Langevin Methods Burkhard Dünweg Max Planck Institute for Polymer Research Ackermannweg 1 D 55128 Mainz Germany
Motivation Original idea: Fast and slow degrees of freedom Example: Brownian motion Replace fast degrees by friction and random noise Conceptually and technically simpler 11111111 1 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
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 )
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!
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 )
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 )
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 <!!
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.
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) =... =.
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)
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)
A Simple Example d = 1 diffusion with constant drift. D (1) = const., D (2) = const. Trajectories: x 16 14 12 1 8 6 4-2 2 2 4 6 8 1 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).3.25.2.15.1.5 t=1 t=3-5 5 1 15 x t=7
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!
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.
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 )
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) +...
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
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
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
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
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)
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)
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)
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
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).
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