1. Topics in Molecular Modeling: IV. Coarse-graining techniques
Xiantao Li
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: 𝑥 =𝑓 𝑥 ,𝑥 0 = 𝑥 0 . dim 𝑥 =𝑁.
Wronskian: 𝑊 𝑡 = 𝜕𝑥(𝑡) 𝜕 𝑥 0 .
Variational equation: 𝑊 𝑡 =𝐴 𝑡 𝑊 𝑡 , 𝐴 𝑡 :=∇𝑓 𝑥 𝑡 , 𝑊 0 = 𝐼.
Fundamental solution: 𝑦 =𝐴 𝑡 𝑦, 𝑦 0 = 𝑦 0 𝑦 𝑡 =𝑊 𝑡 𝑦 0 .
In particular, If 𝑦 𝑡 =𝑓 𝑥 𝑡 , then 𝑦 =𝐴 𝑡 𝑦, 𝑦 0 =𝑓 𝑥 0 .
Therefore, 𝑓 𝑥 𝑡 =𝑊 𝑡 𝑓 𝑥 0 = 𝜕𝑥 𝑡 𝜕 𝑥 0 𝑓( 𝑥 0 ).

4. Effective dynamics
Observable: 𝐴 𝑡 ≔𝐴 𝑥 𝑡 . 𝐴≔𝐴 0 . (notations from stat. mech.)
Time derivative: 𝐴 =𝑓 𝑥 𝑡 𝑇 ∇𝐴 𝑥 𝑡 =𝑓 𝑥 0 ⋅ ∇ 𝑥 0 𝐴 𝑥 𝑡 =:𝐿𝐴.
L is a differential operator w.r.t. 𝑥 0
Time evolution: 𝐴 =𝐿𝐴 𝐴 𝑡 = 𝑒 𝑡𝐿 𝐴 0 .
The equation is not closed.

5. Choice of the coarse-grain variables
𝐴(𝑥) is known as coarse-grain variables. It usually corresponds to slow variables that are sufficient to describe the overall dynamics.
Specific choices
𝑥= 𝑥 1 , 𝑥 2 ,⋯, 𝑥 𝑛 , 𝑥 𝑛+1 ,⋯ 𝑥 𝑁 = 𝑥 , 𝑥 . 𝐴= 𝑥 .
The first few Fourier or generalized Fourier modes 𝑥= 𝑖 𝑞 𝑖 𝜙 𝑖 + 𝑖 𝜉 𝑖 𝜓 𝑖 . 𝐴=𝑞.
Center of mass. 𝐴 𝛼 = 𝑖∈ 𝑆 𝛼 𝑚 𝑖 𝑥 𝑖 . 𝑆 𝛼 is a subset of atoms.
Average local momentum.
Reaction coordinates (dihedral angles).
Local energy: 𝐴 𝛼 = 𝑖∈ 𝑆 𝛼 𝑚 𝑖 𝑥 𝑖 2 + 𝑉 𝑖 .

6. Projection operators
The purpose of the projection operators is to separate out the contributions that can be represented in terms of 𝐴 (therefore, they are in principle computable), and the terms that can not be resolved.
Neglecting fine-scale components: 𝑃𝑔 𝑥 =𝑃𝑔 𝑥 , 𝑥 =𝑔 𝑥 .
Orthogonal projection: 𝑃𝑔 𝑥 = 𝑔, 𝐴 𝑇 𝐴, 𝐴 𝑇 −1 𝐴.
Conditional expectation: 𝑃𝑔 𝑥 =𝐸 𝑔 𝑥 𝐴 𝑥 =𝐴 = ∫𝑔 𝑥 𝛿 𝐴 𝑥 −𝐴 𝜌 𝑥 𝑑𝑥 ∫𝛿 𝐴 𝑥 −𝐴 𝜌 𝑥 𝑑𝑥 .
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 𝑄=𝐼−𝑃.
Consider the equation: 𝑧 =𝑄𝐿𝑧, 𝑧 0 = 𝑧 0 .  𝑧 𝑡 = 𝑒 𝑡𝑄𝐿 𝑧 0 .
This is known as the orthogonal dynamics.
Let’s rewrite the equation: 𝑧 =𝐿𝑧−𝑃𝐿𝑧𝑧 𝑡 = 𝑒 𝑡𝐿 𝑧 0 − 0 𝑡 𝑒 𝑡−𝑠 𝐿 𝑃𝐿𝑧 𝑠 𝑑𝑠.
At the level of operators: 𝑒 𝑡𝐿 = 0 𝑡 𝑒 𝑡−𝑠 𝐿 𝑃𝐿 𝑒 𝑡𝑄𝐿 𝑑𝑠+ 𝑒 𝑡𝑄𝐿 .
This is the Dyson’s formula.

