Non-Gaussianity and Dynamical Trapping in Locally Activated Random Walks
NNon-Gaussianity and Dynamical Trapping in Locally Activated Random Walks
O. B´enichou, N. Meunier, S. Redner, and R. Voituriez Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, CNRS UMR 7600,case courrier 121, Universit´e Paris 6, 4 Place Jussieu, 75255 Paris Cedex MAP5, CNRS UMR 8145, Universit´e Paris Descartes,45 rue des Saints-P`eres, 75270 Paris Cedex 06, France Center for Polymer Studies and Department of Physics,Boston University, Boston, Massachusetts 02215, USA (Dated: October 31, 2018)We propose a minimal model of locally-activated diffusion , in which the diffusion coefficient of aone-dimensional Brownian particle is modified in a prescribed way — either increased or decreased— upon each crossing of the origin. Such a local mobility decrease arises in the formation ofatherosclerotic plaques due to diffusing macrophage cells accumulating lipid particles. We show thatspatially localized mobility perturbations have remarkable consequences on diffusion at all scales,such as the emergence of a non-Gaussian multi-peaked probability distribution and a dynamicaltransition to an absorbing static state. In the context of atherosclerosis, this dynamical transitioncan be viewed as a minimal mechanism that causes macrophages to aggregate in lipid-enrichedregions and thereby to the formation of atherosclerotic plaques.
PACS numbers: 05.40.Jc, 05.40.Fb
Many-particle systems that consume energy for self-propulsion — active particle systems — have receivedgrowing attention in the last decade, both because of thenew physical phenomena that they display and their widerange of applications. Examples include molecular mo-tors, cell assemblies, and even larger organisms [1]. Theintrinsic out-of-equilibrium nature of these systems leadsto remarkable effects such as non-Boltzmann distribu-tions [2], long-range order even in low spatial dimensions[3] and spontaneous flows [4].At the single-particle level, the active forcing of aBrownian particle leads to non-trivial statistics. For ex-ample, it has been recently shown [5, 6] that a randomwalk which is reset to its starting point at a fixed rate hasa non-equilibrium stationary state, as opposed to stan-dard Brownian motion. Another example is given by self-propelled Brownian particles [7], which can yield sharplypeaked probability densities for the particle velocity.In this letter, we consider a new class of problems inwhich the active forcing of a Brownian particle is localizedin space . While the impact of localized perturbations onrandom walks has been investigated [8], in part becauseof its relevance to a wide range of situations, such aslocalized sources and sinks [9, 10], trapping [11, 12] ordiffusion with forbidden [13], hop-over [14] or defective[15] sites, the role of local activation on Brownian-particledynamics remains open. We present a minimal modelof locally activated diffusion, in which the diffusivity ofa Brownian particle is modified — either increased ordecreased — in a prescribed way upon each crossing ofthe origin.A prototypical example is a bacterium in the presenceof a localized patch of nutrients, which enhances the abil-ity of the bacterium to move, or, alternatively, toxinsthat impair bacterial mobility. This type of localized de-crease of mobility also underlies the dynamics of a cell blood vesselmacrophage lipid-enriched regionvessel wall atherosclerosis plaque (1) (2)(3) tx (1) (2) (3) ab FIG. 1: a. Sketch of the different stages of atherosclerosisplaque formation: (1) rapid diffusion of a “free” macrophagecell; (2) upon entering a localized lipid-enriched region, themacrophage accumulates lipids, and thereby grows and be-comes less mobile; (3) after many crossings of the lipid-enriched region, the macrophage eventually