Heat differentiated by projection from particles' trajectories onto the particle number-density field

Many particles suspended in a solution have two analogous but distinct stochastic descriptions referred to as the Langevin and Dean--Kawasaki equations, which are based on monitoring particles' trajectories and their number-density field, respectively. This article focuses on heat defined in the Langevin equation proposed by Sekimoto and also analogous heat assumed in the Dean--Kawasaki equation. Spatial projection derives the Dean--Kawasaki equation from the Langevin equation, so that the amounts of the heat observed on the two stochastic forms are not generally identical even in the same phenomena. Notably, spatiotemporal resolutions are not altered in the above projection. This difference offers an intriguing quantity reduced to the entropy of the number density. In addition, a many-polymer system is also found to retain the analogous formalisms of a many-colloid system when chain configurations are embedded into hyperdimensions. Furthermore, we develop arguments about the interpretation and applicability of the heat differences.