  The observed correlations, between the characteristic ages and dipole surface
magnetic field strengths of all pulsars, can be well explained by magnetic
field decay with core temperatures of $~2\times10^{8}$ K, $\sim2\times10^{7}$
K, and $\sim10^{5}$ K, for magnetars, normal radio pulsars, and millisecond
pulsars, respectively; assuming that their characteristic ages are about two
orders of magnitude larger than their true ages, the required core temperatures
may be reduced by about a factor of 10. The magnetic decay follows a power-law
and is dominated by the solenoidal component of the ambipolar diffusion mode.
In this model, all NSs are assumed to have the same initial magnetic field
strength, but different core temperature which do not change as the magnetic
field decays. This suggests that the key distinguishing property between
magnetars and normal pulsars is that magnetars were born much hotter than
normal pulsars, and thus have much longer magnetic field decay time scales,
resulting in higher surface magnetic field strength even with the same ages of
normal pulsars. The above conclusion agrees well with the observed correlations
between the surface temperatures of magnetars and other young NSs, which do not
agree with the cooling dominated evolution of neutron stars. This suggests a
possible scenario that heating, perhaps due to magnetic field decay, balances
neutron star cooling for observed pulsars.
