MMortal Brownian motion: three short stories
Baruch Meerson ∗ Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
Mortality introduces an intrinsic time scale into the scale-invariant Brownian motion. This facthas important consequences for different statistics of Brownian motion. Here we are telling threeshort stories, where spontaneous death, such as radioactive decay, puts a natural limit to “lifetimeachievements” of a Brownian particle. In story 1 we determine the probability distribution of amortal Brownian particle (MBP) reaching a specified point in space at the time of its death. Instory 2 we determine the probability distribution of the area A = (cid:82) T x ( t ) dt of a MBP on the line.Story 3 addresses the distribution of the winding angle of a MBP wandering around a reflectingdisk in the plane. In stories 1 and 2 the probability distributions exhibit integrable singularitiesat zero values of the position and the area, respectively. In story 3 a singularity at zero windingangle appears only in the limit of very high mortality. A different integrable singularity appears ata nonzero winding angle. It is inherited from the recently uncovered singularity of the short-timelarge-deviation function of the winding angle for immortal Brownian motion. I. INTRODUCTION
In the recent years there has been growing interest in the effects of a finite lifetime of particles on statisticalproperties of random walk and of its continuous limit, the Brownian motion [1–8]. Practical examples are found inphysics, chemistry and biology and vary from diffusion of radioactive gases such as radon [5, 7, 8] to a variety ofsearch problems such as the search for an oocyte by sperm [4]. Additional motivation comes from non-equilibriumstatistical mechanics: The particle mortality breaks detailed balance and even makes any steady state impossiblewithout a particle source. Last but not least, mortal Brownian motion has long been a classical area of study formathematicians [9].Here we are telling three short, and we hope instructive, stories about different statistics of mortal Brownianmotion. Story 1 deals with the position statistics of a mortal Brownian particle (MBP) at the time of its spontaneousdeath. Story 2 addresses the statistics of a time-integrated quantity: the area under mortal Brownian motion in onedimension. Finally, in story 3 we study the statistics of the winding angle of a MBP around an impenetrable disk onthe plane.In stories 1 and 2 the probability distributions, that we determine, exhibit integrable singularities at zero values ofthe position and the area, respectively. In story 3 a singularity at zero winding angle appears only in the limit of veryhigh mortality. In this limit there is also an integrable singularity at a nonzero winding angle. This singularity has adifferent nature. As we show, it is inherited from a recently uncovered singularity of the short-time large-deviationfunction of the winding angle for immortal
Brownian motion [10].
II. STORY 1: POSITION DISTRIBUTION
This story is quite simple, and it sets the stage. Suppose that a MBP with diffusivity D and the decay rate µ isreleased at the origin at t = 0. At time t = T , when the particle spontaneously dies, it arrives at a point x = X .The Brownian motion and the death are independent random processes. Therefore, the joint probability distribution P ( X , T ) is given by P ( X , T ) = P ( X , T ) P ( T ). Here P is the probability distribution of immortal Brownian motionto arrive at point X at time T : P ( X , T ) = 1(4 πDT ) d/ e − | X | DT , (1)where d is the dimension of space. In its turn, the probability distribution P ( T ) follows the exponential decay law: P ( T ) = µe − µT . (2) ∗ Electronic address: [email protected] a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y As a result, the joint probability is P ( X , T ) = µ (4 πDT ) d/ e − | X | DT − µT . (3)The probability distribution of the lifetime achievement of the MBP in terms of X is obtained by integrating P ( X , T )over all possible death times: p ( X ) = (cid:90) ∞ P ( X , T ) dT = µ (cid:90) ∞ (4 πDT ) − d/ e − | X | DT − µT dT. (4)Notice that, when considered as a function of µ , p ( X ) is equal to µ times the Laplace transform of P ( X , T ), where µ serves as the parameter of the Laplace transform, and the integration is over time. This is a generic feature of awhole class of MBP statistics, including those considered in our three stories.Evaluating