冑冑

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Intermittent search process and teleportation

O. Bénichou, M. Moreau, P.-H. Suet,^a兲and R. Voituriez

Laboratoire de Physique Théorique de la Matière Condensée (UMR 7600), Université Paris 6, 4 Place Jussieu, 75252 Paris, France

共Received 5 March 2007; accepted 26 April 2007; published online 19 June 2007兲

The authors study an intermittent search process combining diffusion and “teleportation” phases in a d-dimensional spherical continuous system and in a regular lattice. The searcher alternates diffusive phases, during which targets can be discovered, and fast phases 共teleportation兲 which randomly relocate the searcher, but do not allow for target detection. The authors show that this alternation can be favorable for minimizing the time of first discovery, and that this time can be optimized by a convenient choice of the mean waiting times of each motion phase. The optimal search strategy is explicitly derived in the continuous case and in the lattice case. Arguments are given to show that much more general intermittent motions do provide optimal search strategies in d dimensions. These results can be useful in the context of heterogeneous catalysis or in various biological examples of transport through membrane pores. ©2007 American Institute of Physics.

关DOI:10.1063/1.2741516兴

I. INTRODUCTION

Chemical kinetics in a dense medium is often controlled by diffusion.¹In fact, the elementary reactive process should be described on the microscopic level, by quantum mechan- ics, but a classical, mesoscopic description is possible in the simple case of diffusion-limited reactions:² the reaction is completed as soon as reactants meet共or with a given probability when they meet兲.

In his pioneering work in 1917, von Smoluchowski³ used diffusion theory to calculate the rate of the diffusion- limited annihilation or capture reaction P +A→A for one spherical, immobile particleA, surrounded by particles P. He found that the reaction ratek共t兲is time dependent and criti- cally depends on the space dimensiond: ford= 3, it tends at large times to the finite rate constantk= 4␲RD, whereas for d= 1 andd= 2 its limit value is 0. These well known results have been extended in various directions,^4–8in particular, in the case ofNmobile P particles共catalysts or predators, in the language of ecology兲and one immobileA particle共target or prey兲with a fluctuating reactivity. These studies confirm that the usual kinetic laws in general only hold in three dimen- sions共or more兲, and that even then they can break down, in particular, in asymptotic regimes.^4,6–10

A striking example where low-dimensional reaction paths drastically modify usual reaction kinetics is given by the problem of the localization of a protein at a specific target site on a DNA molecule. It is well established¹¹ that the reactive trajectories combine one-dimensional diffusion along the DNA molecule and long range bulk excursions due to association-dissociation events共see Ref.12for a review兲.

In good agreement with experimental data,¹³recent stochastic models of this reaction^13,14have put forward the existence of an optimal search strategy, achieved when both regimes

have the same mean duration 共see also Refs.15–19兲, which significantly accelerates the reaction. Such intermittent reaction paths combining two regimes, a “slow”reactivemotion, during which the target can be discovered, and a “fast” but nonreactive motion, during which the searcher is unable to detect the target, have proved to play a crucial role in numer- ous search problems.^4,8,20–29

In practice, the effect of low dimension on reaction kinetics is important if the targetsA are located on the boundary⳵^Vof a domainV. The dynamics of a reactant P can then be described in a very general way as intermittent, similar to the above mentioned case of a protein reacting on a specific site on DNA:P will alternate adsorbed phases on the interface

⳵V, where diffusional transport and reaction can occur, inter- rupted by bulk phases of free motion in the interior of V where targets A are absent. The case of a bidimensional boundary ⳵^V is widely realized in chemistry in the case of heterogeneous catalysis 共see Fig. 1兲, or in the physics of porous media.³⁰ Another example comes from physiology:

mammalian respiration can be described as 0₂molecules共re- actants P兲diffusing in the gas exchange units of lungs, called acini, toward their alveolar membrane, until they are adsorbed through a membrane pore 共a targetA兲. At a smaller scale, cell biology provides further examples of such P +A

→Atrapping reactions. Indeed, molecular共and, in particular, ionic兲 transport across channels of the cell membrane³¹ or even viral infection³² through the cell membrane involve a searcher P 共molecule or virus兲 alternating bulk diffusion in the intercellular medium and adsorbed phases on the cell membrane before entering the cell through a “gate”A共chan- nel or membrane receptor, see Fig.1兲.

In this article, we present a model of intermittent search process which generalizes our one-dimensional model of protein search on DNA,¹⁴later generalized by Eliazaret al.³³ This model applies, in particular, in studying the above mentioned reactions of type P +A→A or the catalytic reaction

a兲Electronic mail: [email protected]

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P +A→C+A, where A are immobile targets located on a d-dimensional space⍀=⳵V. A mobile molecule P diffuses in

