From Quenched Disorder to Continuous Time Random Walk Stanislav Burov∗ Physics Department, Bar-Ilan University, Ramat Gan 5290002, Israel
arXiv:1707.05072v1 [cond-mat.stat-mech] 17 Jul 2017
This work focuses on quantitative representation of transport in systems with quenched disorder. Explicit mapping of the quenched trap model to continuous time random walk is presented. Linear temporal transformation: t → t/Λα for transient process on translationally invariant lattice, in the sub-diffusive regime, is sufficient for asymptotic mapping. Exact form of the constant Λα is established. Disorder averaged position probability density function for quenched trap model is obtained and analytic expressions for the diffusion coefficient and drift are provided.
Properties of transport in disordered environment are objects of intensive research [1–3]. While regular diffusion is vastly observed in many systems, anomalously slow diffusion (i.e. hx2 (t)i ∼ tα where 0 < α < 1) effectively describes motion in complex disordered systems such as living cells [4, 5], blinking quantum dots [6], molecular-motor transport on filament network [7] and photo-currents in amorphous materials [8]. Several theoretical approaches give rise to anomalous diffusion of a particle in disordered media. The Fractional Brownian Motion [9] effectively models the disorder as long-ranged temporal correlations. Another approach attributes the slow-down to presence of obstacles, such as traps and barriers, in the media. For example, random walks (RW) obstructed by traps [4, 7] and barriers [10] were used to model properties of intracellular transport. When the expected local dwell times diverge, the diffusion becomes anomalous [2, 3]. Transport mediated by traps and barriers attracted tremendous attention in Physics and Mathematics. The usual theoretical description consists of a RW on a lattice, where the disorder enters via transition probabilities (and rates) to different lattice sites. Two general disorder types prevail: annealed disorder and quenched disorder. The annealed disorder describes the situation when the disorder is uncorrelated. For each visit to a lattice site new disorder is generated. On the contrary, quenched disorder suggests that the disorder per site stays exactly the same for all visits of the RW. This imposes strong correlations and makes theoretical description highly non-trivial. When using traps as disorder, the dwell time at specific lattice site can be constant (quenched) or generated from a random distribution for each arrival (annealed). The later model is known as continuous time random walk (CTRW) [8] and its behavior is well known [2, 11]. Once the dwell times are quenched, a case known as the quenched trap model (QTM), the renewal property is lost. Scaling arguments and renormalization group approach [2, 12, 13] suggest that for dimension d > 2 QTM behaves qualitatively as CTRW in the sub-diffusive phase. Similar result was suggested by using rigorous mathematical description of QTM on a regular lattice [14, 15]. Simple hand-waving argument behind this convergence is based on the fact that for
d > 2 the probability of RW to return to a specific site is < 1. One can then assert that the correlations imposed by quenched dwell times can be effectively renormalized into uncorrelated times, i.e. CTRW description. Similar argument should also hold for the case of a biased transport, i.e. RW with directional preference. For example the case of directional RW (transitions only in one direction) for QTM in d = 1 [16] is believed to be asymptotically similar to the general biased case. While for directed RW the particle never returns to the same site, for a general biased case the probability of return is < 1. In this manuscript explicit mapping between QTM (quenched disorder) and CTRW (annealed disorder) is provided. It will be shown that for any case when a RW is transient, i.e. the probability of return < 1, the probability density function (PDF) in the sub-diffusive regime takes the form of an appropriate CTRW process. The missing quantitative representation of QTM in terms of CTRW will be provided for any QTM that takes place on translationally invariant