Statistical dynamics of early river networks

Physica A 391 (2012) 4497–4505

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Statistical dynamics of early river networks Xu-Ming Wang ∗ , Peng Wang, Ping Zhang, Rui Hao, Jie Huo School of Physics and Electrical Information Sciences, Ningxia University, Yinchuan, 750021, PR China

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Article history: Received 14 November 2011 Received in revised form 21 January 2012 Available online 8 May 2012 Keywords: River network Evolution Statistical dynamics Probability distribution Power law

abstract Based on local erosion rule and fluctuations in rainfall, geology and parameters of a river channel, a generalized Langevin equation is proposed to describe the random prolongation of a river channel. This equation is transformed into the Fokker–Plank equation to follow the early evolution of a river network and the variation of probability distribution of channel lengths. The general solution of the equation is in the product form of two terms. One term is in power form and the other is in exponent form. This distribution shows a complete history of a river network evolving from its infancy to ‘‘adulthood’’). The infancy is characterized by the Gaussian distribution of the channel lengths, while the adulthood is marked by a power law distribution of the channel lengths. The variation of the distribution from the Gaussian to the power law displays a gradual developing progress of the river network. The distribution of basin areas is obtained by means of Hack’s law. These provide us with new understandings towards river networks. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Power laws, the manners of many distributions in the natural world which indicate an scaling invariance under some variations of scale, have attracted a great deal of attention from scientists for the profound scientific significance behind them, such as the self-organized criticality (distribution of the event sizes) [1,2], the evolution of a complex network (characterized by the so-called degree distribution) [3,4], the selected patterns of nature, for instance, fractal river networks and their scaling behaviors [5–10], and so on. As is well known, a river network is one of the much-studied natural systems for the scaling relations between different parameters and/or the input variables, the probability distributions of channel lengths, basin areas, as well as energy dissipations [11–14]. The groundbreaking field studies on river networks was executed by Leopold, which revealed that the slope, width and depth of a channel respectively relates in power law to the discharge [15,16]. From then, both empirical and theoretical investigations have been conducted vigorously for decades. With the aid of the analysis technique of digital elevation maps for natural basins, the measurement and characterization of river networks on a large scale became reality. RodriguezIturbe et al. demonstrated that a natural river network can manifest a power law distribution of the cumulative contributing areas [14], p(≥ a) ∝ a−α .

(1)

Where α ∼ 0.43, which is supported by many other field evidences [8,12,17], and ranges from 0.41 to 0.45 [8]. On the assumption that the collected precipitations, at a node of a river network, are proportional to the area of the supported basin, they obtained similar cumulative probability distributions that are followed by the discharge or mass injecting into divides of channels. They also argued, based on the inference that the rate of energy expenditure per unit length of channel

∗

Corresponding author. Tel.: +86 951 2061072. E-mail address: [email protected] (X.-M. Wang).

0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.04.029

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scales like power 12 of discharge [17], that the cumulative distribution of energy dissipation in the channels follows a power law. Maritan et al. stated that the cumulative distribution of the length of river channels can be characterized by another power law in a similar form [12], p(≥ l) ∝ l−β .

(2)

Where β is in the range 0.8–0.9. River networks around the world were found to adapt to have values of β ranging from 0.67 to 0.85 with the so-called finite size corrections [18]. Interestingly, a and l governed by the above two power law distributions were found to be correlated by the famous Hack’s law which relates l with a through the relationship l ∝ aγ .

(3)

