Lecture 13. Drunk Man Walks

Transcription

1 Lecture 13 Drunk Man Walks H. Risken, The Fokker-Planck Equation (Springer-Verlag Berlin 1989) C. W. Gardiner, Handbook of Stochastic Methods (Springer Berlin 2004)

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3 Great White Shark swims 12,400 miles, shocks scientists WCS release October 6, 2005

4 Random Walks and Levy flights

5 Random Walk The term random walk was first used by Karl Pearson in He proposed a simple model for mosquito infestation in a forest: at each time step, a single mosquito moves a fixed length at a randomly chosen angle. Pearson wanted to know the mosquitos distribution after many steps. The paper (a letter to Nature ) was answered by Lord Rayleigh, who had already solved the problem in a more general form in the context of sound waves in heterogeneous materials. Actually, the the theory of random walks was developed a few years before (1900) in the PhD thesis of a young economist: Louis Bachelier. He proposed the random walk as the fundamental model for financial time series. Bachelier was also the first to draw a connection between discrete random walks and the continuous diffusion equation. Curiously, in the same year of the paper of Pearson (1905) Albert Einstein published his paper on Brownian motion which he modeled as a random walk, driven by collisions with gas molecules. Einstein did not seems to be aware of the related work of Rayleigh and Bachelier. Smoluchowski in 1906 also published very similar ideas.

6 The simplest possible problem: 1 dimensional Random Walk The walker starts at position x = 0 at the step t = 0; at each time-step the walker can go either forward or backward of one position with equal probabilities 1/2. We ask the probability P(x,t) to find the walker at the position x a the time step t. Number of sequences that take to x in t steps probability of any given sequence of t steps p(x,t) = t! # 1 % & ((t + x) /2)!((t " x) /2)! $ 2' t ( ~ t>>1 "x 2 ) t e 2t 2 (t+x)/2 steps taken in the positive direction (t-x)/2 steps taken in the negative direction

7 Stochastic process - normal noise (finite variance) 1 dimension x(t +1) = x(t) + "(t) Central limit theorem (lecture 2) The sum of independent identically-distributed variables with finite variance will tend to be normally distributed. [lecture 2] x(t) = t # $(") p(x,t) ~ t>>1 " =1 Average traveled distance: % 1 exp $ x 2 2" t # 2 ' & 2t # 2 x t = " = 0 x 2 t " x t 2 ( * ) = # 2 t $ t ( " = 0)

8 Stochastic process - normal noise (finite variance) D dimensions r r (t +1) = r r (t) + r " (t) p( r,t +1) = p( r,t +1) = 0 0 iid D-dimesional vectors isotropic (uniformly distributed in the direction) with moduli distributed accordingly to a given probability density function p(! ). " # p( r $ ) p( r % r r $,t)d D $ " # p( r ' $ ) p( r,t) % r $ & r p( r,t) + 1 r $ & r & r p( r,t) r * $ +... ( ) 2 +, d r D $ p( r,t +1) " p( r,t) = #2 2D $ 2 p( r,t) +... "p "t = #2 2D $ 2 p p(",t) = 2D D / 2 " D#1 ( t) $(D/2) 2 % 2 & 2 ) #D" exp D / 2 ( ' 2 % 2 t + * " = r

9 Random Walk - any dimension p(",t) P(",t) D =1 D =2 " = ( ) # (D +1) 2 ( ) # D 2 2 $ 2 D t p(",t) " " P(",t) D =3 D =10 " 2 = # 2 t p(",t) D =100 " " P(") at t = 0,1,..,9 for <!> =0 and <! 2 > =1 This is not a sphere. " Is this a sphere?

10 ...further jumps

11 Stochastic process with large noise fluctuations (non-finite variance) x(t +1) = x(t) + "(t) We already discussed the 1 dimensional case in the context of the central limit theorem and stable distributions [Lecture 3]. f (x,",#,c,µ) = +% $% p(") # 1 & ( ( dke ikx exp ikµ $ ck "$1 * 1$ i# k * ) ) k " $ p(x,t) " a(t) x # + + '(" $1,k) - -,,

12 Super diffusive behavior Probability of a jump larger or equal to L max $ P r (L " L max ) = % p(#)d# ~ 1 L L &'1 " >1 max max in t steps I have a finite probability of a jump equal to L max if: t P r (L " L max ) ~ 1 Mean Square Displacement: x(t +1) = x(t) + "(t) p(") # 1 " $ L max ~ t 1/"#1 x 2 = t " 2 = t # " 2 p(")d" ~ tl 3$% max ~ t 2 /(%$1) x 2 ~ t 1/ 0.9 = t 1.11 " =1.9 Large jumps dominate the behavior L max L min x ~ t 1/ 0.9 = t 1.11 x 2 x ~ L max 1< " < 3 2 ~ L max