lattice. Transitions between different lattice sites are not restricted to nearest neighbors. The presented approach is based on reformulation of subordination technique for CTRW [2, 17, 18] that was introduced in [19, 20]. The Quenched Trap Model is defined as a random process on a lattice of dimension d. For each site x of the lattice a quenched random variable τx is defined. τx describes the time that the particle spends at site x before moving to some random site x0 . The process starts at time t = 0 when the particle is situated at x = 0. Probability of transition from x to x0 is provided by p(x0 ; x). Due to translational invariance of the lattice p(x0 ; x) is a function of x0 − x, i.e p(x0 − x). The quenched variables {τx } are positive, independent and identically distributed random variables with common −(1+α) PDF ψ(τx ) ∼ τx A/|Γ(−α)| for τx → ∞ (A > 0 and Γ(. . . ) is the Gamma function). The values of α will be restricted to 0 < α < 1 in order to describe the subdiffusve regime of QTM [2]. Local dwell times τx describe for how long the particle is ”trapped” on site x. The physical picture is usually attributed to thermally activated jumps upon random energy potential. Each lattice site is associated with energetic trap with
2
x
and the sum is again over all lattice sites. Sα is a spatial variable which depends solely on various positions of the particle and not the time spent at those sites. For α = 1 Sα is the total number of steps performed. In [19] it was 1/α shown that the random variable η = t/ (Sx ) is distributed according to one-sided L´evy PDF lα,A,1 (η) [18]. The argument is as follows: while averaging the quantity exp(−ηu) (u > 0) over disorder, it occurs that ! D E D X nx τx E α −ηu → e−Au (2) e = exp − 1/α x Sα α
and e−Au is the Laplace pair of lα,A,1 (η). When constraining t to a fixed value, the PDF of Sα is easily obtained from the definition of η 1 t t −1 −α Nt (Sα ) = (Sα ) lα,A,1 . (3) 1/α α Sα Equation (3) defines the distribution of Sα and is a part of transformation from accumulated disorder to real time. The probability of arriving to x at time t can be separated into probability of arriving to x at some Sα and probability of observing this specific Sα , i.e. Nt (Sα ). Sα is operational time of the process and Eq. (3) is the transformation from operational time to real time t. For specific Sα the probability to observe the particle at x for specific Sα is written as PSα (x). Disorder averaged PDF P of position x at time t is provided by hP (x, t)i = Sα PSα (x)Nt (Sα ), where the sum is over all possible Sα s. Notice that PSα is independent of disorder. Sα is positively defined and hP (x, t)i is written as Z ∞ hP (x, t)i ∼ PSα (x)Nt (Sα ) dSα . (4) 0
Since Nt (Sα ) is given by Eq. (3) the problem of determining the propagator hP (x, t)i for QTM boils down to determining PSα (x), which is a property of RW on a lattice.
10-2 2
Sα 2/Sα -1
energy depth E > 0 that is exponentially distributed, i.e. f (E) = exp(−E/Tg )/Tg . One thing to notice about QTM is that if the process is observed as a function of number of performed steps, it behaves like a RW with transition probabilities defined by p(x0 − x). Similar statement is true for CTRW. The “solution” of QTM is then a proper transformation from the number of steps to ordinary time. Time is function of all possible traps that the particle encountered on its P path. In QTM time is provided by t = x nx τx , where nx is the number of visits to site x. The sum follows all different sites on the lattice. Similarly to [19, 20] a random variable Sα is defined X Sα = (nx )α , (1)
10-3
○ ○○○○○○○ ○○○○ ○ ○ ○○ ○ ○ ▽▽▽▽▽▽ ▽ ▽ ○○ ▽▽▽▽▽ ▽ ○○ ▽ ▽ ▽ ▽ ▽ ○○○○
▽▽▽ ▽▽▽▽ ○ ○ ▽ ▽○○○○ ▽▽▽○○○ ▽▽▽▽ ○ ▽○ ▽○ ▽○ ▽○ ▽○ ▽○ ▽○ ▽○ ▽
10-4 10-5 1
10
100
1000 N
104
105
FIG. . 1.2Simulated behavior of fraction of moments of Sα , i.e. Sα2 Sα2 − 1, as function of the number of jumps (N ) of a random walk. are the results for a biased one-dimensional RW on a lattice, the transitions are allowed only to nearest neighbors with probability q = 0.7 to the right and 1 − q to the left. 5 presents the results for 3-dimensional unbiased RW on a cubic lattice where the transitions are allowed only to nearest neighbors.