⍀and interacts with anAas soon as it reaches it. However, at stochastic times, P is desorbed from ⍀ and performs a bulk excursion insideV共where noAis present兲before read- sorbing on ⍀ and diffusion again. The main hypothesis of our model is to treat these bulk excursions as “teleportation”

phases: we assume that P is randomly relocated in⍀after an exponentially distributed time or, more generally, after a stochastic time with a finite average, independent of this dis- placement共see Ref.34for related models兲. Keeping in mind the example of heterogeneous catalysis where ⍀ is a two- dimensional plane, a teleportation is an approximate of a three-dimensional diffusive excursion in the bulk in the fol- lowing limit共Fig.1兲: the volumeVmust be finite to ensure a finite mean return time 共as opposed to power law distribu- tions obtained for infinite V兲, and the typical distance cov- ered during such a bulk phase must be larger than the typical intertarget distance to ensure that return locations are effec- tively uncorrelated. Note that a finite accessible volume V can be realized by a confining attractive potential 共electro- static for instance兲toward the interface ⍀. The net gain of such intermittent behavior is not clear, since teleportation is time consuming but, in principle, unproductive since no reaction can occur during such phases. We shall, however, treat this problem explicitly and show that intermittence permits one to minimize the time of first arrival of P atA共or search time兲and thus, to hasten the reaction and increase the reaction rate. First, we present our d-dimensional continuous model, solve the corresponding equations, and compute the average search time ofA. We treat the optimization problem and show that the search can be made much shorter, thanks to intermittence. Then, we consider a similar lattice model and prove that the optimization is also possible, but that the conditions for obtaining it can be significantly different.

Eventually, we show that the conclusions can be extended to much more general stochastic search processes in an arbi- traryd-dimensional space.

II. CONTINUOUS DIFFUSION IN A SYSTEM WITH SPHERICAL SYMMETRY

A. Model

We consider a point P searching for a given, immobile target. P moves in a finite region of a d-dimensional space,

represented by a d-dimensional sphereB of radius b with a reflecting internal surface. The target is a concentric sphereA of radiusa⬍b共Fig.2兲. However, the searcher can only rec- ognize the target when his sensors are activated. In fact, the searcher undergoes an intermittent movement, alternating between two dynamical regimesi= 1 , 2.

共1兲 During regime 1 共diffusion兲, he performs an isotropic diffusion, and its sensors are activated: he finds the target as soon as he reaches it in this regime.

共2兲 During regime 2 共teleportation兲, the searcher is relocated with uniform probability at any point of sphereB.

During the teleportation, however, his sensors are inac- tivated and the target cannot be found.

The durationTiof each regimeiis a stochastic variable independent of other events, with an exponential law:

P共Ti⬎t兲= exp共−␭it兲, 共1兲

withi= 1 , 2, the frequencies␭ibeing constant parameters.

The question is to find the optimal strategy—if any—to reach the target as rapidly as possible.

during a random waiting time共black path兲, before desorbing back to the bulk. The process is continued until P reaches a targetA共dark circles兲.

FIG. 2. Intermittent search with teleportation. Searcher, point P; target, sphereAwith radiusa; search domain, sphereBwith radiusb.

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B. Mean search time

In this problem, a basic variable is the search timeTx, i.e., the first arrival timeTxof P at the targetAin regime 1, starting from the initial positionx. LetF共t兩x兲be its probabil-

ity density, and F共t兲=具F共t兩x兲典B its average over the uniform initial distribution of P in volume B. It can be shown 共see Sec. IV and Ref. 4兲 that the Laplace transform of F共t兲, Fˆ共s兲=兰₀^⬁dte^−stF共t兲, is given by

Fˆ共s兲= 具j1共␭1+s兩x兲典B+v共A兲/v共B兲

1 −共1 −v共A兲/v共D兲−具j1共␭1+s兩x兲典B兲/共1 +s/␭1兲共1 +s/␭2兲, 共2兲

where A is the space occupied by the target, B the total available space, and v共A兲 and v共B兲 are the respective volumes of these regions.

Here¯jis the Laplace transform of the first arrival density which is given³⁵by

j₁

共␭1+s兩x兲=

冉

^r^a

冊

^␯D^D_␯,−^␯,−共a^共r

^冑冑

^共␭共␭¹1⁺+^s兲/D,bs兲/D,b

^冑冑

^共␭共␭¹1⁺+^s兲/D兲s兲/D兲, 共3兲 wherer=兩x兩 andD_␯,−共x,y兲=I_␯共x兲K_␯−1共y兲+K_␯共x兲I_␯−1共y兲.I_␯共x兲 andK_␯共x兲are modified Bessel functions³⁶ with

␯^{= 1 −}^d

2. 共4兲

After lengthy calculations共see Appendix A兲, we obtain Fˆ共s兲= −k共Y_␯/X_␯兲+a^d/b^d

1 −共1 −a^d/b^d+k共Y_␯/X_␯兲兲/共1 +s/␭1兲共1 +s/␭2兲, 共5兲 with

k= d

b^d

冉

␭1^D+s

冊

^1/2^a^d−1 ^共⁶^兲

and

X¯

s=0

, 共9兲

which yields 具T典=␭₁+␭₂

␭1␭2

关共b/a兲^d− 1兴X¯

␯+共d/a兲

冑

D/␭1Y¯

␯

X¯

␯−共d/a兲

冑

D/␭1Y¯

␯

. 共10兲 This exact, explicit formula共10兲is most useful for minimizing the mean search time as a function of the parameters, which is an important point in most practical cases. Sincea, b, and D are determined by the geometrical and physical properties of the medium, the relaxation frequencies ␭1 and

␭2, or the mean durations␶1= 1 /␭1and␶2= 1 /␭2of phases 1 and 2, are the main adjustable parameters.

