Numerical Methods for Langevin Equations with applications to gravitational systems

W. P. Petersen

Seminar for Applied Mathematics, ETH Zürich http://www.math.ethz.ch/∼wpp/

e-mail: wpp@math.ethz.ch

April 18, 2005

W. P. Petersen Numerical Methods for Langevin Equations

Recall simplest case: Stoke’s law for a particle in fluid,

dv(t) = −γ v(t) dt

where

γ = 6πr

m η,

η = viscosity coefficient.

Langevin’s idea: small particles bounced around by fluid molecules,

dv(t) = −γ v(t) dt + σ dw(t), (LE)

w(t) = Brownian motion, γ = Stoke’s coefficient. σ2 = 2kTγm =Diffusion coefficient.

W. P. Petersen Numerical Methods for Langevin Equations

Figure: Simulation of 1-D Brownian motion

W. P. Petersen Numerical Methods for Langevin Equations

We will come back to the 2-D situation:

Figure: Simulation of 2-D Brownian motion

W. P. Petersen Numerical Methods for Langevin Equations

Properties of w(t)? (Physicists’ notation is often 〈X 〉 = EX )

w(0) = 0

Ew(t) = 0

E(w(t))2 = t

and p.d.f. satisfies Heat equation

∂p(w , t)

∂t =

1

2

∂2p(w , t)

∂w2

Formal solution to LE called an Ornstein-Uhlenbeck process

v(t) = v0e −γt + σe−γt

∫ t 0

eγsdw(s)

W. P. Petersen Numerical Methods for Langevin Equations

Solution to Ornstein-Uhlenbeck LE has properties

Ev(t) = v0e −γt + σe−γt

∫ t 0

eγsEdw(s)

= v0e −γt

and

E(v(t))2 = (v0) 2e−2γt + σ2e−2γt

e2γt − 1 2γ

→ σ 2

2γ as t →∞

Anything familiar about this?

m

2 E(v)2 =

m

2

σ2

2γ

= 1

2 kT

W. P. Petersen Numerical Methods for Langevin Equations

If system is non-isotropic, diffusion coefficient σ may depend on process.

dz = b(z) dt + σ(z) dw(t). (SDE)

Must be careful: (SDE) is shorthand for

z(t) = z0 +

∫ t 0

b(zs) ds +

∫ t 0 σ(zs) dw(s).

The Stieltjes integral is interpreted (Itô rule)

∫ t 0 σ(z(s))dw(s) =

lim ∆t→0

n∑ i=1

σ(z(ti ))(w(ti+1)− w(ti ))

and is called belated, or non-anticipating.

W. P. Petersen Numerical Methods for Langevin Equations

What’s important about Itô rule:

E{ ∫ t

0 σ(z(s))dw(s)} = 0,

and

E{ ∫ t

0 σ(z(s))dw(s)}2 =

∫ t 0

E(σ(z(s)))2ds.

These functional integrals are called martingales.

W. P. Petersen Numerical Methods for Langevin Equations

Connection of B-motion to heat equation, we’ve seen, but there is more: Feynman-Kac formula. Solution to

∂u(x , t)

∂t =

1

2 a(x)

∂2

∂x2 u(x , t)

+b(x) ∂

∂x u(x , t) + c(x)u(x , t),

where u(x , t = 0) = f (x), is

u(x , t) = Ef (z(t)) exp{ ∫ t

0 c(z(s))ds}.

z(t = 0) = x is initial condition for SDE.

W. P. Petersen Numerical Methods for Langevin Equations

What about Dirichlet problem on domain D? c(x) ≤ 0 here

1

2 a ∂2u

∂x2 + b

∂u

∂x + cu = f (x)

with u(x) = g(x) on boundary x ∈ ∂D. F-K solution is

u(x) = Eg(zτ ) exp{ ∫ τ

0 ds c(zs)}

−E ∫ τ

0 dt f (zt) exp{

∫ t 0

ds c(zs)}

Here τ = first exit time, i.e. the first time t that z(t) crosses ∂D. Again, z(t = 0) = x .

W. P. Petersen Numerical Methods for Langevin Equations

Figure: Exit process

W. P. Petersen Numerical Methods for Langevin Equations

Simulation of these things? First, to compute expectations:

Ef [z(t)] ≈ 1 N

N∑ i=1

f [z [i ](t)]

for an N sample of paths z(t), and some functional f . N paths {z [1], z [2], ..., z [N]} are integrated by some rule, e.g., simplest is via Euler (higher order in wpp ’98, Milstein ’95, e.g.)

z [1] t+h = z

[1] t + b(z

[1] t ) h + σ(z

[1] t ) ∆w

[1]

z [2] t+h = z

[2] t + b(z

[2] t ) h + σ(z

[2] t ) ∆w

[2]

· · · z

[N] t+h = z

[N] t + b(z

[N] t ) h + σ(z

[N] t ) ∆w

[N].

Where, ∆w = ξ is approximately gaussian, w. variance h.