Although operational time Sα is defined for a RW without disorder its behavior is quite non-trivial since it is defined by the whole history of a random trajectory. PSα (x) describes a random walk that was stopped at specific Sα while the number of performed steps is arbitrary. In [19] it was shown that for d = 1 and nearest-neighbor jumps of the RW, PSα (x) attains transition from a Gaussian shape (α → 1) to a V shape (α → 0). It is the purpose of this manuscript to show that for any transient RW, PSα (x) is easily obtained from PN (x), i.e. the probability to find the particle at position x after N steps. [21] contains a mathematical proof that for transient RW (on translationally invariant lattice) the fraction of the mo2 ments of Sα , i.e. Sα2 /Sα , converges to 1 as N → ∞. The average ( ) is taken with respect to all possible RW that start at the origin and perform N steps. In Fig. 1 the convergence of fraction of moments is presented for two different cases of transient RWs. It is shown below that in the limit of large N , Sα /N converges to a non2 zero constant. Since Sα2 /Sα → 1, it means that Sα /N converges to a δ-function. By calculation of Sα the deterministic mapping between Sα and N is found. This mapping determines N as a function of Sα , i.e. N (Sα ), and consequently PSα (x) ∼ PN (Sα ) (x). Since PN (x) describes RW on a spatially invariant lattice, its properties are well documented [11]. Calculation of Sα . Let βN (x; k) be a probability that a RW visited site x exactly k times after N steps. Sα is expressed in terms of βN (x; k) as Sα = P Pk=∞ α A closely related quantity is x k=0 k βN (x; k). VN (k), the average number of lattice sites visited ex-
3 actly k times after N steps. VN (1) was first derived in [22] and for general k using the generating function approach [11]. The derivation below follows [11]. By the virtue of probability of first return to x = 0 after N steps, fN (0), we write the probability to reach site x for k’th time after N steps, fN (x; k), as: fN (x; k + 1) = PN This relation holds for any m=0 fm (x; k)fN −m (0). translationally invariantP lattice. The generating function ∞ of fN (x; k), fˆz (x; k) := N =0 z N fN (x; k), is
♢ 4
10
where fˆz (0) is the generating function of fN (0) and fˆz (x) is the generating function of fN (x) (the probability of first arrival to x). Since RW must arrive to site x for kth time after m ≤ N step (and afterwords can’t visit again) βN (x; k) takes the form
βN (0; k) =
N X
[fm (x; k) − fm (x; k + 1)]
1.2
x 6= 0
1.0
N X
0.8
[fm (0; k − 1) − fm (0; k)]
m=1
(6)
1
104 t
100
○
106
○
○
○
0.4
0.5
○
108
○ ○
0.6
0.7
0.8 q
0.9
1.0
1010
fˆz (0) is related to Q0 , the probability of a RW to return P∞ to the origin, since Q0 = N =0 fN (0). By taking the z → 1 limit and applying Tauberian theorem [25], Eq. (8) is transformed to (N → ∞)
(9)
[1 − Q0 ] Li−α (Q0 ) Q0
(10)
where 2
♢
〈X 2 〉
107
○
♢ ♢
4
10
♢ ○
10
♢ ○ ○ ▽ ♢
▽
10
1.1 ♢
(�)
k=0
▽
○ ○○ ○○ ○ ○○ ○ ○ ○ ○ ○ ○
x 6= 0
Since PN (x) can be written in terms of fN (x), PN (x) = PN δN,0 δx,0 + m=1 fk (x)PN −m (0) [23], a known [24] relation holds for generating functions of PN (x) and fN (x), i.e., fˆz (x 6= 0) = Pˆz (x 6= 0)/Pˆz (0); fˆz (0) = P 1 − 1/Pˆz (0). Using these expressions, and the fact that x Pˆz (x) = 1/(1 − z), we obtain for the generating function of averaged operational time " #2 ∞ X ˆ ˆ α 1 − fz (0) fˆz (0)k−1 . (8) Sα (z) = k 1−z
Λ=
▽
▽
▽
▽
○ ▽ ♢
○
0.6
0.0
(7)
Sα ∼ ΛN
▽
○
▽
▽
♢ ○
○
(�)
0.2
By taking z-transform of both sides in Eq. 6 and applying Eq. (5), generating function of βN (x; k) is obtained
♢ ○
♢ ○
1 0.1
○ ○
○
♢
100
m=1
ih ik−1 1 h 1 − fˆz (0) fˆz (0) fˆz (x) βˆz (x; k) = 1−z ih ik−1 1 h . 1 − fˆz (0) fˆz (0) βˆz (0; k) = 1−z
♢ ♢
10
〈X 〉/t α
βN (x; k) =
(�)
1000
(5)
〈X 〉
h ik−1 fˆz (x; k) = fˆz (0) fˆz (x),
P∞ and Lia (b) = k=0 bk /k a is the Polylogarithm function. Eq. (9) holds in the asymptotic limit of large number of steps and only for Q0 < 1, i.e. transient RW. The linear relation between N and Sα , together with the convergence of Sα /N to a constant value [21], enables us to establish the mapping between QTM and CTRW.