This relationship was firstly found by Hack based on the observations in the Shanandoah Valley and the mountains in Virginia [19], and is now well accepted as a basic fact in nature. And moreover, the Hack’s exponent γ , which is empirically found in the range from 0.5 to 0.7 [12,18–20], relates to both of α and β through the relationships α = 1 − γ and β = γ1 − 1 [8,12,21]. How the earth evolves into the present profiles, and is dominated by the aforementioned scaling laws may be amazing. In other words, what creates these scaling relationships? This question may be open, although varied efforts have been made to understand such a perfect selection in nature. Theoretical investigations concerning this question were carried out mainly with models. The pioneering modeling study was suggested by Leopold based on the random walk method to mimic the fluvial process in which the fluid runs in a randomly changing direction on the wander and fluctuant earth surface [22]. The output of the model can effectively describe the observational properties in real basins. The majority of the models, referred to as simplified landform evolution models [13], were constructed based on local erosion rules which lead to the stochastic elongation of river channels according to the nature of the fluid: runs downhill and changes the local height of the landscape. These models can generate various tree-like networks and obtain the basic properties of river networks denoted by Eqs. (1)–(3) and those scaling relationships between parameters of channel and discharge. The local erosion rule in the models seems deterministic, but it takes place on a randomly fluctuant landscape. One thus can say that the essentials of these models lie in two points, randomness and erosion. Obviously, these modeling studies accord, to a certain extent, with the random walk methods. However, both the theoretical and empirical results are not in good agreement with the observational results expressed by Eqs. (1) and (2) at large scale, for instance, the results presented in Refs. [8,12]. Maritan introduced so-called finite size effect ansatz to perfect the description of the probability distributions [12]. Dodds and Rothman stated that the scaling behavior will lapse for small scales due to the so-called linear basins and for large scales due to the inherent discreteness of network structure (the larger the scale is, the scarcer the number of the samples will be). Both their theoretical and empirical investigations stand by this point of view [23–25]. Another modeling study, an optional channel network model, was proposed by Rinaldo et al. based on a principle of minimum dissipated energy [26]. This principle may be regarded as an extended version of the principle of least action which implies that the reality experience of a system is mostly like the one that minimizes the ‘‘action’’ among all possible evolutionary paths. It serves to show important scientific and practical significance. The model postulates that a structure formed by channels is the optional channel network generated by erosion under the minimization of total dissipated energy. The energy here refers to that due to the input of precipitations collected in the whole basin [5,26] or the local basin [27]. Banavar et al. proved that the structures resulted from the minimization of dissipated energy are, to a great extent, equivalent to the networks generated by the landform evolution models owing to the fact that the optional configurations are the stationary solutions of the landform evolution models which are in terms of some equation sets [28]. To all appearances, in the aforementioned modeling studies, randomness, whether it exists in the initial conditions or in the process of simulations, together with erosion plays a key role in generating river networks. In fact, Rinaldo et al. pointed out that both randomness and determinacy are equally essential factors for the dynamic origin of the fractal geometry of river networks [29]. They also stated that scale-free recursive features of a complex system depend on some general properties of dynamics other than the detailed behaviors of the individuals of the system. This implies a basic idea that the scale-invariant behaviors, described by events or sizes, are a ubiquitous phenomenon relating to the theory of nonreductionism. Non-reductionism (for instance, emergence or holism) stresses generality or universality. One may say that the methodology of statistical mechanics is of non-reductionism, and therefore one may have reason to believe that we can establish a statistical theory frame to reproduce the scaling behaviors so as to obtain a greater understanding towards the power law distribution behaviors. Indeed, the works presented in the series of three papers by Dodds and Rothman and that by Banavar et al. can play the exemplary roles [23–25,28]. It should be pointed out that the power law distributions are the characterizations of a nearly stationary state of landscape, evolved for long enough for real basins, and also are the stationary solutions of the theoretical models. So some questions puzzle us: before reaching the steady state or in the process of evolution, is there any law that dominates the distributions of basin areas and stream lengths? Are the observed scaling behaviors or the distributions time-dependent or not? To address these questions so as to investigate the evolution history of a river network, we attempt to establish a theoretical framework to reveal the statistical dynamics of river networks. Starting with this motivation, we construct a

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Fig. 1. Schematic drawing of the initial pattern of a channel, showing the prolongation, dl, in time interval dt.

