Langevin Methods. Burkhard Dünweg Max Planck Institute for Polymer Research Ackermannweg 10 D Mainz Germany

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1 Langevin Methods Burkhard Dünweg Max Planck Institute for Polymer Research Ackermannweg 1 D Mainz Germany

2 Motivation Original idea: Fast and slow degrees of freedom Example: Brownian motion Replace fast degrees by friction and random noise Conceptually and technically simpler Learn the mathematics of random noise! Technical interest: Original system: freedom Deterministic, no additional degrees of Add friction and noise to stabilize equations of motion Permitted if noise does not alter the (relevant) dynamics, or only static properties are sought

3 Markov Processes Continuous (state) space (x, usually multi dimensional) Continuous time (t) Conditional probability for x (t ) x(t): P(x, t x, t ) does not depend on previous history (t < t ) Z dxp(x, t x, t ) = 1 Chapman Kolmogorov: P(x, t x, t ) = δ(x x ) x x t t 1 t P(x, t x, t ) = Z dx 1 P(x, t x 1, t 1 )P(x 1, t 1 x, t )

4 Formal Moment Expansion Consider: p(x) > R dx p(x) = 1 µ n = R dx x n p(x) exists for all x Then p(k) = Z dx exp(ikx)p(x) = X n= (ik) n n! µ n I. e. unique p(x) {µ n } p(x) = X n= «n µ n x n! δ(x) Proof: Both sides produce the same moments!

5 Kramers Moyal Expansion Define µ n (t; x, t ) = (x x ) n (t, t ) Z = dx (x x ) n P(x, t x, t ) (mean displacement, mean square displacement etc.) = P(x, t x, t ) X «n δ(x x ) 1 x n! µ n(t; x, t ) n= particularly good for short times. Chapman Kolmogorov (small τ): = = P(x, t x, t ) Z X dx 1 n= x P(x 1, t τ x, t ) X n= «n δ(x x 1 ) 1 n! µ n(t; x 1, t τ) «n 1 x n! µ n(t; x, t τ) P(x, t τ x, t )

6 Subtract n = term: 1 τ [P(x, t x, t ) P(x, t τ x, t )] = 1 X «n 1 τ x n! µ n(t; x, t τ)p(x, t τ x, t ) n=1 Short time behavior of the moments defines the Kramers Moyal coefficients D (n) (o(τ): Terms of order higher than linear!): (x x ) n (t + τ, t ) = n!d (n) (x, t )τ + o(τ) µ n (t; x, t τ) = n!d (n) (x, t τ)τ + o(τ) = n!d (n) (x, t)τ + o(τ) Similarly P(x, t τ x, t ) P(x, t x, t ) Hence t P(x, t x, t ) = X n=1 «n D (n) (x, t)p(x, t x, t ) x generalized Fokker Planck equation Shorthand: t P(x, t x, t ) = LP(x, t x, t )

7 Only four types of processes: Pawula Theorem 1. Expansion stops at n = : No dynamics 2. Expansion stops at n = 1: Deterministic processes ( Liouville equation) 3. Expansion stops at n = 2: Fokker Planck / diffusion processes 4. Expansion stops at n = Proof: See Risken, The Fokker Planck Equation. Truncation at finite order n > 2 would produce P <!!

8 Proof of Pawula Theorem Define scalar product Z f g = dx P(x, t x, t )f (x)g(x) Moments: µ n = Z dx P(x, t x, t )(x x ) n I. e. µ m+n = (x x ) m (x x ) n Schwarz inequality: µ 2 m+n µ 2mµ 2n D (m+n)2 (2m)!(2n)! [(m + n)!] 2D(2m) D (2n) for m 1, n 1 Suppose D (2N) = D (2N+1) =... = Set m = 1, n = N, N + 1,...: D (N+1) = D (N+2) =... = Zeroing always works except for very small N where no new information is obtained.

9 N = 1: D (2) =... = D (2) =... = N = 2: D (4) =... = D (3) =... = N = 3: D (6) =... = D (4) =... = N = 4: D (8) =... = D (5) =... = etc. Thus: Truncation at any finite order implies D (3) = D (4) =... =.

