PRL 106, 220602 (2011)
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PHYSICAL REVIEW LETTERS
Collision-Number Statistics for Transport Processes A. Zoia,* E. Dumonteil, and A. Mazzolo CEA/Saclay; DEN/DANS/DM2S/SERMA/LTSD; 91191 Gif-sur-Yvette, France (Received 4 March 2011; published 2 June 2011) Physical observables are often represented as walkers performing random displacements. When the number of collisions before leaving the explored domain is small, the diffusion approximation leads to incongruous results. In this Letter, we explicitly derive an explicit formula for the moments of the number of particle collisions in an arbitrary volume, for a broad class of transport processes. This approach is shown to generalize the celebrated Kac formula for the moments of residence times. Some applications are illustrated for bounded, unbounded and absorbing domains. DOI: 10.1103/PhysRevLett.106.220602
PACS numbers: 05.40.Fb, 02.50.r
Many practical problems, encompassing areas as diverse as research strategies, market evolution, percolation through porous media, and DNA translocation through nanopores, to name only a few [1–4], demand assessing the statistics of the random residence time tV spent by a walker inside a given domain V . Indeed, complex physical systems are often described in terms of ‘‘particles’’ undergoing random displacements, resulting either from the intrinsic stochastic nature of the underlying process, or from uncertainty [5,6]. As a particular case, when the particle is lost upon touching the boundary @V of V , the residence time is usually called first-passage time [7]. Fully characterizing tV is an awkward task, since its distribution generally depends on walker dynamics, geometry, boundaries and initial conditions, so that one has often to be content with the mean residence time [8]. This has motivated a large number of theoretical investigations over the last decade, covering both homogeneous and heterogeneous, scale-invariant media [7–13]. In the former case, the dynamics of the walker is usually modeled by regular Brownian motion, whereas in the latter one resorts to anomalous diffusion. A seminal work developed by Kac [14], based on a path integral approach, allows all the moments of the residence times of Brownian particles to be evaluated by resorting to convolutions over the ensemble equilibrium distribution of the walkers, for arbitrary boundary conditions on V [15,16]. However, in many realistic situations, the walker typically undergoes a limited number of collisions before leaving the explored domain, so that the diffusion limit is possibly not attained. Examples are widespread, and arise in, e.g., gas dynamics, neutronics and radiative transfer, electronics, and biology [17–20]. In all such systems, the stochastic path can be thought of as a series of straight-line flights, separated by random collisions, as in Fig. 1, and the dynamics is better described in terms of the Boltzmann equation, rather than the (anomalous) Fokker-Planck equation [17]. A natural variable for describing the walker evolution is therefore the number of collisions nV within the observed volume. Application of the diffusion 0031-9007=11=106(22)=220602(4)
approximation to the characterization of the counting statistics, which amounts to assuming a large number of collisions in V , might lead to inaccurate results [21]. In the present Letter, we address the issue of generalizing Kac approach to random walkers obeying the Boltzmann equation, i.e., not satisfying the diffusion regime, for arbitrary geometries and boundary conditions. We derive an explicit formula for the moments of nV , and illustrate its relation to the equilibrium distribution of the walkers. Knowledge of higher order moments allows estimating the uncertainty on the average, as well as reconstructing the full distribution of the collision-number. We show that when V is large as compared to the typical size of a flight, so that the diffusion limit is reached, Kac formula is recovered. Counting statistics.—Consider the random walk of a particle starting from a point-source located in r0 . At each collision, the particle can be either scattered, with probability p, or absorbed (in which case the trajectory terminates). For the sake of simplicity, we assume that scattering is isotropic. We denote by r the position of the walker entering a collision, as customary. The dynamics is characterized by ðr; r0 Þ, namely, the probability density of performing a displacement from r0 to r, between any two collisions. Define the transport operator ½fðrÞ Z ½fðrÞ ¼ ðr; r0 Þfðr0 Þdr0 ; (1) V
FIG. 1 (color online). A random walk starting from r0 and performing a limited number nV of collisions in a region V (with transparent boundaries), before being absorbed at rn .