▽
○ ○
○
♢ ○
▽
▽
▽
105
1000
▽
▽
107
t
FIG. 2. Moments and pre-factors for 1-dimensional biased QTM as presented by Eq. (15) (lines) and numerical simulations (symbols). (a) presents the first moment behavior as a function of time for three different α: ♦ is α = 0.7, is α = 0.5 and 5 is α = 0.3. (b) presents the behavior of pre-factor hXi/tα = V /(AΓ[1 + α]) for α = 0.5 and various q > 0.5. Simulations were performed up to t = 107 . (c) presents the growth of the second moment with time, the parameters similar to (a). 104 realizations of disorder were used for averaging and A = 1.
Asymptotic mapping to CTRW. hP (x; t)i behavior in the asymptotic limit t → ∞ is achieved by substituting
4
QTM
CTRW
(12) where h. . . iQTM means averaging with respect to quenched disorder of QTM and h. . . iCTRW is averaging with respect to annealed disorder of CTRW. Eq. (12) is the main result of this manuscript, simple linear time transformation, t → t/Λ1/α , between quenched and annealed disorder. The immediate outcome is that many known results for CTRW are naturally transformed to quantitative results for QTM. The only limitation of the transformation is the transience of the spatial RW (Q0 < 1). Computation Dof different positional moments, i.e., E R hxµ (t)i = x xµ P (x; t) dx, becomes quite straightforward in the long time limit. Indeed, by application of Eq. (11) the spatial integrationR is preformed only for xµ Pν (x). In the limit of large ν, x xµ Pν (x) dx ∼ Bµ ν γµ . R∞ We use 0 y q lα,1,1 (y) dy = Γ(1 − q/α)/Γ(1 − q) (for q/α < 1) and obtain γµ Q0 Γ[1 + γµ ] tαγµ . hxµ (t)i ∼ Bµ Γ[1 + αγµ ] A[1 − Q0 ]2 Li−α (Q0 ) (13) Constants Bµ , γµ and Q0 depend only on the lattice type and transition probabilities p(x). Since the calculation is performed for large times, Pν (x) usually converges to Gaussian or L´evy distribution [3] where all the moments and pre-factors like Bµ are known. By the same token, or by simpler scaling arguments, the exponent γµ can be obtained. Return probability Q0 has been successfully computed for quite a long time ago [26] for various lattices, in Appendix of [27] (and references therein) appear numerous exact values for Q0 . Two examples of moment behavior are in place (i) biased RW on symmetric lattice in d = 1 and (ii) non-biased RW on a cubic lattice (d = 3). The biased RW in 1-dimension can perform a unit step to the right with probability q > 1/2 or a unit step to the left with probability 1 − q. For large ν, Pν (x) →
p 8πq(1 − q)ν, exp −(x − (2q − 1)ν)2 (8q(1 − q)ν) i.e. the diffusional limit. Bµ and γµ Rare obtained by ∞ performing the Gaussian integration −∞ xµ Pν (x) dx. The return probability for such RW is [11] [28] Q0 = 1− lim
1
z→1 1 2π
Rπ
dy −π 1−z[qeiy +(1−q)e−iy ]
= 2(1−q). (14)
Eventually, from Eq. (13), the first two moments for a biased 1-dimensional RW are hx(t)i ∼
1 V tα AΓ[1 + α]
,
hx(t)2 i ∼
2 V 2 t2α A2 Γ[1 + 2α] (15)