C. Optimization of the mean search time

The optimization with respect to␭2is obvious: the time lost in teleportation should be as short as possible, since the result is independent of it, so that ␭2 should be as high as possible. We now consider the minimization of 具T典with respect to␭1. From expression共10兲, the mean search time can be written as

具T典=␭₁+␭2

␭1␭2

关共b/a兲²− 1兴+ 2共Z共x兲/x兲

1 − 2共Z共x兲/x兲 , 共11兲

withZ共x兲= −K_␯−1共x兲/K_␯共x兲andx=a

冑

_␭₁/D.

The analysis of the exact expression 共11兲 has been treated numerically in the important cases of two and three dimensions共Figs.3and4兲.

It is seen that two cases can occur: for small values of

␭2,具T典increases with␭1, and the minimal value is obtained by choosing ␭1= 0, or ␶1=⬁, thus having an uninterrupted diffusive regime. Then, intermittence is not favorable to the search and should be avoided. On the other hand, for large values of ␭2, 具T典 decreases with ␭1 for small ␭1, it has a minimum for a finite value␭¯ of1 ␭1, and it increases for␭1

⬎␭¯ . Then, intermittence is favorable to the search and it1

allows reducing the search time substantially.

The critical value ␭2c separating these regimes can be obtained explicitly, depending on the dimensiond. It is given by simple expressions in the usual case b/aⰇ1. In two dimensions,

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␭2c⬃ 96Dln共b/a兲

7b² whenbⰇa. 共12兲

In three dimensions,

␭2c⬃ 35D

12ab whenbⰇa. 共13兲

If␭2⬎␭2c, the optimal value␭1of ␭1 can be computed numerically. It is also possible to obtain simple approximations in specific situations. Such a situation occurs in the important case of the small density limit, which is currently observed.

Small density limit. We now assume that the volume of the search region B is much larger than the volume of the

FIG. 3.具T典as a function of␭1in the two-dimensional case.共a兲具T典minimum for␭1= 0 witha= 1,b= 10³,d= 2,D= 1, and␭2= 10⁻⁵.共b兲具T典minimum for a finite␭1witha= 1,b= 10³,d= 2,D= 1, and␭2= 10.

FIG. 4. 具T典as a function of␭1in the three-dimensional case.共a兲具T典minimum for␭1= 0 witha= 1,b= 10³,d= 3,D= 3, and␭2= 10⁻⁵.共b兲具T典minimum for a finite␭1witha= 1,b= 10³,d= 3,D= 3, and␭2= 10.

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target A:b^dⰇa^d. Furthermore, we suppose that b²ⰇD/␭1, which implies that the average fraction ofBexplored during a single diffusive phase is very small. Then,

Z共x兲 ⬃−K_␯−1共x兲

K_␯共x兲 . 共14兲

On the other hand, we may assume thatbis large enough to satisfy

冉

^b^a

冊

^b^Ⰷ^d

冏

^Z共x兲^x

冏

^. ^共15兲

Then, formula共10兲simplifies very much and allows for a simple analytical optimization of具T典 in the cases of short and long waiting times. As we proceed to show, condition 共12兲or共13兲is compatible with both cases, leading to a non- trivial optimization of the search time for␭2fixed.

Short waiting times. Let us first consider the case x

=a

冑

_␭₁/DⰇ1, or ␶1Ⰷ␶⬅a²/D. Then,Z共x兲⬃−1, inequality 共15兲 clearly holds, and the exact formula 共10兲 can be approximated by

具T典 ⬃1

2

冉

a^b

冊

^d^␭␭¹1⁺␭^␭2²

x

x+d. 共16兲

It is easily found that this expression reaches its minimal value if

␭1=␭2

冋

^{1 +}²^d

^冑

^␭^D¹^a²

册

^. ^共17兲

Then, the assumption x=a

冑

_␭₁/DⰇ1 implies that a

冑

_␭₂/DⰇ1 and the optimal value␭1of ␭1 is

␭1⬃

冉

²^d

冊

²^a^D²^␭²²^. ^共18兲

In other words, this case applies when the waiting times

␶1and␶2are both much smaller than the characteristic time

␶⬅a²/D: when the optimal value of␶1scales as␶₂²^{, we have}

␶1Ⰶ␶2Ⰶ␶^and

␶1

␶ ^⬃

冉

^d2

冊

²

冉

^␶␶²

冊

²^. ^共19兲

It should be remarked that in this case, the searcher spends more time in teleportation, although he cannot find the target in this regime, than in diffusion. This counterintui- tive result had also been noticed in the case of alternating diffusion with ballistic motion.²⁰Then, the minimum search time is simply

具T典min=1

2

冉

^ba

冊

^d^␶²^. ^共20兲

These results hold in any dimensiond. Thus,具T典minhas its effective smallest value when␶2has its minimum possible value ␶2min, as it has been already pointed out.

