Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics

Transcription

1 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 1/2 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics Peter R. Kramer Department of Mathematical Sciences Rensselaer Polytechnic Institute, Troy, NY and Institute for Mathematics and Its Applications, Minneapolis, MN Supported by NSF CAREER Grant DMS and CMG OCE

2 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 2/2 Overview Stochastic Drift-Diffusion Parameterization of Water Dynamics near Solute with Adnan Khan and Shekhar Garde Statistical mesoscale modeling for oceanic flows with Banu Baydil and Shafer Smith (Courant, CAOS) Poisson Vortex Streamlines, λ ~ = 1 y 2 y 1

3 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 3/2 Biological Disclaimer ( S. Garde et al)

4 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 4/2 Stochastic Parameterization of Water Dynamics near Solute Simplified statistical description of water dynamics as possible basis for implicit solvent method to accelerate molecular dynamics simulations for proteins, etc. (with Adnan Khan (Lahore) and Shekhar Garde (Biochemical Engineering)) As a first step, we explore stochastic parameterization of water near C 60 buckyball molecule. isotropic, chemically simple

5 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 5/2 Molecular Dynamics Snapshot of Buckyball Surrounded by Water

6 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 6/2 Statistical dynamics encoded in biophysical literature in terms of a diffusion coefficient: (Makarov et al, 1998; Lounnas et al, 1994,... ) X(t + 2τ) X(t) 2 D B (r) 6τ X(t) = r X(t + τ) X(t) 2 6τ X(t) = r But this seems to mix together inhomogeneities in mean and random motion.

7 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 7/2 Drift-Diffusion Framework We explore capacity of models of the form dx = U(X(t)) dt + Σ(X(t)) dw (t), for water molecule center-of-mass position X(t). drift vector coefficient U(r) diffusion tensor coefficient D(r) = 1 2 Σ(r)Σ (r) For isometric solute (buckyball): U(r) = U ( r )ˆr, D(r) = D ( r )ˆr ˆr + D ( r )(I ˆr ˆr), for position r = r ˆr relative to center of symmetry.

8 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 8/2 Physically Inspired (DD-I) Model In analogy to Brownian dynamics simulations, take U(r) = γ 1 φ(r), D(r) = D 0 I.

9 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 8/2 Physically Inspired (DD-I) Model In analogy to Brownian dynamics simulations, take U(r) = γ 1 φ(r), D(r) = D 0 I. Potential of mean force obtained from measuring concentration c(r) and Boltzmann distribution c(r) exp( φ(r)/k B T). Diffusivity unchanged from bulk value. Friction coefficient from Einstein relation γ = k B T/D 0.

10 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 8/2 Physically Inspired (DD-I) Model In analogy to Brownian dynamics simulations, take U(r) = γ 1 φ(r), D(r) = D 0 I. 14 Potential of mean force φ (kj/ mol) r (nm)

11 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 8/2 Physically Inspired (DD-I) Model In analogy to Brownian dynamics simulations, take U(r) = γ 1 φ(r), D(r) = D 0 I. 700 Mean Force (filtered) F MD (kj/ (mol nm)) r (nm)

12 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 9/2 Systematic, Data-Driven Parameterization (DD-II) Model Parametrize drift and diffusion functions from mathematical definitions: U ( r ) = X(t + τ) X(t) lim ˆr τ 0 τ X(t) = r, D ( r ) = (X(t + τ) X(t)) ˆr U (r)τ 2 lim τ 0 2τ X(t) = r,

13 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 9/2 Systematic, Data-Driven Parameterization (DD-II) Model Parametrize drift and diffusion functions from mathematical definitions: D ( r ) = 1 lim τ 0 4τ (X(t + τ) X(t)) (I ˆr ˆr) 2 X(t) = r. Obtain statistical data from MD simulations.

14 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 10/2 Time Difference τ must be chosen carefully Taking τ = t (time step of MD simulation) may not be appropriate Limit τ 0 implicitly refers to times large enough for drift-diffusion approximation to be valid. Must choose T v τ T x, where: T v is time scale of momentum. T x is time scale of position. See also Pavliotis and Stuart (2007) about need to undersample. How choose τ in practice?

15 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 11/2 Explore simple Ornstein-Uhlenbeck (OU) model dx = V dt, mdv = γv dt αx dt + 2k B Tγ dw(t)

16 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 11/2 Explore simple Ornstein-Uhlenbeck (OU) model dx = V dt, mdv = γv dt αx dt + 2k B Tγ dw(t) Forces: friction, potential, and thermal.

