Thermodynamics of Reductions in Multiscale Dynamics

Let L1, L2, L3 be three well established (i.e. well tested with experimental observations) levels of description, ordered from the most micro- scopic to the least microscopic, on which mesoscopic dynamics of macro- scopic systems is formulated. Let Eqs1; Eqs2; Eqs3 be the time evolution equations on the three levels. By comparing solutions to these three systems of equations we find reductions L1 → L2 → L3 and L1 → L3 consisting of: (i) relations Eqs1 → Eqs2 → Eqs3 and Eqs1 → Eqs3, (ii) relations P1 → P2 → P3 and P1 → P3, where P stands for material parameters, i.e. the parameters with which the individual nature of the system under consideration is expressed in the time evolution equations, and (iii) six entropies, namely s^(1→2), s^(1→3), s^(2→3) and S^(3←1), S^(3←2) ,S^(2←1). The entropies s^(i→j); i < j are potentials generating the approach of the level Li to the level Lj and S^(i←j); i > j are the entropies s^(j→i) evaluated at the states on the level Li that are reached in the approach Lj → Li. These six entropies represent the multiscale thermodynamics corresponding to the sequence of levels L1, L2, L3. In the particular case when L3 is the level used in the classical equilibrium thermodynamics then S^(3←2) and S^(3←1) are the classical equilibrium entropies. I will illustrate such multiscale thermodynamics (and provide some of its applications) on the example of L1  level of description used in the Catteneo heat conduction theory, L2  level of description used in the Fourier heat conduction theory, and L3  level of description used in the classical equilibrium thermodynamics.