**1. Introduction**

The problem of model reduction is ubiquitous in physics and mathematics. Consider a system (physical or mathematical) that can be regarded on two levels of description, upper (more detailed level) and lower (less detailed). The state variables on the lower level contain less information than state variables on the upper level. A projection from the upper level to the lower level is necessary to state the problem of model reduction correctly. Assume, moreover, that dynamics of the state variables on the upper level is granted, but one wishes to see evolution of the lower variables. The reason can be for instance the complexity of the detailed evolution, availability of experimental observations or simplicity of the lower description. A reduction of dynamics from the upper level to the lower (less detailed) level is called model reduction.

There is no general model reduction technique applicable to all systems. However, many physically based (we do not focus on formal mathematical expansion methods although we make a certain comparison below) methods have been developed, such as the Chapman–Enskog expansion [1] or other series expansions [2], projector operator techniques [3,4], or the method of natural projector, invariant slow manifolds, entropic scalar product and Ehrenfest reduction [5–8]. A common feature of the reduction techniques is the recognition of entropy, since entropy (measuring unavailable information) grows during the passage from the more detailed level to the less detailed. In particular, states on the higher level corresponding to maximum entropy are referred to as the quasi-equilibrium manifold. This manifold is constructed by maximization of entropy on the upper level while knowing

the result of the projection of the state variables to the lower level. This is the principle of Maximum Entropy (MaxEnt), see e.g., [9,10].

Apart from the relations between state variables on the two levels of description, it is necessary to find relations between vector fields generating evolution on the two levels. Indeed, evolution equations on the higher level can be seen as a motion along a given vector field (following arrows of the field) while the corresponding vector field on the lower level is to be found. For instance, in the Ehrenfest reduction, the vector field on the higher level is first prolonged by Taylor series and subsequently projected to the lower level, using MaxEnt, and closed. See [7] for various geometric techniques on how to obtain the vector field on the lower level of description.

Our approach is different. First, we recognize not only the state variables, but also conjugate variables as independent quantities, being motivated by the formulation of thermodynamics in contact geometry [11–13]. Eventually, conjugate variables become related to the state variables through derivatives of a thermodynamic potential (e.g., energy or entropy), but they are considered independent from direct variables in contact formulation and exhibit evolution towards that relation while approaching a given level. State variables between the levels can be related by MaxEnt as in [7,9,14], but conjugate variables can be exploited to find the approximation of the vector field on the higher level of description such that evolution on the lower level becomes closed while extracting key features ("measured" by entropy potential) of the upper level dynamics. In other words, the vector field on the lower level becomes tangent to the quasi-equilibrium manifold, see [15] or [14]. We refer to this method as Dynamic MaxEnt (DynMaxEnt).

Novelty of this paper lies in the following points: (i) The DynMaxEnt is introduced in the energetic representation, which simplifies the calculations, (ii) a chain or higher order DynMaxEnt approximations is identified, (iii) DynMaxEnt is compared to asymptotic expansions and (iv) it is applied to a reduction of various complex continuum dynamics to classical irreversible thermodynamics (conformation tensor, Reynolds stress, hyperbolic heat conduction and magnetohydrodynamics).

In Section 2, we first recall the usual (static) Maximum Entropy principle (MaxEnt). In Section 3, we introduce the method of Dynamic MaxEnt, which serves as a reduction from the more detailed evolution equations to less detailed, and then we demonstrate it on damped particle dynamics. In Section 4, we use the DynMaxEnt method for reduction of dynamics of complex fluids, hyperbolic heat conduction and electromagnetohydrodynamics to the Navier–Stokes–Fourier system and magnetohydrodynamics. Finally, in Section 5, we present a geometric motivation for the DynMaxEnt method in the framework of contact geometry.