gets trapped, re-sulting in the formation of an atherosclerotic plaque. b. Sketch of a one-dimensional particle trajectory of the modelof locally decelerated random walk. (e.g., a macrophage) that grows by accumulating smallerand spatially localized particles, such as lipids (Fig. 1)[16, 17]. As the cell grows, its ability to move decreasesand the ultimate result is the formation of an atheroscle-rotic plaque [18]. The spatial localization arises fromthe presence of lipids at specific points in the arterialnetwork; these lipids can be located, as is now well ac- a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r cepted, by the properties of the blood flow [19]. Obser-vations show that macrophages that have accumulatedlipids move more slowly. Eventually the macrophagestops in an lipid-enriched region, resulting in the for-mation of an atherosclerotic plaque [20, 21]. Here wepropose a simple model to account for this local mobil-ity decrease and address the particular questions of (i)the potential trapping of cells in locally lipid-enrichedregions, and (ii) the kinetics of the resulting segregationprocess when it exists.Our formalism allows us to describe both the situationsof decreased and increased localized mobility changes.We show that this type of perturbation has remarkableconsequences on the diffusion process at all scales. Westress that the diffusion coefficient of the active particleat any time depends on the entire history of the tra-jectory. Thus the evolution of the particle position isintrinsically non-Markovian [22–25]. Our main findingsare: (i) The probability distribution of the position has anon-Gaussian tail. (ii) For local acceleration, a diffusingparticle is repelled from the origin, so that the maximumin the probability distribution is at non-zero displace-ment. (iii) For local deceleration, a dynamical transitionto an absorbing state occurs. For sufficiently strong de-celeration, the particle can get trapped at the origin at afinite time. The exact time dependence for the particlesurvival probability is determined explicitly. Conversely,if the deceleration process is sufficiently weak, the parti-cle never gets trapped. This dynamical transition to anabsorbing state provides a minimal mechanism that couldhelp understand the formation kinetics of atheroscleroticplaques. The Model.
A one-dimensional diffusing particle is ac-celerated or decelerated whenever it crosses the origin x = 0 according to the following Langevin equations:˙ x = √ D ξ ( t ) , ˙ D = f ( D ) δ ( x ) , (1)where ξ is a Gaussian white noise of intensity one, D theparticle diffusion coefficient, x the particle position, δ ( x )the Dirac distribution, and f ( D ) an arbitrary prescribedfunction that accounts for the local activation. For sim-plicity, we assume that the particle is initially at x = 0with D = D >
0. Note that: (i) both the position x andthe diffusion coefficient D are random variables; (ii) asmentioned previously, the evolutions of x or D alone arenon-Markovian; (iii) the function f ( D ) can be positive(local acceleration) or negative (local deceleration), butwith f (0) = 0 so that D remains non negative.Following standard steps, the corresponding Fokker-Planck equation [26] for the joint distribution of position x and diffusion coefficient D at time t , P ( x, D, t ), is: ∂P∂t = D ∂ P∂x − δ ( x ) ∂ [ f ( D ) P ] ∂D − λ ( t ) δ ( x ) δ ( D ) , (2)where the last term of the right side accounts for the ab-sorbing state at ( x = 0 , D = 0). The explicit expressionfor λ ( t ) is determined demanding that P is normalized, from which we obtain λ ( t ) = lim D → (cid:2) f ( D ) P (0 , D, t ) (cid:3) . (3)When f ( D ) is positive, then D is always non zero. Inthis case, the particle is never trapped and λ ( t ) = 0 atall times. While intuitively obvious