generalized Langevin equation, in a proper form, to express random elongation of river channels in the early evolutionary stage. Then this stochastic dynamical equation is transformed into the Fokker–Plank equation to study the change of river channel’s distribution in the evolutionary process of networks. The solutions, transition probability or probability density function take the product form of an exponential function and a power function. One interesting fact about these solutions is that they can exhibit a historical picture of a river network. During the initial evolution period, the distribution of channel lengths is dominated by the Gaussian distribution which may imply that the river network is developing randomly; However, the distribution gradually loses the feature of the Gaussian distribution—the symmetry of the function curve is broken down; And finally the distribution transforms into a power law distribution with a long deviating tail. The power law distribution may be the mark of the fully developed or matured river network. The transition from the Gaussian distribution to the power law distribution, in fact, is the self-organized evolution of a river network from disorder(more homogenous distribution) to order(scale-free distribution). By a similar method to that adopted by Maritan et al. [12], we take Hack’s law as the connection between the distribution of channel lengths and that of basin areas, and then the distribution of channel lengths can be translated into the distribution of basin areas. The detailed discussions will provide us with more understanding towards the nature of river networks. 2. Stochastic dynamical equation for the evolution of river channels The evolution of a river network is actually the process in which channels randomly elongate while bifurcating. As discussed in the Introduction, the elongation of channels is only due to erosion performed by the collected rainfall, and the randomness is owing to the fluctuations in rainfall or discharge, the status of the earth’s surface such as vegetation and geology, and other related factors. In view of this, the elongation of channels is a typical stochastic dynamical process. It may be described by a so-called generalized Langevin equation in a general form, dl = f (l, t ) + g (l, t )ξ (t ). (4) dt The first term on the right hand side is deterministic corresponding to erosion and the second stochastic corresponding to the Langevin force induced by the fluctuations mentioned above, which is apparently an external, multiplicative noise. Variable ξ (t ) denotes the Gaussian white noise that is characterized by mean value

⟨ξ (t )⟩ = 0

(5)

and the second-order correlation function

⟨ξ (t )ξ (t ′ )⟩ = 2Dδ(t − t ′ )

(6)

where D is the intensity of the Gaussian noise. Now let us derive f (l, t ) from the erosion mechanism. A channel at the preliminary stage can be conceptually depicted by Fig. 1 where we postulate that an initial channel takes the form of a triangle with a fixed width w shown by the bottom

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edge, and length l. After a time interval, dt, the length becomes l + dl. The length increment, dl, results from erosion caused by the collected precipitations. The area, S, of land that effectively collects precipitation to contribute to the evolution of the channel may be difficult to determine. Here we suppose that it is approximatively fan-shaped, and it takes the middle point of the bottom as the center and kl (k is a proportionality factor) as radius, so we have S = k′ k2 l2 ,

(7)

′

′

2

′

where k is another coefficient. Since k is also related to the shape of the area of precipitation collection, k and k can be incorporated into a new parameter, which might as well be denoted by k, satisfying S = kl2 .

(8)

Here we address it as the shape factor of the area of precipitation collection. Undoubtedly, it is difficult to determine due to the fact that the shape of the area of precipitation collection changes randomly in the evolutionary process. Nevertheless, the precipitations gathered in this area can be theoretically written as Q = qS ,

(9)

where q stands for the rainfall within a unit time per unit area. Meanwhile, the extended part of the channel in time interval dt can be denoted by the increment of area, dA =

1 2

wdl.

(10)

A key point that is related to f (l, t ) may be the erosion quantity, Qe , caused by the collected rainfall. A relationship describing the sediment Qs carried by the collected rainfall or discharge Q can be used for reference to express Qe [30], that is, Qs = ηQ δ ,

(11)

where η is sediment carrying coefficient, and δ exponent. This relationship can also be deduced by equation, Qs = kf ′

Q m r n , presented in Ref. [31] with hypothesis that Q depends nonlinearly on the channel slope r, i.e., Q ∼ r n . Sedimentation may be neglected in the development process of a river network, Qs ≈ Qe holds at least for the stream flowing near the head of channel, so one can obtain Qe by inferring from Eq. (11), Qe = eQ δ ,

(12)

where e is erosion coefficient that associates with the property of soil. The equation denotes erosion quantity within a unit time interval, and the product Qe dt should be equal to the lost soil, dAh, that corresponds to the extended channel with a mean depth h. Based on this discussion and synthesizing Eqs. (7)–(12), one obtains dl dt

