[Paper Review] The probability distribution of the average relative distance between two points in a dynamical chain

Franco Ferrari, Jarosław Paturej+2 more|ArXiv.org|Sep 12, 2008

Force Microscopy Techniques and Applications5 references3 citations

TL;DR

This paper derives the probability distribution $ Z(\mathbf{r}_{12}) $ for the average relative distance between two points on a dynamically fluctuating chain under rigid constraints, using a linearized Gaussian approximation of the generalized nonlinear sigma model (GNLSM). The key result shows that thermal fluctuations induce a narrow Gaussian peak centered on the classical relative position $ \mathbf{r}_{cl} $, with peak width controlled by a time-interval-dependent parameter $ \kappa $, indicating weak fluctuations at short times and increasing broadening with longer averaging windows.

ABSTRACT

Subject of this letter is the dynamics of a chain obtained performing the continuous limit of a system of links and beads. In particular, the probability distribution of the relative position between two points of the chain averaged over a given interval of time is computed. The physical meaning of the obtained result is investigated in the limiting case of a stiff chain.

Motivation & Objective

• To compute the probability distribution $ Z(\mathbf{r}_{12}) $ for the average relative position between two points on a chain over a finite time interval $ \Delta t $.
• To investigate the role of thermal fluctuations in shaping the relative distance distribution in a chain with rigid bond-length constraints.
• To analyze the stability of classical chain configurations under thermal fluctuations, especially in the stiff chain limit.
• To establish a connection between the GNLSM framework and the dynamics of flexible polymers under continuous limit and constraints.
• To explore how the time-averaging window $ \Delta t $ affects the broadening of the relative distance distribution due to fluctuations.

Proposed method

• Linearize the generalized nonlinear sigma model (GNLSM) via a Gaussian approximation of the functional Dirac delta function enforcing constant bond length $ |\mathbf{R}'| = \ell $.
• Apply a background field method, treating the classical chain configuration $ \mathbf{R}_{cl} $ as fixed and expanding around it to compute fluctuation contributions.
• Use path integral techniques to compute the partition function and the relative distance distribution $ Z(\mathbf{r}_{12}) $, incorporating time-averaged relative position via a delta function constraint.
• Perform Fourier transformation in momentum space to evaluate the fluctuation contributions, retaining only dominant terms that scale with $ \nu\ell $.
• Derive an approximate closed-form expression for $ Z(\mathbf{r}_{12}) $ in the stiff chain limit, where exponential decay terms from high-frequency modes are neglected.
• Identify the effective variance parameter $ \kappa $, which governs the width of the Gaussian peak in $ Z(\mathbf{r}_{12}) $, and show it increases linearly with $ \Delta t = t_2 - t_1 $.

Experimental results

Research questions

• RQ1 How does the probability distribution of the average relative distance between two points on a chain depend on the time-averaging window $ \Delta t $?
• RQ2 What is the role of thermal fluctuations in distorting the classical relative position $ \mathbf{r}_{cl} $ between two points on a chain with rigid constraints?
• RQ3 How does the stiffness of the chain—quantified by large $ \nu\ell $—affect the broadening of the relative distance distribution?
• RQ4 Why is the classical configuration stable against fluctuations, and under what conditions do fluctuations become significant?
• RQ5 What is the functional form of $ Z(\mathbf{r}_{12}) $ in the asymptotic stiff-chain limit, and how does it relate to the underlying GNLSM?

Key findings

• The probability distribution $ Z(\mathbf{r}_{12}) $ takes the form of a Gaussian centered at the classical relative position $ \mathbf{r}_{cl} $, with width controlled by the parameter $ \kappa $.
• The parameter $ \kappa $ increases linearly with the time-averaging interval $ \Delta t = t_2 - t_1 $, leading to broader distributions for longer averaging windows.
• The coefficient $ \kappa $ is inversely proportional to $ \nu\ell $, indicating that stronger rigidity (larger $ \nu\ell $) leads to narrower, more sharply peaked distributions.
• Fluctuations are negligible at short times because no zeroth-order contribution exists in $ \nu\ell $, making classical configurations highly stable against thermal noise.
• The dominant contributions to $ \kappa $ arise from terms proportional to $ A $ and $ B $, which depend on the spatial separation $ \ell_2 - \ell_1 $, showing that the distribution depends on the relative positions of the two points.
• Exponential decay terms in the fluctuation expansion are negligible when $ \Delta t \gg \frac{L}{\pi} \sqrt{\frac{c}{2\nu\ell}} $, justifying the use of the simplified Gaussian form in Eq. (29).