Entropic tightening of knots in granular chains under vibration

Knots are common in long chains, from DNA and proteins to everyday strings, yet direct observation of knot dynamics is difficult. Using granular chains on vibrating platforms, we directly visualize knot dynamics and find that knot cores spontaneously tighten, a behavior observed across different knot types. We quantitatively explain spontaneous knot tightening using knot-tube theory: segments confined within the knot core experience entropic loss and tend to escape the core to gain conformational entropy. Although driven granular chains are not in thermodynamic equilibrium, their steady-state knot-size statistics are approximately captured by equilibrium statistical mechanics and Monte Carlo simulations. Analyses of state-transition probabilities and time-reversal asymmetry suggest a weak violation of detailed balance when the granular dynamics are projected onto the knot-size coordinate. Finally, we discuss the possible relevance of knot tightening to topoisomerase-mediated DNA unknotting.