=

keqδ 2δ l . wh

(13)

The right hand side term is actually f (l, t ) in which each variable or coefficient can be regarded as a mean value. If the fluctuations are taken into account, an instantaneous form equation corresponding to Eq. (13) can be written as dl dt

=

(k ± ∆k)(e ± ∆e)(q ± ∆q)δ 2δ l . (w ± ∆w)(h ± ∆h)

(14)

Where ∆k, ∆e, ∆q, ∆w and ∆h are the fluctuation amplitudes of k, e, q, w and h, respectively. Carrying out a Taylor expansion of each of terms, for instance, w±1∆w , in the neighborhood of w , and neglecting the higher-order terms, we change the above equation to dl dt

=

keqδ 2δ l wh

By setting ε = dl dt

∆k k



+

 1±

∆e e

∆k k

+

+ δ ∆qq −

∆e e

∆w w

+δ

−

∆h h

∆q q

−

∆w ∆h − w h



.

(15)

keqδ

, M = wh and κ = 2δ , the equation is rewritten as

= Mlκ ± M ε lκ .

(16)

Where ±ε denotes pure noise caused by the relative fluctuations, and the noise is naturally in terms of ±M ε . Easily, one can obtain each of terms in Eq. (4) via comparing them with Eq. (16), that is, f (l, t ) = Mlκ ,

g (l, t ) = lκ .

Thus the generalized Langevin equation to describe the random elongation of river channels is established.

(17)

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3. Probability distribution of channel lengths From the Langevin equation, one can derive n-order moment for any possible n. As shown by Eq. (6), since the correlation function is nought for the Gaussian white noise ξ (t ), the moments expressing l(t + τ ) and l(t ) are presented as follows:

 ⟨l(t + τ ) − l(t )⟩ = (f (l, t ) + Dg ′ (l, t )g (l, t ))τ + O(τ 2 ), ⟨(l(t + τ ) − l(t ))2 ⟩ = 2Dg 2 (l, t )τ + O(τ 2 ),  ⟨(l(t + τ ) − l(t ))n ⟩ = 0, n ≥ 3.

(18)

Where g ′ (l, t ) is the derivative of g (l, t ) with respect to l. Substituting the equations above into Kramers-Moyal expansion (see the details in Ref. [32]) holding over the linear terms (stopping after the second term), the stochastic dynamical equation is then transformed into the Fokker–Plank equation describing the time evolution of the probability distribution of river lengths,

∂ ∂2 ∂ p(l0 , l; t ) = − (f (l, t ) + Dg ′ (l, t )g (l, t ))p(l0 , l; t ) + D 2 g 2 (l, t )p(l0 , l; t ) (19) ∂t ∂l ∂l where p(l0 , l; t ) denotes the transition probability function, which means the probability of a channel with length l0 at t = 0 becoming a channel with length l after t. Here we focus on the original evolutionary process that means l0 = 0 at the beginning of evolution, t = 0. So p(l0 , l; t ) becomes, in substance, p(l, t ) that also satisfies Eq. (19). Inserting the expressions of f (l, t ) and g (l, t ) into this equation, one can obtain its time-dependent solution [32]   l−κ (l1−κ − Mt + Dκ lκ−1 t )2 p(l, t ) = √ exp −M κ lκ−1 t + Dκ(2κ − 1)l2κ−2 t − . (20) 4Dt 2 π Dt In the following discussions, the values of parameters are chosen as κ = 1.25, M = 0.09, D = 0.0049. Obviously, this solution is characterized by a product of two functions. One is in power form and the other in exponent form. We address this distribution as generalized exponential power distribution (GEPD). Many observations in real systems are governed by a kind of distribution laws in similar forms [33–36], so-called exponential power distributions (EPDs), for instance, the degree distributions in some complex networks [33]. The EPD has two equivalent forms. One is known as the shift power law (SPL) that is in terms of p(x) ∝ (x + a)−µ [34], the other is named the stretched exponential distribution µ (SED), expressed as p(x) = µ(xµ−1 /x0 ) exp(−(x/x0 )µ ) [35]. The characteristic parameter for the former is a and for the latter µ. As a = 0 (µ → 0) SPL (SED) tends to be a typical power law, while a → ∞(µ = 1) SPL (SED) reduces to an usual exponent distribution. As Lerrènette and Sornete stated, these distribution laws may interpret the deviation from a power law more physically other than finite size effect corrections [36]. Now a question occurs, that is, is there a similar parameter by which different extreme forms of the GEPD described by Eq. (20) are characterized? And if so, what should be the form of the extremes? To answer these questions, we present Fig. 2 to depict the variation of the probability distribution with time. In the beginning, in terms of time t, the probability is dominated by a Gaussian-like distribution, which bears the characteristic of single-peak and symmetry with respect to the peak point. However, it soon has its symmetry broken down with the evolution of time. Compared with the Gaussian distribution,

