Dynamical phase transition in the first-passage probability of a Brownian motion
Benjamin Besga, Felix Faisant, Artyom Petrosyan, Sergio Ciliberto, Satya N. Majumdar
DDynamical phase transition in the first-passage probability of a Brownian motion
B. Besga, F. Faisant, A. Petrosyan, S. Ciliberto ∗ Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS,Laboratoire de Physique, UMR 5672, F-69342 Lyon, France
Satya N. Majumdar † LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, UMR 8626, 91405 Orsay, France (Dated: February 16, 2021)We study theoretically, experimentally and numerically the probability distribution F ( t f | x , L ) of the firstpassage times t f needed by a freely diffusing Brownian particle to reach a target at a distance L from the initialposition x , taken from a normalized distribution ( / σ ) g ( x / σ ) of finite width σ . We show the existence of acritical value b c of the parameter b = L / σ , which determines the shape of F ( t f | x , L ) . For b > b c the distribution F ( t f | x , L ) has a maximum and a minimum whereas for b < b c it is a monotonically decreasing function of t f .This dynamical phase transition is generated by the presence of two characteristic times σ / D and L / D , where D is the diffusion coefficient. The theoretical predictions are experimentally checked on a Brownian bead whosefree diffusion is initialized by an optical trap which determines the initial distribution g ( x / σ ) . The presence ofthe phase transition in 2d has also been numerically estimated using a Langevin dynamics. First-passage properties of stochastic processes are funda-mental to understand many important phenomena in natureand have wide ranging applications across fields [1, 2]. Theseinclude estimating reaction rates in chemical processes [3, 4],understanding persistence properties in nonequilibrium sys-tems [5, 6], computing efficiencies of search algorithms [7, 8],estimating the statistics of extreme events [9] and records ina time series [10–12], numerous applications in biology [13–15], astrophysics [16] and computer science [17].In particular, the first-passage properties of a simple ran-dom walk or a Brownian motion have been widely studied,not only as a simple solvable example, but due to its plethoraof applications. One recent application that has created muchinterest is in the context of a random walk or a Brownian mo-tion subjected to resetting to its initial starting point, eitherat random times [18–23] or periodically [24, 25]. Repeatedresetting to its starting position of a freely diffusing Brown-ian particle has two major effects: (i) it drives the particle toa nonequilibrium steady state so that its position distributionbecomes stationary at long times (ii) the mean first-passagetimes to a fixed target becomes finite. Moreover, an optimalresetting rate was found that makes the mean first-passagetime minimal, thus rendering a diffusive search an efficientsearch process via resetting [18]. This led to an enormousrecent activities in the field, both theoretically [8] and morerecently, experimentally [26, 27].There are however two ways in which realistic situationsdiffer from the assumptions used in these theoretical models:(a) it is impossible to reset the particle to its starting point‘instantaneously’ as was assumed in the original models (b)it is physically impossible to reset the particle exactly to itsstarting point. The latter situation arises in particular in ex-periments conducted with optical tweezers [26, 27], where aparticle is usually trapped in an external confining potential(optical trap), typically harmonic. At thermal equilibrium, thestationary position distribution of the particle is thus a Gaus-sian with a finite width σ (which depends on the temperature T and the stiffness of the laser trap κ as σ = k B T / κ ). Theparticle is initially prepared in thermal equilibrium in the trapand then the trap is switched off and the particle undergoesfree diffusion during a certain period. After this period, thetrap is again switched on and the particle is allowed to relaxto its thermal equilibrium before the trap is switched off again.The