On the Asymptotic Behavior of the Kernel Function in the Generalized Langevin Equation: A One-dimensional lattice model

Abstract: We present some estimates for the memory kernel function in the generalized Langevin equation, derived using the Mori-Zwanzig formalism from a one-dimensional lattice model, in which the particles interactions are through nearest and second nearest neighbors. The kernel function can be explicitly expressed in a matrix form. The analysis focuses on the decay properties, both spatially and temporally, revealing a power-law behavior in both cases. The dependence on the level of coarse-graining is also studied.

• The paper derives explicit operator-level formulas for the memory kernel in a one-dimensional harmonic lattice using the Mori-Zwanzig formalism.
• It demonstrates that kernel magnitude decays as O(M⁻¹) for piecewise constant and O(M⁻²) for piecewise linear coarse-graining, reflecting the influence of block size and weighting.
• The analysis reveals a temporal decay scaling of O(t⁻¹/2), indicating persistent non-Markovian effects that challenge efficient Markovian approximations.

Asymptotic Analysis of the Memory Kernel in the Generalized Langevin Equation for 1D Lattice Systems

Introduction

This work investigates the asymptotic properties of the memory kernel in the generalized Langevin equation (GLE) arising from Mori-Zwanzig projections applied to a one-dimensional lattice model with harmonic, first- and second-neighbor interparticle interactions. The kernel, which encodes non-Markovian friction, is studied both analytically and numerically with a focus on its spatial and temporal decay, and the dependence on coarse-graining strategies. The rigorous operator-level treatment, using block-structured Toeplitz forms and their Fourier analysis, yields explicit and asymptotic results relevant to coarse-grained molecular models.

Model Formulation and Projection Framework

The system considered is an infinite one-dimensional harmonic lattice, with atoms coupled via both nearest and next-nearest neighbor linear springs. The equations of motion are linearized in the usual harmonic approximation, leading to a discrete constant-coefficient system for atomic displacements. Coarse-graining is introduced by partitioning the lattice into blocks of $M$ atoms and forming local averages via suitable weighting functions (piecewise constant or piecewise linear).

The Mori-Zwanzig formalism, projecting full lattice dynamics onto a subspace spanned by these local averages, yields a GLE for the coarse-grained variables. The resulting memory kernel admits an explicit form in terms of the block structure and the harmonic lattice operator. The paper derives operator-level formulas for the kernel and its entries, enabling precise asymptotic analysis.

Dependence on Coarse-Graining and Weighting

A central result is the quantitative characterization of how the block size $M$ and weighting function $\Phi$ controlling coarse-graining influence the kernel’s magnitude:

• Piecewise constant weighting: For block size $M$, the diagonal entry of the kernel at $t=0$ satisfies $0 \leq \Theta_{0,0}(0) \leq C_1 M^{-1}$, where $C_1$ depends on force constants. This result demonstrates the variance of the projected random force (as dictated by the second fluctuation-dissipation theorem) decays as the coarse-graining scale increases, with a rate $\mathcal{O}(M^{-1})$.
• Piecewise linear weighting: Introducing overlaps between neighboring blocks (emulating hat functions in FEM) leads to a more rapid decay, with $\Theta_{0,0}(0) \leq C_2 M^{-2}$. Here, off-diagonal operators and nontrivial local overlap structure further suppress the kernel, reducing memory and noise intensity at coarse scales much more aggressively.

Analytical estimates for both cases are supported by large-scale numerical simulations, validating the scaling regimes and confirming the utility of the operator framework.

Spatial Decay of the Kernel

The off-diagonal kernel entries $\Theta_{0,J}(0)$ quantify correlations between well-separated coarse-grained blocks. Using Fourier representation, repeated integration by parts, and smoothness of the kernel's spectral density, it is shown that $\Theta_{0,J}(0)$ decays faster than any polynomial in $|J|$. Numerical experiments reveal that this decay is in fact (super)exponential for realistic parameter values and block sizes.

This supports common physical assumptions underlying spatially local Markovian closures and the introduction of cutoff radii in coarse-grained models. The strong spatial localization of the kernel justifies neglecting long-range block interactions in practical implementations.

Temporal Decay and Long Memory

The most rigorous and consequential result concerns the time dependence of the diagonal kernel for fixed coarse-graining: $\Theta_{0,0}(t) = \mathcal{O}(t^{-1/2})$ as $t\to\infty$. Applying stationary phase approximations to the explicit Fourier integral of the kernel, the analysis exploits smooth, nondegenerate phase structure of the lattice dynamical spectrum. The absence of eigenvalue crossing or degeneracy in the relevant matrices is proven constructively for $M > 4$, ensuring the validity of the asymptotic approach.

The $t^{-1/2}$ scaling is robust and matches previous analytical solutions for related harmonic systems and experimental observations in stochastic protein dynamics. It reveals fundamentally slow memory loss, reflecting the intrinsic persistence of harmonic vibrations even after coarse-graining—a significant constraint for Markovianization schemes.

Implications and Future Directions

The results demonstrate quantitatively how the scale and form of coarse-graining affect both the magnitude and persistence of non-Markovian memory effects in projected dynamics. The spatially rapid but temporally slow decay of the kernel suggests that, although coarse-graining decreases the noise and friction strength, memory effects remain substantial over long timescales and cannot be ignored except at extremely coarse resolutions or with further approximations.

From a simulation perspective, these findings challenge the construction of efficient and accurate numerical solvers for GLEs in mesoscale modeling where storing and computing with long temporal histories is computationally prohibitive. Approximating power-law kernels by sums of exponentials offers a promising direction, though error control for such approximations in physical systems remains a significant open question.

The approach and results generalize conceptually to higher-dimensional periodic systems, where block-Toeplitz and Fourier techniques can again be brought to bear. The specific temporal decay rates and cross-over scales, however, are anticipated to be dimension-dependent and warrant separate in-depth analysis.

Conclusion

This work provides a thorough operator-theoretic analysis of the memory kernel in Mori-Zwanzig GLEs derived from one-dimensional harmonic lattice models. The rigorous asymptotic estimates elucidate how both coarse-graining scale and functional form affect the strength and persistence of emergent non-Markovian noise and friction. The methods and conclusions have direct implications for coarse-grained molecular simulations, stochastic modeling, and theoretical understanding of memory in reduced descriptions of complex dynamical systems.