8. Mori-Zwanzig Equation
We start with 𝐴 𝑡 =𝐿𝐴 𝑡 = 𝑒 𝑡𝐿 𝐿𝐴 0 .
Apply the projection operators: 𝐴 𝑡 = 𝑒 𝑡𝐿 𝐿𝐴 0 = 𝑒 𝑡𝐿 𝑃𝐿𝐴 0 + 𝑒 𝑡𝐿 𝑄𝐴 0 .
Apply the Dyson’s formula to the last term:
𝐴 𝑡 = 𝑒 𝑡𝐿 𝑃𝐿𝐴 𝑡 𝑒 𝑡−𝑠 𝐿 𝑃𝐿 𝑒 𝑠𝑄𝐿 𝑄𝐿𝐴(0)𝑑𝑠+ 𝑒 𝑡𝑄𝐿 𝑄𝐿𝐴(0).
This is the Mori-Zwanzig equation.
The first two terms are in principle functions of 𝐴 𝑠 , 0≤𝑠≤𝑡.
The last term 𝐹 𝑡 =𝑒 𝑡𝑄𝐿 𝑄𝐿𝐴 0 is often regarded as random noise
The actual form will depend on the specific choice of the projection operator.

9. 𝜃 𝑡 =− 𝐿𝐹 𝑡 ,𝐴 0 𝑇 𝐴 0 ,𝐴 0 𝑇 −1 = 𝐹 𝑡 ,𝑄𝐿𝐴 0 𝐴 0 ,𝐴 0 𝑇 −1
Mori’s projection
Markovian term: 𝑒 𝑡𝐿 𝑃𝐿𝐴 0 = 𝐿𝐴, 𝐴 𝑇 𝐴, 𝐴 𝑇 −1 𝐴 𝑡 =:Ω𝐴 𝑡 .
The memory term: 0 𝑡 𝑒 𝑡−𝑠 𝐿 𝑃𝐿𝐹 𝑠 𝑑𝑠= 0 𝑡 𝑒 𝑡−𝑠 𝐿 𝐿𝐹 𝑠 ,𝐴 0 𝐴(0),𝐴 0 𝑇 −1 𝐴𝑑𝑠=:− 𝟎 𝒕 𝜽 𝒔 𝑨(𝒕−𝒔)𝒅𝒔 .
The memory term becomes a linear convolution, with memory kernel,
𝜃 𝑡 =− 𝐿𝐹 𝑡 ,𝐴 0 𝑇 𝐴 0 ,𝐴 0 𝑇 −1 = 𝐹 𝑡 ,𝑄𝐿𝐴 0 𝐴 0 ,𝐴 0 𝑇 −1
= 𝐹 𝑡 ,𝐹 0 𝑇 𝐴 0 ,𝐴 0 𝑇 −1
Here we used the fact that 𝐹 is in the range of 𝑄, 𝑄 2 =𝑄, and F 0 =𝑄𝐿𝐴 0 .
The GLE: 𝐴 =𝛺𝐴− 0 𝑡 𝜃 𝑠 𝐴 𝑡−𝑠 𝑑𝑠 +𝐹(𝑡).
The second fluctuation-dissipation theorem:
𝐹 𝑡 ,𝐹 0 𝑇 =𝜃(𝑡) 𝐴 0 ,𝐴 0 𝑇

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 𝐴 =0, then 𝐹 𝑡 =0. Together with the FDT, the random noise is stationary. By taking the average, one gets, ⟨𝐴⟩ =𝛺⟨𝐴⟩− 0 𝑡 𝜃 𝑠 ⟨𝐴⟩ 𝑡−𝑠 𝑑𝑠 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: 𝐹 𝑡 = 𝑗≥0 𝑐 𝑗 𝑡 𝐿 𝑗 𝐴(0) .
ODEs
The moments: 𝑀 𝑗 =⟨ 𝐴 𝑗 0 ,𝐴 0 ⟩.
The kernel function 𝜃 𝑡 =− 𝑗≥0 𝐶 𝑗 0 𝑀 𝑗+1 𝑀 −1 .
While the values of 𝜃 𝑡 for t>0 might not be accessible, the values of 𝜃 (𝑗) 𝑡 can be computed directly from the moments (short-time statistics).

