  Active matter systems are inherently out of equilibrium and break the
detailed balance (DB) at the microscopic scale, exhibiting vital collective
phenomena such as motility-induced phase separation (MIPS). Here, we introduce
a coarse-grained mapping method to probe DB breaking in the density-energy
phase space, which allows us to reveal the dynamic and thermodynamic origins of
MIPS based on nonequilibrium potential and flux landscape theory. Hallmarks of
nonequilibrium properties are manifested by identifying the visible probability
flux in the coarse-grained phase space. Remarkably, the flux for the system
with the activity lower than the MIPS threshold tends to ``tear up" the single
potential well of the uniform-density phase to create two wells of phases with
different densities, presenting directly that the nonequilibrium flux is the
dynamic origin of MIPS. Moreover, we find that the obtained entropy production
rate (EPR) of the system undergoes a transition from nearly independent of
activity to increasing proportionally as activity increases after the single
well is "teared up". The transition of EPR's scaling behavior might provide a
hint of the thermodynamic origin of MIPS in the coarse-grained space. Our
findings propose a new route to explore the nonequilibrium nature of active
systems, and provide new insights into dynamic and thermodynamic properties of
MIPS.