for local acceleration( f ( D ) > λ ( t ) can equal zero forlocal deceleration processes. Local Acceleration, f ( D ) > . Laplace transformingEq. (2) gives − s (cid:98) P + D ∂ (cid:98) P∂x = δ ( x ) (cid:104) ∂ ( f (cid:98) P ) ∂D − δ ( D − D ) (cid:105) , (4)where (cid:98) P = (cid:98) P ( x, D, s ) is the Laplace transform of theprobability distribution. For x (cid:54) = 0, the solution is (cid:98) P ( x, D, s ) = A ( D, s ) e −| x | √ s/D , (5)where the coefficient A ( D, s ) is determined by integrat-ing Eq. (4) across x = 0 to obtain the jump of the firstderivative of (cid:98) P with respect to x at this point: D (cid:34) ∂ (cid:98) P∂x (cid:12)(cid:12)(cid:12) x =0+ − ∂ (cid:98) P∂x (cid:12)(cid:12)(cid:12) x =0 − (cid:35) = ∂ [ f (cid:98) P ( x = 0)] ∂D − δ ( D − D ) . Using Eq. (5), we have f ∂A∂D + (cid:104) f (cid:48) + 2 √ sD (cid:105) A = δ ( D − D ) . (6)When f ( D ) is positive, then A ( D, s ) = 0 for
D < D ,while for D > D the solution to (6) is A = B ( s ) f ( D ) f ( D ) e −√ s F , (7)with F ( D ) ≡ (cid:90) DD √ D (cid:48) | f ( D (cid:48) ) | d D (cid:48) . The unknown function B ( s ) is determined by the jumpof A at D : A ( D +0 , s ) − A ( D − , s ) = 1 f ( D ) , which finally yields (cid:98) P ( x, D, s ) = Θ( D − D ) 1 f ( D ) e − Z √ s , (8)where Θ is the Heaviside step function and Z ≡ | x |√ D + 2 F ( D ) . Laplace inverting this expression, we obtain the joint dis-tribution P ( x, D, t ) = Θ( D − D ) Zf ( D ) √ πt e − Z / t . (9)The marginal distribution with respect to x , that is,the probability distribution of positions, is obtained byintegrating Eq. (9) over all D in the range [ D , ∞ ]. Whileit does not seem possible to evaluate this integral ana-lytically, the large- x behavior can be obtained by theLaplace method. For the illustrative case where f ( D ) isa constant (that we define as a ), this method gives P ( x, t ) ∼ t (cid:114) | x | a exp (cid:20) − | x | / √ a t (cid:21) x → ∞ , (10)which we numerically checked is close to the exact value P ( x, t ) over a wide spatial range. We wish to empha-size two important features of this result for P ( x, t ) thatare in marked contrast with the Gaussian propagator ofthe usual Brownian motion: (i) P ( x, t ) generally has anon-Gaussian tail; (ii) P ( x, t ) reaches its maximum ata non-zero displacement. Equation (10) shows that thelocation of this maximum asymptotically grows as t / when f ( D ) = a . Thus local acceleration pushes a diffus-ing particle away from the origin.From the general expression (9), the marginal distri-bution with respect to D can also be easily obtained byintegration over x . We find P ( D, t ) = Θ( D − D ) 2 √ Df ( D ) √ πt e − F /t . (11)In the particular case of f ( D ) = a , Eq. (11) shows thatthe diffusion coefficient of the particle asymptoticallygrows as t / .As a byproduct, Eq. (11) also provides the distributionof the local time τ ( t ) spent by the particle in the activezone (the origin for the present case) up to time t . Usingthe second of Eqs. (1), this basic observable in the the-ory of diffusion, which has dimension of time per unit oflength [27], is related to the diffusion coefficient at time t by τ ( t ) ≡ (cid:90) t δ ( x ( t (cid:48) )) d t (cid:48) = (cid:90) DD d D (cid:48) f ( D (cid:48) ) . (12)Thus the distribution of the local time, defined as P ( τ, t ),is given by P ( τ, t ) = f ( D ) P ( D, t ), with P ( D, t ) givenby Eq. (11) and D implicitly defined as a function of τ in Eq. (12). For the illustrative case of f ( D ) = a , thedistribution of the local time at time t therefore is P ( τ, t ) = 2 √ aτ + D √ πt exp (cid:40) − (cid:2) ( aτ + D ) / − D / (cid:3) a t (cid:41) . (13)This result strongly contrasts with the Gaussian distri-bution that arises in the case of Brownian motion, whichcan be recovered from Eq. (13) in