17 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 11/2 Explore simple Ornstein-Uhlenbeck (OU) model dx = V dt, mdv = γv dt αx dt + 2k B Tγ dw(t) Nondimensionalize: dx = V dt, dv = av dt ax dt + a dw (t)

18 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 11/2 Explore simple Ornstein-Uhlenbeck (OU) model dx = V dt, mdv = γv dt αx dt + 2k B Tγ dw(t) Nondimensionalize: dx = V dt, dv = av dt ax dt + a dw (t) where a = γ 2 /(mα) is ratio of position to momentum time scale.

19 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 11/2 Explore simple Ornstein-Uhlenbeck (OU) model dx = V dt, mdv = γv dt αx dt + 2k B Tγ dw(t) Nondimensionalize: dx = V dt, dv = av dt ax dt + a dw (t) Exact drift-diffusion coarse-graining when a 1: dx = X dt + dw(t)

20 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 11/2 Explore simple Ornstein-Uhlenbeck (OU) model dx = V dt, mdv = γv dt αx dt + 2k B Tγ dw(t) Nondimensionalize: dx = V dt, dv = av dt ax dt + a dw (t) What if we try to obtain this from analysis of trajectories with finite but large a?

21 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 12/2 Drift and diffusion coefficients of exact OU solution sampled with finite time difference τ. 0.3 Longitudinal Drift Coefficient for OU, a= Blue : Exact Red : Asymptotic 0.5 U (x) x/ x τ

22 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 12/2 Drift and diffusion coefficients of exact OU solution sampled with finite time difference τ. Longitudinal Diffusivity for OU, a= Blue : Exact Red : Asymptotic D τ

23 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 13/2 Inferences from OU model Good choice of τ may be the one which maximizes drift magnitude and diffusivity. Beginning estimate obtained from OU model with same a value.

24 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 14/2 OU Model Insights MD Data Parameterization in DD-II Model To obtain time scales, approximate main well in potential of mean force by quadratic. 8 Potential of mean force 6 Red : Buckyball Potential Green : Quadratic Fit to a characteristic part of the potential 4 φ (kj/mol) r (nm) This gives a = 132.

25 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 14/2 OU Model Insights MD Data Parameterization in DD-II Model Examine drift and diffusivity computed from various choices of τ.

26 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 14/2 OU Model Insights MD Data Parameterization in DD-II Model Examine drift and diffusivity computed from various choices of τ. 0.1 Longitudinal Drift 0.05 U (nm/ps) Blue : τ=20fs Green : τ=40 fs Red : τ=60 fs Magenta : τ=80 fs Cyan : τ=100 fs r (nm)

27 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 14/2 OU Model Insights MD Data Parameterization in DD-II Model Examine drift and diffusivity computed from various choices of τ x 10 3 Longitudinal Diffusivity D (nm 2 /ps) Blue :τ=60 fs Green :τ=80 fs Magenta :τ=120 fs Cyan :τ=160 fs Black :τ=200 fs Yellow :τ=240 fs Red :τ=280 fs r (nm)

28 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 14/2 OU Model Insights MD Data Parameterization in DD-II Model Examine drift and diffusivity computed from various choices of τ. Both desiderata about drift and diffusivity behavior not simultaneously satisfiable. Correct bulk diffusivity behavior more important We choose τ = 0.2 ps = 200 fs.

29 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 15/2 Parameterization Used in DD-II Model Longitudinal Drift Blue : DD II output Red : DD II input from MD data U (nm/ps) r (nm)

30 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 15/2 Parameterization Used in DD-II Model x Longitudinal Diffusivity 6 5 D (nm 2 /ps) Blue : DD II output Red : DD II input from MD Data r (nm)

31 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 15/2 Parameterization Used in DD-II Model x Lateral Diffusivity 7 6 D (nm 2 /ps) Blue : DD II output Red : DD II input from MD data r (nm)

32 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 16/2 Compare Predictions of Biophysical Diffusivity Formula D B (r) X(t + 2τ) X(t) 2 6τ X(t) = r X(t + τ) X(t) 2 6τ X(t) = r

33 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 16/2 Compare Predictions of Biophysical Diffusivity Formula 1.5 D B (r) for DD I Model 1 D B /D D DD I D MD r (nm)

34 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 16/2 Compare Predictions of Biophysical Diffusivity Formula D B (r) for DD II Model D B /D D MD r (nm)

35 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 17/2 Future Work Next steps anisotropies chemical heterogeneity

36 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 18/2 Unresolved Mesoscale Turbulence in Ocean Circulation Models Computational ocean models for climate prediction have resolution 100 km: does not adequately resolve mesoscale turbulent structures on length scales 100 km even smaller scales 100 m : Kolmogorov turbulence We focus on representing turbulent transport by unresolved mesoscale turbulence. Wind-driven double-gyre 2000 km basin-scale computational simulation, 5 km resolution, by Shafer Smith.

