Stochastic dynamics of the resistively shunted superconducting tunnel junction system under the impact of thermal fluctuations

The paper correctly formulates the RSJ SDE ẍ + βJ ẋ + sin x = κ + √(2D) η(t) and its first-order form ẋ = v, v̇ = κ − βJ v − sin x (with ξ := βJ v) (see Eqs. (4.14)–(4.17) ) and states a phase-space Fokker–Planck equation consistent with the conservative Kramers form when expanded (their Eq. (5.19)) . It also reports a clockwise I–V hysteresis with a vertical segment at κ = 1 for 0 < ⟨ξ⟩ < 1, an ohmic branch κ = ⟨ξ⟩ for κ > 1, and a return current κc ≈ 0.6965 on the decreasing branch, consistent with standard RSJ phenomenology (Fig. 5 and associated text) . However, several key steps are asserted but not proved: (i) the Fokker–Planck derivation is sketched via a Liouville balance with an explicit noise term and then deferred to external references, rather than derived by an Itô/Kramers recipe; the displayed result is correct but the derivation as written is not sound , (ii) the existence and asymptotic stability of the running limit cycle are shown only numerically (including a plot in (sin x, v), i.e., a cylindrical projection), without a Poincaré-map or contraction argument; the text even describes it as a planar limit cycle without clarifying the S1 × R geometry , and (iii) the paper defines ⟨v⟩ as the average of the long-time max and min rather than a time average, which is nonstandard and obscures the exact averaged identity κ = ⟨ξ⟩ + ⟨sin x⟩ (not stated) . It also mixes dimensionless variables with a physical gap voltage 2Δ/e without supplying a normalization map, which is dimensionally inconsistent in the presented coordinates . By contrast, the model’s solution gives the standard Kramers/FP form, the correct deterministic structure (centers/saddles at D = 0; sink–saddle for |κ| < 1, βJ > 0), the averaged identity κ = ⟨ξ⟩ + ⟨sin x⟩, a clear explanation of the hysteretic branches, and a rigorous cylinder-limit-cycle existence/stability argument via a Poincaré section and phase-space contraction. It also notes the classical large-Q retrapping estimate κc ≈ (4/π) βJ, which the paper does not provide. Overall: the paper’s conclusions are qualitatively consistent with the model, but the paper lacks several key arguments and contains presentation inconsistencies; the model solution is correct and more complete.