  The aim of this paper is twofold: the first is to formulate and validate a
multi-scale discrete Boltzmann method (DBM) based on density functional kinetic
theory for thermal multiphase flow systems, ranging from continuum to
transition flow regime; the second is to present some new insights into the
thermo-hydrodynamic non-equilibrium (THNE) effects in the phase separation
process. Methodologically, DBM includes three main pillars: (i) the
determination of the fewest kinetic moment relations, which are required by the
description of significant THNE effects beyond the realm of continuum fluid
mechanics, (ii) the construction of appropriate discrete equilibrium
distribution function recovering all the desired kinetic moments, (iii) the
detection, description, presentation and analysis of THNE based on the moments
of the non-equilibrium distribution ($f-f^{(eq)}$). The incorporation of
appropriate additional higher-order thermodynamic kinetic moments considerably
extends the DBM's capability of handling larger values of the liquid-vapor
density ratio, curbing spurious currents, and ensuring mass-momentum-energy
conservation. Compared with the DBM with only first-order THNE (Gan et al. Soft
Matter 11,5336), the model retrieves kinetic moments beyond the third-order
super-Burnett level, and is accurate for weak, moderate, and strong THNE cases
even when the local Knudsen number exceeds $1/3$. Physically, the ending point
of the linear relation between THNE and the concerned physical parameter
provides a distinct criterion to identify whether the system is near or far
from equilibrium. Besides, the surface tension refrains the local THNE around
the interface, but expands the THNE range and strengthens the THNE intensity
away from the interface through interface smoothing and widening.
