Modeling the evolution and distribution of the frequency's second derivative and the braking index of pulsar spin

We model the evolution of the spin frequency's second derivative $\ddot\nu$ and the braking index $n$ of radio pulsars with simulations within the phenomenological model of their surface magnetic field evolution, which contains a long-term power-law decay modulated by short-term oscillations. For the pulsar PSR B0329+54, a model with three oscillation components can reproduce its $\ddot\nu$ variation. We show that the "averaged" $n$ is different from the instantaneous $n$, and its oscillation magnitude decreases abruptly as the time span increases, due to the "averaging" effect. The simulated timing residuals agree with the main features of the reported data. Our model predicts that the averaged $\ddot\nu$ of PSR B0329+54 will start to decrease rapidly with newer data beyond those used in Hobbs et al.. We further perform Monte Carlo simulations for the distribution of the reported data in $|\ddot\nu|$ and $|n|$ versus characteristic age $\tau_{\rm c}$ diagrams. It is found that the magnetic field oscillation model with decay index $\alpha=0$ can reproduce the distributions quite well. Compared with magnetic field decay due to the ambipolar diffusion ($\alpha=0.5$) and the Hall cascade ($\alpha=1.0$), the model with no long term decay ($\alpha=0$) is clearly preferred for old pulsars by the p-values of the two-dimensional Kolmogorov-Smirnov test.