√

p(x, σ ) = (1/ 2π σ 2 ) exp(−(x − µ)2 /(2σ 2 )),

(21)

√ the GEPD is led by factor, 1/ 4π Dt exp(−(l1−κ − Mt + Dκ lκ−1 t )2 /(4Dt )), and modulated jointly by factors l−κ and exp(−M κ lκ−1 t + Dκ(2κ − 1)l2κ−2 t ). The leading factor determines the single-peak, and the two modulating factors destroy the symmetry of the peak. Fig. 2 also shows that the deformation of the probability curve gradually nibbles away at the left side of the peak. Finally, the probability distribution transforms into the one that decreases monotonously. It should be pointed out that the distributions presented by Fig. 2 are not normalized. In many statistical-based or empirical studies, the cumulative probability distribution is often considered to avoid oversize error due to the limited data, so it is in the empirical investigations of river networks. In order to compare our theoretical results with the observed cumulative probability distribution of channel lengths, integration performed for Eq. (20) from l to infinity yields p(≥ l, t ) =

∞



p(l, t )dl.

(22)

l

The case at t = 20 is depicted by Fig. 3(a) in log–log coordinates, and suggests a power law distribution ′ p(≥ l, t ) ∝ l−β

(t = 20)

(23)

with β = 0.89, which is in the reasonable range of β of Eq. (2) stated by Banavar et al. [8]. In fact, this scaling law has been established as time t is beyond 16. As shown in Fig. 3(b), the fitted straight line parts of the curves corresponding to different times, from t = 17 to t = 21, nearly stay parallel to each other, the slope slightly increases with time, which ′

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a

b

c

d

Fig. 2. The variation of the GEPD with time: (a) t = 1; (b) t = 5, (c) t = 10, (d) t = 20.

a

b

0

Slope=-0.89

-1 ln P(>=l)

ln P(>=l)

-5

t=5 t=10 t=15 t=20

-10

t=17 t=18 t=19 t=20 t=21 t=22

-2

-3

-15 -4

-3

-2

-1 ln l

0

1

-3

-2

-1

0

ln l

Fig. 3. Cumulative Probability of channel lengths: (a) shows scaling law; (b) the parallel relationship among the partially magnified parts of the curves drawn at different times. Notice that symbol ‘‘>=’’ here actually stands for symbol ‘‘≥’’ in the text.

means that once the scaling law has been established, the exponent β ′ will approximately hold a fixed value. However, the curve at t = 22 loses monotonicity. And from then, the curve will stray from the scaling law. The fitted straight line part of the curve gradually shortens and even disappears. This departure from the power law distribution seems a weakness of our work. However, what we want to bring to our reader’s attention is that our discussions, especially in constructing the Langevin equation, are for describing a developing river network from its infancy to the adulthood marked by the power law distribution of channel lengths having been established, not for a developed river. So the incorrectness of the description for a long term may consist in the triangle-like channel related to Eq. (10), the area collecting rainfall expressed by Eq. (8) and the erosion relation, Eq. (12). They all together determine the Langevin equation to be unsuitable for describing the elongation of the developed river channel. This unworthiness is translated into the Fokker–Plank equation and its solution, and results in the deformation of the distribution and finally completely losing the characteristics of a power law.