relaxation to thermal equilibrium where the particle isdriven towards the trap center mimics the ‘resetting’ (whichis thus non-instantaneous). However, under this mechanism,the particle never goes back exactly to its starting position, butits new starting position for the subsequent diffusive phase iseffectively chosen from a Gaussian distribution with a finitewidth σ . The case σ = σ is always finite.Several recent theoretical studies have addressed the issue(a) that a physical resetting is always non-instantaneous andthe effect of a finite duration of the resetting period is well un-derstood [28–33]. On the experimental side the developmentof protocols accelerating the dynamics of optically trappedcolloids [34, 35] could lead to a drastic reduction of the re-setting period. However, equally important is issue (b), i.e,how a finite width σ in the initial position distribution mayaffect the first-passage properties of the Brownian motion un-der resetting? Indeed, in the experiment reported in Ref. [27],the presence of a finite σ was found to alter substantially themean first-passage time to the target.The purpose of this Letter is to demonstrate that a finitewidth σ in the initial position distribution of a Brownianparticle profoundly affects its first-passage time distribution(FPTD), even in the absence of resetting! We first consideran extremely simple system: just a free Brownian motion in1d with a diffusion constant D , starting from x , with a targetlocated at L (see Fig. 1a). The FPTD F ( t | x , L ) to the targetfor fixed x is well known [1, 6]. Now we just average overthe initial position x drawn from, say a Gaussian distribution P ( x ) = e − x / σ / √ πσ with a finite σ . We show that just a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b x L Space T i m e ( m s ) a. First passage time t (ms) P r o b a b ili t y d e n s i t y b. FIG. 1. a. Experimental trajectories of a Brownian motion on a linestarting at x with a fixed target at L (parameters : L − x =
50 nmand D = . × − m .s − ). b. First-passage probability density F ( t | , L ) in Eq.(5) plotted as a function of t (red continuous line)for L =
50 nm and D = . × − m .s − corresponding to theexperimental data (blue dots) obtained for 1 . × first passages. this simple averaging leads to a profound change in the FPTD.We have implemented an experiment where we can follow thetrajectories of a free Brownian particle from an initial gaus-sian distribution as a result of the optical trapping. The ex-perimental setup is detailed in [27] and consists in an infraredlaser beam tightly focused into a microfluidic chamber to trapa silica micro-sphere of radius R = µ m in water. The posi-tion of the Brownian particule is read from the deviation of ared laser on a quadrant photodiode at 50 kHz. The stiffnessof the trap is chosen by changing the trapping laser intensitypower thanks to an electro-optical modulator. In particularafter equilibration the trap is switched off to follow the parti-cle free diffusion (see Fig. 1a) and the protocol is repeated toacquire statistics on the first passage times.Our main results are summarised as follows. A finite σ in-troduces a new time scale t ∗ ∼ O ( σ / D ) and for t ≤ t ∗ theaveraged FPTD develops an anomalous regime. First, the av-eraged FPDT diverges as t − / as t →
0. Secondly, as timeincreases, the FPTD decreases, achieves a minimum at t ∼ t ∗ ,then increases and achieves a maximum at t ∗ ∼ O ( L / D ) , be-fore finally decaying as t − / when t (cid:29) O ( L / D ) . Thirdly,and most remarkably, there exists a critical value σ = σ c suchthat for σ > σ c , the minimum at t ∗ and the maximum at t ∗ both disappear and distribution decays monotonically to 0 as t → ∞ . We derive this result analytically, and demonstrate thatboth the simulation and the experimental data match perfectlyour theoretical predictions. We also provide a physical mean-ing of this anomalous regime and the associated transition at σ = σ c : we show that for σ < σ c when t ∗ (cid:28) t ∗ , the anoma-lous early time regime in FPTD is caused by rare trajectoriesthat start very close to the target at L . For t ∗ (cid:28) t ∗ , such rareatypical trajectories are well separated in time scales from typ-ical trajectories that start close to the origin. The transition at σ = σ c occurs when these two time scales t ∗ and t ∗ merge withother. In this sense, this phase transition is ‘dynamical’. Wethen show that this phase transition is robust and happens forany unbounded initial distribution with a finite width σ , not necessarily a Gaussian. Furthermore, we argue and verify nu-merically that this transition is not limited to one dimension,and occurs even in higher dimensions. Our results are partic-ularly striking since the underlying system and its associatedphysics is really very simple.We start with a Brownian particle on a line with diffusionconstant D , in the presence of a fixed target at L . The parti-cle starts at the initial position x (which can be on either sideof L ). Let F ( t | x , L ) dt denote the probability that the parti-cle finds the target for the first time in [ t , t + dt ] , given fixed x and L . This FPTD F ( t | x , L ) can be computed very sim-ply [1, 6]. Let P ( x , x , t ) denote the probability density thatthe particle reaches x at time t , starting from x and does notcross L in time t . Let us consider x ≤ L (the case x ≥ L canbe similarly computed). Then P ( x , x , t ) satisfies the standardFokker-Planck equation, ∂ t P = D ∂ x P , for x ≤ L with absorb-ing boundary condition at the target P ( x = L , x , t ) =
0. Thesolution can be obtained using the method of images [1] P ( x , x , t ) = √ π Dt (cid:104) e − ( x − x ) / Dt − e − ( x + x − L ) / Dt (cid:105) . (1)Integrating over the final position x ∈ [ − ∞ , L ] gives the classi-cal result for the survival probability (i.e., the probability thatthe target is yet to be found till t ) S ( t | x , L ) = (cid:90) L − ∞ P ( x , x , t ) dx = erf (cid:18) | L − x |√ Dt (cid:19) , (2)where erf ( z ) = √ π (cid:82) z e − u du is the error function. The FPTDis then simply F ( t | x , L ) = − ∂ t S ( t | x , L ) = | L − x |√ π Dt e − ( L − x ) / Dt . (3)Using Eq. (1) and comparing with Eq. (3), it is further easy tosee that F ( t | x , L ) = − D ∂ x P ( x , x , t ) (cid:12)(cid:12)(cid:12) x = L . (4)This has a nice physical interpretation: one can interpret eachtrajectory of the particle as an independent Brownian motionand F ( t | x , L ) denotes the flux of such independent particlesthat arrive, all starting from x , for the first time at time t atthe target (sink) located at L .Suppose that x is fixed, say at x =
0. Then Eq. (3) gives F ( t | , L ) = L √ π Dt e − L / Dt . (5)As a function of t (see Fig. 1b) for a plot), F ( t | , L ) has verydifferent behavior across the time scale t ∗ = L / D . It decaysalgebraically as t − / for t (cid:29) t ∗ = L / D . This is caused bytrajectories that start at 0, but typically diffuse away in theopposite direction and finally reaches L at times t (cid:29) t ∗ . Incontrast, for t (cid:28) t ∗ , it vanishes extremely rapidly in an essen-tial singular way ∼ e − L / Dt as t →
0. This behavior is alsoeasy to understand physically. The trajectories that reach L atearly times, starting from 0, are those that move ballistically from 0 to L in time t . Indeed, the statistical weight of a Brown-ian trajectory is ∝ exp [ − / ( Dt ) (cid:82) t ( dx / d τ ) d τ ] . For ballistictrajectories, dx / d τ = L / t and hence the flux of such trajecto-ries contribute ∝ e − L / Dt which exactly reproduces the small t behavior of F ( t | , L ) in Eq. (5).Experimentally we compile the first passage times of theBrownian particule at a target situated at L − x =