The efficiency of intermittence can be characterized by the ratio E=具T典diff/具T典min, 具T典diff being the search time in a purely diffusive regime 共Appendix C兲. It is found that if b Ⰷa, in one, two, and three dimensions,Ehas the respective values

E=

冦

^{共2/3兲共ab/D}^␶²^兲⁼共2/3兲共b/a兲共␶/␶2兲 ifd= 1 共a²/D␶2兲ln共b/a兲= ln共b/a兲共␶^/␶2兲 ifd= 2

共2/d共d− 2兲兲共␶^/␶2兲 ifd艌3.

冧

^共21兲

Thus,E is much larger than 1 if␶2Ⰶ␶and, in one and two dimensions,b/aⰇ1. The latter enhancement factor dis- appears in three or higher dimensions. As a conclusion, in the limit of short waiting times␶2, or high frequencies␭₂, the efficiency is proportional to ␭2 in any dimension, as con- firmed by numerical analysis 共Fig. 5兲. It is clear that the efficiency of intermittence depends on the minimum possible value ␶2min, and that it decreases as dimension d increases.

Nevertheless, intermittence can always be a favorable search

FIG. 5. Plots of efficiencyE=具T典diff/具T典minas a function of␭2.共a兲Two dimensions witha= 1,b= 10³,d= 2, andD= 1.共b兲Three dimensions witha= 1,b

= 10³,d= 2, andD= 1.

FIG. 6. One-dimensional lattice.

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strategy if␶2can be made sufficiently small, and if the minimum realizable value of ␶1共␶1min兲 allows one to reach the optimal value␶1.

Long waiting times. The opposite limit of long waiting times, when both ␶1 and ␶2 are much larger than ␶^{, is in} principle less favorable, and conditions共12兲and共13兲impose upper bounds on␶2for intermittence to be favorable. Never- theless, these conditions can be satisfied even with ␶2Ⰷ␶^, and the case of long waiting times may allow one to minimize the search time for a finite value of ␶1. This case is observed in practice if the characteristic time ␶^{, which is} imposed by physical conditions, is much shorter than ␶1min

and ␶2min. It is shown in Appendix C that in this situation intermittence again allows one to reduce the search time significantly in one and two dimensions, but this strategy loses its interest in higher dimensions. More precisely, in one di- mension, the optimal value of␭1 is

␭1⬃␭2 共22兲

so that␶1⬃␶2Ⰶ␶: both waiting times should be equal, which was indeed the result found in Ref.20concerning the search of specific DNA site by a protein.

Furthermore,

具T典min⬃ 2b

冑

D

冑

␶2 共23兲

and the efficiency is

E=2 3

冑

Db␶2

=2 3 b a

冑

␶^␶2

. 共24兲

Here 共␶^/␶2兲^1/2Ⰶ1, but the large factor b/a can allow the efficiency to be much larger than 1. InTwo dimensions, the optimal value of␭1 is

␭1= ␭2

ln共␥兲

冋

^{1 −}^ln共^␥^␥^兲

册

^, ^共25兲

with␥=兩ln共a²␭2/D兲兩, corresponding to the minimum search time

具T典min⬃ b⁴

4D关␥+ ln共␥兲兴, 共26兲

and the efficiency of intermittence is

E⬃ln共b²/a²兲

ln共␶2/␶兲 . 共27兲

It can still be large for large values ofb/a.

However, in three or more dimensions, it is shown that the efficiency cannot be significantly larger than 1 if␶2⬎␶: then, intermittence is only an interesting search strategy if

␶2Ⰶ␶.

Part of the conclusion of this section is extended quali- tatively to more general situations in Sec. IV.

III. DISCRETE SYSTEMS

We now assume that the searcher P moves on a regular lattice.

A. One-dimensional lattice

Consider N= 2L+ 1 equally spaced lattice points on an axis 0x. 共Fig.6兲 Between coordinates −L and L a target is located at coordinate 0. The searcher P now switches between two dynamic regimes.

共i兲 During regime 1, P performs a continuous time random walk between points −LandLwhich are reflecting points, whereasAis absorbing. The transition rate from one lattice pointxto any of its next neighbours isp/ 2.

共ii兲 During regime 2, or teleportation, P is randomly relocated on any lattice points in共−L,L兲.

共iii兲 The durationT_iof regimei共i= 1 , 2兲is an exponential, independent, variable.

Mean search time. Formula共2兲still applies and gives the Laplace transform of the search time density:

FIG. 7.具T典as a function of␭1in the one-dimensional lattice case.共a兲具T典withL= 100,p= 1, and␭2= 0.0001.共b兲具T典withL= 100,p= 1, and␭2= 0.1.共c兲具T典 withL= 100,p= 1, and␭2= 1.

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Fˆ共s兲= 具j1共s兩x兲典+ 1/共2L+ 1兲

1 −共1 − 1/共2L+ 1兲−具j1共s兩x兲典兲/共1 +s/␭1兲共1 +s/␭2兲, 共28兲 where具j₁共s兩x兲典 is the Laplace transform of the search time density in regime 1, average on the initial positionx. Using Montroll’s paper³⁷and Hughes’ book³⁸in continuous time, it is found that

具j1共s兩x兲典= 1

N

冋

^{共1 −}^p/共p⁺s兲兲P共0,p/共p¹ +s兲兲− 1

册

^, ^共29兲

whereP共0 ,z兲 is the generating function for all walks which start and end at the origin.