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Ó 2011 American Physical Society
PRL 106, 220602 (2011)
over a d-dimensional volume V . We can then express the propagator ðr; njr0 Þ, i.e., the probability density of finding a particle in r at the nth collision (starting from r0 ), as ðr; njr0 Þ ¼ pn1 n ½ðr; r0 Þ, where n ½fðrÞ is the nth iterated operator Z Z ðr; rn Þ ðr2 ; r1 Þfðr1 Þdr1 drn ; n ½fðrÞ ¼ V
V
(2) and ¼ ðr r0 Þ is a shorthand notation for the initial point-source condition. The probability of performing nV collisions in the volume V is related to the propagator by Z Z drðr;nV jr0 Þ drðr;nV þ1jr0 Þ: P ðnV jr0 Þ ¼ V
(3) The moments
nV ¼1
ðrjr0 Þ ¼ lim
N!1
nm P ðnV jr0 Þ V
(4)
N X
ðr; njr0 Þ;
(5)
n¼1
which intuitively represents the equilibrium (stationary) particle distribution [17]. It follows immediately that R hn1V iðr0 Þ ¼ V drðrjr0 Þ, i.e., the integral of the collision density over a volume V gives the mean number of collisions within that domain, hence the name given to ðrjr0 Þ. Higher order moments of nV can be obtained as follows. Define the operator Z ðrjr0 Þfðr0 Þdr0 : (6) ½fðrÞ ¼ V
By making use of the Neumann series
n¼1
hnm iðr0 Þ ¼ V
pn1 n ½fðrÞ ¼
½fðrÞ; 1 p
(7)
½fðrÞ; ½fðrÞ ¼ 1 p
V
(8)
(11)
(12)
which are defined as k-fold convolutions of the collision density ðrjr0 Þ with itself [14,15]. It follows the equivalence Z drk ½ðr; r0 Þ; (13) C k ðr0 Þ ¼ k! V
with Ckþ1 ðr0 Þ ¼ ðk þ 1Þ½Ck ðr0 Þ and C0 ðr0 Þ ¼ 1r0 2V , 1 being the characteristic function. The convergence of the integrals Ck ðr0 Þ depends on the features of the underlying stochastic process as well as on boundary conditions. For instance, the persistence property of walks in d 2 implies diverging Ck ðr0 Þ for transparent V in absence of absorption [15]. We finally obtain the central result of this Letter, i.e., the explicit formula for the moments of the collision number in V hnm iðr0 Þ ¼ V
m 1 X s pk Ck ðr0 Þ: p k¼1 m;k
(14)
When the underlying dynamics ðr; r0 Þ is known, Eq. (14) provides exact estimates of the collision statistics for transport-dominated processes. Thanks to linearity, Eq. (14) allows expressing hnm iðr0 Þ V as a combination of m Kac integrals, k ¼ 1; . . . ; m, each given from Eq. R (12). In particular, for m ¼ 1 we recover hn1V iðr0 Þ ¼ V drðrjr0 Þ, since s1;1 ¼ 1. Furthermore, for m ¼ 2, s2;1 ¼ s2;2 ¼ 1, so that Z Z dr2 dr1 ðr2 jr1 Þðr1 jr0 Þ þ hn1V i: (15) hn2V i ¼ 2p V
we have then
m X 1Z dr k!s pk k ½ðr; r0 Þ: p V k¼1 m;k
We introduce then the repeated Kac integrals Z Z Ck ðr0 Þ ¼ k! drk ... dr1 ðrk jrk1 Þ...ðr1 jr0 Þ; V
þ1 X
depend on the boundary conditions on @V , which affect the functional form of the propagator. The absence of boundary conditions corresponds to defining a fictitious (‘‘transparent’’) volume V , where particles can indefinitely cross @V back and forth. On the contrary, the use of leakage boundary conditions leads to the formulation of first-passage problems [7]. We introduce now the collision density
1 X
1Z drLim ðpÞð1 pÞ½ðr; r0 Þ; (9) p V P k s where Lis ðxÞ ¼ 1 k¼1 x =k is the polylogarithm function [22]. When m is a non-negative integer, the polylogarithm is a rational function, namely, m X x kþ1 Li m ðxÞ ¼ k!smþ1;kþ1 ; (10) 1x k¼0 P where the coefficients sm;k ¼ k!1 ki¼0 ð1Þi ðkiÞðk iÞm are the Stirling numbers of second kind [22]. Thanks to the recurrence properties of the Stirling numbers, Eq. (9) gives hnm iðr0 Þ ¼ V
V
hnm iðr0 Þ ¼ V
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PHYSICAL REVIEW LETTERS
V
Let Gðzjr0 Þ be the moment generating function of P ðnV jr0 Þ. By definition we have the moment expansion
and in particular ðrjr0 Þ ¼ ½ðr; r0 Þ. Now, combining Eqs. (3) and (4), we get 220602-2
Gðzjr0 Þ ¼
1 X m¼0
hnm iðr0 Þ V
zm : m!