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0.6

the Gaussian distribution GEPD

P(l,t=1)/P(x)

0.4

0.2

0.0 0

2

4

6

8

10

l/x Fig. 4. Comparison between the GEPD and the Gaussian distribution indicates that the Gaussian distribution dominates the initial evolution of river networks.

Let us focus again on the initial distribution, the Gaussian-like one, before the power law distribution is established: what actually is the Gaussian-like distribution  ∞ in the early time of evolution? Fig. 4 shows the distribution at time t = 1 normalized with a factor A = 3.72 × 1013 (A 0 p(l, t )dl = 1). It is in good agreement with the normal Gaussian distribution expressed by Eq. (21) with µ = 3.34 and σ = 0.7. This demonstrates that it is the Gaussian distribution that dominates the probability distribution of river channel lengths in the early evolutionary process. So far, one can present a panorama of the evolutionary process of a river network. In the initial stage of the evolutionary process, the Gaussian distribution leads the developing process of a river network, and implies that the channels of the river network are developing in a disorderly manner. The occurrence of the distribution losing the characteristics of the Gaussian distribution marks that the river network is evolving from disorder to order. Finally, the establishment of the power law distribution corresponds to the beginning of the period where river network has evolved into an order, a self-organized state, and also indicates that the river network has become a developed or matured one. Re-plotting the result shown by Fig. 2(d) in dual-logarithm coordinates (the figure is not presented here to save space) will give the following probability density scaling with the channel length p(l, t ) ∝ l−η

(t = 20),

(24)

and is in accordance with that derived by differentiating with respect to the cumulative probability, that is, we have the same p(l, t = 20) satisfying p(l, t ) =

d dl

′ p(≥ l) ∝ l−(1+β )

(t = 20).

(25)

Theoretically, η = 1 + β ′ always holds as soon as the river network has been developed. 4. Probability distribution of drainage areas In the development process of a river network, the basin enlarges with elongation of the streams. As discussed above, when the river network has evolved into a developed one, the scaling law governing the distribution of stream lengths is established. So, one can say that the corresponding scaling law holding responsible for the distribution of basin areas is also set up at the same time. In other words, in this situation the relatively stable relationship between the length and area, in a statistical sense, has been established. From the viewpoint of real evidence, it is only Hack’s law that can act as a relationship relating the basin area with the channel length. This relationship was introduced to be the conversion function between the distribution of lengths and that of areas [12]. With the same idea in mind, we define the same conversion function, f (l, a) = δ(l − aγ ),

(26)

with which one can obtain one distribution from another, between the distribution of lengths and that of areas, by conducting the following integration, p(a, t ) =



p(l, t )f (l, a)dl

  (a(1−κ)γ − Mt + Dκ a(κ−1)γ t )2 a−κγ = √ exp −M κ a(κ−1)γ t + Dκ(2κ − 1)a2(κ−1)γ t − 4Dt 2 π Dt

(t ≥ tc ).

(27)

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0

slope=-0.43

ln P(>=a)

-4

-8

-12 -3

-2

-1

1

2

Fig. 5. Cumulative probability of basin areas suggests a scaling law. Symbol ‘‘>=’’ here also stands for ‘‘≥’’ in the text.

Where tc (here equal to 17) denotes the time at which the river network has developed or evolved into a matured one. To compare with the cumulative probability distribution of areas discussed in the first section, Introduction, we integrate Eq. (27) from a to ∞, yielding p(≥ a) =

∞



p(a, t )da

(t ≥ tc ).

(28)

a

The result at t = 20 is depicted in Fig. 5 and suggests a cumulative power law distribution of basin areas, p(≥ a, t ) ∝ a−α′

(t = 20).