37 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 19/2 Mathematical Framework for Transport T(x,t) t + u(x,t) T(x,t) = κ T(x,t), T(x,t = 0) = T in (x) Passive scalar field T(x,t) Velocity field u = V + v: large-scale mean flow + small-scale fluctuations Molecular diffusion coefficient κ

38 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 20/2 Parameterization Problem Obtain an equation for coarse-grained T : T t + V T = κ T F T (x,t = 0) = T in (x), Turbulent Flux F = v T Problem of Parametrization: F = F( T,V ) where F only involves a few parameters.

39 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 21/2 Parameterization of Small Scales One parameterization used in AOS and engineering (Gent and McWilliams 1990, Griffies 1998, Middleton and Loder 1989): F = K (x,t) T, with generally nonsymmetric K = S + A. Symmetric part S : variably enhanced diffusion Antisymmetric part A : effective drift U = A relative to mean flow

40 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 22/2 Parameterization Approaches Within the class of effective diffusivity or eddy diffusivity schemes, the way in which K is modeled differs. Some examples: Mixing-length theory: K lv. K ε theory: parameterize based on local energy and energy dissipation rate Gent-McWilliams: related to slope of surfaces of constant potential density These are empirical, with varying degrees of success. Ocean circulation models often employ even simpler practice of choosing K as constant multiple of identity, tuned a posteriori.

41 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 23/2 Homogenization-Based Parameterization Under assumption of scale separation (not so crazy for ocean), homogenization theory provides rigorous support and formula for effective diffusivity (Avellaneda, Majda, McLaughlin, Papanicolaou, Varadhan, Fannjiang, Pavliotis & K): where χ τ K (x,t) = κ (I v χ )) + (V + v) y χ κ y χ = v, is cell problem, solved on sub-grid scale with frozen mean flow V. denotes subgrid-scale average.

42 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 24/2 Computational Homogenization Heterogenous multiscale methods (HMM) natural, but perhaps too complex a modification to existing codes. More negotiable: parameterization formula for K based on statistical subgrid-scale model with small number of nondimensional parameters Our approach: build up from simple models, combine numerical solutions with new and existing asymptotic results (Avellaneda, Majda, Vergassola, Bonn, McLaughlin, Kesten, Papanicolaou, etc.)

43 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 25/2 Example: Dynamical Poisson Vortex Model (Cinlar, Caglar,... ) Stream function: ψ(y,t) = n ψ vor (y Y (n) c (t ξ (n) ),t ξ (n) )

44 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 25/2 Example: Dynamical Poisson Vortex Model (Cinlar, Caglar,... ) Stream function: ψ(y,t) = n ψ vor (y Y (n) c (t ξ (n) ),t ξ (n) ) composed of simple vortex blobs, e.g., [ ] ψvor / y v(y,t) = 2, ψ vor / y 1 [ 1 ψ vor (y,t = 0) = 12 v vor l vor y l vor y ] l vor 2 ] for y l vor

45 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 25/2 Example: Dynamical Poisson Vortex Model (Cinlar, Caglar,... ) Stream function: ψ(y,t) = n ψ vor (y Y (n) c (t ξ (n) ),t ξ (n) ) with centers Y c (n) (t) obeying certain simple dynamical law Brownian motion intervortical advection and ψ vor (y,t) 0 for t,

46 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 25/2 Example: Dynamical Poisson Vortex Model (Cinlar, Caglar,... ) Stream function: ψ(y,t) = n ψ vor (y Y (n) c (t ξ (n) ),t ξ (n) ) vortices nucleated according to space-time Poisson process (η (n),ξ (n) ) of prescribed intensity λ, with Y c (n) (0) = η (n)

47 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 26/2 Sample Snapshot of Poisson Vortex Field 10 Poisson Vortex Streamlines, λ ~ = y y 1

48 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 27/2 Uncertainty Modeling Issues How choose the statistical model parameters for dynamical parameterization? for observed subgrid-scale data, can try parametric statistical fitting but want statistical subgrid-scale model to represent unobserved small scales, with only coarse-scale data available so how predict small-scale statistical parameters from large-scale observations?

49 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 28/2 References regarding Statistical Modeling in Molecular Dynamics T. Schlick, Molecular Modeling and Simulation: An Interdisciplinary Guide, V. A. Makarov et al, Biophys J., 75 (1998) biophysical modeling framework for water near surface G. Pavliotis and A. M. Stuart, J. Stat. Phys. 124 (2007) mathematical framework for computing coarse-grained coefficients statistically from microscale data

50 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 29/2 References regarding Statistical Modeling in Ocean Turbulence Mathematics of Subgrid-Scale Phenomena in Atmospheric and Oceanic Flows, Institute for Pure and Applied Mathematics, A. J. Majda and P. R. Kramer, Phys. Rep. 314 (1999) review of some statistical flow models M. Çaglar, et al, J. Atmos. Ocean. Tech. 23 (2006) parameterically fitting observational data to statistical vortex model