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The rheological behavior of a material is strongly related to the energydissipation, and the understanding and modeling of dissipation is importantfrom the view point of rheology. To study rheological properties with somemesoscopic and macroscopic dynamics models, the modeling of dynamic equationwhich appropriately incorporates the dissipation is important. Although thereare several methods to construct mesoscopic and macroscopic dynamic equations,such as the Onsager's method, their validity is not fully clear. In this work,we theoretically analyze the dissipation in a mesoscopic Langevin equation indetail, from the view point of stochastic energetics. We show that thedissipative heat flow from the heat bath to the system plays an important rolein the mesoscopic dynamics. The dissipative heat flow is unchanged under thevariable transform, and thus it is a covariant quantity. We show that we canconstruct the Langevin equation and also perform a coarse-graining based on thedissipative heat flow. We can derive the mobility tensor from the dissipativeheat flow, and construct the Langevin equation by combining it and the freeenergy. Our method can be applied to various systems, such as the dumbbellmodel and the diffusion type equation for the density field, to give thecoarse-grained dynamic equations for them.

Dissipation in Langevin Equation and Construction of Mobility Tensor from Dissipative Heat Flow