P共0,z兲= 1 N

兺

k=0

N−1 1

1 −zcos共2␲k/N兲=1 +x^N 1 −x^N

冑

1 −1 z², 共30兲 with

x=1

z关1 −

冑

1 −z²兴. 共31兲

Eventually, after straightforward but lengthy calculations,具T典 writes

具T典=␭1+␭2

␭1␭2

2L共␭1+p−␣兲共␣^2L+1⁺^p^2L+1兲−p共p+␣兲共p^2L−␣^2L兲

共2p+␭1兲共p^2L+1−␣^2L+1兲 , 共32兲

with␣⁼␭1+p−

冑

_␭₁

冑

_␭₁+ 2p.

Optimization of the mean search time. It can be seen that 具T典 can have three possible behaviors as a function of ␭1

共Fig.7兲.

共i兲 Ifp/␭2⬍2L/共2L+ 1兲,具T典increases continuously with

␭1, and具T典 is minimum if␭1= 0.

共ii兲 If 2L/共2L+ 1兲⬍p/␭2⬍共2 / 15兲共3 +L+L²兲,具T典is minimum for a finite value␭1of␭1.

共iii兲 Ifp/␭2⬎共2 / 15兲共3 +L+L²兲,具T典decreases with␭1and the minimization of the search time requires choosing

␭1as large as possible.

It may be noticed that with condition共iii兲,具T典decreases with␭1to a lower bound. Then, intermittence is always favorable. This is the main difference with the continuous model where this situation cannot occur. We will see in Sec.

V that this behavior can be observed in more general intermittent systems with teleportation, but not if the search regime 1 is diffusive.

Small density limit. ForL→⬁,具T典can be approximated by

具T典= 2␭1+␭2

␭1␭2

冑

1 + 21共p/␭1兲L 共33兲 In this limit, condition共iii兲cannot be satisfied, but one can obtain a minimum search time for a finite value of␭1, if p⬎␭2.

Then, the optimal value of␭1 is given by

␭1= p p−␭2

␭2. 共34兲

IfpⰇ␭2, the random walk is approximately a diffusion and one finds that␭1⬃␭2, as in the one-dimensional continuous case.

B. Two-dimensional lattice

For a simple random walk on a square lattice ofm⫻m

=N points, we have^37,39 P共0,z兲= 1

m²_k

兺

1=0 m−1_k

兺

2=0

m−1 1

1 −共1/2兲共ck₁+ck₂兲, 共35兲 withck= cos共2␲^k/m兲0艋z艋1,

or

P共0,z兲= 1 m

兺

k=0

m−1 1

1 −共1/2兲zck

1 1 −␳k2

1 +x_k^m 1 −xk

m, 共36兲

where␳k=z共2 −zc_k兲 andx_k=共1 −共1 −␳k 2兲^1/2兲␳k

−1.

It is found numerically that the three situations described in Sec. III A can occur 共Fig.8兲.

共i兲 For very small values of ␭2 共with respect to p兲,具T典 continuously increases with␭1, and it is minimum for

␭1= 0: intermittence is not favorable.

共ii兲 For intermediary values of ␭2,具T典 is minimum for a finite value␭1of ␭1.

共iii兲 For large values of␭2,具T典decreases with␭1 and the search is optimized by taking␭1as large as possible.

Because of the complexity of analytical formulas, it is difficult to give precise conditions for observing these situations.

In case共ii兲, numerical calculations of the optimal value

␭1 show that it can be approximately estimated by formula 共25兲of the two-dimension case whenp→⬁.

IV. MEAN SEARCH TIME: GENERAL RESULTS

The intermittent motion of a point searcher alternating a slow motion共regime 1兲, which allows for the detection of an immobile target, and a fast motion共regime 2兲without detec-

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tion can be formulated more generally when the search region is anyd-dimensional finite volumeBand the target is a subvolume A included into B. General conclusions can be obtained although specific models are necessary to get explicit formulas. We now describe this general formalism.

Search regime 1. During this regime, the target is found as soon as P is inB.q₁共t兩x兲is the survival probability of the target at timet under regime 1, starting fromx.

The durationT₁of regime 1 is an exponential stochastic variable, independent of other events,

P共T1⬎t兲=e^−␭¹^t. 共37兲

Thus, the probability density that regime 1 finishes at timetwithout discovering the target is

¯p₁共t兩x兲 ⬅␭1e^−␭¹^tq₁共t兩x兲. 共38兲

Move regime 2. During this regime, P is redistributed in volumeBwith a given stationary densityp₀共y兲, independent of its initial position x and of the duration of this regime.

Teleportation corresponds to the special case of a uniform stationary density. The durationT₂of regime 2 is a stochastic variable, independent of other events, not necessarily exponential in a general case. We denote its probability density

␸2共t兲= −d␺2共t兲/dt,␺2共t兲being the survival probability of regime 2 after a timet.