10 Goal Langevin simulation Generation of stochastic trajectories for a process of type 3 with a discretization time step τ Physical input: Drift coefficient D (1) ( deterministic part) Diffusion coefficient D (2) ( stochastic part)

11 We know for the displacements: Euler Algorithm x i = D (1) i (x, t)τ + o(τ) x i x j = 2D (2) ij (x, t)τ + o(τ) ( x) n = o(τ) n 3 Satisfied by x i (t + τ) = x i (t) + D (1) i r i random variables with: τ + 2τ r i r i = r i r j = D ij All higher moments exist Some distribution, e. g. Gaussian or uniform (B. D. & W. Paul, Int. J. Mod. Phys. C 2, 817 (1991)) Stochastic term dominates for τ Large number of independent kicks Central Limit Theorem Gaussian behavior Gaussian white noise Often: D ij has a simple structure (diagonal, constant, or both) Non diagonal D ij for systems with hydrodynamic interactions (correlations in the stochastic displacements): Ermak & MacCammon, J. Chem. Phys. 69, 1352 (1978)

12 A Simple Example d = 1 diffusion with constant drift. D (1) = const., D (2) = const. Trajectories: x t continuous but not differentiable! Solution of the Fokker Planck equation: P(x, t, ) = 1 4πD (2) t exp (x! D(1) t) 2 4D (2) t P(x,t) t=1 t= x t=7

13 Langevin Equation Formal way of writing the Euler algorithm (stochastic differential equation): d dt x i = D (1) i + f i (t) f i (t) Gaussian white noise with properties f i = f i (t)f j (t ) = 2D ij δ(t t ) Thus x i x j = Z τ dt Z τ dt f i (t)f j (t ) = 2D ij τ Higher order moments: R τ dt f i(t) is Gaussian!

14 Thermal Systems: The Fluctuation Dissipation Theorem Langevin dynamics to describe: Decay into thermal equilibrium Thermal fluctuations in equilibrium D (1), D (2) do not explicitly depend on time Equilibrium state: Hamiltonian H(x) β = 1/(k B T) Z = R dx exp( βh) ρ(x) = Z 1 exp( βh) Necessary: P(x, t x, ) ρ(x) for t L exp( βh) = Balance between drift and diffusion defines temperature.

15 Definition of the process via Ito vs. Stratonovich Langevin equation plus interpretation of the stochastic term! So far: Ito interpretation Other common interpretation: Stratonovich: Assume that the trajectories are differentiable, and take the limit of vanishing correlation time at the end! Consider d x = F(x) + σ(x)f(t) dt F deterministic part σ(x) noise strength (multiplicative noise) f = f(t)f(t ) = 2δ(t t )

16 d x = F(x) + σ(x)f(t) dt Ito: Z τ dt σ(x(t))f(t) σ(x()) fiz τ fl dt σ(x(t))f(t) Z τ = dt f(t) Stratonovich: Z τ dt σ(x(t))f(t) σ(x()) = σ(x()) fiz τ Z τ Z τ dt f(t) + dσ dx dt f(t) + σ dσ dx fl dt σ(x(t))f(t) = + σ dσ dx τ + o(τ) spurious drift Z τ Z τ dt x(t)f(t) +... dt Z t dt f(t )f(t) +...

17 Brownian Dynamics System of particles Coordinates r i Friction coefficients ζ i Diffusion coefficients D i Potential energy U ( H) Forces F i = U r i d dt r i = 1 Fi + δ i ζ i D E δi = D δi (t) δ j (t )E = 2D i 1 δ ij δ(t t ) I. e. L = X i r i 1 ζ i F i + X i «2 D i r i X i r i L exp( βh) =» 1 H H βd i exp( βh) = ζ i r i r i D i = k BT ζ i Einstein relation

18 Stochastic Dynamics Generalized coordinates q i Generalized canonically conjugate momenta p i Hamiltonian H Hamilton s equations of motion, augmented by friction and noise terms: d dt q i = H p i d dt p i = H q i ζ i H p i + σ i f i ζ i = ζ i ({q i }) σ i = σ i ({q i }) f i = fi (t)f j (t ) = 2δ ij δ(t t ) L = L H + L SD

