Stochastic Particle Dynamics for Unresolved Degrees of Freedom. Sebastian Reich in collaborative work with Colin Cotter (Imperial College)

Transcription

1 Stochastic Particle Dynamics for Unresolved Degrees of Freedom Sebastian Reich in collaborative work with Colin Cotter (Imperial College)

2 1. The Motivation Classical molecular dynamics (MD) as well as particle methods for numerical weather prediction (NWP) lead to large systems of conservative Newtonian equations of motion.

3 1. The Motivation Classical molecular dynamics (MD) as well as particle methods for numerical weather prediction (NWP) lead to large systems of conservative Newtonian equations of motion. Underresolved simulations may occur for two reasons:

4 1. The Motivation Classical molecular dynamics (MD) as well as particle methods for numerical weather prediction (NWP) lead to large systems of conservative Newtonian equations of motion. Underresolved simulations may occur for two reasons: certain degrees of freedom are ignored and absent in the model,

5 1. The Motivation Classical molecular dynamics (MD) as well as particle methods for numerical weather prediction (NWP) lead to large systems of conservative Newtonian equations of motion. Underresolved simulations may occur for two reasons: certain degrees of freedom are ignored and absent in the model, the equations of motion are integrated with time steps large compared to the fastest time-scales present in the system.

6 1. The Motivation Classical molecular dynamics (MD) as well as particle methods for numerical weather prediction (NWP) lead to large systems of conservative Newtonian equations of motion. Underresolved simulations may occur for two reasons: certain degrees of freedom are ignored and absent in the model, the equations of motion are integrated with time steps large compared to the fastest time-scales present in the system. One might want to model the effect of unresolved motion using stochastic models.

7 2. The Plan We start with the stiff spring pendulum and compare

8 2. The Plan We start with the stiff spring pendulum and compare the conservative model,

9 2. The Plan We start with the stiff spring pendulum and compare the conservative model, the conservative model plus Langevin dynamics,

10 2. The Plan We start with the stiff spring pendulum and compare the conservative model, the conservative model plus Langevin dynamics, the dissipative particle dynamics (DPD) model,

11 2. The Plan We start with the stiff spring pendulum and compare the conservative model, the conservative model plus Langevin dynamics, the dissipative particle dynamics (DPD) model, the extended DPD model.

12 2. The Plan We start with the stiff spring pendulum and compare the conservative model, the conservative model plus Langevin dynamics, the dissipative particle dynamics (DPD) model, the extended DPD model. We then discuss these algorithms in the context of large time step methods for MD, unresolved wave motion in NWP.

13 3. The Stiff Spring Pendulum We use the stiff spring pendulum equations ṙ = p, ṗ = g K r ( r L) r as our model problem. Here r, p R 3, g = (0, 0, 1) T, K = 1000 is the spring constant, and L = 1 is the length of the pendulum.

14 3. The Stiff Spring Pendulum We use the stiff spring pendulum equations ṙ = p, ṗ = g K r ( r L) r as our model problem. Here r, p R 3, g = (0, 0, 1) T, K = 1000 is the spring constant, and L = 1 is the length of the pendulum. The stiff pendulum equations are Hamiltonian with conserved energy where we introduced r = r. E = 1 2 pt p + K 2 (r L)2 r T g,

15 Furthermore, because of the relatively large spring constant K = 1000, the (oscillatory) energy in the spring is an adiabatic invariant, i.e., J = 1 2 p2 r + K 2 (r L)2 with p r = r T p/r, is approximately conserved over long intervals of time.

16 Furthermore, because of the relatively large spring constant K = 1000, the (oscillatory) energy in the spring is an adiabatic invariant, i.e., J = 1 2 p2 r + K 2 (r L)2 with p r = r T p/r, is approximately conserved over long intervals of time. In the following figure we plot E(t), J(t), and the slow rotational energy E s (t) = E(t) J(t) from a simulation using the symplectic Störmer-Verlet method.

17

18 4. The Langevin Model for the Stiff Spring Pendulum We next replace the Hamiltonian equations of motion by the Langevin model ṙ = p, ṗ = g K r r (r L) γp + γẇ(t), with γ = 0.1, σ = 0.2, k B T = 1, and W(t) a vector of independent Wiener processes.

