Coarse-Graining Techniques in Molecular Modeling: Theory and Applications

1. Topics in Molecular Modeling: IV. Coarse-graining techniques Xiantao Li (xiantao.li@gmail.com) Department of Mathematics, Pennsylvania State University

2. Outline The free energy The Mori-Zwanzig formalism The approximation of the memory term Fluctuation-dissipation theorem Data assimilation techniques

3. Part I: Mori-Zwanzig formalism Nonlinear dynamical system: . Wronskian: . Variational equation: Fundamental solution: In particular, If then Therefore,

4. Effective dynamics Observable: (notations from stat. mech.) Time derivative: L is a differential operator w.r.t. Time evolution: The equation is not closed.

5. Choice of the coarse-grain variables is known as coarse-grain variables. It usually corresponds to slow variables that are sufficient to describe the overall dynamics. Specific choices The first few Fourier or generalized Fourier modes . Center of mass. is a subset of atoms. Average local momentum. Reaction coordinates (dihedral angles). Local energy: .

6. Projection operators The purpose of the projection operators is to separate out the contributions that can be represented in terms of (therefore, they are in principle computable), and the terms that can not be resolved. Neglecting fine-scale components: Orthogonal projection: Conditional expectation: . The orthogonal projection leads to the Mori’s formulation and the conditional expectation leads to the Zwanzig’s formulation. We will compare them later.

7. Dyson’s equation We define Consider the equation:. This is known as the orthogonal dynamics. Let’s rewrite the equation: At the level of operators: This is the Dyson’s formula.

8. Mori-Zwanzig Equation We start with Apply the projection operators: Apply the Dyson’s formula to the last term: This is the Mori-Zwanzig equation. The first two terms are in principle functions of The last term is often regarded as random noise The actual form will depend on the specific choice of the projection operator.

9. Mori’s projection Markovian term: The memory term: The memory term becomes a linear convolution, with memory kernel, = Here we used the fact that is in the range of , , and The GLE: . The second fluctuation-dissipation theorem:

10. Mori’s projection (cont’d) The projection leads to a linear Markovian and a memory term in the form of a linear convolution. If then Together with the FDT, the random noise is stationary. By taking the average, one gets, There is a close relation between the Mori’s approach and the linear response theory. The memory function can be approximated in terms of its Laplace transform. Approximations of the memory term typically leads to linear SDEs, for which Gaussian statistics is expected, unless multiplicative noise is introduced.

11. Mori’s projection (cont’d) The random noise can be embedded into an infinite system of ODEs Direct observations suggest that: . ODEs The moments: . The kernel function While the values of for t the values of can be computed directly from the moments (short-time statistics).

12. Zwanzig’s projection General form: If from molecular dynamics, then the free energy is given by, The mean force: For linear dynamical systems, the memory term becomes a convolution and the kernel function is a matrix function. Even in this case, the Mori’s projection and Zwanzig’s projections yield different equations.

13. Zwanzig’s projection (cont’d) Memory term Fluctuation-dissipation theorem Approximation of the memory term For the r.h.s. function f(A) is a polynomial, this term can be explicitly expressed.

14. Example: harmonic bath Full problem . The GLE from Zwanzig’s projection:

15. Extension 1: oblique projection Suppose that Then we can define a generalized force , and the projection, . The GLE equation, . For example: Markovian approximation  Stochastic gradient system White-noise approximation: .

16. Extension 2: conservation laws Let be a conserved quantity In order to obtain a closed mode, we need a constitutive relation: For example, we pick the orthogonal projection, and follow the Dyson’s equation, This becomes a stochastic constitutive relation.

17. Approximation of the kernel function ¼ Suppose that we have the GLE: . The memory kernel is usually not directly accessible for It has slow decay in time. In terms of the time correlations, it satisfies an integral equation of the first kind Laplace transform Markovian approximation (zeroth order): It is computable if the data are available.

18. Approximation of the kernel function 2/4 In the time domain: The noise is chosen s.t. First order model: Rational function: Hermite interpolation: and The second condition is moment matching: . They can be obtained from data.

19. Approximation of the kernel function ¾ The first order model in the time domain The noise can be chosen such that the probability density of A is correct. Higher order models (up to 4): • How good are the rational approximations?

20. Approximation of the kernel function 4/4 In certain cases, this is equivalent to a Galerkin projection to a Krylov subspace. Full problem: Subspaces: The Galerkin projection is equivalent to the rational function approximation matching one long time statistics and short time statistics. The Lanczos algorithm makes it more robust.

21. Application to heat conduction Motivations: Fourier’s Law 𝑞=−𝑘𝛻𝑇 breaks down at small scales 〖10〗^(−6)~〖10〗^(−9) m heat pulse experiments can’t be described by Fourier’s Law (Both, et al. 2015) thermal conductivity depends on the system size (Győry & Márkus, 2014) thermal fluctuations play a more important role Direct molecular dynamics (MD) simulations are not affordable How do we derive a generalized constitutive relation for the many-particle description?

22. Example:1dchainmodel NearestneighborinteractionwithpairpotentialisFermi-Pasta-Ulam(FPU)potential. Local energy:

23. Energy conservationandheatflux Closure Problem:Canweexpresslocalenergyfluxinaclosedmodel? Continuousenergyconservation: Discreteenergyconservation: Energyflux:

24. Orthogonal and oblique projection • --Mori’sprojection • --non-localheatequation • --negativevariationof SIAMCONFERENCEONANALYSISOFPARITALDIFFERENTIALEQUATIONS,2017@BALTIMORE

25. Observations of the non-Gaussianstatistics NormalizedhistogramofandfittingLaplacedistribution NormalizedhistogramofandfittingGammadistribution .

26. withGaussianadditivenoise • Zerothorder–stochasticheatequation • Firstorder–stochasticdampedwaveequation • Secondorder • Higherorder (up to 4th order) SIAM CONFERENCE ON ANALYSIS OF PARITAL DIFFERENTIAL EQUATIONS, 2017@BALTIMORE

27. with additive noise • Markovian model • Generalized constitutive relation • Implications • Temperature-dependent conductivity • Traveling solutions • Stochastic phase-field crystal? • First order model • Generalized constitutive relation • Implications • Propagation of temperature (second sound speed) • relaxation SIAMCONFERENCEONANALYSISOFPARITALDIFFERENTIALEQUATIONS,2017@BALTIMORE

28. Consistency with the exact statistics 1d chain Nano-tube