19 L H = X i H + X q i p i i H p i q i = X i H p i q i + X i H q i p i L H exp( βh) = o.k. L SD = X i p i» ζ i H p i + σ 2 i p i L SD exp( βh) = X i p i» ζ i H p i βσ 2 i H e βh = p i σ 2 i = k BTζ i Simple recipe for MD with hard potentials & weak noise: Standard velocity Verlet Add friction and noise at those instances where forces are calculated Symplectic algorithm in the ζ = limit

20 Dissipative Particle Dynamics (DPD) Disadvantages of SD: v = reference frame is special Galileo invariance is broken Global momentum is not conserved No proper description of hydrodynamics Idea: Dampen relative velocities of nearby particles Stochastic kicks between pairs of nearby particles satisfying Newton III Result: Galileo invariance Momentum conservation Locality Correct description of hydrodynamics No profile biasing in boundary driven shear simulations

21 In practice: Define ζ(r) (relative) friction for particles at distance r σ(r) noise strength for particles at distance r r ij = r i r j = r ijˆr ij Friction force along interparticle axis: F (fr) i = X j ζ(r ij ) [( v i v j ) ˆr ij ] ˆr ij Pi F (fr) i = (antisymmetric matrix in ij) Stochastic force along interparticle axis: F (st) i = X j σ(r ij ) η ij (t) ˆr ij η ij = η ji η ij = ηij (t)η kl (t ) = 2(δ ik δ jl + δ il δ jk )δ(t t ) Pi F (st) i = (similar)

22 d dt r i = 1 m i p i d dt p i = F i + F (fr) i + F (st) i L = L H + L DPD L DPD = X ij X i j + X i = X i ζ(r ij )ˆr ij p i σ 2 (r ij ) ˆr ij p i X j( =i) ˆr ij + σ 2 (r ij )ˆr ij σ 2 (r ij ) X p i j( =i)» H ˆr ij H p i p j «ˆr ij " p i «ˆr ij p i «2 ζ(r ij )ˆr ij p j «# p j «H H «p i p j Fluctuation dissipation theorem: σ 2 (r) = k B Tζ(r)

23 Force Biased Monte Carlo Idea: Use a BD step (with large τ) as a trial move for Monte Carlo. Accept / reject with Metropolis criterion correct Boltzmann distribution without discretization errors Just d = 1, set γ = τ/ζ. Algorithm: Start position x Calculate energy U = U(x) Calculate force F = F(x) = U/x Trial move: Generate x = x + γf + p 2k B Tγ ρ ρ Gaussian with ρ = D ρ 2E = 1 Hence, w ap (x x ) = 1! exp ρ2 2π 2 = w 1 Calculate energy U = U(x ) Calculate U = U U Calculate force F = F(x ) Calculate ρ = (2k B Tγ) 1/2 `x x γf (random number needed to go back)

24 Hence, w ap (x x) = 1! exp ρ 2 2π 2 = w 2 Calculate w acc Accept with probability w acc w acc =? Detailed balance: w ap (x x ) w acc (x x ) w ap (x x) w acc (x x) = exp( β U) I. e. standard Metropolis with exp( β U) exp( β U) w 2 w 1

25 Higher Order Algorithms Additive noise: Systematic approach via operator factorization. Idea: Fokker Planck equation: t P = LP P = exp(lt)δ(x x ) Factorize the exponential, each factor such that the result is known. Example (2nd order): L = L det + L stoch (deterministic propagation, stochastic diffusion) exp(lt) = exp(l stoch t/2) exp(l det t) exp(l stoch t/2) + O(t 3 ) exp(l stoch t/2) acting on δ(x x ): Exactly known solution Gaussian distribution / solution of the diffusion equation exp(l det t): Just deterministic propagation. Can be done up to any desired accuracy with known methods (e. g. Runge Kutta) State of the art: Fourth order: H. A. Forbert, S. A. Chin, Phys. Rev. E 63, 1673 (2).

26 Multiplicative noise: Schemes of higher order than Euler are very difficult to construct and apply (Greiner, Strittmatter, Honerkamp, J. Stat. Phys. 51, 95 (1988)). No known general solution for a diffusion equation of type t P = 2 x 2D(x)P