12. Zwanzig’s projection General form:
𝐴 𝑡 = 𝑒 𝑡𝐿 𝑃𝐿𝐴 𝑡 𝑒 𝑡−𝑠 𝐿 𝑃𝐿 𝑒 𝑠𝑄𝐿 𝑄𝐿𝐴(0)𝑑𝑠+ 𝑒 𝑡𝑄𝐿 𝑄𝐿𝐴(0).
If 𝐴= 𝑞 𝑝 , 𝑞=𝐵𝑥, 𝑝=𝐵𝑣 from molecular dynamics, then the free energy is given by, 𝑊 𝑞, 𝑘 𝐵 𝑇 =− 𝑘 𝐵 𝑇𝑙𝑛∫ 𝑒 −𝛽𝑉 𝑥 𝛿 𝑞−𝐵𝑥 𝑑𝑥.
The mean force: −∇𝑊 𝑞(𝑡), 𝑘 𝐵 𝑇 =𝐸 −𝐵∇𝑉 𝐵𝑥=𝑞(𝑡) = 𝑒 𝑡𝐿 𝑃𝐿𝑝 0 .
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
0 𝑡 𝜃 𝐴 𝑠 ,𝑡−𝑠 𝜇(𝐴 𝑠 )𝑑𝑠
Fluctuation-dissipation theorem
𝐸 𝐹 𝑡 ,𝐹 0 𝑇 𝐴 = 𝑘 𝐵 𝜃 𝐴,𝑡 .
Approximation of the memory term
0 𝑡 𝑒 𝑡−𝑠 𝐿 𝑃𝐿 𝑒 𝑠𝑄𝐿 𝑄𝐿𝐴 0 𝑑𝑠≈𝑡 𝑒 𝑡𝐿 𝑃𝐿𝑄𝐿𝐴 0 .
For the r.h.s. function f(A) is a polynomial, this term can be explicitly expressed.

14. Example: harmonic bath
Full problem
𝑥 𝐼 =𝑓 𝑥 𝐼 − 𝐴 𝐼,𝐼𝐼 𝑥 𝐼𝐼
𝑥 𝐼𝐼 =− 𝐴 𝐼𝐼,𝐼𝐼 𝑥 𝐼𝐼 − 𝐴 𝐼𝐼,𝐼 𝑥 𝐼 . dim 𝑥 𝐼𝐼 ≫dim⁡( 𝑥 𝐼 ).
The GLE from Zwanzig’s projection:
𝑥 𝐼 =𝑓 𝑥 𝐼 − 𝐴 𝐼,𝐼𝐼 𝐴 𝐼𝐼,𝐼𝐼 −1 𝐴 𝐼𝐼,𝐼 𝑥 𝐼 − 0 𝑡 𝐴 𝐼,𝐼𝐼 𝐴 𝐼𝐼,𝐼𝐼 −1 cos 𝜏 𝐴 𝐼𝐼,𝐼𝐼 𝐴 𝐼𝐼,𝐼 𝑥 𝐼 𝑡−𝜏 𝑑𝜏+𝐹(𝑡)

15. Extension 1: oblique projection
Suppose that
𝐴∼𝜌 𝐴 ∝ 𝑒 −𝑊(𝐴)
Then we can define a generalized force 𝐵=− 𝛿𝑊(𝐴) 𝛿𝐴 , and the projection,
𝑃𝑔= 𝑔, 𝐵 𝑇 𝐵, 𝐵 𝑇 −1 𝐵.
The GLE equation,
𝐴 =𝛺𝐵− 0 𝑡 𝜃 𝑠 𝐵 𝑡−𝑠 𝑑𝑠 +𝐹(𝑡).
For example: Markovian approximation  Stochastic gradient system
𝐴 =−𝐷∇𝑊 𝐴 +𝐹 𝑡 .
White-noise approximation: 𝐹 𝑡 =𝜎𝜉(𝑡), 𝜎 𝜎 𝑇 =2symm(𝐷).

16. Extension 2: conservation laws
Let 𝐴 be a conserved quantity 𝐴 + ∇ ℎ ⋅ 𝐽 𝐴 =0. In order to obtain a closed mode, we need a constitutive relation: 𝐽 𝐴 =ℎ 𝐴 , ∇𝐴 ,⋯ . For example, we pick the orthogonal projection, and follow the Dyson’s equation, 𝐽 𝐴 𝑡 =Ω𝐴 𝑡 − 0 𝑡 𝜃 𝑠 𝐴 𝑡−𝑠 𝑑𝑠 +𝐹(𝑡) This becomes a stochastic constitutive relation.