the integral in Eq. (4), we obtain p ( X ) = (cid:114) µ D e − √ µD | X | , d = 1, (5) µ πD K (cid:18)(cid:114) µD | X | (cid:19) , d = 2, (6) µ πD | X | e − √ µD | X | , d = 3, (7)where K ( . . . ) is the modified Bessel function of the second kind. In all dimensions p ( X ) is isotropic, and it decaysexponentially at large | X | . The distribution p ( X ) exhibits the scaling behavior p ( X ) = 1 (cid:96) d Φ d (cid:18) | X | (cid:96) (cid:19) . (8)The intrinsic time scale T ∼ /µ and the diffusivity D of mortal Brownian motion determine the characteristic decaylength (cid:96) = (cid:112) D/µ . As expected on the physical grounds, a higher mortality leads to a stronger localization of thedistribution near the origin.Noticeable in Eqs. (5)-(7) is a singularity of p ( X ) at | X | = 0. In one dimension p ( X ) is bounded at | X | = 0, but itsfirst derivative with respect to X is discontinuous there. In two dimensions p ( X ) diverges logarithmically at | X | = 0,while in three dimensions it diverges as 1 / | X | . These singularities, however, are integrable.Interestingly, the same results (5)-(7) are given by the normalizable solutions of the stationary diffusion-decayequation with a d -dimensional delta-function particle source at X = 0: ∇ p ( X ) − µD p ( X ) = − µD δ ( X ) , (9)The normalizable solutions automatically obey the normalization condition (cid:82) p ( X ) d X = 1. Equation (9) is apparentlyvalid for any d . III. STORY 2: AREA DISTRIBUTION
Here a MBP is released at the origin at t = 0 and allowed to move on the line −∞ < x < ∞ . This time we areinterested in the statistics of a time-integrated quantity: the area A = (cid:90) T x ( t ) dt (10)that the particle accumulates by the time of its spontaneous death at t = T , see Fig. 1. The joint probabilitydistribution P ( A, T ) = P ( A, T ) P ( T ) is again a product of two distributions. P ( A, T ) is the probability distributionof immortal Brownian motion accumulating the area A by time T , whereas P ( T ) is the probability distribution ofthe particle death at t = T , given by Eq. (2). Now, x ( t ) is a Gaussian random variable with zero mean, and the sameis true for A = A ( T ) from Eq. (10). Therefore, the distribution P ( A, T ) can be calculated in a straightforward wayby computing the second moment (cid:104) A (cid:105) (see the Appendix), and we obtain P ( A, T ) = √ √ πDT e − A DT . (11) P ( A, T ) is a simple member of a whole family of area distributions of immortal Brownian motion. Other membersof this family (the Brownian bridge, the Brownian excursion, the positive part of the Brownian motion, the absolutevalue of the Brownian motion, etc .) are subject to additional constraints, see Ref. [11] and references therein. Allthese distributions exhibit the same scaling behavior as the simple Brownian motion that we are dealing with: P ( A, T ) = 1 √ DT f (cid:18) A √ DT (cid:19) , (12)as to be expected from dimensional analysis and scale invariance of immortal Brownian motion. T - - - x FIG. 1: A realization of mortal Brownian motion with diffusivity D = 1. The particle died at t = T = 10 . We study thestatistics of the area (10) of the shaded region. Areas under the t -axis are negative. Using Eqs. (2) and (11), we obtain the joint probability P ( A, T ) = √ µ √ πDT e − A DT − µT . (13)Now we turn to the probability distribution p ( A ) of the lifetime achievement of the MBP. The presence of the intrinsictime scale T ∼ µ − of mortal Brownian motion implies the following scaling behavior of p ( A ): p ( A ) = 1 A µ F (cid:18) AA µ (cid:19) , where A µ = D / µ − / . (14)This is indeed what the calculation shows. The distribution p ( A ) is given by the integral over death times p ( A ) = (cid:90) ∞ P ( A, T ) dT = (cid:90) ∞ √ µ √ πDT e − A DT − µT dT, (15)which can be evaluated exactly. The result can be written as Eq. (14), where the scaling function is F ( w ) = 2 π · / Ai (cid:0) e i π | w | / (cid:1) Ai (cid:0) e − i π | w | / (cid:1) | w | / , (16)and Ai( . . . ) is the Airy function. Note that the product Ai( z )Ai( z ∗ ), where z is any complex number, is a real number.The function F ( w ) has the following asymptotics: F ( w ) (cid:39) π / [Γ(2 / | w | / , | w | (cid:28)
1, (17)12 (cid:112) | w | e − (cid:113) | w | , | w | (cid:29)