19 4. The Langevin Model for the Stiff Spring Pendulum We next replace the Hamiltonian equations of motion by the Langevin model ṙ = p, ṗ = g K r (r L) γp + γẇ(t), r with γ = 0.1, σ = 0.2, k B T = 1, and W(t) a vector of independent Wiener processes. It is apparent from the following figure that none of the energy contributions is conserved (even approximately) any longer. Note that, by definition, J 0, E r 1, and E s r 1.

20

21 5. Dissipative Particle Dynamics (DPD) The DPD model (Hoogerbrugge & Koelman, Espanol & Warren) leads to a stochastic coupling only in the direction of the stiff spring and the equations of motion become ṙ = p, ṗ = g r [ K(r L)+γpr + γẇ (t) ], r with the parameters values as before and W (t) a scalar Wiener process.

22 5. Dissipative Particle Dynamics (DPD) The DPD model (Hoogerbrugge & Koelman, Espanol & Warren) leads to a stochastic coupling only in the direction of the stiff spring and the equations of motion become ṙ = p, ṗ = g r [ K(r L)+γpr + γẇ (t) ], r with the parameters values as before and W (t) a scalar Wiener process. Indeed, the simulation results, as displayed in the following figure, reveal that the slow energy is now approximately conserved while both the total and the oscillatory energy drift.

23 5. Dissipative Particle Dynamics (DPD) The DPD model (Hoogerbrugge & Koelman, Espanol & Warren) leads to a stochastic coupling only in the direction of the stiff spring and the equations of motion become ṙ = p, ṗ = g r [ K(r L)+γpr + γẇ (t) ], r with the parameters values as before and W (t) a scalar Wiener process. Indeed, the simulation results, as displayed in the following figure, reveal that the slow energy is now approximately conserved while both the total and the oscillatory energy drift. This demonstrates the capability of the DPD model to thermostate degrees of freedom in a rather targeted manner.

24

25 6. The Extended DPD Model (Cotter & Reich, 2003) Finally, we replace the DPD model by the extended version ṙ = p, ṗ = g r [K(r L) + s], r εṡ = s + γp r + σẇ (t) for various values of the correlation parameter ε. The parameters γ and σ are left as before and W (t) is again a Wiener process.

26 6. The Extended DPD Model (Cotter & Reich, 2003) Finally, we replace the DPD model by the extended version ṙ = p, ṗ = g r [K(r L) + s], r εṡ = s + γp r + σẇ (t) for various values of the correlation parameter ε. The parameters γ and σ are left as before and W (t) is again a Wiener process. The fluctuation-dissipation variable s satisfies an Ornstein- Uhlenbeck process, which has exponential decay of correlation K(τ) ε 1 e τ /ε.

27 6. The Extended DPD Model (Cotter & Reich, 2003) Finally, we replace the DPD model by the extended version ṙ = p, ṗ = g r [K(r L) + s], r εṡ = s + γp r + σẇ (t) for various values of the correlation parameter ε. The parameters γ and σ are left as before and W (t) is again a Wiener process. The fluctuation-dissipation variable s satisfies an Ornstein- Uhlenbeck process, which has exponential decay of correlation K(τ) ε 1 e τ /ε. The extended model reduces to the standard DPD model for ε 0.

28 The rapid change in energy behavior shown in the following figure for parameter values from an interval 1 ε 0.01 is quite striking and indicates that the model s response to added fluctuationdissipation can depend crucially on the decay rate ε 1 of the associated auto-covariance matrix.

29

30 7. Conservative Large Time-Step Methods Consider a spiced-up stiff spring pendulum with Hamiltonian H = 1 2 pt p + K 2 (r L)2 + V short (r) + V long (r) where V short is some short-ranged potential energy function while V long is long-ranged and expensive to compute.

31 7. Conservative Large Time-Step Methods Consider a spiced-up stiff spring pendulum with Hamiltonian H = 1 2 pt p + K 2 (r L)2 + V short (r) + V long (r) where V short is some short-ranged potential energy function while V long is long-ranged and expensive to compute. Then symplectic multiple-time-stepping is a good idea but prone to numerical resonance instabilities. It was suggested by Garcia- Archilla, Sanz-Serna & Skeel to mollify the long range potential energy V long to avoid these instabilities.