. where V = 2(1 − q) [(2q − 1))Li−α (2[1 − q])]. Comparison between theoretical result and simulations of QTM is presented in Fig. 2. The response to bias is nonlinear in time but also in q, as is seen from the form of V (Fig. 2 (b)). In the limit of q → 1/2 the response in q is: V ∼ (2q − 1)α , this non-linear scaling was previously predicted in [29] by scaling arguments and in [30] for very small α. Notice also that hX 2 i − hXi2 ∼ (V /A)2 2/Γ[1 + 2α] − 1/Γ2 [1 + α] t2α and behaves super-diffuseivily for α > 1/2. Such superdiffusive behavior has been observed in quite a few studies of disordered systems [31–36]. ♢
104 ♢ ♢
100 10 1○ ▽ ♢
○
♢
1000 〈X 2 〉
PN (Sα ) (x) into Eq. (4) instead of PSα (x). The t → ∞ regime makes sure, by the means of Nt (Sα ), that sufficient amount of steps has been performed and Sα ∼ Sα . Further, a change of variables in Eq. (4), Sα → Λν, leads to ! Z ∞ [t Λ1/α ] [t Λ1/α ] dν. hP (x; t)i ∼ Pν (x) −( 1 +1) lα,A,1 ν 1/α αν α 0 (11) For CTRW there are no correlations between different waiting times and each site is considered as a new one, from the dwell time perspective. The operational time Sα for CTRW is then simply N and the position PDF is provided by Eq. (4) [2, 18]. From Eq. (11), and the mentioned representation of CTRW, follows that D E D E P (x; t) ∼ P (x; t Λ1/α ) (t → ∞) ,
♢ ○ ♢ ○
▽
10
○ ○
○
○
▽
▽
▽
▽
▽
▽
105
1000
107
t
FIG. 3. Second moment of position as function of time for unbiased QTM on a cubic lattice. Lines are analytical predictions provided by Eq. (16) and symbols are numerical simulations. ♦ is α = 0.7, is α = 0.5 and 5 is α = 0.3. 104 realizations of disorder were used for averaging and A = 1.
The second example is of a non-biased RW on a cubic lattice that can perform 6 different unitary steps, two for every dimension. Any transition of the form x = (x, y, z) → (x ± 1, y, z) has probability 1/6 (similarly in y and z directions). We again take the asymptotic limit of large number of steps and .p (2πν/3)3 . Due Pν (x) → exp −3(x2 + y 2 + z 2 ) 2ν to the symmetry of the process, the first moment is strictly 0 and the second moment is dictated by the
5 R fact that x (x2 + y 2 + z 2 )Pν (x) dx ∼ ν. The return probability for a cubic lattice was already calculated.in [26] while the analytic expression Q0 = 1 − √ 32π 3 ( 6Γ[1/24]Γ[5/24]Γ[7/24]Γ[11/24]) ≈ 0.34057... was provided in [37]. According to Eq. (13) the second moment is h|x|2 i ∼
0.783 tα , AΓ[1 + α]Li−α (0.3405)
(16)
where we explicitly used the numerical value of Q0 . The comparison to simulations is presented in Fig. 3. The presented quantitative representation of QTM in terms of CTRW (as described by Eq. (12)) is applicable in any situation where Q0 is less than 1. Specifically this occurs for systems with dimension > 2 or any driven system [31, 35, 36, 38] with quenched trapping disorder. Additionally, the mapping will be of value for disentangling the nature of observed anomalous diffusion [39– 41]. While the simple temporal mapping covers a broad range of disordered systems, possible generalizations of the method are in place. This includes 2-dimensional systems. Existent duality [42] between trap and barrier models suggests that some variation of the mapping can be applicable to the general case of transport on random potential landscape [43]. This work was partially supported by the Pazy Foundation. I thank E. Barkai for many discussions.
