Random Walk on Circle Imagine a Markov process governing the random motion of a particle on a circular

Transcription

1 Random Walk on Circle Imagine a Markov process governing he random moion of a paricle on a circular laice: 1 2 γ γ γ The paricle moves o he righ or lef wih probabiliy γ and says where i is wih probabiliy 1 2γ.

2 Random Walk on Circle (cond.) The random walk can be defined as follows: p +1 (i) = N 1 j=0 p +1 (i j)p ( j) where (1 2γ) if i = j p +1 (i j) = γ if i = j ± 1 mod N 0 oherwise. and i, j {0,1,...,N 1}.

3 Random Walk on Circle (cond.) Because Markov processes are linear, he disribuion a ime +1 can be compued from he disribuion a ime by mari vecor produc: (+1) = P (). Because he random walk is shif-invarian, he ransiion mari P is circulan: (1 2γ) γ γ P = γ (1 2γ) γ γ γ (1 2γ)

4 Diffusion in he Frequency Domain Since P is circulan, i is diagonalized by he DFT: P = WΛW where he mari Λ conains he eigenvalues of P on is diagonal: λ Λ = 0 λ λ N 1

5 Diffusion in he Frequency Domain (cond.) Muliplying boh sides of his epression by W yields P = WΛW W P = W WΛW W P = ΛW W p 0 = Λw 0 where p 0 and w 0 are he firs columns of P and W. Since w 0 = 1 N and Λ is diagonal, i follows ha W (1 2γ) γ 0. 0 γ = 1 N = 1 N λ N λ 0 λ 1. λ N 1. 0 λ N λ N 1

6 Diffusion in he Frequency Domain (cond.) We see ha he eigenvalues are N imes he DFT of P s firs column: λ m = γe j2πm 1 N + (1 2γ) + γe j2πm (N 1) N. Because (N 1) mod N and 1 = (N 1) mod N are conjugae frequencies e j2πm(n 1) N +e j2πm 1 N = 2cos ( 2πm (N 1) N I follows ha he eigenvalues of P are real. Since γ < 1/2 and ( ) (N 1) 0 < cos 2πm < 1 N for 0 < m < N 1, i follows ha λ 0 = 1 and 0 < λ m < 1 for m > 0. ).

7 Diffusion in he Frequency Domain (cond.) The updae equaion for he Markov process looks like his: (+1) = WΛW (). Because Λ is diagonal, higher powers of P are easy o compue: where Λ = P = WΛ W λ λ λ N 1. Significanly, given an iniial disribuion, (0), he disribuion a any fuure ime, (), can be compued by evaluaing: () = WΛ W (0).

8 Limiing Disribuion of Diffusion Process Taking he limi as goes o infiniy yields lim () = lim WΛ W (0) = lim ( N 1 λ mw m w H m m=0 ) (0) where H is conjugae ranspose. Since λ 0 = 1 and lim λ m = 0 for m 0 i follows ha lim () = w 0 w H 0 (0) = 1 N 1 because w 0 = 1 N 1, and n 1 n=0 n = 1. We see ha probabiliy mass is uniformly disribued among he sies in he ring.

9 Diffusion Equaion The following epression for P +1 in erms of P, P+1, and P 1 is ermed he maser equaion for he diffusion process: P +1 = P 2γP + γp 1 + γp +1 where 2γP is he probabiliy mass which leaves P in one sep and γp 1 + γp +1 is he probabiliy mass which eners P in one sep.

10 Diffusion Equaion (cond.) The above epression for = = 1 can be generalized for arbirary and by defining γ = D ( ) 2 : P + = P 2DP +DP }{{ ( ) } 2 ( ) + DP 2 + }{{ ( ) } 2 ou in where D is ermed he diffusion consan. Solving for ( ) P + P / yields: ( ) P + P / ) /( ) 2 = ( DP+ 2DP + DP = ( ) DP+ DP + DP DP /( ) 2

11 Diffusion Equaion (cond.) ( P + P ) / = D ( ) P+ P + P P /( ) 2 = D [ (P+ P ) ( )] P P /( ) 2 which can be rewrien as follows: P + P = D [ P + P P P ].

12 Diffusion Equaion (cond.) Taking he limi as = 0: ( ) P + P lim = 0 [ ( P+ P ) lim D ( ] P P ) 0 yields a parial differenial equaion (PDE): P = P D 2 2 which is known as he diffusion equaion.

