Stochastic heat differences between many-particle and density-field descriptions

This article investigates spatiotemporally discrete or continuous stochastic descriptions, where we focus on differences in heat naturally defined between the particle level and the density field. Both descriptions are found to generally make the heat differences by the entropic term expressed just with the number density through spatial projection from the many particles' positions onto the density field. The transformation from the Langevin to Dean-Kawasaki equations is considered as the projection in the continuous descriptions, where the emergent heat differences undergo little temporal variations due to the sparse distributions of the point particles. On the other hand, the analogous formalisms constructed in the discrete models may exhibit the explicit temporal evolutions of the entropic term. Furthermore, we develop arguments about the interpretation and applicability of the heat differences as well as the perspectives to a many-polymer system.