the limit a → P BM ( τ, t ) = 2 √ D √ πt e − D τ /t . (14)Notice, in particular, the typical local time for an accel-erated particle with f ( D ) = a grows as t / instead of t / in the case of Brownian motion. It is worth noting an intriguing dichotomy with adiscrete-time version of local acceleration — the “greedy”random walk [28]. In this discrete model, the step length (cid:96) k after the k th return of a random walk to the origin isgiven by (cid:96) k = k α . To match with the continuous modelwith f ( D ) = D α , one must choose the value α = 1 / P ( x, t ) ∝ x / /t exp[ − x / /t ], which isdifferent from (10). The source of this discrepancy is thatthe probability of being at the origin is not affected bythe enhancement mechanism of greedy walks [28], whilethis return probability is fundamentally modified in thecase of locally-activated random walks, as seen explicitlyfrom the distribution of the local time (13). Thus ourlocally activated diffusion model cannot be viewed as thecontinuous limit of the greedy random walk. However, itcan be shown that it is the continuous limit of a discretespace and continuous time random walk whose jump fre-quency is modified at each visit of the active site, whichis intrinsically different from the greedy random walk. Local Deceleration, f ( D ) < . Following the same anal-ysis as that used for local acceleration, the Laplace trans-form of the joint distribution is (cid:98) P = Θ( D − D ) | f ( D ) | e − Z √ s − (cid:98) λ ( s ) s δ ( x ) δ ( D ) , (15)where (cid:98) λ is the Laplace transform of λ ( t ) defined inEqs. (2) and (3). Using these defining relations for λ ( t ),Eq. (15) gives (cid:98) λ ( s ) = lim D → (cid:2) f ( D ) (cid:98) P (0 , D, s ) (cid:3) = − e −√ s F , (16)where we define F ( D ) ≡ (cid:90) D √ D (cid:48) | f ( D (cid:48) ) | d D (cid:48) . In this result for (cid:98) λ , we have used δ ( D ) f ( D ) = 0, since f (0) = 0 by the definition of our model. The importantfeature of Eq. (16) is that (cid:98) λ ( s ) = 0 as soon as F diverges.Thus our final result is (cid:98) P ( x, D, s ) = Θ( D − D ) | f ( D ) | e − Z √ s + δ ( x ) δ ( D ) s e −√ s F , (17)which gives, after Laplace inversion, P ( x, D, t ) = Θ( D − D ) | f ( D ) | Z e − Z / t √ πt + T ( t ) δ ( x ) δ ( D ) . (18)Here T ( t ) = erfc (cid:16) √ t (cid:90) D √ D (cid:48) | f ( D (cid:48) ) | d D (cid:48) (cid:17) (19)is the trapping probability, namely, the probability thatthe particle becomes stuck at x = 0 by time t becausethe diffusion coefficient has reached zero. As a corol-lary, the survival probability is given by S ( t ) = 1 − T ( t ),and we have obtained this quantity for an explicitly non-Markovian process. We also mention that, as in the caseof local acceleration, the joint distribution easily gives themarginal distributions of the position and the diffusioncoefficient, as well as the local time.A fundamental consequence of the local decelerationof a Brownian particle is that two different dynamicalregimes emerge. We illustrate these regimes for the par-ticular case where f ( D ) = − D β as D →
0. If the decel-eration is sufficiently strong, which occurs when β < / S ( t ) ∼ D / − β √ πt (3 − β ) → t → ∞ . (20)Thus in this regime of strong deceleration, the survivalprobability has the same scaling with time as in the caseof a usual Brownian particle in the presence of a perfecttrap. In the opposite case of β ≥ /