Overall survival probability. The overall survival prob- ability of P at timet, starting at time 0 from positionx苸B

−A, and knowing that P has previously experimented 2n regime changes at times 0⬍t₁⬍t₂⬍¯⬍t_2n⬍t, is clearly given by

S_2n共t兩x,t1, . . . ,t_2n兲=

冕

x_2k苸B−A

dx₂¯dx_2n,

¯p₁共␶1兩x兲p¯₁共␶3兩x2兲p0共x2兲. . .q₁共␶兩x2n兲p0共x2n兲, 共39兲 with␶1=t₁,␶i=ti−t_i−1, and␶=t−t_2n.

Multiplying byp₀共x兲and integrating over the initial po- sitionxyields the average survival probability,

S_2n共t兩t1¯t_2n兲=

冋

^1艋

^兿

^k^艋ⁿ^¯^p¹^共^␶^2k−1^兲

册

^q¹^共^␶^兲, ^共40兲

where ¯p₁共␶兲=兰x苸B−Adxp¯₁共␶兩x兲p0共x兲 and q₁共␶兲

=兰x苸B−Adxq₁共␶兩x兲p0共x兲.

Thus, the average probability that P survives at time t after 2n regime changes is

S_2n共t兲=

冕

_␶₁_+¯+␶_2n+1=t

兿

1艋k艋n

关d␶2ke^−␭¹^␶¹兴

⫻_1艋

兿

_k_艋_n^关d^␶^2k^␸²^共^␶^2k^兲兴e⁻^␭¹^␶^2k^S²ⁿ^共t兩t¹^{, . . . ,}^t²ⁿ^兲, ^共41兲

and its Laplace transform is

˜S

2n共s兲=关␭1˜q₁共s+␭1兲兴ⁿ˜q₁共s+␭1兲关␸^˜2共s兲兴ⁿ, 共42兲 where˜q₁共s兲and␸^˜2共s兲are the Laplace transforms ofq₁共t兲and

␸2共t兲. Similarly, it is found that the Laplace transform of the survival probability after 2n+ 1 regime changes is

˜S

2n+1共s兲=关␭1˜q₁共s+␭1兲兴ⁿ⁺¹␺^˜2共s兲, 共43兲 with ␺^˜2共s兲=共1 −␸^˜2共s兲兲/s being the Laplace transform of

␺2共t兲, the survival probality of regime 2.

Summing Eqs.共42兲 and共43兲 overn yields the Laplace transform of the overall survival probability of particle P:

˜S共s兲=˜q₁共s+␭1兲关␭1␺^˜2共s兲+ 1兴

1 −␭1˜q₁共s+␭1兲␸^˜2共s兲 . 共44兲 In the case of a uniform stationary probability density 共teleportation兲, we have

˜q₁共s兲= 1

v共B兲

冕

x苸B−A

dx1 s关1 −˜j

1共s兩x兲兴=1 s关␤⁻具j˜

1共s兲典B兴, 共45兲 where˜j

1共s兩x兲 is the Laplace transform of the search time densityduring regime 1, and具j˜

1共s兲典Bis its average on region B. Furthermore,v共A兲andv共B兲are the volumes ofA andB, and␤= 1 −␣=共v共A兲−v共B兲兲/v共B兲.

FIG. 8.具T典as a function of␭1in the two-dimensional lattice case.共a兲具T典withL= 150,p= 2, and␭2= 0.002.共b兲具T典withL= 150,p= 2, and␭2= 0.2.共c兲具T典 withL= 150,p= 2, and␭2= 200.

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As a result, if the initial density is uniform and ifT₂is exponential, the Laplace transformF˜共s兲 of the mean search time density −⳵^S共t兲/⳵^t is given by

F˜共s兲= 具˜j

1共s兩x兲典B+␣ 1 −共1 −具˜j

1共s兩x兲典B−␣兲/共1 +s/␭1兲共1 +s/␭2兲. 共46兲 Mean search time. The mean search time具T典is obtained by takings= 0 in formula共44兲:

具T典= ˜q₁共␭1兲

1 −␭1q˜₁共␭1兲

冉

^{1 +}^␭␭¹2

冊

^, ^共47兲

where we have noticed that ␸^˜2共0兲= 1 if waiting time T₂ is finite with probability 1 and that ␺^˜2共0兲=␶2⬅1 /␭2 is the mean duration of regime 2. If␶2is infinite, the mean search time is also infinite. Similar, more precise results are given in Ref.33but are not used here.

Let us assume that the mean duration of regime 2 is finite. 具T典 is obviously a decreasing function of ␭2, or an increasing function of ␶2, so that ␶2 should be as small as possible. It is clear that in practice the duration of regime 2 has some minimum realizable value␶2mindue to finite times necessary for all actual operations. Thus, we henceforth assume that the mean duration of regime 2 is ␶2 min, and we now consider 具T典 as a function of␶1.

Asymptotic behavior of具T典 when␭1→+⬁. It is easily seen that

␭1˜q共␭1兲=␤关1 −具j˜

1共␭1兩x兲典B−A兴, 共48兲

where␤⁼兰x苸B−Adxp₀共x兲 is the stationary probability that P 苸B−A and 具j˜

1共␭1兩x兲典B−A= 1 /␤兰x苸B−Adxj˜

1共␭1兩x兲p0共x兲 is the stationary average of˜j

1共␭1兩x兲normalized overB−A.

