Single file dynamics of tethered random walkers

Area of Science:

• Statistical Mechanics
• Soft Matter Physics
• Condensed Matter Physics

Background:

• Random walkers are fundamental models in statistical mechanics.
• Understanding multi-particle interactions is crucial for complex systems.
• Previous studies on random walkers often assume no interaction or infinite separation.

Purpose of the Study:

• To investigate the dynamics of N identical random walkers with a maximum separation threshold (Δ).
• To characterize the diffusional relaxation and equilibrium properties of these "tethered walkers."
• To compare the behavior of tethered walkers with the unconstrained case (Δ = ∞).

Main Methods:

• Development of an approximate analytical approach for the N-particle probability distribution.
• Derivation of one-particle distribution functions for central and edge particles.
• Exact solution for N=2 to validate approximations.
• Monte Carlo simulations to confirm analytical results.

Main Results:

• Characterization of diffusional relaxation with a characteristic time scaling as (NΔ)²/D.
• One-particle positional moments derived as power expansions in Δ/(4Dt).
• Edge particles exhibit an effective long-time diffusivity of D/N for finite Δ.
• Equilibrium system length distribution and entropic spring constant (6kBT/(NΔ²)) determined.

Conclusions:

• Tethered walkers exhibit distinct collective dynamics compared to unconstrained walkers.
• The system's behavior, particularly its entropic elasticity, resembles that of an ideal polymer.
• The finite separation constraint significantly alters the system's diffusive and equilibrium properties.