Time-Non-Local Stationary-Action Framework

• The framework is a variational formalism for coarse-grained models that captures memory effects through time-non-local action functionals.
• It systematically derives Generalized Langevin Equations with data-driven memory kernels, ensuring both equilibrium structure and dynamic fidelity.
• Validated on coarse-grained water, the approach outperforms traditional models by accurately reproducing time-dependent correlations.

The time-non-local stationary-action framework is a variational formalism for coarse-grained (CG) models that accounts for memory effects arising from integrating out microscopic degrees of freedom. By expressing the effective dynamics of CG variables through a stationarity principle applied to a non-local action functional, this approach facilitates the systematic derivation and parametrization of Generalized Langevin Equations (GLEs) with data-driven memory kernels. The framework addresses fundamental dynamical shortcomings of traditional CG methodologies that rely on local-in-time action principles, providing a route to models that maintain both equilibrium structure and realistic time-dependent correlations.

1. Motivation for Time-Non-Local Action Functionals

Standard classical mechanics postulates a time-local action,

$S[q] = \int_0^T dt\, L(q(t), \dot{q}(t)),$

whose stationarity under variations of $q(t)$ yields ordinary Euler–Lagrange equations:

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0.$$

However, when constructing CG models of soft matter by partitioning the system into “slow” CG variables $R(t)$ and “fast” microscopic variables $q(t)$, the projection formalism of Mori–Zwanzig leads to effective equations of motion with memory:

$$M \ddot{R}(t) = -\nabla U(R(t)) - \int_0^t \! ds\, K(t-s)\, \dot{R}(s) + \xi(t),$$

where $K(\tau)$ is a friction (memory) kernel, and $\xi(t)$ is temporally correlated noise consistent with fluctuation–dissipation constraints, $$\langle \xi(t+\tau)\, \xi(t) \rangle = k_B T\, K(\tau)$$. Since no time-local action functional $S[R]$ can generate the memory (non-Markovian) term via its Euler–Lagrange equations, a variational formulation must explicitly incorporate the full past history of $R$. This necessitates the development of time-non-local (“history-dependent”) action functionals to recover GLEs from stationary-action principles.

2. Construction of the Time-Non-Local Action Functional

The effective time-non-local action functional employed in the stationary-action framework is formulated as:

$$\Sigma[ R, \dot{R}; \zeta ] = \int_0^T dt\, [ K_R(\dot{R}(t)) - U(R(t)) ] + \sum_{I=1}^{N_B} \int_0^T dt \;\dot{R}_I(t) \cdot \int_0^t ds\, Q(t-s; \zeta)\, \dot{R}_I(s) + \sum_{I=1}^{N_B} \int_0^T dt\, \xi_I(t; \zeta) \cdot R_I(t).$$

where:

• $K_R(\dot{R}) = \sum_I \frac{M}{2}|\dot{R}_I|^2$ is the CG kinetic energy,
• $U(R) = \sum_{I<J} U_2(|R_I - R_J|)$ is a two-body CG potential typically constructed via iterative Boltzmann inversion (IBI) from all-atom statistics,
• $Q(\tau; \zeta)$ is a memory-potential, with $dQ/d\tau = K(\tau; \zeta)$,
• $\xi_I(t;\zeta)$ is a realization of colored noise.

In condensed notation,

$$\Sigma[R] = \int_0^T dt\, (K_R(\dot{R}) - U(R)) + \frac{1}{2} \int_0^T dt \int_0^T ds\, \dot{R}(t) \cdot Q(t-s; \zeta) \cdot \dot{R}(s) + \int_0^T dt\, \xi(t; \zeta) \cdot R(t),$$

highlighting the locality of the kinetic and potential terms and the non-locality of the dissipative and noise contributions.

3. Derivation of the Generalized Langevin Equation

Stationarity of the time-non-local action is formulated via a generalized Euler–Lagrange equation (cf. Ferialdi–Bassi, 2012):

$$\frac{\delta \Sigma}{\delta R_I(\sigma)} - \frac{d}{d\sigma}\left[ \frac{\delta \Sigma}{\delta \dot{R}_I(\sigma)} \right] = 0,$$

for every $\sigma \in [0, T]$. Key steps in the derivation are:

1. Variation of $$R_I(\sigma) \to R_I(\sigma) + \epsilon\, \eta(\sigma)$$ and expansion of $\Sigma$ to first order in $\epsilon$;
2. Extraction of a local term yielding $M\ddot{R}_I + \partial_I U$;
3. Reorganization of the non-local double integral, contributing $\int_0^\sigma ds\, K(\sigma - s)\, \dot{R}_I(s)$;
4. Contribution of the colored noise term as $+\xi_I(\sigma)$.

