Numerical Methods for Langevin...

Numerical Methods for Langevin Equationswith applications to gravitational systems

W. P. Petersen

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

e-mail: [email protected]

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

mη,

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.


Figure: Simulation of 1-D Brownian motion


We will come back to the 2-D situation:

Figure: Simulation of 2-D Brownian motion


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

∫ t0

eγsdw(s)


Solution to Ornstein-Uhlenbeck LE has properties

Ev(t) = v0e−γt + σe−γt

∫ t0

eγsEdw(s)

= v0e−γt

and

E(v(t))2 = (v0)2e−2γt + σ2e−2γt

e2γt − 12γ

→ σ2

2γas t →∞

Anything familiar about this?

m

2E(v)2 =

m

2

σ2

2γ

=1

2kT


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

dz = b(z) dt + σ(z) dw(t). (SDE)

Must be careful: (SDE) is shorthand for

z(t) = z0 +

∫ t0

b(zs) ds +

∫ t0σ(zs) dw(s).

The Stieltjes integral is interpreted (Itô rule)

∫ t0 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.


What’s important about Itô rule:

E{∫ t

0σ(z(s))dw(s)} = 0,

and

E{∫ t

0σ(z(s))dw(s)}2 =

∫ t0

E(σ(z(s)))2ds.

These functional integrals are called martingales.


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

∂u(x , t)

∂t=

1

2a(x)

∂2

∂x2u(x , t)

+b(x)∂

∂xu(x , t) + c(x)u(x , t),

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

u(x , t) = Ef (z(t)) exp{∫ t

0c(z(s))ds}.

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


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

1

2a∂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

0ds c(zs)}

−E∫ τ

0dt f (zt) exp{

∫ t0

ds c(zs)}

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


Figure: Exit process


Simulation of these things? First, to compute expectations:

Ef [z(t)] ≈ 1N

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 viaEuler (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.


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.


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 thelast Jeans’ equation

∂ρv̄

∂t+∇x(ρvv)−∇(ρΦ) = 0


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.


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

vi∂2f

∂vj∂vkd3v = −δij

∫∂f

∂vkd3v = 0

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

Figure: Impact parameter model


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

∆v ≈ 2Gmbv

and∆v

v≈ ψ ∼ 1

N2/3

where N = number of stars in the system. Langevin equation isequivalent to Fokker-Planck equation. For example, Balescu’sbook.


Stochastic Dyer-Roeder equation: start with Sachs’ equationsfor shear (σ), ray separation θ, in free space with scatteredpoint-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

dzln(A)

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

dσ

ds+ 2

1

D

dD

dsσ = F

1

D

d2D

ds2+ |σ|2 = 0.


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

F = 1c2

(1 + z)2d2Φ

dx2.

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


Figure: 2-D character of light scattering


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


More useful form for 1st:

D2σ =

∫ s0

D2(s ′)F(s ′)ds ′.

Expressing the affine parameter in terms of the redshift

s =

∫ z0

dξ

(1 + ξ)3√

1 + Ωξ)

Yields a generalized Dyer-Roeder eq.

(1 + z)(1 + Ωz)d2D

dz2

+(7

2Ωz +

Ω

2+ 3)

dD

dz

+|σ(z)|2

(1 + z)5D = 0.


Shear can be well approximated by

σ(z) = γ3Ω

8π(D(z))2×∫ z

0(D(ξ))2(1 + ξ)(1 + Ωξ)−

12 dw(ξ)

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


Figure: Shear free Dyer-Roeder D(z)


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


Comments:

I Long ago, a 2-D version of Fokker-Planck eq. for f (E , J) wasused by BGK to get King Model (1965), Lightman andShapiro (1977) etc. Since late 1980s, SDE (Langevin)simulation methods are much improved.

I SDE simulations will not yield details on clustering.I However, they will be useful as initial conditions for N-body

I power spectrum is easy to implement: |v | ∝ |k|.I the local temperature T ∝ |v |2.

I M-C procedures will complement N-body simulations, notreplace.

I SDE simulations are particularly useful in high dimensions, i.e.D ≥ 3, particularly when there are many species.

I SDEs get distributions right, not local details.


References:

I W. C. Saslaw, Gravitational Physics of Stellar and GalacticSystems, Cambridge, 1987.

I R. Balescu, Equilibrium and Nonequilibrium Stat. Mechanics,Wiley Interscience, 1975.

I G. Milstein, Numerical Itegration of Stochastic DifferentialEquations, Kluwer, 1995.

I C.C. Dyer and R.C. Roeder, Astrophysical J., 180, pp.L31-L34, 1972.

I S. Seitz and P. Schneider, Astronomy and Astrophysics, 287,pp. 349-360, 1994.

I H.M.P. Couchman, et al, MNRAS, 308, pp. 180-201, 1998.

I W. Petersen, SINUM, 35, no. 4, pp. 1439-1451, 1998.

I W. Petersen, Stoch. Analysis and Appl., 22, no. 4, pp.989-1008, 2004.


Numerical Methods for Langevin...

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