum of the potential and its highestmaximum, provided the minimum occurs before the maximum. We can thereforethink of ¯ T ( x ) being determined by a ‘flooding’ model, according to which theprobability density for locating the particle occupies a region which is constrained looding dynamics of diffusive dispersion in a random potential 11 by a potential barrier which can trap a particle for time ¯ T . As ¯ T increases, higherbarriers are required.In terms of the original problem, discussed in section 2, N is the number ofextrema of the potential before we reach position x . The arguments of section3 imply that the upper octile of the mean-first-passage time, ¯ T ( x ), satisfies anequation similar to (48). We define logarithmic variables η = ln (cid:16) x ˜ x (cid:17) , τ = ln ¯ T ( x ) (52)where ˜ x is the mean separation of minima of V ( x ). In terms of these logarithmicvariables, the dispersion is described by η = J ( Dτ / . (53)which is valid up to τ ∗ , which is defined by the conditiond η d τ (cid:12)(cid:12)(cid:12)(cid:12) τ ∗ = 12 . (54)Equation (53) is our principal result. It applies to any percentile of the distributionwhich remains fixed when we take the limits N → ∞ and ǫ →
0. When τ is largecompared to τ ∗ , equation (53) is replaced by a linear relation, with an effectivediffusion coefficient D eff η ∼ τ + ln( D eff /D ) , D eff ∼ D exp (cid:2) J ( Dτ ∗ / − τ ∗ (cid:3) . (55)An important example is the case where V has a Gaussian distribution, so that J ∼ V / C (0). In terms of the diffusion coefficient D , equations (53)-(55) give η ∼ D τ C (0) τ < τ ∗ = 2 C (0) D η ∼ τ − C (0)2 D τ > τ ∗ (56)and using (55) we find D eff ∼ D exp[ − C (0) /D ], in agreement with (5). A sketchof the dependence of η upon τ for the Gaussian case is shown in Fig. 4. We performed a variety of numerical investigations, using Gaussian distributedrandom variables f j to test the theory of extreme-weighted sums, and a Gaussianrandom function V ( x ) to test the analysis of continuous potentials. In both casesthe Gaussian variables had zero mean and unit variance. In the case of the randompotential, we also used a Gaussian for the correlation function, with a correlationlength of order unity: h V ( x ) V ( x ′ ) i = exp[ − ( x − x ′ ) / . (57) η = ln( x/ ˜ x ) τ = ln ¯ T ( x ) η ∗ τ ∗ flooding diffusion Fig. 1
The dynamics of a typical realisation, characterised by the median ¯ T of the mean-first-passage time, shows a crossover from sub-diffusive ‘flooding’ dynamics to slow diffusion. f . By substituting (33) into (32),we find ¯ˆ f ≈ p N − ln ln N − ln (2 π ) − . (58)The expression for the median approaches that for the mean S N ∼ N exp (cid:16) ǫ (cid:17) (59)at large values of N when ¯ˆ f exceeds f ∗ = 1 /ǫ .For very large N and very small ǫ , the medians of S N and T N are estimatedby simplified expressions, relating σ = ln ¯ S N and τ = ln ¯ T N to η = ln N . In theGaussian case, these equations (47), (49) and (50) give σ = (cid:26) √ ηǫ < η < η ∗ η − η ∗ + σ ∗ η ≥ η ∗ (60)and τ = (cid:26) √ ηǫ < η < η ∗ η − η ∗ ) + τ ∗ η > η ∗ (61)where η ∗ = 12 ǫ , σ ∗ = 1 ǫ , τ ∗ = 2 ǫ . (62)These equations imply that, in the limit as ǫ →
0, if we plot y = σ/η ∗ as functionof x = η/η ∗ , the numerical data for ¯ S N should collapse onto the function y = f ( x ) = (cid:26) √ x < x < x + 1 x > . (63)Similarly, y ′ = τ /η ∗ plotted as a function of x = η/η ∗ should collapse to y ′ = 2 f ( x ). looding dynamics of diffusive dispersion in a random potential 13 We computed M ∈ { , , } realisations of the sums S N and T N , for ǫ ∈ { / , / , / , / } and N ≤ (except for M = 1000, in which case N ≤ × ). We evaluated the sample average h S N i M , the sample median, ¯ S N | M, and the sample upper octile ¯ S N | M, . We also computed the same statistics for the T N .Figure 2 plots ln ¯ S N , and ln h S N i as a function of η = ln N , for different samplesizes, for ǫ = 1 / a ) and ǫ = 1 / b ). We compare with the theoretical predic-tion, obtained from (41) and (58) (for the median) and (59) (for the mean). Theagreement with the theory for the median is excellent. Note that the convergenceof the mean value for different sample sizes is very poor when η < η ∗ = 1 / ǫ (thisis especially apparent for smaller values of ǫ ). σ η ( a ) ε =1/4 Mean: M =10 Mean: M =100 Mean: M =1000 Mean: theoryMedian: M =10 Median: M =100 Median: M =1000 Median: theory 0 5 10 15 20 25 30 0 2 4 6 8 10 12 σ η ( b ) ε =1/6 Mean: M =10 Mean: M =100 Mean: M =1000 Mean: theoryMedian: M =10 Median: M =100 Median: M =1000 Median: theory Fig. 2