17. Approximation of the kernel function ¼
Suppose that we have the GLE: 𝐴 =𝛺𝐵− 0 𝑡 𝜃 𝜏 𝐵 𝑡−𝜏 𝑑𝑠 +𝐹(𝑡). The memory kernel 𝜃 𝑡 is usually not directly accessible for 𝑡>0. It has slow decay in time. In terms of the time correlations, it satisfies an integral equation of the first kind 𝐴 𝑡 ,𝐴 0 =Ω 𝐵 𝑡 ,𝐴 0 −𝜃⋆ 𝐵 𝑡 ,𝐴 0 Laplace transform 𝑠 𝐶 − 𝐴,𝐴 = Ω−Θ 𝐷 𝑠 , 𝐶 𝑡 = 𝐴 𝑡 ,𝐴 0 ,𝐷 𝑡 = 𝐵 𝑡 ,𝐴 0 . Markovian approximation (zeroth order): Θ 𝑠 ≈Θ 0 + =Ω+ 𝐴,𝐴 [ 0 +∞ 𝐵 𝑡 ,𝐴 0 𝑑 𝑡] −1 It is computable if the data 𝐴 𝑡 ,𝐵 𝑡 0≤𝑡≤𝑇 are available.

18. Approximation of the kernel function 2/4
In the time domain:
𝑑𝐴=𝛤𝐵 𝐴 𝑑𝑡+𝜎 𝐴 𝑑 𝑊 𝑡
The noise is chosen s.t. 𝑝 𝑒𝑞 𝐴 ∝ 𝑒 −𝑊 𝐴 .
First order model:
Rational function: 𝚯(𝐬)≈ 𝑹 𝟏 𝒔 = 𝒔𝑰− 𝑩 𝟏 −𝟏 𝑨 𝟏
Hermite interpolation: Θ 𝑠 ≈Θ and lim 𝜆→ d d𝜆 Θ= lim 𝜆→ d d𝜆 R 1 with 𝜆= 𝑠 −1 .
The second condition is moment matching: Θ=𝜆 𝑀 0 + 𝜆 2 𝑀 1 +⋯
𝑨 𝟏 = 𝑴 𝟐 = 𝐴 , 𝐴 . 𝐵 1 = −𝐴 1 Θ(0). They can be obtained from data.

19. Approximation of the kernel function ¾
The first order model in the time domain 𝐴 =𝛺𝐵−𝑧, 𝑧 = 𝐵 1 𝑧− 𝐴 1 𝐵+𝜎 𝑊 The noise can be chosen such that the probability density of A is correct. Higher order models (up to 4): 𝑨 𝒕 = 𝑩 𝟏 𝑨 𝒕 +𝑩 𝟐 𝑨 𝒕 + 𝑨 𝟏 𝑩 𝒕 + 𝑨 𝟐 𝑩 𝒕 +𝜎 𝑊
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:
𝑥 𝐼 =𝑓 𝑥 𝐼 + 𝐴 𝐼,𝐼𝐼 𝑥 𝐼𝐼 + 𝜎 𝐼 𝑊 𝐼 , 𝑥 𝐼𝐼 = 𝐴 𝐼𝐼,𝐼𝐼 𝑥 𝐼𝐼 + 𝐴 𝐼𝐼,𝐼 𝑥 𝐼 + 𝜎 𝐼𝐼 𝑊 𝐼𝐼
Subspaces:
𝑋= 𝐾 ℓ 𝐴 𝐼𝐼,𝐼𝐼 , 𝐴 𝐼𝐼,𝐼 , 𝑌= 𝐾 ℓ 𝐴 𝐼𝐼,𝐼𝐼 𝑇 , 𝐴 𝐼𝐼,𝐼𝐼 −𝑇 𝐴 𝐼𝐼,𝐼 .
The Galerkin projection is equivalent to the rational function approximation matching one long time statistics and 2ℓ+1 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: 1d chain model
Nearest neighbor interaction with pair potential 𝜑 𝑟 is Fermi-Pasta-Ulam (FPU) potential .
𝑉= 𝑖 𝜑 𝑥 𝑖−1 − 𝑥 𝑖 𝜑 𝑥 𝑖+1 − 𝑥 𝑖 .
Local energy:
𝐸 𝐼 ℎ 𝑡 = 1 ℎ 𝑖∈ 𝑆 𝐼 𝑚 𝑣 𝑖 𝜑 𝑥 𝑖−1 − 𝑥 𝑖 𝜑 𝑥 𝑖+1 − 𝑥 𝑖 .
𝐸 1 ℎ
𝐸 𝐼 ℎ
𝐸 2 ℎ
𝐸 𝑑 ℎ

23. Energy conservation and heat flux
Continuous energy conservation: 𝐸 +𝛻⋅𝑞=0
Discrete energy conservation: 𝐸 ℎ + 𝛻 ℎ ⋅ 𝑞 ℎ =0
Energy flux: 𝑞 𝐼+1/2 ℎ = 𝑣 𝑛𝐼 + 𝑣 𝑛𝐼+1 𝜑′ 𝑥 𝑛𝐼 − 𝑥 𝑛𝐼+1
𝑞 1 2
𝐸 1 ℎ
𝐸 𝐼 ℎ
𝐸 2 ℎ
𝐸 𝑑 ℎ
𝑞 3 2
𝑞 𝐼− 1 2
𝑞 𝐼+1 2
This equation is exact, but the model is not closed. Can we express the energy flux in terms of the local energy?
Closure Problem: Can we express local energy flux in a closed model?