The behavior of˜j

1共␭1兩x兲 when ␭1→⬁ depends on the behavior of j₁共t兩x兲 when t→0, and it is found that if the search time density in regime 1 is finite for t= 0, j₁共0兩x兲

=a₁共x兲, then␭1q˜₁共␭1兲⬃␤关1 −具a1典B−A/␭1兴when ␭1→⬁and 具T典 ⬃ 1

␭2

␤

1 −␤

冋

^{1 −}^␭¹1

冉

^具a^{1 −}¹^典^B−A␤ ⁻^␭²

冊

⁺ ^¯

册

^. ^共⁴⁹^兲

Thus, when␭1→⬁the mean search time具T典tends to the finite limit 共␤/共1 −␤兲兲␶2. Furthermore, unless a₁共x兲= 0 for any x苸B−A, 具T典 tends to this limit by lower values pro- vided that 1 −␤⬍具a1共x兲典B−A␶2共which is more easily satisfied as the stationary probability 1 −␤of the target is very small兲. Case of diffusive regime 1. Formula共49兲does not apply if the search time density is not analytic att= 0, which is the case for a diffusive regime 1. If, for instance, during regime 1 P performs a one-dimensional diffusion between an absorbing pointa共the target兲and a reflecting pointb, we have 具˜j

1共␭1兲典B−A⬃C␭₁^−1/2, where C can be a positive constant.

Then, when␭1→⬁ 具T典 ⬃ 1

␭₂

␤

1 −␤

冋

^{1 −}

^冑

¹^␭1

C

1 −␤⁺ ^¯

册

^共⁵⁰^兲

and具T典always tends to its limit by lower values. This can be generalized to d-dimensional diffusion 共see Appendix C兲.

Behavior of 具T典when␭1→0. Expression共47兲of 具T典 can be written as

具T典= ␤˜r₁共␭1兲

1 −␭₁␤^˜^r1共␭₁兲

冉

^{1 +}␭^␭¹2

冊

^, ^共51兲

where˜r₁共␭1兲 is the Laplace transform of 具q1共t兲典B−A, the average of theq₁共t兲 overB−A. When ␭1→0, we have

具T典=␤t₁

冋

^{1 −}^␭¹^t¹

^冉

¹²

^冉

^␴^t1¹

冊

²⁺¹² ⁻^␤⁻^␶t₁²

冊册

^, ^共52兲

wheret₁⬅具t1共x兲典B−Ais the mean search time of the targetA in regime 1, averaged over B−A. Similarly, ␴1

2=具t₁²共x兲典B−A

−t₁²is the corresponding variance.

When␭1= 0,具T典 is just the mean search time␤t₁if regime 1 is maintained permanently. Thus, intermittence is surely favorable either if ␤^t1⬎␶2共␤^/共1 −␤兲兲 or if 共⳵具T典/⳵␭1兲_␭₁=0⬍0.

The first condition is verified if

␶2

t₁ ⬍1 −␤^, 共53兲

which is always possible if the minimum possible value␶2min

is small enough. Condition共⳵具S典/⳵␭1兲_␭₁=0⬍0 implies

␶2

t₁ ⬍1

2

冉

^␴^t1¹

冊

²⁻^␤^. ^共54兲

Since␴1/t₁⬎1, condition共54兲can surely be realized for small ␶2 if ␤⬍¹₂, but in general, if the target is small, ␤

⬃1. Then, this condition implies that the fluctuations of the search time in regime 1 should be large enough for the right hand side of Eq.共54兲to be positive. If this is the case, intermittence is surely favorable is␶2 minis sufficiently small.

If regime 1 is a three-dimensional diffusion, it can be shown^40,41 that共␴1/t₁兲²⬃2, so that conditions共53兲and共54兲 are identical. Assuming that Eq.共50兲holds共Appendix C兲, we conclude that if the minimum value␶2minis small enough to satisfy Eq. 共54兲, the search time is minimized for a finite value of ␶1. If␶2 exceeds the critical value ␶2crit=共1 −␤兲t1, intermittence is no longer generically favorable, and the search time increases with ␭1 共unless some atypical minimum exists far from␭1= 0兲.

If the search time of regime 1 is regular att= 0, formula 共49兲 allows one to anticipate other behaviors, with, for instance, 具T典 continuously decreasing with ␭1, as it is found possible in the lattice model of Sec. III. These conclusions show that the results of the previous sections can be qualita- tively extended to much more general cases.

V. CONCLUSION

We have shown that a searcher, an animal or a molecule, searching for a target which does not allow for remote detection can have a reason to alternate regimes of slow, care- ful scanning, with periods of random relocation, when it is able to perform them. As in other intermittent behaviors, alternating slow scanning periods with fast, nonreactive dis- placements, the interest of such a strategy is not obvious and it depends on the situation. We have obtained exact results

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ACKNOWLEDGMENT

The authors wish to acknowledge J. Klafter for stimulating discussions.

APPENDIX A: CALCULATION OFŠj˜„␭1+s円x…‹B

We proceed here to the calculation of具j˜共␭₁+s兩x兲典Bfrom formulas 共2兲and共3兲.

