  This study investigates the impact of molecular thermal fluctuations on
compressible decaying isotropic turbulence using the unified stochastic
particle (USP) method, encompassing both two-dimensional (2D) and
three-dimensional (3D) scenarios. The findings reveal that the turbulent
spectra of velocity and thermodynamic variables follow the wavenumber scaling
law of ${k}^{(d-1)}$ for different spatial dimensions $d$ within the high
wavenumber range, indicating the impact of thermal fluctuations on small-scale
turbulent statistics. With the application of Helmholtz decomposition, it is
found that the thermal fluctuation spectra of solenoidal and compressible
velocity components (${\vec{u}}_{s}$ and ${\vec{u}}_{c}$) follow an energy
ratio of 1:1 for 2D cases, while the ratio changes to 2:1 for 3D cases.
Comparisons between 3D turbulent spectra obtained through USP simulations and
direct numerical simulations of the Navier-Stokes equations demonstrate that
thermal fluctuations dominate the spectra at length scales comparable to the
Kolmogorov length scale. Additionally, the effect of thermal fluctuations on
the spectrum of ${\vec{u}}_{c}$ is significantly influenced by variations in
the turbulent Mach number. We further study the impact of thermal fluctuations
on the predictability of turbulence. With initial differences caused by thermal
fluctuations, different flow realizations display significant disparities in
velocity and thermodynamic fields at larger scales after a certain period of
time, which can be characterized by "inverse error cascades". Moreover, the
results suggest a strong correlation between the predictabilities of
thermodynamic fields and the predictability of ${\vec{u}}_{c}$.