32 A simple mollifier is Equilibrium where H = 1 2 pt p + K 2 (r L)2 + V short (r) + V long ( r) and r(r) is defined by the equations r = (1 + λ)r, r = L.

33 A simple mollifier is Equilibrium where H = 1 2 pt p + K 2 (r L)2 + V short (r) + V long ( r) and r(r) is defined by the equations r = (1 + λ)r, r = L. It has been shown by Izaguirre, Reich & Skeel that Equilibrium allows step-size of up to 7 fs for flexible water molecules. The energy drift observed for step-sizes between 7 fs and 15 fs is rather mild. This suggests to...

34 8. Stochastic Large Time-Step Methods... add a Langevin term to the equations of motion (see Izaguirre, Catarello, Wozniak & Skeel). But that approach couples noise to all degrees of freedom and in all directions.

35 8. Stochastic Large Time-Step Methods... add a Langevin term to the equations of motion (see Izaguirre, Catarello, Wozniak & Skeel). But that approach couples noise to all degrees of freedom and in all directions. More recently Izaguirre & Ma suggested Targeted MOLLY (TM), which combines mollification of the long range potential with a DPD-type fluctuation-dissipation on the highly oscillatory degrees of freedom.

36 8. Stochastic Large Time-Step Methods... add a Langevin term to the equations of motion (see Izaguirre, Catarello, Wozniak & Skeel). But that approach couples noise to all degrees of freedom and in all directions. More recently Izaguirre & Ma suggested Targeted MOLLY (TM), which combines mollification of the long range potential with a DPD-type fluctuation-dissipation on the highly oscillatory degrees of freedom. For flexible water this approach allow for step-sizes up to fs without affecting diffusion constants etc.

37 9. Particle Methods for NWP The rotating two-dimensional shallow-water equations (SWEs) describe a shallow layer of fluid subject to gravity and rotation.

38 9. Particle Methods for NWP The rotating two-dimensional shallow-water equations (SWEs) describe a shallow layer of fluid subject to gravity and rotation. The Lagrangian formulation of the SWEs is ẍ = fẋ c 2 0 xh(x),

39 9. Particle Methods for NWP The rotating two-dimensional shallow-water equations (SWEs) describe a shallow layer of fluid subject to gravity and rotation. The Lagrangian formulation of the SWEs is ẍ = fẋ c 2 0 xh(x), where x = (x, y) T are the particle positions (a continuous function of both space and time),

40 9. Particle Methods for NWP The rotating two-dimensional shallow-water equations (SWEs) describe a shallow layer of fluid subject to gravity and rotation. The Lagrangian formulation of the SWEs is ẍ = fẋ c 2 0 xh(x), where x = (x, y) T are the particle positions (a continuous function of both space and time),c 0 = gh, g is the gravitational constant, H is the mean layer-depth,

41 9. Particle Methods for NWP The rotating two-dimensional shallow-water equations (SWEs) describe a shallow layer of fluid subject to gravity and rotation. The Lagrangian formulation of the SWEs is ẍ = fẋ c 2 0 xh(x), where x = (x, y) T are the particle positions (a continuous function of both space and time),c 0 = gh, g is the gravitational constant, H is the mean layer-depth,f is twice the (constant) angular velocity of the fluid, and ẋ = ( ẏ, ẋ) T.

42 Frank, Gottwald & Reich suggested a Hamiltonian particle-mesh (HPM) method for the solution of the two-dimensional shallowwater equations.

43 Frank, Gottwald & Reich suggested a Hamiltonian particle-mesh (HPM) method for the solution of the two-dimensional shallowwater equations. The HPM method may be viewed as an accurate numerical discretisation of the regularised fluid equations: ẍ = fẋ c 2 0 x [A h(x)] where A is a smoothing operator with some smoothing length α.

44 Frank, Gottwald & Reich suggested a Hamiltonian particle-mesh (HPM) method for the solution of the two-dimensional shallowwater equations. The HPM method may be viewed as an accurate numerical discretisation of the regularised fluid equations: ẍ = fẋ c 2 0 x [A h(x)] where A is a smoothing operator with some smoothing length α. We denote the numerically unresolved part of the layer-depth by η = h A h.