2, then S ( t ) = 1for all t > x = 0 , D = 0) as the deceleration strength increases.Mathematically, this transition occurs at the point where F is no longer divergent.In conclusion, we introduced a minimal model of lo-cally activated diffusion, in which the diffusion coeffi-cient of a Brownian particle is modified in a prescribedway at each crossing of the origin. In one dimension, apurely diffusing particle hits the origin of the order of √ t times after a time t . Consequently, the local activation mechanism is repeatedly invoked during the trajectoryof a Brownian particle. Thus the asymptotic dynam-ics of a Brownian particle is globally affected, leading tomarkedly different behavior than that of pure diffusion.Since the unusual properties of local activation rely onthe recurrence of Brownian motion, we anticipate thatqualitatively similar, but quantitatively distinct, behav-ior would arise in two dimensions.Our model encompasses both the situations where theBrownian particle is locally accelerated or decelerated.For local acceleration, the probability distribution is non-Gaussian and multi-peaked, with maxima away from theorigin no matter how weak the acceleration. For suffi-ciently weak local deceleration, a Brownian particle man-ages to avoid getting trapped at the origin in spite of itsrecurrence. However, for strong deceleration, there is adynamical transition to an absorbing state in which theparticle ultimately gets trapped at the origin.In the context of atherosclerosis mentioned initially,the dynamical transition to an absorbing state can beviewed as a minimal mechanism that leads to the segre-gation of macrophages in lipid-enriched regions and thusto the formation of atherosclerotic plaques. Our modelsuggests that even in absence of chemical signals (suchas chemokines or cytokines) that can bias the motion ofcells, there exists a critical intensity of the mobility de-crease, which depends on the local lipid concentration,beyond which an atherosclerotic plaque will occur. Ourmodel can also help understand the kinetics of this plaqueformation.OB is supported by the ERC starting grant FPTOpt-277998. NM is supported by the European ProjectARTreat FP7 - 224297. SR gratefully acknowledges fi-nancial support from NSF Grant No. DMR-0906504 andthe hospitality of UPMC where this work was initiated. [1] J. Toner, Y. Tu, and S. Ramaswamy, Annals of Physics , 170 (2005).[2] A. Puglisi, V. Loreto, U. Marini Bettolo Marconi, andA. Vulpiani, Phys. Rev. E , 5582 (1999).[3] J. Toner and Y. Tu, Phys. Rev. Lett. , 4326 (1995).[4] R. Voituriez, J. F. Joanny, and J. Prost, EurophysicsLetters , 404 (2005).[5] M. R. Evans and S. N. Majumdar, Phys. Rev. Lett. ,160601 (2011).[6] M. R. Evans and S. N. Majumdar, ArXiv e-prints (2011),1107.4225.[7] P. Romanczuk and L. Schimansky-Geier, Phys. Rev.Lett. , 230601 (2011).[8] B. D. Hughes, Random Walks and Random Environ-ments (Oxford Press, 1995).[9] S. Redner and K. Kang, Phys. Rev. A , 3362 (1984).[10] S. N. Majumdar, Physica A: Statistical Mechanics andits Applications , 207 (1990).[11] G. Weiss, Aspects and Applications of the Random Walk (Amsterdam, Netherlands: North-Holland, 1994).[12] V. Tejedor, O. B´enichou, R. Voituriez, and M. Moreau, Physical Review E , 056106 (2010).[13] C. K. L., Markov Chains with Stationary TransitionProbabilities (Springer, Berlin, 1960).[14] R. K. P. Zia and Z. Toroczkai, Journal of Physics A:Mathematical and General , 9667 (1998).[15] O. B´enichou and G. Oshanin, Phys Rev E Stat NonlinSoft Matter Phys , 031101 (2002).[16] V. Calvez, A. Ebde, N. Meunier, A. Raoult, and G. Ruz-nakova, ESAIM Proc. , 1 (2008).[17] V. Calvez, J. Houot, N. Meunier, A. Raoult, and G. Ruz-nakova, ESAIM Proc. , 1 (2009).[18] M. P. J. de Winther and M. H. Hofker, The Journal ofClinical Investigation , 1039 (2000).[19] C. G. Caro, Arterioscler. Thromb. Vasc. Biol. , 158(2009).[20] L. Yvan-Charvet, M. Ranaletta, N. Wang, S. Han,N. Terasaka, R. Li, C. Welch, and A. Tall, The Journalof Clinical Investigation , 3900 (2007).[21] D. Siegel-Axel, K. Daub, P. Seizer, S. Lindemann, andM. Gawaz, Cardiovascular Research , 8 (2008).[22] N. V. Kampen, Stochastic Processes in Physics and
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