Plot of ln ¯ S N and ln h S N i , as a function of η = ln N , for ǫ = 1 / a ) and ǫ = 1 / b ). In figure 3, we plot y = σ/σ ∗ ( a ) and y ′ = τ /τ ∗ ( b ) as a function of x = η/η ∗ ,for all of the values of ǫ in our data set, using the largest sample size ( M = 1000) ineach case, comparing with the theoretical scaling function (63). We see convergencetowards the function (63) as ǫ →
0. In panels ( c ) and ( d ), we make the samecomparison using the upper octile rather than the median. σ / η * η / η *( a ) S N ε =1/3 ε =1/4 ε =1/6 ε =1/8 theory 0 1 2 3 4 5 6 0 0.5 1 1.5 2 τ / η * η / η *( b ) T N ε =1/3 ε =1/4 ε =1/6 ε =1/8 theory 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 σ / η * η / η *( c ) S N ε =1/3 ε =1/4 ε =1/6 ε =1/8 theory 0 1 2 3 4 5 6 0 0.5 1 1.5 2 τ / η * η / η *( d ) T N ε =1/3 ε =1/4 ε =1/6 ε =1/8 theory Fig. 3
Plot of σ/η ∗ ( a ) and y ′ = τ/τ ∗ ( b ) based upon median values, as a function of η/η ∗ ,compared with the theoretical prediction for the ǫ → c ) and ( d ) weshow similar plots for the upper octile data. x , in order to make a comparison with theory. The density D of zerosof V ′ ( x ) may be determined by the approach developed by Kac [17] and Rice [18].If P ( V, V ′ , V ′′ ) is the joint PDF of V ( x ) and its first two derivatives, evaluated atthe same point, we find that D = Z ∞−∞ d V Z ∞−∞ d V ′′ P ( V, , V ′′ ) | V ′′ | . (64)By noting that the vector ( V, V ′ , V ′′ ) has a multivariate Gaussian distribution,and expressing P ( V, V ′ , V ′′ ) in terms of the correlation function of the elements ofthis vector, we obtain D and hence the separation of minima ˜ x for the potentialsatisfying (57): 2˜ x ≡ D = √ π . (65)First we investigated whether the mean-first passage time can be accurately repre-sented by sums over maxima and minima of the potential. In figure 4, we comparethe numerical evaluation of the integrals S ( x ) ( a ) and T ( x ) ( b ), given by Eq. (16)and Eq. (19), respectively, with the approximations which estimate the integralsusing maxima and minima, Eq. (20). looding dynamics of diffusive dispersion in a random potential 15 Eq. (19)Eq. (20)10 Eq. (16)Eq. (20) (a) (b)
Fig. 4
Plots of S ( x ) ( a ) and T ( x ) ( b ), for one realisation of the smooth random potential,with D = 1 /
8. The numerically evaluated integrals (solid curves) are in good agreement withapproximations based on sums over maxima and minima (dashed lines).
We evaluated the median ¯ T ( x ) and mean h T ( x ) i of the mean first passagetime T ( x ) for 1000 realisations of the potential V ( x ), up to x max = 10 , for D ∈{ / , / , / , / , / } . According to the discussion in section 4, we expect that τ = ln ¯ T ( x ) and η = ln( x/ ˜ x ) are related by η = J ( Dτ /
2) = D τ /
8, up to amaximum value of η , given by η ∗ = 1 / D . In figure 5 we plot Y med = 2 D ln ¯ T ( x )and Y av = 2 D ln h T ( x ) i as a function of X = 2 D ln( x/ ˜ x ) for different values of D ,and compare with the theoretical scaling function, given by equation (63). (a) (b) Fig. 5
Numerical results on the median ¯ T ( x ) of the mean first-passage time to reach x , for dif-ferent values of the diffusion coefficient D . We plot logarithmic variables Y med = D ln ¯ T ( x ) / a ) and Y av = D ln h T ( x ) i ( b ) as a function of X = 2 D ln( x/ ˜ x ) (where ˜ x is the mean distancebetween maxima of V ( x )). The numerical results converge towards the theoretically predictedscaling function, equation (63), as D → In his analysis of equation (1), Zwanzig considered the mean-first-passsage time T ( x ) to reach displacement x . Computing the expectation value h T ( x ) i over dif-ferent realisations of the random potential, he showed [4] that h T ( x ) i ∼ x , which is consistent with a diffusive dispersion, with an effective diffusion coefficient D eff .The effective diffusion coefficient vanishes in a highly singular manner as D → h T ( x ) i being dominated by rareevents, where an unusually large fluctuation of the potential V ( x ) acts as a bar-rier to dispersion. The central limit theorem is applicable to this problem, and atsufficiently large values of x the ratio T ( x ) / h T ( x ) i is expected to approach unity,for almost all realisations of V ( x ). However, at values of x which are of practicalrelevance, most realisations T ( x ) will