13 Finie Difference Approimaion of P The value of he funcion, P, a he poin, (+,), can be epressed as a Taylor series epansion abou he poin, (,), as follows: P+ = P + P + ( )2 2 P, 2! 2 + O[( ) 3 ]., By rearranging he above, we derive he forward difference approimaion for P P+ P = P + O[ ].,,:

14 Backward Difference Approimaion of P The value of he funcion, P, a he poin, (,), can be epressed as a Taylor series epansion abou he poin, (,), as follows: P = P P + ( )2 2 P, 2! 2 + O[( ) 3 ]., By rearranging he above, we derive he backward difference approimaion for P,: P P = P + O[ ].,

15 Cenered Difference Approimaion of P P + ( )2, 2! P + = P + 2 P 2 + ( )3, 3! 3 P 3 +O[( ) 4 ], P = P P + ( )2 2 P, 2! 2 + ( )3, 3! 3 P 3 Subracing P from P + yields: P+ P = 2 P 2 ( )3 3! +, 3 P 3 + O[( ) 4 ]., +O[( ) 4 ],

16 Cenered Difference Appro. of P (cond.) This can be rearranged o yield he cenered difference approimaion for P P + P 2 = P + O[( ) 2 ]., : Noice ha he cenered difference approimaion is second order accurae.

17 Finie Difference Approimaion of 2 P 2 The value of he funcion, P/, a he poin, ( +,), can be epressed as a Taylor series epansion abou he poin, (,), as follows: P = P +, +, 2 P 2 + ( )2 3 P, 2! 3 + O[( ) 3 ]., Given he above we can derive he forward difference approimaion for 2 P 2, : P +, P, = 2 P 2 + O[ ].,

18 Finie Difference Appro. of 2 P 2 (cond.) For reasons of symmery, we approimae P +, and P, using backward differences: [ P ] + P P P = 2 P 2 + O[ ]., Combining erms yields he following epression for 2 P 2, : P+ 2P + P = ( ) 2 2 P 2 + O[ ].,

19 Diffusion Equaion (reprise) Applying he finie difference approimaions we ve derived o he diffusion equaion: P = P D 2 2 yields P + P ( P + = D 2P + P ( ) 2 which can be re-arranged o yield: P + P = D [ P + P P P which (we recall) is equivalen o he maser equaion: P 2DP ( ) 2 +DP P + = ( ) +DP 2 + ] ) ( ) 2.

20 Wave Equaion The parial differenial equaion governing wave moion is: 2 P = P 2 c Applying he finie difference approimaions for 2 P 2, and 2 P 2, yields: P + 2P + P c 2 ( ) 2 Solving for P + updae formula: 2 [ 1 c 2 ( P + ) 2 ] ( P + 2P + P ( ) 2 gives he following = P + P +c 2 ( ). ) 2 ( P + + P ).

21 Firs Order in Time Unforunaely, his formula is secondorder in ime. To derive a formula which is firs-order in ime, we recall ha 2 P 2, = P,+ P, + O[ ]. P Replacing P,+ wih P + and using he resuling epression for 2 P 2, and a cenered difference approimaion for 2 P 2, in he wave equaion yields: P + P P, ( P c 2 + 2P + P ( ) 2 Muliplying boh sides by : P + P Ṗ c 2 ( P ( ) 2 + 2P + P ). ).

22 Firs Order in Time (cond.) Muliplying boh sides by again, and hen adding P and Ṗ o boh sides yields: P + P + Ṗ ) 2 ( ) P + 2P + P ( +c 2 which can be rearranged o give an updae equaion for P which is firs-order in ime: P + = [ ( ) ] 2 1 2c 2 P + Ṗ ( ) 2 ( +c 2 P + + P ).

23 Firs Order in Time (cond.) To derive an updae equaion for Ṗ which is also firs-order in ime, we once again begin wih 2 P P,+ P, 2, = + O[ ]. Using he above and a cenered difference approimaion for 2 P 2, in he wave equaion resuls in: P,+ P, ( P c 2 + 2P + P ( ) 2 ). Wriing Ṗ for P, yields he following updae equaion for Ṗ: Ṗ + = Ṗ +c 2 ( P ( ) 2 + 2P + P We observe ha he updae equaions for boh P and Ṗ are firs-order in ime. ).