13 Diffusive behavior ( jumps with finite variance) " = 3 x 2 ~ t 1/ 2 = t 0.5 x ~ t 1/ 2 = t 0.5 Super-diffusive behavior x 2 ~ t 1/1.5 = t 0.66 " = 2.5 x ~ t 1/1.5 = t 0.66 x 2 = " 2 t ~ t 0.5 x " t = t 0.5 " =1.9 x 2 ~ t 1/ 0.9 = t 1.11 x ~ t 1/ 0.9 = t 1.11

14 Scale free and Self similarity "x ~ ("t) 2 "x ~ ("t) 1/# " =

15 Higher dimensions: Levi Flights r (t +1) = r (t) + r "(t) ˆ p ( r k ) ~ exp("c k # ) p( r " ) ~ 1 " d +# " =1.1

16 Sub-diffusive behaviors x(t + ") = x(t) + #(t) " 2 with finite, but with power law distributed waiting times: p(") ~ 1 " # In n steps the mean square displacement will grow as x 2 = n " 2 the total time elapsed is t ~ n " Case " > 2 t ~ n x 2 ~ t diffusive Case 1 < " < 2 t ~ " max ~ n 1/(#$1) (same reasoning as for L max in previous slide) x 2 ~ t "#1 sub-diffusive!

17 Random walk on graphs Which is the probability to find the walker at distance j after t steps? The probability to be at time t+1 at a given geodesic distance j form the starting point is given by the probability that at time t the walker is probability to be at distance j and stay there P( j,t +1) = p stay ( j)p( j,t) + p out ( j "1)P( j "1,t) + p in ( j +1)P( j +1,t) probability to be at distance j-1 and move forward probability to be at distance j+1 and move inward with and P(0,t +1) = p in (1)P(1,t) P( j,0) = " j,0

18 Random walk on graphs - continuous limit P( j,t +1) = p stay ( j)p( j,t) + p out ( j "1)P( j "1,t) + p in ( j +1)P( j +1,t) the equation for P(j,t) can be re-written as P( j,t +1) " P( j,t) = 1 [ 2 p ( j +1) + p ( j +1) in out ]P( j +1,t) + 1 [ 2 p ( j "1) + p ( j "1) in out ]P( j "1,t) + [ p in ( j) + p out ( j) ]P( j,t) + 1 [ 2 p in( j +1) " p out ( j +1) ]P( j +1,t) - 1 [ 2 p in( j "1) " p out ( j "1)]P( j "1,t) the continuous limit (j #"; t # $) gives (Fokker-Plank Equation) "P(#,$) "t = " 2 % p out (#) + p in (#)( "# 2 & ' 2 ) * P(#,$) + " [ "# p (#) + p (#) out in ]P(#,$)

19 Random walk on graphs - shell map Each one of the probabilities p in ( j) p out ( j) p stay ( j) is proportional to the relative number of paths that take the walker inwards, outwards or within the shell j square lattice (j > 1): - inward 4(j - 1); - outward 4(j+1); - stay 0 J+1 j J-1 For a broad class of graphs holds (j>>1): p out ( j) + p in ( j) ~ Const p out ( j) " p in ( j) # V ( j +1) "V( j) V ( j)

20 Random walk on graphs - continuous asymptotic solutions p out (") + p in (") ~ Const = # 2 Finite dimensions: V (") ~ " D#1 P(",t) = D p out (") # p in (") = $ 2 p out (") # p in (") ~ DV (") 2D D / 2 " D#1 &#D"2 ) $(D/2) ( 2% 2 D / 2 exp ( + t) ' 2% 2 t * %V (") %" Hyperbolic spaces: V (") ~ exp(") p out (") # p in (") ~ Const which leads to an equation of the form (D #1)$ 2 D" D#1 " D#2 ~ 1 " "P(#,$) " = %D 1 "t "# P(#,$) + D " 2 2 "# P(#,$) 2 and the solution for large times is a density wave which moves ballistically outwards: <"> ~ t, < " 2 > ~ t 2 and < " 2 > - < "> 2 ~ t T. Aste, "Random walks on disordered networks'', Phys. Rev. E 55 (1997), p

21 Will the walker ever return to the origin?

22 Will the walker ever return to the origin? The mean time spent in the origin is: " # t= 0 P(0,t) The probability to return to the origin is (Polya theorem) " =1# $ % t= 0 1 P(0,t) Which can be computed using the characteristic function +$ 1 P(0,t) = % P ˆ ( k r,t)d D k r P ( 2" ) ˆ ( k r,t) = [ p ˆ ( k r )] t D " #$ +" " 1 # P(0,t) = #[ p ( k r )] t d r +" 1 & D k = & 2$ 2$ 1% p ˆ ( k r ) d r D k t= 0 ˆ ( ) D %" t= 0 1 ( ) D %"

