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 $q(t)$0. 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:

$q(t)$1

where:

• $q(t)$2 is the CG kinetic energy,
• $q(t)$3 is a two-body CG potential typically constructed via iterative Boltzmann inversion (IBI) from all-atom statistics,
• $q(t)$4 is a memory-potential, with $q(t)$5,
• $q(t)$6 is a realization of colored noise.

In condensed notation,

$q(t)$7

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):

$q(t)$8

for every $q(t)$9. Key steps in the derivation are:

1. Variation of $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0.$$0 and expansion of $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0.$$1 to first order in $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0.$$2;
2. Extraction of a local term yielding $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0.$$3;
3. Reorganization of the non-local double integral, contributing $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0.$$4;
4. Contribution of the colored noise term as $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0.$$5.

Assembling these, the resulting equation of motion for each $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0.$$6 becomes:

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0.$$7

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

4. Mori–Zwanzig Projection Formalism and Action Equivalence

Mori–Zwanzig theory employs projection operators $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0.$$8 and $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0.$$9 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:

$R(t)$0

Within the stationary-action construction, the “memory functional” $R(t)$1 and the colored noise $R(t)$2 correspond to the back-reaction of the fast variables as projected by $R(t)$3. 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 $R(t)$4 is parameterized as

$R(t)$5

with parameter set $R(t)$6. The action $R(t)$7 is discretized along mapped CG trajectories $R(t)$8 produced from atomistic molecular dynamics in the NVE ensemble. The optimization objective is to enforce

$R(t)$9

subject to:

• non-negativity, $q(t)$0;
• preservation of the Green–Kubo integral for the diffusion coefficient, $q(t)$1;
• optional smoothness/Tikhonov regularization for $q(t)$2.

Practically, a derivative-free optimizer, such as Nelder–Mead, is employed to solve for $q(t)$3. The optimized kernel $q(t)$4 and noise $q(t)$5 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 $q(t)$6 is calibrated by Iterative Boltzmann Inversion (IBI) utilizing the all-atom center-of-mass radial distribution function $q(t)$7. The parameters $q(t)$8 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) $q(t)$9, 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 $$M \ddot{R}(t) = -\nabla U(R(t)) - \int_0^t \! ds\, K(t-s)\, \dot{R}(s) + \xi(t),$$0;
• 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.