∗
[email protected] [1] S. Alexander, J. Bernasconi, W. R. Schneider, and R. Orbach, Rev. Mod. Phys 53, 175 (1981). [2] J. P. Bouchaud and A. Georges, Phys. Rep. 195, 127 (1990). [3] R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000). [4] E. Barkai, Y. Garini, and R. Metzler, Phys. Today 65, 29 (2012). [5] S. M. A. Tabei et al., Proc. Natl. Acad. Sci. U. S. A. 110, 4911 (2013). [6] F. D. Stefani, J. P. Hoogenboom, and E. Barkai, Phys. Today 62, 34 (2009). [7] M. Scholz et al., Phys. Rev. X 6, 011037 (2016). [8] H. Scher and E. W. Montroll, Phys. Rev. B 12, 2455 (1975). [9] B. B. Mandelbrot and J. W. Van-Ness, SIAM Rev. 10, 422 (1968). [10] D. S. Novikov, E. Fiermans, J. H. Jensen, and J. A. Helpern, Nat. Phys. 7, 508 (2011). [11] G. H. Weiss, Aspects and Applications of the Random Walk (North-Holland, Amsterdam, 1994).
[12] J. Machta, J. Phys. A. 18, L531 (1985). [13] C. Monthus, Phys. Rev. E 68, 036114 (2003). ˇ [14] G. B. Arous, J. Cern´ y, and T. Mountford, Probab. Theory Related Fields 134, 1 (2006). ˇ [15] G. B. Arous and J. Cern´ y, Ann. Probab. 35, 2356 (2007). [16] C. Aslangul, M. Barthelemy, N. Pottier, and D. SaintJames, J. Stat. Phys. 59, 11 (1990). [17] H. Fogedby, Phys. Rev. E 50, 1657 (1994). [18] E. Barkai, Phys. Rev. E 63, 046118 (2001). [19] S. Burov and E. Barkai, Phys. Rev. Lett. 106, 140602 (2011). [20] S. Burov and E. Barkai, Phys. Rev. E 86, 041137 (2012). [21] S. Burov, In preparation. [22] P. Erd¨ os and S. J. Taylor, Acta Math. Acad. Sci. 11, 137 (1960). [23] δa,b is the Kronecker delta function. [24] S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, 2001). [25] W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, Eastern New Delhi, 1969) vol. 2. [26] G. N. Watson, Quarterly Journal of Mathematics 10, 266 (1939). [27] B. D. Hughes, Random Walks and Random Environments (Clarendon Press, Oxford, 1995) vol. 1. [28] Note that the calculation in [11] is erroneous due to existence of additional pole. [29] E. M. Bertin and J. P. Bouchaud, Phys. Rev. E 67, 065105 (2003). [30] C. Monthus, Phys. Rev. E 69, 026103 (2004). [31] S. F. E. Schroer and A. Heuer, Phys. Rev. Lett. 110, 067801 (2013). [32] S. Leitmann and T. Franosch, Phys. Rev. Lett. 118, 018001 (2017). [33] G. Gradenigo, E. Bertin, and G. Biroli, Phys. Rev. E 93, 060105 (2016). [34] D. Winter, J. Horbach, P. Virnau, and K. Binder, Phys. Rev. Lett. 108, 028303 (2012). [35] M. Khoury, A. M. Lacasta, J. M. Sancho, and K. Lindenberg, Phys. Rev. Lett. 106, 090602 (2011). [36] O. B´enichou et al., Phys. Rev. Lett. 111, 260601 (2013). [37] M. L. Glasser and I. J. Zucker, Proc. Natl. Acad. Sci. U. S. A. 74, 1800 (1977). [38] J. R. Gomez-Solan, A. Blokhuis, and C. Bechinger, Phys. Rev. Lett. 116, 138301 (2016). [39] S. Burov et al., Proc. Natl. Acad. Sci. U. S. A. 112, 123 (2013). [40] Y. Meroz and I. M. Sokolov, Phys. Rep. 573, 1 (2015). [41] F. Thiel, F. Flegel, and I. M. Sokolov, Phys. Rev. Lett. 111, 010601 (2013). [42] R. L. Jack and P. Sollich, J. Phys. A: Math. Theor 41, 1 (2008). [43] F. Camboni and I. M. Sokolov, Phys. Rev. E. 85, 050104 (2012).