1. (18)Interestingly, the scaling function F ( w ) and, as a result, the distribution p ( A ), diverges at w = 0 (correspondingly, at A = 0), see Eq. (17). This singularity is integrable, so that (cid:82) ∞−∞ F ( w ) dw = 1 as it should be. The asymptotic (17)can be obtained directly from Eq. (15), by neglecting µT (cid:28) T in Eq. (15) by the Laplacemethod. Figure 2 shows the scaling function F ( w ). - - - F ( w ) FIG. 2: The scaling function F ( w ) of the area distribution p ( A ) of mortal Brownian motion. See Eqs. (14) and (16) fordefinitions. What can be said about the lifetime achievements in terms of the area distribution of other variants of mortalBrownian motion? Here we will briefly consider the absolute value | x ( t ) | of mortal Brownian motion on the line. Tostart with, it has the same scaling form (14) as that of mortal Brownian motion itself. Furthermore, the large- A tailsof the area distributions of immortal Brownian motion and of its absolute value on any finite time interval 0 < t < T coincide: both behave as exp[ − A / (4 DT )] up to pre-exponential factors. As a consequence, the Laplace methodleads to identical results, up to pre-exponential factors, for the stretched exponential tails | A | /A µ (cid:29) ∼ exp( − (cid:112) | w | / w = A/A µ = D − / µ / A .The coincidence of the tails in the immortal case is explained by the fact that, for very large | A | , the probabilityof observing a given | A | is dominated by a single Brownian trajectory x ( t ). This optimal trajectory can be found, inthe spirit of geometrical optics, by minimizing the Wiener’s action (see, e.g. Ref. [12]) S = − D (cid:90) T ˙ x dt (19)subject to the initial condition x (0) = 0 and the following constraints: (cid:82) T x ( t ) dt = A for the Brownian motion, and (cid:82) T | x ( t ) | dt = A for its absolute value. The constraints can be accommodated via Lagrangian multipliers, and the“lacking” boundary condition at t = T becomes, in both cases, ˙ x ( T ) = 0 [13]. For the Brownian motion the optimaltrajectory, for any sign of A , is x ( t ) = 3 A T t (2 T − t ) , ≤ t ≤ T. (20)For the absolute value of the Brownian motion the optimal trajectory is exactly the same, except that A mustbe positive. The action (19) along this trajectory is S = 3 A / (4 DT ). It leads to identical distribution tails P ( | A | (cid:29) √ DT ) ∼ exp[ − A / (4 DT )] (up to pre-exponential factors) in the two problems. We refer the readerto Refs. [10, 14–16] for other applications of geometrical optics of constrained immortal Brownian motion. IV. STORY 3: WINDING ANGLE DISTRIBUTION
Suppose that a MBP is released at t = 0 at a distance L from the center of a reflecting disk with radius R < L in the plane. We are interested in the probability distribution p ( θ ) of the winding angle θ of the particle around thedisk at the time of particle death, see Fig. 3.The joint probability of the winding angle θ and of the death time T is P ( θ, T ) = P ( θ, T ) P ( T ), where P ( θ, T )is the probability distribution of reaching the winding angle θ at time T for immortal Brownian motion, and P ( T )is given by Eq. (2). The distribution P ( θ, T ) has been studied in many works [10, 14, 17–20]. In principle, it canbe determined exactly by solving the diffusion equation subject to the reflecting boundary condition on the disk anda delta-function initial condition. However, the resulting exact expressions for P ( θ, T ), obtained in Refs. [14, 17],include triple integrals of combinations of Bessel functions and trigonometric and/or exponential functions, and theyare too cumbersome for our purposes. Here we will confine ourselves to evaluating the joint distribution P ( θ, T ) andthe achieved winding angle distribution p ( θ ) in the limits of very low and very high mortality, where we can rely onthe previously obtained long- and short-time asymptotics of P ( θ, T ). − R R x − R R y ( ,L ) θ FIG. 3: A realization of mortal Brownian motion around a reflecting disk with radius R . We study the distribution of thewinding angle θ accumulated by the time of particle death. The long-time asymptotic of P ( θ, T ) corresponds to the strong inequalities ( DT ) / (cid:29) R, L . In this limit P ( θ, T )does not depend on L [14, 17–20]: P ( θ, T ) (cid:39) π DTR sech (cid:32) πθ ln DTR (cid:33) , √ DT (cid:29) R, L. (21)The short-time asymptotic of P ( θ, T ) holds when ( DT ) / (cid:28) R, L − R . It has been determined recently in thegeometrical-optics approximation [10]. At such short times a sizable winding angle is a large deviation, and theprobability distribution P ( θ, T ) can be written, up to a pre-exponential factor, as − ln P ( θ, T ) (cid:39) R DT g (cid:18) θ, RL (cid:19) , √ DT (cid:28) R, L − R, (22)where g ( θ, w ) = w − sin θ, | θ | ≤ arccos w , (23) (cid:16) | θ | + (cid:112) /w − − arccos w (cid:17) , | θ | ≥ arccos w , (24)and 0 < w <
