Fig. 2: The topological fluctuation theorem for a single vortex.

a The exact winding number distribution for a free vortex exhibits a power law scaling for large positive n values. b At all times t and initial radial position r₀, the ratio of negative to positive winding number distributions maps onto the theoretical prediction (3) (inset) resulting in an exponential scaling of p(n, t∣r₀) for large negative windings (caption in panel a). c In the presence of boundaries restricting the particle motion with diffusivity D inside a disk of radius R and outside the vortex center, the winding angle distribution shows exponential tails at small times (red curve, Dt/R² = 0.1), while it becomes Gaussian for larger t (blue curve, Dt/R² = 10). The theorem holds irrespectively of the shape of the distribution (inset, the orange curve corresponds to Dt/R² = 1). d, e Representative trajectories corresponding to cases II and III discussed in the text. f The probability ratio (14) averaged over initial positions r₀ and trajectory times t as function of the total winding angle Δϕ for the three cases (case I corresponding to isotropic confinement) discussed in the text and Δr = Δz = 0. In a, b γ = 2π, while in c–e γ = 0.1 × 2π. In d–f we used k_BT/k = 10, while for case II (resp. III) α = 0.5(1) and x_c = 0(4).