45 The HPM method uses a regular grid x kl = (k x, l y) T, moving particles x i = (x i, y i ) T, and grid-centred basis functions ψ kl (X).

46 The HPM method uses a regular grid x kl = (k x, l y) T, moving particles x i = (x i, y i ) T, and grid-centred basis functions ψ kl (X). The finite-dimensional Hamiltonian equations of motion are given by ẍ i = fẋ i c 2 0 k,l xi ψ kl (x i )ĥ kl.

47 The HPM method uses a regular grid x kl = (k x, l y) T, moving particles x i = (x i, y i ) T, and grid-centred basis functions ψ kl (X). The finite-dimensional Hamiltonian equations of motion are given by ẍ i = fẋ i c 2 0 k,l xi ψ kl (x i )ĥ kl. The numerically unresolved gravity waves η can be modelled by a generalized Langevin process.

48 The HPM method uses a regular grid x kl = (k x, l y) T, moving particles x i = (x i, y i ) T, and grid-centred basis functions ψ kl (X). The finite-dimensional Hamiltonian equations of motion are given by ẍ i = fẋ i c 2 0 k,l xi ψ kl (x i )ĥ kl. The numerically unresolved gravity waves η can be modelled by a generalized Langevin process. This idea can be mathematically motivated by representing η as the solution of a linear wave equation coupled to the particle system and subsequent reduction following the Kac-Zwanzig approach.

49 This leads to the extended DPD equations given by ẍ i = fẋ i c 2 ] 0 [ĥkl + η kl Xi ψ kl (X i ), k,l ε η kl = η kl + γḣ kl + σẇ kl.

50 This leads to the extended DPD equations given by ẍ i = fẋ i c 2 ] 0 [ĥkl + η kl Xi ψ kl (X i ), k,l ε η kl = η kl + γḣ kl + σẇ kl. It should be noted that h kl = ĥ kl + η kl is ment to be a better approximation to the true layer-depth h at the grid point x kl than ĥ kl alone.

51 This leads to the extended DPD equations given by ẍ i = fẋ i c 2 ] 0 [ĥkl + η kl Xi ψ kl (X i ), k,l ε η kl = η kl + γḣ kl + σẇ kl. It should be noted that h kl = ĥ kl + η kl is ment to be a better approximation to the true layer-depth h at the grid point x kl than ĥ kl alone. The unresolved waves η kl are modeled by Ornstein-Uhlenbeck processes. Whether this is reasonable in terms of real fluid motion remains an open question. But it can be tested numerically (time consuming...).

52 10. The HPM Method on the Sphere Lately we had great fun implementating the particle method for spherical geometry. A common test case is provided by a five day forecast for 21st December 1978 (500 mb geopotential height field).

53

54

55

56 11. Conclusion

57 11. Conclusion We started by looking at a simple stiff spring pendulum and the effect of various stochastic modifications on the dynamics.

58 11. Conclusion We started by looking at a simple stiff spring pendulum and the effect of various stochastic modifications on the dynamics. DPD and its extended version appear as more sensitive than straightforward Langevin dynamics which is still widely used in MD.

59 11. Conclusion We started by looking at a simple stiff spring pendulum and the effect of various stochastic modifications on the dynamics. DPD and its extended version appear as more sensitive than straightforward Langevin dynamics which is still widely used in MD. Two particular examples for which this is certainly true are (i) stablized long time step methods and (ii) particle methods for fluid dynamics.

60 11. Conclusion We started by looking at a simple stiff spring pendulum and the effect of various stochastic modifications on the dynamics. DPD and its extended version appear as more sensitive than straightforward Langevin dynamics which is still widely used in MD. Two particular examples for which this is certainly true are (i) stablized long time step methods and (ii) particle methods for fluid dynamics. The material presented in this talk is part of a survey article written by us for a handbook on nanoscale computation techniques (American Scientific Publisher).

61 11. Conclusion We started by looking at a simple stiff spring pendulum and the effect of various stochastic modifications on the dynamics. DPD and its extended version appear as more sensitive than straightforward Langevin dynamics which is still widely used in MD. Two particular examples for which this is certainly true are (i) stablized long time step methods and (ii) particle methods for fluid dynamics. The material presented in this talk is part of a survey article written by us for a handbook on nanoscale computation techniques (American Scientific Publisher). THE END