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We test models for the evolution of neutron star (NS) magnetic fields (B).Our model for the evolution of the NS spin is taken from an analysis of pulsartiming noise presented by Hobbs et al. (2010). We first test the standard modelof a pulsar's magnetosphere in which B does not change with time and magneticdipole radiation is assumed to dominate the pulsar's spin-down. We find thismodel fails to predict both the magnitudes and signs of the second derivativesof the spin frequencies ($\ddot{\nu}$). We then construct a phenomenologicalmodel of the evolution of $B$, which contains a long term decay (LTD) modulatedby short term oscillations (STO); a pulsar's spin is thus modified by itsB-evolution. We find that an exponential LTD is not favored by the observedstatistical properties of $\ddot{\nu}$ for young pulsars and fails to explainthe fact that $\ddot{\nu}$ is negative for roughly half of the old pulsars. Asimple power-law LTD can explain all the observed statistical properties of$\ddot{\nu}$. Finally we discuss some physical implications of our results tomodels of the B-decay of NSs and suggest reliable determination of the trueages of many young NSs is needed, in order to constrain further the physicalmechanisms of their B-decay. Our model can be further tested with the measuredevolutions of $\dot{\nu}$ and $\ddot{\nu}$ for an individual pulsar; the decayindex, oscillation amplitude and period can also be determined this way for thepulsar.

Testing Models of Magnetic Field Evolution of Neutron Stars with the Statistical Properties of Their Spin Evolutions