1. The rate function g ( θ, w ) and its first θ -derivative are continuous functions, but the second derivative ∂ θ g is discontinuous at θ = θ c = arccos w . This singularity of the rate function (which appears only in the limit of T →
0, and is smoothed at finite T ) can be interpreted as a second-order dynamical phase transition [10].Using Eqs. (2), (21) and (22), we obtain the long- and short-time asymptotics, respectively, of the joint distribution P ( θ, T ). Let us introduce a dimensionless parameter (cid:15) = µR / (4 D ). At very low mortality, (cid:15) (cid:28)
1, the dominatingcontribution to the integral p ( θ ) = (cid:82) ∞ P ( θ, T ) dT comes from long times, where we can use the asymptotic (21) of P ( θ, T ). Upon a change of variables y = µT in the integral, we obtain p ( θ ) (cid:39) π (cid:90) ∞ dy ln y(cid:15) sech (cid:18) πθ ln y(cid:15) (cid:19) e − y . (25)The characteristic integration length, O (1), of this integral is determined by the exponential e − y . On this scale thefunction ln( y/(cid:15) ) changes very little. Therefore, we can evaluate it at any y = O (1) and obtain, with logarithmicaccuracy, p ( θ ) (cid:39) π (cid:15) sech (cid:18) πθ ln (cid:15) (cid:19) , (cid:15) = µR D (cid:28) . (26)As one can see, the µ -dependence is only logarithmic.The situation is very different at high mortality, (cid:15) (cid:29)
1. Using Eq. (22) we obtain, up to a pre-exponential factor, p ( θ ) ∼ (cid:90) ∞ e − R DT g ( θ, RL ) − µT dT (27)Evaluating the integral by the Laplace method, we arrive at p ( θ ) ∼ exp (cid:20) − √ (cid:15) h (cid:18) θ, RL (cid:19)(cid:21) , (cid:15) = µR D (cid:29) , (28)where h (cid:18) θ, RL (cid:19) ≡ (cid:115) g (cid:18) θ, RL (cid:19) = L | sin θ | R , | θ | ≤ arccos RL , (29) | θ | + (cid:114) L R − − arccos RL , | θ | ≥ arccos RL . (30) θ h ( θ , / ) FIG. 4: The rate function h ( θ, R/L = 1 /
2) of the winding angle distribution p ( θ ) of mortal Brownian motion as a function of θ . Only nonnegative θ are shown. See Eqs. (28)-(30) for definitions. There is a second-order dynamical phase transition at θ = arccos(1 / Note that the high-mortality rate function h ( θ, R/L ) is everywhere non-convex, see Fig. 4. This is a relatively raresituation. In addition to the corner singularity at θ = 0, h ( θ, R/L ) exhibits a discontinuity in the second derivative ∂ θ h at θ = arccos( R/L ) which is inherited from the singularity of the short-time large-deviation function (22). Thetail of p ( θ ) is exponential. Finally, the µ dependence here is much stronger than in the low-mortality limit. V. DISCUSSION
The model of mortal Brownian motion is relevant for a number of practical issues, including diffusion of radioactivematerials [5, 7, 8]. But it is also interesting in a more general context. That a finite lifetime puts a natural upper limiton the achievements of any individual or a collective of individuals is almost a truism, and a convenient mathematicalframework for this general phenomenon can be useful. Here we have considered three very basic settings – whichwe called three short stories – dealing with different statistics of a mortal Brownian particle (MBP). The “lifetimeachievements” of the MBP – in terms of (1) the final position, (2) the accumulated area, and (3) the final winding anglearound an obstacle – are totally determined by chance. As we observed, the corresponding probability distributionsare nontrivial. In stories 1 and 2 they exhibit singularities at zero values of the position and the area, respectively.In story 3 we observed a singularity at zero winding angle only in the limit of very high mortality. In this limit thereis also a singularity of a different type. It appears at a nonzero winding angle and results from the singularity of theshort-time large-deviation function of the winding angle for immortal Brownian motion [10]. It would be interesting toextend some of our analysis to ensembles of MBPs and, more generally, to ensembles of interacting diffusing particleswhich can be modelled as lattice gases [3]. The macroscopic fluctuation theory (see Ref. [21] for a general review andRef. [3] for some examples) provides an efficient and versatile starting point for such extensions.