(16)
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The small-z expansion of Gðzjr0 Þ reads then Gðzjr0 Þ ’ 1 þ hn1V iðr0 Þz, which for the Tauberian theorems corresponds to the large-nV behavior. It follows the exponential tail 1
P ðnV jr0 Þ ’ enV =hnV iðr0 Þ ;
Cm ðr0 Þ ; Dm
5
0
(17)
provided that C1 ðr0 Þ is finite. Diffusion limit and Kac formula.—Suppose for the sake of simplicity that the walker moves at constant speed v, and p ¼ 1. Then, the time spent between any two collisions is ti ¼ jri ri1 j=v. For isotropic walks, ðr; r0 Þ ¼ ð‘ ¼ jr r0 jÞ thanks to the spherical symmetry. Then, flight times are identically distributed, and obey ti wðti Þ, R where wðti Þ ¼ d ‘d1 ð‘Þðti ‘=vÞd‘, d being the surface of the unit sphere. The diffusion limit of the transport process described above is obtained by letting the typical flight length (the standard deviation of jump sizes, as customary) and the average intercollision time ¼ hti i shrink to zero in such a way that the ratio D ¼ 2 = converges to a constant, namely, the diffusion coefficient. When and vanish, the collision PnV number in V diverges, whereas the quantity tV ¼ i¼1 ti converges to the residence time in the volume. Actually, tV should take into account also additional terms due to boundary conditions. However, as ! 0 and nV ! 1, the trajectory will almost surely have a turning point touching the boundary, so that corrections can be safely neglected. Let now QðtV jr0 Þ be the distribution of the residence times. Under the previousR assumptions, in the Laplace space we have Qðsjr0 Þ ¼ expðstV ÞQðtV jr0 ÞdtV ¼ wðsÞnV . Any arbitrary wðsÞ with finite has an expansion wðsÞ ’ 1 s when ! 0. Then we have Qðsjr0 Þ ’ enV s , which implies QðtV jr0 Þ ’ ðtV nV Þ for small . It follows that iðr0 Þ ’ m hnm iðr0 Þ, with hnm iðr0 Þ given by Eq. (14). If htm V V V we rescale the space variable r by , each of the terms of the sum in Eq. (14) carries a contribution 2k . In the diffusion limit, we therefore recover the celebrated Kac formula [14,15] for the moments of the residence times of Brownian motion htm iðr0 Þ ’ V
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PHYSICAL REVIEW LETTERS
(18)
because all other terms in the sum vanish when ! 0 and ! 0, and sm;m ¼ 1. Moreover, we have the recursion mþ1 iðr0 Þ ¼ ðm þ 1Þ½htm iðr0 Þ, starting property DhtV V 0 from htV iðr0 Þ ¼ 1r0 2V . As for the residence time distribution, from Eq. (17) at long times we have QðtV jr0 Þ ’ expð DtV =C1 ðr0 ÞÞ. Discussion and perspectives.—For simple geometries and ðrjr0 Þ, formula (14) is amenable to analytical solutions. Here we illustrate some significant cases, V being a d sphere of radius R centered in 0. Isotropic gamma flights with kernel ð‘Þ ¼ ‘d expð‘Þ=d ðÞ, > 0, are a
0
0.5
1
1.5
2
0
1
2
3
4
1
0.5
0
FIG. 2 (color online). Moments hn1V iðr0 Þ (stars) and hn2V iðr0 Þ (circles) for 3d ¼ 2 gamma flights with scattering probability p ¼ 1 in a volume V with transparent boundaries. Theoretical predictions from Eq. (14): solid lines; Monte Carlo simulations: symbols.
widespread transport process and describe, among others, search strategies in biology [23]. When ¼ 2, the 3d scattering collision density assumes a simple form, ðrjr0 Þ ¼
1 : 4jr r0 j
(19)
It follows 8 2 2 < 3R r0 ; r < R 0 hn1V iðr0 Þ ¼ R3 6 : ; r 0 R 3r0
(20)
and hn2V iðr0 Þ
8 4 < 25R 10R2 r20 þr40 þ hn1 iðr Þ; r < R 0 0 V ¼ 4 R5 60 1 : (21) : þ hn iðr Þ; r R 0 0 15 r0 V
Comparisons with Monte Carlo simulations with 105 particles are shown in Fig. 2. Exponential flights ( ¼ 1) arise when the scattering centers are uniform, so that intercollision distances obey a Poisson distribution. Such a process is crucial for understanding, e.g., radiation propagation [17,18]. The transport kernel reads ð‘Þ ¼ ‘1d expð‘Þ=d . In 1d systems, the collision density for absorbing V and transparent boundaries reads pffiffiffiffiffiffiffi e 1pjrr0 j pffiffiffiffiffiffiffiffiffiffiffiffiffi : ðrjr0 Þ ¼ 2 1p
(22)
The moments hnm iðr0 Þ are again easily obtained from V Eq. (14). The resulting formulas are rather cumbersome and will not be presented here. Comparisons of exact
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available, as an estimator to infer the equilibrium distribution of the underlying stochastic path, which is often not directly accessible.
