Fractional Brownian Polymers

• Fractional Brownian polymers are models that use fractional Brownian motion with a tunable Hurst parameter to introduce anomalous spatial and temporal correlations.
• They generalize classical Rouse and Edwards frameworks by incorporating self-avoidance and long-memory effects, allowing for a unified description of subdiffusive, persistent, and intermediate dynamics.
• Analytical and numerical studies reveal key scaling laws and dynamic exponents that help predict mean-square displacements, mode relaxations, and non-equilibrium kinetics like zipping and translocation.

Fractional Brownian polymers comprise polymer models in which chain conformations or critical coordinates are described by fractional Brownian motion (fBm) with Hurst parameter $H \in (0,1)$. These models generalize classical Brownian (Rouse or Edwards) models to incorporate both anomalous spatial and temporal correlations, enabling a unified description of subdiffusive, persistent, or intermediate chain statistics as well as non-Markovian dynamic processes. fBm-based frameworks have been developed for both equilibrium and non-equilibrium polymer statistics (e.g., self-repelling fBm, fractional Edwards models), for random environments, and for critical polymer kinetics such as zipping or translocation. This entry surveys the definitions, mathematical formulations, scaling properties, dynamical aspects, and key results on fractional Brownian polymers.

1. Fractional Brownian Motion and Polymer Models

Fractional Brownian motion $B_H(t)$ is a zero-mean Gaussian process with covariance

$$\mathbb{E}[B_H^i(s) B_H^j(t)] = \delta^{ij} \frac{\sigma^2}{2} \left( s^{2H} + t^{2H} - |t-s|^{2H} \right),$$

where $H \in (0,1)$ is the Hurst parameter, and $\sigma^2$ sets the unit of step size (Eleutério et al., 2024, Grothaus et al., 2010, Bock et al., 2019). For $H=1/2$, $B_H$ is standard Brownian motion; $H<1/2$ yields anti-persistent trajectories, while $H>1/2$ corresponds to persistent, long-range memory.

A fractional Brownian polymer substitutes the Gaussian Rouse chain’s monomer positions (open or closed chain) by samples of fractional Brownian paths. For open chains, bead coordinates are set as $X_k = B^H(k)$, $B_H(t)$0; for loops, periodic fBm is constructed via geodesic distances on the circle, but positive definiteness of the covariance restricts this to $B_H(t)$1 (Bock et al., 2019, Bock et al., 2019). Equilibrium measures and observables follow directly from the joint fBm distribution, potentially modified by self-avoidance or additional interactions.

For dynamics, fBm can also emerge as an effective slow variable (“collective coordinate”) describing anomalous Markovian or non-Markovian subchains within polymer systems, as in the case of critical zipping coordinates (Walter et al., 2011).

2. Fractional Edwards Model and Self-Avoidance

The Edwards model for self-avoiding polymers can be generalized to fBm via a Gibbs weight penalizing self-intersections (Bornales et al., 2011, Grothaus et al., 2010, Bock et al., 2019, Eleutério et al., 2024). The (formal) polymer measure is

$B_H(t)$2

with $B_H(t)$3 a self-repulsion parameter and $B_H(t)$4 the fBm covariance. The self-intersection local time $B_H(t)$5 requires regularization since $B_H(t)$6 is singular; mollification by a heat kernel ($B_H(t)$7) and subsequent $B_H(t)$8-almost sure convergence defines the model for $B_H(t)$9, with Varadhan renormalization required at $$\mathbb{E}[B_H^i(s) B_H^j(t)] = \delta^{ij} \frac{\sigma^2}{2} \left( s^{2H} + t^{2H} - |t-s|^{2H} \right),$$0 (Grothaus et al., 2010, Bock et al., 2019).