be much smaller than h T ( x ) i .In order to give a description of the dynamics of (1) which is both empiri-cally useable and analytically tractable, we considered the median (with respectto different realisations of the potential) of the mean-first-passage time. In thelimit where the diffusion coefficient D is small, the integrals which appear in theexpression for the first passage time, equation (9), are dominated by maxima andminima of the potential, described by equations (17) and (20). This observationled us to consider the statistics of sums of exponentials of random variables, equa-tions (8) and (21). We gave a quite precise estimate, equation (41), for the medianof (8) and also derived simple relations describing the asymptotic behaviour ofthese sums, equations (47) and (49).It is these expressions which enable us to formulate a concise asymptotic de-scription of the dynamics of (1) in the limit as D →
0, in terms of the largedeviation rate function of the potential, J ( V ). We argued that at very long lengthscales ¯ T ( x ) approaches the expectation value h T ( x ) i , and that the dispersion isdiffusive, in accord with the theory of Zwanzig [4]. On shorter timescales ¯ T ( x ) isdetermined by a ‘flooding’ model, according to which the probability density forlocating the particle occupies a region which is constrained by a potential barrierwhich can trap a particle for time ¯ T . As ¯ T increases, higher and higher barriersare required. For a Gaussian distribution of barrier heights, equation (56) impliesthat the dispersion is described as sub-diffusive, of the form x ∼ ˜ x exp (cid:18) D (ln ¯ T ) C (0) (cid:19) (66)which is distinctively different from the power-law anomalous diffusion which hasbeen reported by some authors [6, 7, 8]. Our numerical investigations of the dy-namics of equation (1) for different values of D , illustrated in figure 5, show a datacollapse which is in excellent agreement with equation (63), verifying (66). Acknowledgments . We thank Baruch Meerson for bringing [4] to our notice,and for interesting discussion about the statistics of barrier heights. MW thanksthe Chan-Zuckerberg Biohub for their hospitality.
References
1. Havlin, S. and Ben-Avraham, D.,
Diffusion in disordered media
Advances in Physics , Principles of Condensed Matter Physics , Cambridge: Uni-versity Press, (1995).3. Saxton, M. J.,
A Biological Interpretation of Transient Anomalous Subdiffusion. I. Quali-tative Model
Biophysical Journal , Diffusion in a rough potential , PNAS , , 2029-30.5. De Gennes, P. G., 1975, Brownian motion of a classical particle through potential barriers.Application to the helix-coil transitions of heteropolymers , J. Stat. Phys. , , 463?81.6. Khoury, M., Lacasta, A.M., Sancho, J.M. and Lindenberg, K., Weak Disorder: Anomaloustransport and diffusion are normal yet again , Physical Review Letters , , 090602, (2011).7. Simon, M. S., Sancho, J. M. and Lindenberg, K., Transport and diffusion of overdampedBrownian particles in random potentials , Physical Review E , , 062105, (2013).8. Goychuk, I., Kharchenko, V. O. and Metzler, R., 2017, Persistent Sinai-type diffusion inGaussian random potentials with decaying spatial correlations , Phys. Rev. E , , 052134.9. Sinai, G. Ya., 1982, Theor. Prob. Appl. , , 256.10. Comtet, A. and Dean, D. S., 1998, Exact results on Sinai’s diffusion , J. Phys. A: Math.Gen. , , 8595-8605.11. Le Doussal, P., Monthus, C. and Fisher, D. S., 1999, Random walkers in one-dimensionalrandom environments: Exact renormalization group analysis , Phys. Rev. E , , 4795-4840.12. Kramers, H. A., Brownian motion in a field of force and the diffusion model of chemicalreactions , Physica , , 284-304, (1940).13. Redner, S., 2001, A guide to first-passage processes , Cambridge, University press, ISBN0-521-65248-0.14. Lifson, S. and Jackson, J. L., 1962,
On self-diffusion of ions in a polyelectrolyte solution , J. Chem. Phys. , , 2410-14.15. M. Romeo, V. Da Costa, and F. Bardou, Broad distribution effects in sums of lognormalrandom variables , Eur. Phys. J. B , , 513-525, (2003).16. M. Pradas, A. Pumir, and M. Wilkinson, Uniformity transition for ray intensities in ran-dom media , J. Phys. A: Math. Teor. , , 155002, (2018).17. Kac, M, 1943, On the average number of real roots of a random algebraic equation , Bull.Am. Math. Soc. , , 314-20.18. Rice, S O, 1945, Mathematical analysis of random noise , Bell Syst. Tech. J. ,23