4 2 0
0
0.5
1
1.5
2
2.5
3
3.5
4
2 1.5 1 0.5 0 0
0.5
1
1.5
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2.5
3
3.5
4
FIG. 3 (color online). Moments hn1V iðr0 Þ (stars) and hn2V iðr0 Þ (circles) for 1d exponential flights with scattering probability p ¼ 0:5 in a volume V with transparent boundaries. Theoretical predictions Eq. (14): solid lines; Monte Carlo simulations: symbols.
formulas and Monte Carlo simulations with 105 particles are shown in Fig. 3. Analogous findings for exponential flights have been obtained also for leakage boundary conditions, where the collision density reads ðrjr0 Þ ¼
jr0 r=re re r0 =r0 j jr r0 j ; 2
(23)
with re ¼ R þ 1 [24]. In principle, the integrals appearing in Eq. (14) can be carried out numerically for arbitrary complex volumes and ðrjr0 Þ. Moreover, the isotropy hypothesis can be possibly relaxed by replacing r with a state variable y ¼ fr; g accounting for the scattering angle. Actually, the approach presented in this Letter is fairly broad, in that it relies on a minimal number of hypotheses on the underlying displacement kernel ðr; r0 Þ. Though we have used here the language specific to transport phenomena, the same formalism could be rephrased in terms of a semi-Markov renewal process for an arbitrary state variable q evolving in the phase space according to ðq; q0 Þ. In this respect, Eq. (14) would provide the counting statistics for the events falling in V , i.e., satisfying a given ‘‘condition’’, when the diffusion regime is not attained, and the number of such events is small. Finally, observe that a nontrivial application of Eq. (14) would be to make use of the knowledge on the number of collisions in a given domain, when
*
[email protected] [1] M. F. Shlesinger, Nature (London) 443, 281 (2006). [2] J. Ph. Bouchaud and A. Georges, Phys. Rep. 195, 127 (1990). [3] J. Ph. Bouchaud and M. Potters, The Theory of Financial Risk and Derivative Pricing (Cambridge University Press, Cambridge, England, 2003). [4] A. Zoia, A. Rosso, and S. N. Majumdar, Phys. Rev. Lett. 102, 120602 (2009). [5] B. D. Hughes, Random Walks and Random Environments Vol. I (Clarendon Press, Oxford, 1995). [6] G. H. Weiss, Aspects and Applications of the Random Walk (North Holland Press, Amsterdam, 1994). [7] S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, England, 2001). [8] S. Condamin et al., Nature (London) 450, 77 (2007). [9] D. ben Avraham and S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (CUP, UK, 2000). [10] R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000). [11] S. Condamin, O. Be´nichou, and M. Moreau, Phys. Rev. Lett. 95, 260601 (2005). [12] O. Be´nichou et al., J. Stat. Phys. 142, 657 (2011). [13] E. Barkai, J. Stat. Phys. 123, 883 (2006). [14] M. Kac, Probability and Related Topics in Physical Sciences, Lectures in Applied Mathematics (Wiley, New York, 1957). [15] A. M. Berezhkovskii, V. Zaloj, and N. Agmon, Phys. Rev. E 57, 3937 (1998). [16] D. S. Grebenkov, Phys. Rev. E 76, 041139 (2007). [17] C. Cercignani, The Boltzmann Equation and its Applications (Springer, New York, 1988). [18] M. Weinberg and E. P. Wigner, The Physical Theory of Neutron Chain Reactors (UCP, Chicago, 1958). [19] C. Jacoboni and P. Lugli, The Monte Carlo Method for Semiconductor Device Simulation (Springer, New York, 1989). [20] G. Le Cae¨r, J. Stat. Phys. 140, 728 (2010). [21] S. Blanco and R. Fournier, Phys. Rev. Lett. 97, 230604 (2006). [22] A. Erde´lyi et al., Higher Transcendental Functions (Krieger, NY, 1981). [23] F. Bartumeus et al., J. Theor. Biol. 252, 43 (2008). [24] A. Zoia, E. Dumonteil, and A. Mazzolo, Phys. Rev. E 83, 041137 (2011).
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