  Secular oscillations in multi-planet systems can drive chaotic evolution of a
small inner body through non-linear resonant perturbations. This "secular
chaos" readily pushes the inner body to an extreme eccentricity, triggering
tidal interactions or collision with the central star. We present a numerical
study of secular chaos in systems with two planets and test particles using the
ring-averaging method, with emphasis on the relationship between the planets'
properties and the time-scale and efficiency of chaotic diffusion. We find that
secular chaos can excite extreme eccentricities on time-scales spanning several
orders of magnitude in a given system. We apply our results to the evolution of
planetary systems around white dwarfs (WDs), specifically the tidal disruption
and high-eccentricity migration of planetesimals and planets. We find that
secular chaos in a planetesimal belt driven by large ($\gtrsim 10 M_{\oplus}$),
distant ($\gtrsim 10 \, {\rm au}$) planets can sustain metal accretion onto a
WD over Gyr time-scales. We constrain the total mass of planetesimals initially
present within the chaotic zone by requiring that the predicted mass delivery
rate to the Roche limit be consistent with the observed metal accretion rates
of WDs with atmospheric pollution throughout the cooling sequence. Based on the
occurrence of long-period exoplanets and exo-asteroid belts, we conclude that
secular chaos can be a significant (perhaps dominant) channel for polluting
solitary WDs. Secular chaos can also produce short-period planets and
planetesimals around WDs in concert with various circularization mechanisms. We
discuss prospects for detecting exoplanets driving secular chaos around WDs
using direct imaging and microlensing.
