Fluctuation theorems for the undriven Duffing oscillator

DEZA, ROBERTO RAÚL; WIO, HORACIO S.; DEZA, IGNACIO

Sitges, Barcelona

Conferencia; XXVI Sitges Conference on Statistical Mechanics "New Trends in Statistical Physics"; 2019

Universidad de Barcelona

Undamped autonomous nonlinear systems with one degree of freedom are described by a Hamiltonian $$H(x,p)=\frac1{2m}[p^2+V(x)].$$When their damping is of the form $\gamma\,p$, we prove by two methods that $H(x,p)/\gamma$ is a global Lyapunov function. We exploit this result to verify the Jarzynski relation, past the pitchfork bifurcation of the undriven Duffing oscillator, to within an error of $\sum(\mathrm{lhs}-\mathrm{rhs})^2/n\sim\mathcal{O}(10^{-7})$.Along the first method, we also obtain the explicit form of a detailed fluctuation theorem,$$\gamma D\left[\ln(p_r^\mathrm{F})-\ln(p_r^\mathrm{B})\right]=\left[\frac{\dot{x}_1^2}2+V\right]_{t_f}-\left[\frac{\dot{x}_1^2}2+V\right]_{t_i},$$analog to $T\Delta S=\Delta U$. Here $D$ is the noise intensity, and $p_r^\mathrm{F}$, $p_r^\mathrm{B}$ the forward and backward probabilities of a given trajectory in configuration space. We verify numerically this theorem for the undriven Duffing oscillator, also within an error of $\sum(\mathrm{lhs}-\mathrm{rhs})^2/n\sim\mathcal{O}(10^{-7})$.