(29)

Where α = 0.43 as γ = 0.7, which is in exact agreement with the observational one [12], and within the general empirical statistical results ranging from α = 0.41 to −0.45 [8] presented in the first section. However, in the initial stage, the probability distribution in the manner of the Gaussian for lengths lasts longer than that for areas. It may be due to the fact that the basin extends transversely as it prolongs longitudinally, which determines that the disorderly distribution of areas will be depressed more early than that of lengths. Being analogous to the method to derive the probability density function from the cumulative probability of lengths, the derivative of Eq. (29) with respect of a will give the power law distribution of areas. It is the same of that obtained directly from Eq. (27) as t ≥ tc . Now we can answer the question appearing in the Introduction. The probability distributions for channel lengths are indeed time-dependent. In the beginning, the distribution is dominated by the Gaussian distribution. Then it is replaced by the Gaussian-like distributions that are characterized by the single-peak and the absence of symmetry. This distribution tends to get far away from the Gaussian distributions, and finally transforms into the power law distribution. As far as the distribution of basin areas is concerned, we may guess that it varies time-dependently in a similar way. It is obvious that time t is the characteristic parameter of the GEPD. As t → 0, the GEPD tends to be the Gaussian distribution, while as t ≥ tc , the GEPD becomes a typical power law distribution (which may be extrapolated to the extreme as t → ∞). ′

5. Conclusion and discussion In this article, we construct a generalized Langevin equation to express the stochastic dynamical process of the prolongation of river channels, the initially evolving process of a river network. The deterministic part is derived based on the erosion rule, and the stochastic part is determined by the random fluctuations of rainfall, rain-harvesting acreage, geology and parameters of the channels including width and depth. This equation can be transformed into the Fokker–Plank equation to describe the probability distribution of channel lengths. The time-dependent solution of the Fokker–Plank equation, so-called GEPD, takes on a product form of a power function and an exponential function. It shows the transient changes of the distribution of channel lengths in the originally developing process. It starts with the Gaussian distribution, and then gradually becomes a deformed Gaussian distribution having its symmetry broken down (the left side of the distribution curve gets nibbled little by little in the process). Finally, it completely transmits into a power law distribution with a deviating tail for large scale. Using the link between channel lengths and basin areas in a statistical sense, Hack’s law, we derived the distribution of basin areas from the power law distribution of the channel lengths which marks that the river network has been a developed or matured one. As the power index for the distribution of channel lengths takes a reasonable value, the obtained result for the distribution of basin areas is also in the reasonable range. As far as the significance is concerned, the power law distribution here is the signature of river networks that have developed into an orderly phase, that is, a self-organized state, while the Gaussian-like distribution is a symbolic

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representation of river networks that are disorderly and making for the orderly phase. It is especially worth noting that the Gaussian distribution implies that in the initial phase of evolution, a river network may be completely stochastic or disorderly. What is particularly important is that these results imply that the power law distributions of channel lengths and basin areas of river networks are the fruit of the randomness and determinacy of the evolutionary dynamics. This is in accordance with the idea that both randomness and determinacy are the general dynamical origin of scale-free geometry in nature [29]. From this point of view, the method of statistical dynamics, which is described by the generalized Langevin equation and the Fokker–Plank equation here, is actually within the non-reductionism paradigm of complex science. This approach may be applied to investigate the dynamics of a kind of system such as the process of cracks propagating on a material surface, the formations of oilfields and coalfields, the growth of fractal pore networks in porous media and so forth. The common characteristics of these systems are open, self-organized and scale-free, and maybe far from equilibrium. This investigation revealed that both power law distributions of channel lengths and basin areas bear long deviating tails at large scale, which were well studied by many scientists [12,23,36]. This deviation was conventionally attributed to the finite size effect caused by the decrease in the number of samples. Since such a cut-off is the nature of the GEPD, we could draw a conclusion that the long deviating tail may be an inherent quality of river networks. It is in agreement with the findings in the deviation of Hack’s law by Dodds and Rothman [23] who stated that the deviation at small scale is due to the linear connection between the length and basin area, while the deviation at large scale is caused by the discrete substructures of river networks. It is also in accordance with the statement that the deviation from the power law is fundamental and not only the finite size effect correction [36]. These understandings may be important for both theoretical work and practical statistical work. Acknowledgment This study is supported by the National Natural Science Foundation of China under Grant Nos. 10965004 and 10565002. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]

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