This construction has been established both for line polymers and for more complex architectures (loops, starbursts), with explicit integrability domains for the Gibbs measure in terms of $$\mathbb{E}[B_H^i(s) B_H^j(t)] = \delta^{ij} \frac{\sigma^2}{2} \left( s^{2H} + t^{2H} - |t-s|^{2H} \right),$$1 and $$\mathbb{E}[B_H^i(s) B_H^j(t)] = \delta^{ij} \frac{\sigma^2}{2} \left( s^{2H} + t^{2H} - |t-s|^{2H} \right),$$2 (Bock et al., 2019). For $$\mathbb{E}[B_H^i(s) B_H^j(t)] = \delta^{ij} \frac{\sigma^2}{2} \left( s^{2H} + t^{2H} - |t-s|^{2H} \right),$$3, the Gibbs measure is well-defined for any coupling $$\mathbb{E}[B_H^i(s) B_H^j(t)] = \delta^{ij} \frac{\sigma^2}{2} \left( s^{2H} + t^{2H} - |t-s|^{2H} \right),$$4; at $$\mathbb{E}[B_H^i(s) B_H^j(t)] = \delta^{ij} \frac{\sigma^2}{2} \left( s^{2H} + t^{2H} - |t-s|^{2H} \right),$$5, only small $$\mathbb{E}[B_H^i(s) B_H^j(t)] = \delta^{ij} \frac{\sigma^2}{2} \left( s^{2H} + t^{2H} - |t-s|^{2H} \right),$$6 are permitted.

The Flory-type argument for the mean-square end-to-end distance yields

$$\mathbb{E}[B_H^i(s) B_H^j(t)] = \delta^{ij} \frac{\sigma^2}{2} \left( s^{2H} + t^{2H} - |t-s|^{2H} \right),$$7

for $$\mathbb{E}[B_H^i(s) B_H^j(t)] = \delta^{ij} \frac{\sigma^2}{2} \left( s^{2H} + t^{2H} - |t-s|^{2H} \right),$$8, with a crossover to Gaussian scaling $$\mathbb{E}[B_H^i(s) B_H^j(t)] = \delta^{ij} \frac{\sigma^2}{2} \left( s^{2H} + t^{2H} - |t-s|^{2H} \right),$$9 for $H \in (0,1)$0 (Bornales et al., 2011). This precisely recovers the classical Flory result for $H \in (0,1)$1 and provides a tunable route to compact or swollen statistics via $H \in (0,1)$2.

3. Static and Dynamic Properties of fBm Polymer Chains

Discretized fBm chains possess a Gaussian structure with explicit pairwise monomer–monomer interactions. For open chains, the quadratic form in bond variables induces couplings between all pairs of beads: $H \in (0,1)$3 with $H \in (0,1)$4 inherited from the inverse covariance of the bond increments (Bock et al., 2019). The sign of $H \in (0,1)$5 changes with $H \in (0,1)$6. For $H \in (0,1)$7, all $H \in (0,1)$8 (purely attractive), while for $H \in (0,1)$9, long-range repulsion emerges ($\sigma^2$0 for distant pairs). The exact scaling

$\sigma^2$1

is preserved for the free (non-interacting) fBm chain (Bock et al., 2019, Eleutério et al., 2024).

For fBm-based ring polymers, geodesic distances on the circle define the covariance. Positive definiteness constraints require $\sigma^2$2 for periodic fBm, but more general cyclic bead–spring models can interpolate to $\sigma^2$3 with prescribed long-range stiffness, conditional on positive-definiteness of the spring constant matrix (Bock et al., 2019, Bock et al., 2019). Mode relaxation spectra interpolate between Rouse-like ($\sigma^2$4) and stiffer ($\sigma^2$5) decay as $\sigma^2$6 varies.

4. Kinetic Fractional Brownian Coordinates—Anomalous Zipping, Translocation

Fractional Brownian motion with $\sigma^2$7 governs the slow coordinate dynamics of various non-equilibrium polymer processes. At the critical temperature $\sigma^2$8 for zipping two complementary Rouse-like strands, the zipping coordinate $\sigma^2$9 (number of closed base pairs) behaves as a fBm with $H=1/2$0 (Walter et al., 2011). The mean first-passage time to zip or unzip scales as

$H=1/2$1

outpacing the standard Rouse time $H=1/2$2 and indicating subdiffusive, anti-persistent dynamics.

Key stochastic properties for the zipping coordinate include:

• Mean-square displacement: $H=1/2$3,
• Velocity autocorrelation: $H=1/2$4 (anti-persistent),
• Survival statistics: $H=1/2$5; first-passage distributions lack power-law tails seen in fractional Fokker–Planck models,
• Boundary behavior: near absorbing walls, $H=1/2$6 with $H=1/2$7.