50 nm awayfrom its initial position and find a very good agreement withthe theoretical FPTD F ( t | , L ) (see Fig. 1b).What happens when the initial position x is not fixed,but drawn from a distribution P ( x ) ? Averaging the FPTD F ( t | x , L ) in (4) over x , we get F ( t | σ , L ) = (cid:90) ∞ − ∞ F ( t | x , L ) P ( x ) dx . (6)For the Gaussian distribution P ( x ) = e − x / σ / √ πσ , theintegration in (6) can be performed explicitly. In terms of thedimensionless variables τ = Dt σ ; and b = L σ , (7)the averaged FPTD in (6) can be expressed in the scaling form F ( t | σ , L ) = D σ Φ (cid:18) Dt σ = τ , L σ = b (cid:19) , (8)where the scaling function Φ ( τ , b ) = π √ τ ( + τ ) (cid:34) e − b / + (cid:115) π b τ ( + τ ) e − b / ( + τ ) ×× erf (cid:32)(cid:115) b τ ( + τ ) (cid:33) (cid:35) (9)The average FPTD is plotted in Fig. (2) vs. time for dif-ferent values of b . In (9), for any fixed b , the function Φ ( τ , b ) diverges as τ − / as τ → b c = . · · · such that for b > b c ,there are two time scales τ ∗ ∼ O ( ) and τ ∗ ∼ O ( b ) (in origi-nal time t they correspond to t ∗ ∼ O ( σ / D ) and t ∗ ∼ O ( L / D ) respectively). The FPTD scaling function decreases with in-creasing τ and achieves a minimum at τ ∗ , then increases andachieves a maximum at τ ∗ before finally decaying as τ − / for τ (cid:29) τ ∗ , thus creating a horizontal S -shaped curve visi-ble in Fig. (2) for F ( t | σ , L ) . As b → b c , the two time scalesmerge and for b > b c , the scaling function decays monoton-ically with increasing τ . Thus just a simple averaging overthe initial condition leads to a rather rich FPTD, including a‘dynamical’ phase transition at b = b c caused by the mergingof two time scales. The critical value b c can be precisely de-termined as follows. If we plot the derivative ∂ τ Φ ( τ , b ) as afunction τ or b > b c , it vanishes at the two roots τ ∗ and τ ∗ ,corresponding respectively to the minimum and maximum inFig. (2). As b → b c , the two roots approach each other and at b = b c , they merge. Consequently at b = b c , both the first and First passage time t (ms) P r o b a b ili t y d e n s i t y FIG. 2. Theoretical (red lines) an experimental (dots) FPTD F ( t | σ , L ) in (8), for Gaussian initial condition ( σ =
36 nm), plot-ted as a function of t for b = , .
6, 2 . . . × , 7 . × , 9 . × and 1 . × first pas-sages time measurements respectively. For any finite b the scalingfunction diverges as t − / as t →
0. For b > b c = . t increases and achieves a minimum at t ∗ ∼ O ( σ ) .It then increases and achieves a maximum at t ∗ ∼ O ( L ) and thendecays algebraically as t − / for t (cid:29) t ∗ (not visible in the figure).As b → b c the two time scales merge and for b < b c , the functiondecreases monotonically with increasing t . the second derivatives of Φ ( τ , b ) (with respect to τ ) vanishat τ ∗ = τ ∗ = τ ∗ . Solving these two equations (using Math-ematica software) for the two unknowms τ ∗ and b c , we get b c ≈ . σ =
36 nm. We release the particlefrom the trap 5 × times and measure the average FPTDfor different target positions. Our experimental results matchvery well the theoretical predictions (see Fig. (2)) and the dy-namical phase transition associated.The physics behind this rather striking behavior of theFPTD can be understood as follows. We can think of the tra-jectories of the single Brownian particle as an assembly ofindependent Brownian particles with different starting points x and different histories. Let us first assume that L (cid:29) σ , i.e., b (cid:29) τ ∗ ∼ O ( ) and τ ∗ ∼ O ( b ) are well separated with τ ∗ (cid:28) τ ∗ . We consider the differentparts of the horizontal S -shaped FPTD in Fig. (2).• Anomalous regime 0 ≤ τ ≤ τ ∗ : when time τ is small,the particles that arrive at L for the first time in [ τ , τ + d τ ] are the ones that diffuse from the starting points inthe vicinity of the target L . Essentially, the particlesthat initially are in the region [ L − √ Dt , L + √ Dt ] will contribute to this flux at L at time t . Thus in-tegrating Eq. (6) over this region, it is easy to seethat one gets Φ ( τ , b ) ∼ e − b / / √ τ . The weight fac-tor e − b / = e − L / σ is just the probability of having aparticle at x = L in the initial condition. Hence in thisregime, the first term in (9) dominates.• When τ ∗ ≤ τ ≤ τ ∗ : As time exceeds τ ∗ ∼ O ( ) , the dif-fusive particles in the vicinity of L have already reached L . So, the particles that contribute to the flux at L atthis time are the ones that start from the center of thetrap x = L ballistically. The weight ofsuch trajectories ∼ e − L / Dt ∼ e − b / τ makes the min-imum around τ = τ ∗ ∼ O ( ) . In this regime where1 (cid:28) τ (cid:28) b , the second term in (9) dominates.