The worm-like chain models an inextensible filament with bending energy and thermal tangent fluctuations.
The Kuhn length is the link length of an equivalent long ideal chain with the same contour length and mean squared end-to-end distance. A three-dimensional worm-like chain satisfies for , hence .
A strongly stretched inextensible worm-like chain has two small transverse tangent components. Its quadratic energy is . For a long bulk chain, the equipartition theorem assigns each Fourier mode covariance . Integrating over and summing both components gives , hence the extension deficit is half that value. With persistence length , the deficit is . It requires small slopes and negligible stretching of the backbone.
The bulk tangent covariance is twice the inverse Fourier transform of , one term for each transverse direction. Equivalently the inverse of is . This gives the displayed correlation and identifies a force-dependent correlation length, different from the zero-force persistence length. Finite chains require endpoint conditions or a periodic convention.
Periodizing the bulk transverse tangent correlation of a stretched worm-like chain gives , where is the distance to zero on the periodic interval. Summing the geometric series yields the displayed result. At zero separation its finite-length factor is ; for the bulk result is recovered. Periodic tangent fluctuations are a mathematical boundary convention, not a claim that a pulled physical chain forms a closed loop.
The persistence length is the decay length of tangent correlations. A common three-dimensional convention is ; a strictly planar chain has tangent-correlation length .
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A worm-like chain (WLC) is a theoretical model used in polymer physics and biophysics to describe the conformational properties of long, flexible polymers. The model helps in understanding the behavior of macromolecules such as DNA, proteins, and synthetic polymers. Key characteristics of the worm-like chain model include: 1. **Continuous Chain**: The worm-like chain is often represented as a continuous chain of tangent segments, where each segment maintains a certain angle with respect to its neighbors.