[Paper Review] Dynamical Properties of Gaussian Chains and Rings with Long Range Interactions

Wolfgang Böck, Jinky B. Bornales+1 more|arXiv (Cornell University)|Jul 23, 2019

Advanced Thermodynamics and Statistical Mechanics4 references4 citations

TL;DR

This paper investigates Gaussian chains and rings with long-range interactions using fractional Brownian motion (fBm), showing that monomer interactions are governed by harmonic forces whose nature—attraction or repulsion—depends on the Hurst index H. For H < 1/2, all pairwise forces are attractive; for H > 1/2, distant monomers experience repulsion, with a critical H ≈ 0.75964 where third-nearest neighbors turn attractive. The study derives coupling constants and energy spectra, and proposes modified periodic models for H > 1/2 where standard fBm fails.

ABSTRACT

Various authors have invoked discretized fractional Brownian (fBm) motion as a model for chain polymers with long range interaction of monomers along the chain. We show that for these, in contrast to the Brownian case, linear forces are acting between all pairs of constituents, attractive for small Hurst index H and mostly repulsive when H is larger than 1/2. In the second part of this paper we extend this study to periodic fBm and related models with a view to ring polymers with long range interactions.

Motivation & Objective

• To clarify the nature of effective interactions between monomers in Gaussian chains and rings with long-range correlations modeled by fractional Brownian motion (fBm).
• To derive explicit expressions for the harmonic coupling constants $ g_{kl} $ between monomers as a function of the Hurst index $ H $.
• To extend fBm-based models to periodic systems (ring polymers), addressing the failure of standard fBm for $ H > 1/2 $ due to non-positive definite covariance.
• To construct alternative stiff ring models with long-range repulsive interactions for $ H > 1/2 $, ensuring physical consistency.
• To compute the spectrum of the quadratic energy form $ ig angle x, \mathcal{H}x \big\rangle $ and relate it to the underlying force structure.

Proposed method

• Model chains and rings as discrete Gaussian processes derived from fBm via $ X_k = B^H(k) $, with increments $ Y_k = X_{k+1} - X_k $.
• Use the covariance matrix $ R_{ik} = \mathbb{E}[Y_i Y_k] $ to define the inverse precision matrix $ A = R^{-1} $, leading to a quadratic energy form $ (y, A y) $.
• Apply Lemma 1 to rewrite the energy as a sum of pairwise harmonic potentials: $ (y, A y) = \sum_{k,l} g_{kl} (x_k - x_l)^2 $, with $ g_{kl} $ derived from $ A $'s entries.
• For periodic systems, use geodesic distance on the circle to define a modified covariance $ \mathbb{E}[|B^H(t) - B^H(s)|^2] = d(t-s)^{2H} $, valid only for $ H \leq 1/2 $.
• Construct alternative periodic models for $ H > 1/2 $ by controlling the energy eigenvalues to ensure physical consistency and long-range repulsion.
• Use circulant matrix theory and discrete Fourier transforms to compute eigenvalues and eigenvectors of the interaction matrix $ \mathcal{H} $, enabling spectral analysis.

Experimental results

Research questions

• RQ1 How do the effective forces between monomers in a Gaussian chain with long-range interactions depend on the Hurst index $ H $? Specifically, are they attractive or repulsive?
• RQ2 Why does standard fractional Brownian motion fail to define a valid periodic process for $ H > 1/2 $, and what conditions ensure positive semi-definiteness of the covariance?
• RQ3 Can a physically consistent model for ring polymers with long-range interactions be constructed when $ H > 1/2 $, where standard fBm fails?
• RQ4 What is the exact value of $ H $ at which third-nearest neighbor interactions in a chain transition from repulsive to attractive?
• RQ5 How can the energy form $ \sum g_{kl} (x_k - x_l)^2 $ be expressed as a quadratic form $ \langle x, \mathcal{H} x \rangle $, and what are the spectral properties of $ \mathcal{H} $?

Key findings

• For $ H < 1/2 $, all pairwise interactions in a Gaussian chain are attractive, as confirmed by positive coupling constants $ g_{kl} > 0 $ for all $ k \neq l $, with a finite-size effect at chain ends.
• For $ H > 1/2 $, nearest neighbors remain attracted ($ g_{kl} > 0 $), but monomers further apart experience repulsion, with $ g_{kl} < 0 $ for $ |k-l| \geq 2 $.
• At $ H \approx 0.75964 $, the coupling constants for third-nearest neighbors (e.g., $ g_{31,34} $ and $ g_{31,28} $) become zero, marking a transition point where these interactions vanish.
• For $ H > 1/2 $, standard periodic fBm cannot be constructed due to non-positive definite covariance; however, the paper constructs alternative models that preserve long-range repulsive interactions.
• The matrix $ \mathcal{H} $, representing the energy quadratic form, has a zero eigenvalue corresponding to uniform translation, and its spectrum is computed via circulant matrix theory and discrete Fourier transforms.
• The analysis shows that for long-range couplings $ g_k \sim -c k^{-\gamma} $, the model remains valid if $ \gamma > 3 $, ensuring convergence and physical consistency.