23 He might never get back home p( r k ) ~ exp("c k # ) ~ 1" c k # 1 $ 1" p( k r ) d r D k ~ +# "# +# $ "# 1 k d r D % k ~ +# $ "# k D"1 k % d k " = $ $ 1 for D # 2 % (& <1 for D > 2 % ( $ 1 for D # ' % &( & <1 for D > ' Finite variance Power Law, Non defined variance! < 2 Recurrent " = 1 (he always get back) Transient " < 1 (he might never get back) P(",t) D =1 P(",t) D =2 P(",t) D =100 P(") at t = 0,1,..,9 for <!> =0 and <! 2 > =1 " " "

24 Are sharks intelligent mathematicians? G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince, H. E. Stanley, Levy flight search patterns of wandering albatrosses, Nature 381 (1996) A.M. Reynolds, Cooperative random Levy flight searches and the flight patterns of honeybees, Physics Letters A 354 (2005)

25 maximally searchable graphs (lecture 9) We want to be able to navigate efficiently with local information only (greedy algorithm). Let us put the vertices on a lattice and choose shortcuts between lattice vertices with probability p (i, j) " d #$ s i, j with d i,j the distance between the vertices on the lattice. It results that any target vertex from any random starting vertex is found within a time ~ (Log V ) 2 only when: " = space dimension J. M. Kleinberg, Navigation in a small world, Nature, 406 (2000), p. 845.

26 Two dimensions: p s (r) " r #2 Shortfin Mako Shark Jul 2007 to 23 Mar 2008 Shortfin Mako Shark Jul 2007 to 9 Mar 2008 Shortfin Mako Shark Jul 2007 to 23 Mar 2008

27 Supplemetary material

28 Weierstrass Random Walk p(n) ~ p n p(") = ˆ p (k) = +$ p!=b!=b n % p(") e ik" = (1# p) p m cos(kb m ) #$ p n " = ±b n n = log(" ) b (1# p) p(") ~ 1 " 1+# 2 m= 0 Characteristic function $ [ ] % p n &(" # b n ) + &(" + b n ) $ & m= 0 Levy stable The walker starts at position x = 0 at step t = 0; at each time-step the walker can go either forward or backward b n positions with probability ~ p n. p(") ~ 1 " 1+# ~ exp(#c(() k ( ) k '$ P(x,t) ~ 1 x 1+" " = # log ( p ) log(b)

29 +$ +$ p(x,t) = % d" 1...% d" t p(" 1 )...p(" t )&(" " t # x) ˆ p (k,t) = solution #$ #$ +# +# dxe ikx $ $ d% 1... $ d% t p(% 1 )...p(% t )&(% % t " x) "# "# +# "# p ˆ (k,t) = p ˆ (k) Diffusive behavior % = 2 "P(x,t) "t = D eff " 2 ˆ P (k,t) "x 2 ( ) [ ] t ~ exp "D eff t k # Non-equally distributed time-steps can lead to the same kind of fat-tailed jump probabilities. " # "x := $ 1 # 2% +& " ˆ P (k,t) "t = #D eff k $ ˆ P (k,t) Fractional diffusion behavior "P(x,t) "t = D eff " # ˆ P (k,t) "x # ' dk k # e $ikx Fractional calculus $&

30 Probabulity to return to the origin, on graphs P( j,t +1) = p stay ( j)p( j,t) + p out ( j "1)P( j "1,t) + p in ( j +1)P( j +1,t) From the above relation one can calculate the mean time spent in the origin: " # P( j,t) = t= 0 " # j= 0 1 # of paths between shell j and shell j +1 % V ( j) for k 2 finite (# of paths between shell j and shell j +1 ~ k 2 j k j V ( j) ~ " =1# $ % t= 0 1 P(0,t) V ( j) ~ j D"1 T. Aste, "Random walks on disordered networks'', Phys. Rev. E 55 (1997), p ' &[ V ( j) ] 2 "#1 scale free with 1 < " < 2 ' [ V ( j) ] " "#1 (' scale free with 2 $ " < 3 ( 1 for D & 2 and finite k 2 * * 1 for scale free with 2 & ' < 3 and D < 2 - ~ ) (3 - ') 2 * 1 for scale free with 1< ' < 2 and D < 2-1 * ' + < 1 otherwhise

31 Brownian motion is a fractal with dimension 4/3 G.F. Lawler, W. Werner (1999), Intersection exponents for planar Brownian motion, Ann. Probab. 27,