24. Orthogonal and oblique projection 𝑢≔𝐴≔𝐸−⟨𝐸⟩
𝑏=𝑢 -- Mori’s projection
𝜕 𝑡 𝑢+ 0 𝑡 𝜃 𝑡−𝑠 𝑢 𝑠 𝑑𝑠+𝐹 𝑡 =0.
𝑏=− 𝛻 ℎ 𝑢 -- non-local heat equation
𝜕 𝑡 𝑢+ 𝛻 ℎ ⋅ 0 𝑡 𝜃 𝑡−𝑠 𝛻 ℎ 𝑢(𝑠)𝑑𝑠+ 𝛻 ℎ ⋅𝐹 𝑡 =0.
𝑏=− 𝛿𝑆 𝑢 𝛿𝑢 -- negative variation of 𝑆
𝜕 𝑡 𝑢+ 0 𝑡 𝜃 𝑡−𝑠 𝑏 𝑢(𝑠) 𝑑𝑠+𝐹 𝑡 =0.
SIAM CONFERENCE ON ANALYSIS OF PARITAL DIFFERENTIAL EQUATIONS,

25. Observations of the non-Gaussian statistics
Normalized histogram of 𝑢 1 and fitting Gamma distribution
Normalized histogram of 𝐹 1 and fitting Laplace distribution
𝜌 𝑢 1 ,⋯, 𝑢 𝑑 ∝ 𝑖 (𝑢 𝑖 + 𝛼 𝑖 / 𝛽 𝑖 ) 𝛼 𝑖 −1 𝑒 − 𝛽 𝑖 ( 𝑢 𝑖 + 𝛼 𝑖 / 𝛽 𝑖 ) .

26. 𝑏=− 𝛻 ℎ 𝑢 with Gaussian additive noise
Zeroth order – stochastic heat equation
𝑢 𝑡 = 𝛻 ℎ ⋅ 𝜅𝛻 ℎ 𝑢 𝑡 + 𝛻 ℎ ⋅𝜉 𝑡
First order – stochastic damped wave equation
𝑢 𝑡 =𝜏 𝑢 𝑡 + 𝛻 ℎ ⋅ 𝜅 𝛻 ℎ 𝑢 𝑡 + 𝛻 ℎ ⋅𝜉 𝑡
Second order
𝑢 𝑡 = 𝜏 1 𝑢 𝑡 + 𝜏 2 𝑢 𝑡 + 𝛻 ℎ ⋅ 𝜅 1 𝛻 ℎ 𝑢 𝑡 + 𝜅 2 𝛻 ℎ 𝑢 𝑡 + 𝛻 ℎ ⋅𝜉 𝑡
Higher order (up to 4th order)
SIAM CONFERENCE ON ANALYSIS OF PARITAL DIFFERENTIAL EQUATIONS,

27. 𝑏=− 𝛿𝑆 𝑢 𝛿𝑢 with additive noise
Markovian model
𝑢 𝑡 = 𝛻 2 𝜅 𝛿𝑆(𝑢) 𝛿𝑢 +𝛻⋅𝜎𝜉(𝑡)
Generalized constitutive relation
𝑞=−∇𝜅 𝛿𝑆 𝑢 𝛿𝑢 +𝜎𝜉(𝑡)
Implications
Temperature-dependent conductivity
Traveling solutions
Stochastic phase-field crystal?
First order model
𝜏𝑢 𝑡𝑡 + 𝑢 𝑡 = 𝛻 2 𝜅 𝛿𝑆(𝑢) 𝛿𝑢 +𝛻⋅𝜎𝜉(𝑡)
Generalized constitutive relation
𝜏 𝑞 +𝑞=−∇𝜅 𝛿𝑆 𝑢 𝛿𝑢 +𝜎𝜉(𝑡)
Implications
Propagation of temperature (second sound speed)
relaxation
SIAM CONFERENCE ON ANALYSIS OF PARITAL DIFFERENTIAL EQUATIONS,

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