Ac-driven periodic systems in a non-chaotic regime: Can the diffusion coefficient be reconstructed from Lyapunov exponents?

We consider an ac-driven particle moving in a spatially periodic and symmetric potential. In the zero-temperature limit, for the analyzed parameter set, its dynamics is non-chaotic and the particle does not manifest diffusive properties. At non-zero temperatures, the asymptotic long-time motion follows normal (Brownian) diffusion. Recent studies have shown that within tailored parameter regimes, the diffusion coefficient is a quasiperiodic function of the external driving amplitude [Marchenko et al., Chaos 32, 113106 (2022)]. Although no general relation between Lyapunov exponents and the diffusion coefficient exists, we demonstrate that its quasiperiodic behavior at non-zero temperature can be accurately reconstructed from the maximal Lyapunov exponent of the corresponding deterministic system (at vanishing temperature). We propose an approximate formula for this purpose, which shows good agreement with numerical simulations, although some discrepancies are detected in the vicinity of the local maxima of the diffusion coefficient. Finally, we examine the robustness of the correlation between diffusion and the Lyapunov exponent under variations of the system parameters.