W. P. Petersen Numerical Methods for Langevin Equations

Gravitational systems - starting with Boltzmann’s equation.

f (x, v, t) = prob. density in 6-D x, v space

If phase-space is incompressible,

d

dt

∫ f d3x d3v = 0

or

∂f

∂t + v̇ · ∇v f + ẋ · ∇x f = 0

∂f

∂t −∇xΦ · ∇v f + v · ∇x f = 0 (B-T)

this is the collisionless Boltzmann eq.

W. P. Petersen Numerical Methods for Langevin Equations

Now include probability conservation

∂f

∂t +∇x · (vf ) +∇v · (v̇f ) = 0 (P-C)

and the mass density

ρ(x) =

∫ f d3v

Multiplying (B-T) by vector v and integrating over d3v , we get the last Jeans’ equation

∂ρv̄

∂t +∇x(ρvv)−∇(ρΦ) = 0

W. P. Petersen Numerical Methods for Langevin Equations

Here,

v̄ = ρ−1 ∫

v f d3v

vivj = ρ −1

∫ vivj f d

3v

but what about collisions? One simple model is

dx = vdt

dv = −∇Φ(x)dt + σ(x)dw(t)

inserting this in (B-T), and taking fluctuation averages,

∂f

∂t + v̇ · ∇v f −∇Φ · ∇x f +

1

2 (σ · ∇v )2f = 0.

W. P. Petersen Numerical Methods for Langevin Equations

Now, integrate over d3v , notice if σ(x) depends only on x, therefore ∫

vi ∂2f

∂vj∂vk d3v = −δij

∫ ∂f

∂vk d3v = 0

we recover Jeans’ equation, even with collisions. What do these collisions look like?

Figure: Impact parameter model

W. P. Petersen Numerical Methods for Langevin Equations

Velocity distributions are given by Fokker-Planck eq. (Spitzer and Härn, ’58), and kicks ψ are typically very small:

∆v ≈ 2Gm bv

and ∆v

v ≈ ψ ∼ 1

N2/3

where N = number of stars in the system. Langevin equation is equivalent to Fokker-Planck equation. For example, Balescu’s book.

W. P. Petersen Numerical Methods for Langevin Equations

Stochastic Dyer-Roeder equation: start with Sachs’ equations for shear (σ), ray separation θ, in free space with scattered point-like particles:

dσ

ds + 2θσ = F

dθ

ds + θ2 + |σ|2 = 0

σ is complex, F is the Weyl term, and s is an affine parameter - related to redshift z .

θ = 1

2

d

dz ln(A)

where A ∝ D2 is the beam area, get two eqs.,

dσ

ds + 2

1

D

dD

ds σ = F

1

D

d2D

ds2 + |σ|2 = 0.

W. P. Petersen Numerical Methods for Langevin Equations

In Lagrangian coordinates (contract with redshift z), the Weyl term to 1st order has derivatives of the gravitational potential Φ(x , y), with x = x + i y :

F = 1 c2

(1 + z)2 d2Φ

dx2 .

Light ”sees” shearing forces orthogonal to congruence and problem is essentially 2-D.

W. P. Petersen Numerical Methods for Langevin Equations

Figure: 2-D character of light scattering

W. P. Petersen Numerical Methods for Langevin Equations

Correlation length is about 7 cells, i.e. ∼ 7 Mpc at z = 0. Softened (2-3 cells) shears are normal in < 128 Mpc.

Figure: Shearing forces, from H. Couchman’s code

W. P. Petersen Numerical Methods for Langevin Equations

More useful form for 1st:

D2σ =

∫ s 0

D2(s ′)F(s ′)ds ′.

Expressing the affine parameter in terms of the redshift

s =

∫ z 0

dξ

(1 + ξ)3 √

1 + Ωξ)

Yields a generalized Dyer-Roeder eq.

(1 + z)(1 + Ωz) d2D

dz2

+( 7

2 Ωz +

Ω

2 + 3)

dD

dz

+ |σ(z)|2

(1 + z)5 D = 0.

W. P. Petersen Numerical Methods for Langevin Equations

Shear can be well approximated by

σ(z) = γ 3Ω

8π(D(z))2 ×∫ z

0 (D(ξ))2(1 + ξ)(1 + Ωξ)−

1 2 dw(ξ)

where w(z) is a complex (2-D) B-motion. Constant γ ≈ 0.62 was determined by N-body simulations.

W. P. Petersen Numerical Methods for Langevin Equations

Figure: Shear free Dyer-Roeder D(z)

W. P. Petersen Numerical Methods for Langevin Equations

Figure: D(z) histograms at 0 ≤ z ≤ 5. Non-linear integration. Scales for the abscissas are: 10−6 for z = 1/2, 10−5 for z = 1, 2, 3, 4, 5.

W. P. Petersen Numerical Methods for Langevin Equations

Comments:

I Long ago, a 2-D version of Fokker-Planck eq. for f (E , J) was used by BGK to get King Model (1965), Lightman and Shapiro (197