• When τ ≥ τ ∗ : For t ∼ L / D , i.e., τ ∼ τ ∗ ∼ O ( b ) ,the particles that hit L for the first time are the typical trajectories that start from the most populated initial re-gion near x = L . Finallywhen τ (cid:29) τ ∗ , the first-passage flux to L are caused bytrajectories that start from the trap center but take muchlonger times to reach L due to their sojourns in the di-rection opposite to L .When b → b c from above, the two time scales τ ∗ and τ ∗ merge, the atypical ballistic trajectories disappear and the av-eraged FPTD is controlled entirely by diffusion — causingthus a dynamical phase transition at b = b c where the two timescales merge.How robust is this dynamical phase transition? Does it oc-cur for generic initial conditions, or is it something special forthe Gaussian case? In fact, consider a generic initial condi-tion with a finite width σ such that, P ( x ) = ( / σ ) g ( x / σ ) ,where g ( y ) is assumed to have an unbounded support on y ∈ [ − ∞ , ∞ ] . Then the integral in (6) can still be expressedin the scaling form in (8), with the scaling function given by Φ ( τ , b ) = √ √ π τ (cid:90) ∞ − ∞ dz | z | e − z g (cid:16) b − √ τ z (cid:17) . (10)In the limit τ →
0, we get Φ ( τ , b ) ≈ √ g ( b ) √ π τ as τ → . (11)Thus, for generic unbounded initial condition, the scalingfunction diverges universally as τ − / as τ →
0. Furthermore,it is not difficult to see that for any such initial condition g ( y ) ,the dynamical transition at some critical b c will also exist. Asan example, we consider g ( y ) = e −| y | / Φ ( τ , b ) = e − b (cid:34)(cid:114) πτ + e τ / erf (cid:18)(cid:114) τ (cid:19) − e τ / (cid:18) erfc (cid:18) b − τ √ τ (cid:19) + e b erfc (cid:18) b + τ √ τ (cid:19)(cid:19)(cid:21) . (12)When plotted again τ (not shown here), the scaling functionagain exhibits a horizontal S -shaped form as in Fig. (2) witha minimum at τ ∗ ∼ O ( ) and a maximum at τ ∗ ∼ O ( b ) for b > b c ≈ . b = b c again First passage time t (ms) − P r o b a b ili t y d e n s i t y FIG. 3. 2d numerical simulation of the average FPTD for b = , . , . , b c ≈ . a = .
5. Plotted for the experi-mental values D = . × − m .s − and σ =
36 nm. causing a dynamical phase transition. Hence we conclude thatthis transition is robust and occurs for any generic unboundedinitial condition.Finally, is this transition restricted only to one dimension?From the general physical picture of the problem, it is clearthat this transition should exist even in higher dimensions.In d >
1, it is necessary to have a target of a finite ‘toler-ance’ size R tol , because Brownian trajectories will surely missa point target for d >
1. Thus we have an additional timescale t ∗ = R / D . But for fixed R tol , the dynamical transi-tion at some σ c should exist. While the averaged FPTD in d > d =
2, by integratingthe Langevin equation :˙ x ( t ) = √ D η x ( t ) , ˙ y ( t ) = √ D η y ( t ) (13)where η m ( m = x , y ) are white noises with (cid:104) η m ( t ) η m (cid:48) ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) δ mm (cid:48) . The simulation starts with { x ( ) , y ( ) } ran-domly extracted from a 2d Gaussian distribution of standarddeviation σ and the first passage time is computed when theparticle arrives within R tol of the target. We fixed a = R tol / L = . × trajectories with an integration stepof ≈ µ s. The average FPTD F d ( t | , b , a = . ) is com-puted from ≈ first passage times and plotted in Fig. 3. Asin the 1d case the average FPTD changes with b from a mono-tonically decreasing function at b < b c to a distribution with aminimum and a maximum for b > b c with b c ≈ . b c of b , such that for b > b c the distribution has aminimum and a maximum and for b < b c it is a monotonicallydecreasing function. The experimental data agree with the 1dtheoretical predictions. The numerical simulations show thatthis transition is present in 2d. This is an important result thatneeds to be taken into account in optimal search strategies. ∗ E-mail me at: [email protected] † E-mail me at: [email protected][1] S. Redner,
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