Numerical Methods for Langevin Langevin’s idea: small particles bounced around by fluid...

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Transcript of Numerical Methods for Langevin Langevin’s idea: small particles bounced around by fluid...

  • 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

    = 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:

    ds + 2θσ = F

    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.,

    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

    (1 + ξ)3 √

    1 + Ωξ)

    Yields a generalized Dyer-Roeder eq.

    (1 + z)(1 + Ωz) d2D

    dz2

    +( 7

    2 Ωz +

    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