The Smoluchowski-Kramers Approximation: What model describes a Brownian particle?

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1 The Smoluchowski-Kramers Approximation: What model describes a Brownian particle? Scott Hottovy shottovy@math.arizona.edu University of Arizona Applied Mathematics October 7, 2011

2 Brown observes a particle 1820s, botanist Robert Brown observes irregular motion of particle Want to measure velocity.

3 Brown observes a particle 1820s, botanist Robert Brown observes irregular motion of particle (Pictured, 1-D Wiener process W t (ω) or Brownian motion, E[W t ] = 0, E[(W t W s ) 2 ] = t s ).

4 A dynamical theory for Brownian motion Einstein and Smoluchowski derived the motion to be diffusive. Langevin and later Ornstein and Uhlenbeck (1930s) have dynamical theory. Newton s second law dx m t = v m t dt, x m 0 = x mdv m t = b(x m t ) γv m t dt + σ dw t v m 0 = v. b and γvt m are the drift forces, σ diffusion (noise) coefficient. Shorthand for x m t = x + v m t = v + t vs m 0 t 0 ds, b(x m s ) γv m s m t ds + 0 σ m dw s.

5 (en.wikipedia.org) Kramers

6 Hendrick Kramers took m = 0 to simplify calculations to chemical reaction rates. dx t = b(x t) γ dt + σ γ dw t. Called Smoluchoski-Kramers approximation.

7 Hendrick Kramers took m = 0 to simplify calculations to chemical reaction rates. dx t = b(x t) γ dt + σ γ dw t. Called Smoluchoski-Kramers approximation. Nelson showed x m t x t almost surely for t [0, T ], T <. How do we integrate t 0 f (s, ω) dw s?

8 (conservapedia.com) Riemann Sums

9 To integrate φ t = over [0, t] take partial sums. Stochastic Integral t 0 f (s, ω) dw t

10 To integrate Stochastic Integral φ t = over [0, t] take partial sums. t lim φ N(α) t = lim N N W t is the Wiener process. 0 f (s, ω) dw t N f (ti, ω)(w ti W ti 1 ), i=1 t i = αt i + (1 α)t i 1, for all 0 α 1

11 To integrate Stochastic Integral φ t = over [0, t] take partial sums. t lim φ N(α) t = lim N N W t is the Wiener process. 0 f (s, ω) dw t N f (ti, ω)(w ti W ti 1 ), i=1 t i = αt i + (1 α)t i 1, for all 0 α 1 t 0 W s d α W s, = 1 ( ) 1 2 W t 2 2 α t Special cases α = 0 Itô, α = 1/2 Stratonovich, α = 1 anti-itô.

12 Stochastic Integral For f smooth (E[ f (s, ω) f (t, ω) 2 ] k s t 1+ɛ ), φ(α) t = t 0 f (s, ω) dw s, for all α [0, 1].

13 Stochastic Integral For f smooth (E[ f (s, ω) f (t, ω) 2 ] k s t 1+ɛ ), φ(α) t = t 0 f (s, ω) dw s, for all α [0, 1]. For Kramers equation dxt m = vt m dt mdvt m = b(xt m ) γvt m dt + σ dw t. Used the S-K approximation dx t = b(x t) γ dt + σ γ dw t. Since σ and σ/γ constant, the integral is the same for all interpretations.

14 (courtesy Giovanni Volpe) Measure the drift forces Experiment

15 Model dxt m = 1 v m m t dt ( b(x dvt m m = t ) γ(x m ) t ) m m v t m dt + σ(x m t ) m dw t. Approximated by the Smoluchowski-Kramers approximation or dx t = b(x t) γ(x t ) dt + σ(x t) γ(x t ) dw t, [ b(xt ) dx t = γ(x t ) + ασ(x t) d γ(x t ) dx t ( )] σ(xt ) γ(x t ) dt + σ(x t) γ(x t ) dw t, with the stochastic integral of the Itô form.

16 Results Freidlin studies γ(x t ) = γ (purely mathematical?). dx t = b(x t) γ + σ(x t) γ dw t, (in Probability). Volpe/Wehr et al. use fluctuation dissipation theorem (γ(x t ) = cσ(x t ) 2 ), [ ] b(xt ) dx t = cσ(x t ) 2 dt + 1 cσ(x t ) dw t. Measurements of forces not coinciding.

17 Results Freidlin studies γ(x t ) = γ (purely mathematical?). dx t = b(x t) γ + σ(x t) γ dw t, (in Probability). Volpe/Wehr et al. use fluctuation-dissipation relation (γ(x t ) = cσ(x t ) 2 ), [ b(xt ) dx t = cσ(x t ) ( )] d 1 dt+ 1 cσ(x t ) dx t cσ(x t ) cσ(x t ) dw t, (in L 2 ). α = 1, anti-itô!

18 PDE Recall that Einstein and Smoluchowski derived diffusion equation. Probability transition function of (x t, v t ), defined P((x t, v t ) A) = p(t; (x, v ), (x, v)) dxdv, p is the density. A

19 Connection to PDE dx m t = v m t dt mdv m t = b(x m t ) γ(x m t )v t dt + σ(x m t ) dw t, Density p m satisfies the Fokker-Planck (or forward Kolmogorov) equation p m = 1 2 ( ) σ(x) 2 t 2 v 2 m p m x (vp m) ( ) (b(x) γ(x)v) p m v m = L x,v p, Also, for all (x, x ) R 2, p m satisfies the backward Kolmogorov equation p m = L x t,v p m.

20 Heuristic calculation Two scales, x m t of order 1 and v m t of order 1/ m.

21 Heuristic calculation Two scales, x m t of order 1 and v m t of order 1/ m. For p m the solution to the backward Kolmogorov equation, p m = p 0 + mp 1 + mp

22 Heuristic calculation Two scales, x m t of order 1 and v m t of order 1/ m. For p m the solution to the backward Kolmogorov equation, p m = p 0 + mp 1 + mp σ(x) 2 2 ( ) p 0 b(x) 2γ(x) 2 x 2 + γ(x) σ(x)2 d(γ(x)) p0 2γ(x) 3 dx x = p 0 t. the BK equation satisfied by p 0. Compare to the PDE of SK σ(x) 2 2 ( [ p b(x) 2γ(x) 2 x 2 + γ(x) + α σ(x) ]) γ(x) 3 γ (x) σ(x)2 p γ(x) 2 σ (x) x = p t.

23 α = α(x t ) = α constant iff γ(q) = cσ(q) λ. Equation for α γ (x t )σ(x t ) 2(γ (x t )σ(x t ) γ(x t )σ (x t )). α = λ 2(λ 1). This coincides with Freidlin (λ = 0 = α = 0) and Wehr/Volpe (λ = 2 = α = 1).

24 Further work α = Properties of v t as ɛ 0. λ 2(λ 1). Consider the color noise case (W t replaced by a differentiable process). Extend to n dimensions. λ = 1 and its applications. dx t = ( ) b(xt ) cσ(x t ) 1 2c 2 σ(x t ) dt + 1 c dw t.

25 Further work α = Properties of v t as ɛ 0. λ 2(λ 1). Consider the color noise case (W t replaced by a differentiable process). Extend to n dimensions. λ = 1 and its applications. Questions? dx t = ( ) b(xt ) cσ(x t ) 1 2c 2 σ(x t ) dt + 1 c dw t.