  Particle-in-cell simulations have unveiled that shock-accelerated electrons
do not follow a pure power-law distribution, but have an additional low-energy
"thermal" part, which owns a considerable portion of the total energy of
electrons. Investigating the effects of these thermal electrons on gamma-ray
burst (GRB) afterglows may provide valuable insights into the particle
acceleration mechanisms. We solve the continuity equation of electrons in the
energy space, from which multi-wavelength afterglows are derived by
incorporating processes including synchrotron radiation, synchrotron
self-absorption, synchrotron self-Compton scattering, and gamma-gamma
annihilation. First, there is an underlying positive correlation between
temporal and spectral indices due to the cooling of electrons. Moreover,
thermal electrons would result in the simultaneous non-monotonic variation in
both spectral and temporal indices at multi-wavelength, which could be
individually recorded by the 2.5-meter Wide Field Survey Telescope and Vera
Rubin Observatory Legacy Survey of Space and Time (LSST). The thermal electrons
could also be diagnosed from afterglow spectra by synergy observation in the
optical (with LSST) and X-ray bands (with the Microchannel X-ray Telescope on
board the Space Variable Objects Monitor). Finally, we use Monte Carlo
simulations to obtain the distribution of peak flux ratio ($R_{\rm X}$) between
soft and hard X-rays, and of the time delay ($\Delta t$) between peak times of
soft X-ray and optical light curves. The thermal electrons significantly raise
the upper limits of both $R_{\rm X}$ and $\Delta t$. Thus the distribution of
GRB afterglows with thermal electrons is more dispersive in the $R_{\rm X} -
\Delta t$ plane.
