Dynamic Maximum Entropy Reduction

Abstract

Any physical system can be regarded on different levels of description varying by how detailed the description is. We propose a method called Dynamic MaxEnt (DynMaxEnt) that provides a passage from the more detailed evolution equations to equations for the less detailed state variables. The method is based on explicit recognition of the state and conjugate variables, which can relax towards the respective quasi-equilibria in different ways. Detailed state variables are reduced using the usual principle of maximum entropy (MaxEnt), whereas relaxation of conjugate variables guarantees that the reduced equations are closed. Moreover, an infinite chain of consecutive DynMaxEnt approximations can be constructed. The method is demonstrated on a particle with friction, complex fluids (equipped with conformation and Reynolds stress tensors), hyperbolic heat conduction and magnetohydrodynamics.

Multiscale Thermodynamics

“…This new requirement plays now a very important role in determining the potentials appearing in ( 29 ). The precise meaning of maximally invariant (or alternatively “quasi-invariant”) used in ( 30 ) as well as the meaning of “appropriately projected” used in ( 33 ) below remains still a part of the pattern recognition analysis of the upper time-evolution that has to be investigated [ 2 , 19 , 20 , 21 , 22 ].…”

“…The output of the reduction is the lower thermodynamics relation: obtained from ( 29 ) in the same way as ( 21 ) is obtained from ( 19 ) (see more in Section 2.2 ) and the vector field (compare with ( 20 )): which, if appropriately projected on the tangent space of the manifold and pushed forward on by the mapping ( 28 ), becomes the vector field generating the time-evolution at the lower level l . In this paper, we limit ourselves only to recalling the main idea behind the Chapman and Enskog analysis (see more in Section 4.1 and in [ 2 , 18 , 19 , 20 , 21 , 22 , 23 ]).…”

Dynamic and Renormalization-Group Extensions of the Landau Theory of Critical Phenomena

“…The reducing time evolution can be either the time evolution taking place in and approaching an invariant (or in most cases a quasi-invariant) manifold that represents in the state space used on the lower level or it can be the time evolution of vector fields taking the vector field generating the upper time evolution to the vector field generating the lower time evolution. The former viewpoint is discussed for example in [ 11 , 12 , 13 , 14 ]. In this paper we follow the second route, discussed in [ 15 ], since on this route we can directly transpose the 2-level equilibrium thermodynamics introduced in the previous section to 2-level rate-thermodynamics.…”

Multiscale thermodynamics of charged mixtures

“…To this end we recall the method of Dynamic MaxEnt (DynMaxEnt) [4,62]. Extensive treatment of the DynMaxEnt principle in the context of the continuum thermodynamics can be found in [63].…”