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Ribbons are a class of slender structures whose length, width, and thicknessare widely separated from each other. This scale separation gives a ribbonunusual mechanical properties in athermal macroscopic settings, e.g. it canbend without twisting, but cannot twist without bending. Given the ubiquity ofribbon-like biopolymers in biology and chemistry, here we study the statisticalmechanics of microscopic inextensible, fluctuating ribbons loaded by forces andtorques. We show that these ribbons exhibit a range of topologically andgeometrically complex morphologies exemplified by three phases - atwist-dominated helical phase (HT), a writhe-dominated helical phase (HW), andan entangled phase - that arise as the applied torque and force is varied.Furthermore, the transition from HW to HT phases is characterized by thespontaneous breaking of parity symmetry and the disappearance of perversionsthat characterize chirality reversals. This leads to a universal response curveof a topological quantity, the link, as a function of the applied torque thatis similar to magnetization curves in second-order phase transitions.

Statistics and Topology of Fluctuating Ribbons