ACKNOWLEDGMENTS
I am grateful to Tal Agranov and Naftali R. Smith for useful discussions, and to Naftali R. Smith for help withFig. 3. This research was supported by the Israel Science Foundation (grant No. 807/16).
Appendix A: Derivation of Eq. (11)
We start with the stochastic PDE for the immortal Brownian motion, see e.g.
Ref. [12]:˙ x ( t ) = √ Dξ ( t ) , (A1)where ξ ( t ) is a Gaussian white noise with zero mean and (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) . (A2)The solution of Eq. (A1) with initial condition x (0) = 0 is x ( t ) = √ D (cid:90) t ξ ( t (cid:48) ) dt (cid:48) . (A3)We are interested in the probability distribution of A = (cid:90) T x ( t ) dt = √ D (cid:90) T dt (cid:90) t dt (cid:48) ξ ( t (cid:48) ) . (A4)This distribution is Gaussian, therefore it suffices to compute the variance: Var = (cid:104) A (cid:105) − (cid:104) A (cid:105) . Since (cid:104) A (cid:105) = 0, we haveVar = (cid:104) A (cid:105) = 2 D (cid:68) (cid:90) T dt (cid:90) T dt (cid:90) t dt (cid:48) ξ ( t (cid:48) ) (cid:90) t dt (cid:48)(cid:48) ξ ( t (cid:48)(cid:48) ) (cid:69) = 2 D (cid:90) T dt (cid:90) T dt (cid:90) t dt (cid:48) (cid:90) t dt (cid:48)(cid:48) (cid:10) ξ ( t (cid:48) ) ξ ( t (cid:48)(cid:48) ) (cid:11) . (A5)Using Eq. (A2), we obtain Var = 2 D (cid:90) T dt (cid:90) T dt min( t , t ) = 2 DT , (A6)leading to Eq. (11). [1] S. B. Yuste, E. Abad and K. Lindenberg, Phys. Rev. Lett. , 220603 (2013).[2] E. Abad, S. B. Yuste and K. Lindenberg, Phys. Rev. E , 062110 (2013).[3] B. Meerson, J. Stat. Mech. P05004 (2015).[4] B. Meerson and S. Redner, Phys. Rev. Lett. , 198101 (2015).[5] M. P. Silverman, M. P. World J. Nucl. Sci. Techn. , 232 (2016).[6] D. S. Grebenkov and J.-F. Rupprecht, J. Chem. Phys. , 084106 (2017).[7] M. P. Silverman, J. Mod. Phys. , 1809 (2017).[8] M. P. Silverman and A. Mudvari, World J. Nucl. Sci. Techn. , 86 (2018).[9] A. N. Borodin and P. Salminen, Handbook of Brownian Motion - Facts and Formulae (Basel, Birkh¨auser, 2002).[10] B. Meerson and N. R. Smith, arXiv 1901:04209.[11] S. Janson and G. Louchard, Electron. J. Prob. , 1600 (2007).[12] S. N. Majumdar, “Brownian Functionals in Physics and Computer Science”, in “The Legacy of Albert Einstein” , edited byS. R Wadia (World Scientific, Singapore, 2006), Chapter 6, pp. 93-129.[13] L. Elsgolts, Differential Equations and the Calculus of Variations (Mir Publishers, Moscow, 1977).[14] A. Grosberg and H. Frisch, J. Phys. A: Math. Gen. , 4421 (1987).[18] C. B´elisle, Ann. Prob. , 1377 (1989).[19] H. Saleur, Phys. Rev. E , 1123 (1994).[20] A. Kundu, A. Comtet and S. N. Majumdar, J. Phys. A: Math. Theor , 385001 (2014).[21] L. Bertini, A. De Sole, D. Gabrielli, G. Jona Lasinio, C. Landim. Rev. Mod. Phys.87