Comparisons with unbiased translocation show both similarities (subdiffusive entropic slow coordinate, logarithmic free energy) and differences (distinct dynamical exponents and universality classes).

Mode-sum arguments and simulations reinforce that the slow collective coordinate in many critical polymer kinetics emerges as an effective fBm with nontrivial $H=1/2$8 (Walter et al., 2011).

5. Fractional Brownian Polymers in Random Environments

Polymer models in random (quenched) fields have been analyzed in the context of fractional Brownian noise. In the Anderson polymer in a fractional Brownian environment (Kalbasi et al., 2016), a partition function

$H=1/2$9

(where $B_H$0 is a random walk, $B_H$1 independent fBms indexed by $B_H$2) exhibits:

• For $B_H$3, $B_H$4 almost surely and in mean,
• For $B_H$5, exponential growth with sub-exponential corrections: $B_H$6.

The analysis is based on subadditive ergodic theorems and control of fBm regularity; in compact spatial domains, exponential growth remains robust for all $B_H$7 (Kalbasi et al., 2016). A plausible implication is that anomalous memory (persistent fBm increments) softens the exponential growth of typical partition functions in disordered environments compared to standard Brownian polymer models.

Directed polymers in spatially correlated environments with algebraic decay can converge in the intermediate-disorder limit to stochastic heat equations driven by fractional noise, where the limiting Hurst parameter is set by the correlation decay rate (Rang, 2017). This links spatial correlation properties of the environment directly to the disorder-induced fluctuation scaling in the polymer.

6. Analytical, Numerical, and Physical Implications

Fractional Brownian polymer models unify both mathematical and physical aspects of chain statistics under a single exponent $B_H$8:

• Statistical geometry: fBm parameter $B_H$9 directly determines chain roughness (fractal dimension $H<1/2$0) and memory,
• Excluded volume and swelling: For self-avoiding fBm (fractional Edwards model), scaling of the end-to-end distance interpolates between compact and swollen phases via the Flory-type exponent $H<1/2$1,
• Long-range forces: In equilibrium bead-spring chains based on fBm, the sign of induced spring constants and the range of repulsion/attraction depend sharply on $H<1/2$2, generating compact, Rouse, or swollen/extended polymer conformations (Bock et al., 2019, Eleutério et al., 2024),
• Ring architectures and limitations: Positive definiteness requirements restrict naive periodic fBm to $H<1/2$3, but more general stiffness matrices can extend this (Bock et al., 2019, Bock et al., 2019),
• Physical model fitting: Numerical evaluation of end-to-end scaling and comparison with step and Δ-repulsion models allow mapping of $H<1/2$4 to solvent quality and fine-tuning polymer-monomer or monomer-monomer affinities (Eleutério et al., 2024),
• Dynamical universality classes: Non-equilibrium fBm coordinates capture subdiffusive or persistent kinetic regimes in critical polymer dynamics (zipping, translocation), forming distinct universality classes from equilibrium Rouse or classical subdiffusion (Walter et al., 2011).

Simulation methodologies leverage fast evaluation of fBm increments, large-scale Rouse Monte Carlo, and determinant-based Gaussian integration for observables (Eleutério et al., 2024, Walter et al., 2011).

7. Open Problems and Extensions

Key open directions include:

• Systematic extension to ring and network polymers with $H<1/2$5 under strict positive definiteness (Bock et al., 2019, Bock et al., 2019),
• Rigorous determination of critical exponents and scaling relations for complex, branched, or composite architectures,
• Hierarchical fractional models (dynamical RG approaches) for kinetically roughened or collapsed phases,
• Stochastic quantization and dynamical versions (fractional Langevin dynamics, time-reversed fBm polymers),
• Mutual intersection local times and multiple-chain systems with cross-interactions,
• Experimental calibration of $H<1/2$6 and validation of tunable scaling via solvent quality, external fields, or viscoelastic embedding (Eleutério et al., 2024).

The general fBm-based polymer framework provides a controllable and analytically tractable generalization of traditional polymer physics, enabling precise interpolation between Markovian and non-Markovian, compact and extended, purely Brownian and strongly correlated chain regimes.