Assembling these, the resulting equation of motion for each $I$ becomes:

$$M \ddot{R}_I(\sigma) = -\partial_{R_I} U(R(\sigma)) - \int_0^\sigma ds\, K(\sigma-s)\, \dot{R}_I(s) + \xi_I(\sigma),$$

which is the GLE associated with the Mori–Zwanzig projection.

4. Mori–Zwanzig Projection Formalism and Action Equivalence

Mori–Zwanzig theory employs projection operators $P$ and $Q=1-P$ to distinguish CG variables in phase space. The procedure recasts the microscopic dynamics (Liouville equation) into equations for the projected variables, yielding a dynamical equation with explicit memory and a Q-projected orthogonal force:

$$\dot{R}_I = P_I/M,\quad \dot{P}_I = -\partial_I U(R) - \int_0^t ds\, K(t-s)\, P_I(s)/M + \xi_I(t).$$

Within the stationary-action construction, the “memory functional” $\int_0^t ds\, Q(t-s)\dot{R}(s)$ and the colored noise $\xi \cdot R$ correspond to the back-reaction of the fast variables as projected by $Q$. Thus, enforcing stationarity of the time-non-local action is equivalent to enforcing the projected Mori–Zwanzig GLE, establishing a formal equivalence between the variational and projection-operator approaches.

5. Data-Driven Optimization of the Memory Kernel

The memory kernel $K(t; \zeta)$ is parameterized as

$$K(t; \zeta) = g(t; a)\, b(t; \{b_0, ..., b_{M-1}\}),\quad g(t; a) = e^{-a t^2},\quad b(t_k) = b_k,\, t_k = k\Delta t,$$

with parameter set $\zeta = \{a, b_0, ..., b_{M-1}\}$. The action $\Sigma$ is discretized along mapped CG trajectories $\{R_I(n\Delta t)\}_{n=0}^N$ produced from atomistic molecular dynamics in the NVE ensemble. The optimization objective is to enforce

$$\left. \frac{\partial \Sigma}{\partial \zeta_i} \right|_{\zeta = \zeta^*} = 0$$

subject to:

• non-negativity, $K(t_k;\zeta) \geq 0$;
• preservation of the Green–Kubo integral for the diffusion coefficient, $\int_0^\infty dt\, K(t;\zeta) = \text{ const}$;
• optional smoothness/Tikhonov regularization for $\{b_k\}$.

Practically, a derivative-free optimizer, such as Nelder–Mead, is employed to solve for $\zeta^*$. The optimized kernel $K^*(t)$ and noise $\xi^*(t)$ are then used in the CG GLE integrator, ensuring the dynamical consistency of the CG model.

6. Application to Coarse-Grained Water

The framework is applied to a system of 1,001 SPC/E water molecules at 298 K in a periodic cubic box. CG mapping is performed by assigning each molecule to a single bead at its center of mass. The two-body CG potential $U_2(R)$ is calibrated by Iterative Boltzmann Inversion (IBI) utilizing the all-atom center-of-mass radial distribution function $g(R)$. The parameters $(K(t;\zeta),\, \xi(t;\zeta))$ are determined by minimizing the non-local action over the atomistic trajectory mapped to CG variables.

Key outcomes include:

• The CG GLE accurately reproduces the atomistic velocity autocorrelation function (VACF) $C(t)$, including its negative lobe, whereas a traditional Langevin CG model with constant friction does not;
• Both GLE and standard Langevin CG schemes preserve the diffusion constant and static pair distribution $g(R)$;
• Only the GLE captures correct time-dependent dynamical correlations, establishing its superiority for dynamical fidelity.

Elevating the projected Mori–Zwanzig GLE to a time-non-local stationary-action framework and optimizing the parameters on mapped atomistic trajectories yields a CG model that is both structurally and dynamically consistent with its microscopic reference.