具j˜共␭1+s兩x兲典B=

冕

_v共B兲

˜j共␭1+s兩x兲 dr

v共B兲=2共1 −␯兲 b²^共^1−␯兲

1 a^␯

1

I_␯共a˜兲K_␯−1共b˜兲+K_␯共a˜兲I_␯−1共b˜兲

⫻

冋

^K^␯⁻¹^共b^˜^兲

^冕

â^b^drr^1−␯Î^␯^共r^¯^兲⁺Î^␯⁻¹^共b^˜^兲

^冕

^a^b^drr^1−␯^K^␯^共r^¯^兲

册

^.

The latter integrals can be evaluated explicitly,

冕

a b

具˜j共␭1+s兩x兲典B=

冕

_v共B兲

˜j共␭1+s兩x兲 dr

^1/2^K^␯−1_K_␯^共a₋₁^˜_共b^兲I˜^␯−1兲I_␯共a^共b˜^˜兲^兲+⁻K^I^␯−1_␯共a˜^共a^˜兲I^兲K_␯−1^␯−1共b˜^共b兲^˜^兲⬅−kY_␯ X_␯,

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withk=共da^d−1/b^d兲共D/␭1+s兲^1/2.

APPENDIX B: OPTIMIZATION IN LONG WAITING TIME LIMIT

In this limit␶1Ⰷ␶⁼^a²^/^{D, or} ^x⁼^a

具T典 ⬃1

2

冉

^a^b

冊

^1/2^␭␭¹1⁺␭^␭2²

x²兩ln共x兲兩

= b²

2D

冉

^{1 +}^␭␭¹2

冊冏

^ln

冉

^a

^冑

^␭^D¹

冊冏

^. ^共B6兲

If␭2⬎␭2c, the optimal value ␭1 of␭1satisfies

⳵具T典

⳵␭1

⬀ ␭1

␭1+␭2

+ 1

2 ln共x兲= 0, 共B7兲

which implies x²兩ln共x²兲兩=a²␭2

D ⬅c² 共B8兲

andcⰆ1. Eventually, one obtains

␭1⬃ ␭2

兩log共a²共␭2/D兲兲兩

冋

^{1 −}^{log兩log共a}^兩log共a²^共␭²^共␭²^/D兲兲兩²^/D兲兲兩

册

^共B9兲

so that␭1Ⰶ␭2and␶1Ⰷ␶2Ⰷ␶^.

Numerical analysis of the exact formulas shows that these approximations are very accurate in these situations, the relative error being of order 10⁻³.

Inserting this value into Eq. 共B6兲 and taking Eq. 共B8兲 into account, we find the minimum search time

具T典min⬃ b²

4D关␥+ ln共␥兲+ 1 + ¯兴, 共B10兲 with␥⁼兩ln共c²兲兩=兩ln共a²␭2/D兲兩.

From Eqs.共B1兲and共B2兲it is seen that the efficiency of intermittence is now

E⬃2ln共b/a兲

ln共c²兲 =ln共b²/a²兲

ln共␶2/␶兲 . 共B11兲

It is much larger than 1 if condition共15兲is satisfied, since we have

D␶2

a² ⬍ 7 96

b²/a² ln共b/a兲Ⰶb²

a². 共B12兲

Eventually, inthree dimensions, we have Z共x兲 ⬃−K_3/2共x兲

K_1/2共x兲= −x+ 1

x , 共B13兲

which holds for anyx=a

冑

_␭₁/D, and 具T典 ⬃ b³

3Da

冉

^{1 +}^␭␭¹2

冊

1 +a¹

冑

␭1/D. 共B14兲 The optimal value of␭1in three dimensions is then

␭1= D

a²

冋

^{− 1 +}

^冑

^{1 +}^a^D²^␭²

册

²^. ^共^B15^兲

It yields the minimal search time for any␭2or ␶2, 具T典min= b³

^{1 +}␶^␶2

冊

^⬎^1. ^共^B19^兲

Once more, it increases if ␶2 decreases, but if ␶2Ⰷ␶, it is approximately 1.

In any case, the efficiency is not significantly higher than 1 if␶2⬎␶, and intermittence is hardly useful in such a situation. It can be seen similarly that if ␶2⬎␶ intermittence is not a favorable search strategy in dimensionsd⬎3.

APPENDIX C: MEAN SEARCH TIME OF THE TARGET IN A DIFFUSIVE REGIME

Assume point P performs an ordinary diffusion along axis 0x, between the absorbing point x= 0 共target兲 and the

冑 冑

Intermittent search process and teleportation

冉

冊

冑 冑

冑 冑

冉

冊

冉

冑

冊

冉

冑

冊

冉

冑

冊

冉

冑

冊

冉

冑

冊

冉

冑

冊

冉

冑

冊

冉

冑

冊

冉

冊

冑

冑

冑

冉

冊

冏

冏

冑

冉

冊

冋

冑

册

冑

冑

冉

冊

冉

冊

冉

冊

冉

冊

冦

冧

冑

冑

冑

冑

冋

册

冋

册

兺

冑

冑

冑

冑

冑

兺

兺

兺

冕

冋

兿

册

冑冑

^冑冑

^冑冑

^冑

^冑

^冑

^冑

^冑

^冑

^冑

^冑

^冑

^兿

^冑

^冉

^冉

冊册

^冕

^冕

^冉

冊冏

^冑

冊冏