1 Model reduction and uniqueness of thermodynamic projector Alexander Gorban ETH Zurich, Switzerland, and Institute of Computational Modeling Russian Academy of Sciences

2 Plan The closure problem and thermodynamic projector; Preservation of dissipation and Uniqueness theorem; Method of Invariant manifold; Examples: - High-order hydrodynamics from BE, -Chemical kinetics; Conclusion

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8 Tamm-Mott-Smith approximation for shock waves (1950s): f is a linear combination of two Maxwellians (f TMS =af hot +bf cold ) Variation of the velocity distribution in the shock front at M=8,19 (Zharkovski at al., 1997)

9 The projection problem: t a(x,t)=? t b(x,t)=? Coordinate functionals F 1,2 [f(v)]. Their time derivatives should persist (BE t F 1,2 =TMS t F 1,2 ): BE t F 1,2 [f(x,v,t)]= ( F 1,2 [f]/ f){-(v, x f(x,v,t))+Q(f,f)}dv; TMS t F 1,2 [f TMS ]= t (a(x,t)) ( F 1,2 [f]/ f)f hot (v)dv+ t (b(x,t)) ( F 1,2 [f]/ f)f cold (v)dv. There exists unique choice of F 1,2 [f(v)] without violation of the Second Law: F 1 =n= fdv - the concentration; F 2 =s= f(lnf-1)dv - the entropy density. Proposed by M. Lampis (1977). Uniqueness was proved by A. Gorban & I. Karlin (1990).

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21 Bobylev instability of Burnett equations Acoustic dispersion curves for Burnett approximation (dashed line), for first iteration of MIM (solid line), and for regularization of the Burnett approximation via partial summing of the Chapman-Enskog expansion (punctuated dashed line). Arrows indicate an increase of k 2.

22 Negative viscosity for Burnett equations Dependency of viscosity on compression for Burnett approximation (dashed line), for first iteration of MIM (solid line), and for partial summing (punctuated dashed line).

23 Invariant grid for two-dimensional chemical system: A1 + A2 A3 A2 + A4 One-dimensional invariant grid (circles) Projection onto the 3d-space of c oncentrations c 1, c 4, c3. The trajectories of the system in the phase space are shown by lines. The equilibrium point is marked by square. The system quickly reaches the grid and further moves along it.

24 Invariant grid for model Hydrogen burning a) Projection onto the 3d-space of c H, c O, c OH concentrations. b) Projection onto the principal 3D-subspace.

25 Invariant grid as a screen for visualizing dierent functions Model Hydrogen burning

26 Conclusion 1: Three reasons to use the thermodynamic projector It guarantees the persistence of dissipation: All the thermodynamic processes which should product the entropy conserve this property after projecting, moreover, not only the sign of dissipation conserves, but the value of entropy production and the reciprocity relations too;

27 Universality: The coefficients (and, more generally speaking, the right hand part) of kinetic equations are known significantly worse then the thermodynamic functionals, so, the universality of the thermodynamic projector (it depends only on thermodynamic data) makes the thermodynamic properties of projected system as reliable, as for the initial system; It is easy (much more easy than spectral projector, for example).

28 Conclusion 2: To construct slow invariant manifolds for kinetic problems is useful; It is possible indeed to construct slow invariant manifolds for kinetic problems.