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...ogorov’s axiomatics (1933) [97] is now accepted as standard. Thirty years later, the complexity approach to randomness was invented by Solomonoff and Kolmogorov (see the review [149] and the textbook =-=[108]-=-). The rigorous foundation of equilibrium statistical physics of many particles based on the central limit theorems was proposed [96, 30]. The modern development of the limit theorems in high dimensio...

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...0) f (1) M eq = Q−1 M (1 − (DMf M )m)(A(feq M )). A detailed analysis of explicit versions of this formula for the Boltzmann equation and other kinetic equations is presented in many books and papers =-=[24, 79]-=-. Most of the physical applications of kinetic theory, from the transport processes in gases to modern numerical methods (lattice Boltzmann models [137]) give examples of the practical applications an...

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...sary to use the kinetic representation,HILBERT’S 6TH PROBLEM: EXACT HYDRODYNAMICS 45 for rarefied gases the Boltzmann kinetic equation gives the framework for studying the structure of strong shocks =-=[26]-=-. One of the common heuristic ways to use the Boltzmann equation far from local equilibrium consists of three steps: (1) Construction of a specific ansatz for the distribution function for a given phy...

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... (for a review of some further results see [32]). Such terms appear naturally in theories of the phase transitions such as van der Waals liquids [132], Ginzburg–Landau [1] and Cahn–Hilliard equations =-=[17, 16]-=-, and phase fields models [25]. Surprisingly, such terms are also found in the ideal gas dynamics as a consequence of the Chapman–Enskog expansion [134, 133]. In higher-order approximations, the visco...

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...s onto the hydrodynamic fields and the parametrization of this manifold by the hydrodynamic fields are represented. [69, 70]) to ODEs. The recently developed geometric theory of singular perturbation =-=[36, 37, 84]-=- can be considered as a formalization of the Chapman–Enskog approach for the area where complete rigorous theory is achievable. A program of the derivation of (weak) solutions of the Navier–Stokes equ...

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...tion of dissipation seems to be a universal phenomenon [124, 52, 53, 102, 133, 91, 60]. It appears in all exactly solved reduction problems for kinetic equations [91] and in the Bhatnagar–Gross–Krook =-=[7]-=- (BGK) kinetics [88] and is also proven for various regularizations of the Chapman–Enskog expansion [124, 52, 133, 60]. The answer to Hilbert’s 6th Problem concerning transition from the Boltzmann equ...

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...for the creation, validation and destruction of theories for real-world phenomena, where “We are trying to prove ourselves wrong as quickly as possible, because only in that way can we find progress” =-=[38]-=-. There exist no mathematical tools to formalize relations between Theory and Reality in live Physics. Therefore the 6th Problem may be viewed as a tremendous challenge in deep study of ideas of physi...

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...s onto the hydrodynamic fields and the parametrization of this manifold by the hydrodynamic fields are represented. [69, 70]) to ODEs. The recently developed geometric theory of singular perturbation =-=[36, 37, 84]-=- can be considered as a formalization of the Chapman–Enskog approach for the area where complete rigorous theory is achievable. A program of the derivation of (weak) solutions of the Navier–Stokes equ...

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...the collision operator. For the Boltzmann equation, Q is a quadratic operator and, therefore, the notation Q(f, f) is often used. The second group of examples are the systems of Grad moment equations =-=[69, 9, 85, 60]-=-. The system of 13-moment Grad equations linearized near equilibrium is (1.2) (1.3) ∂tρ = −∇ · u, ∂tu = −∇ρ − ∇T − ∇ · σ, ∂tT = − 2 (∇ · u + ∇ · q), 3 ∂tσ = −2∇u − 4 1 ∇q − 5 ǫ σ, ∂tq = − 5 2 ∇T − ∇ ·...

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...the collision operator. For the Boltzmann equation, Q is a quadratic operator and, therefore, the notation Q(f, f) is often used. The second group of examples are the systems of Grad moment equations =-=[69, 9, 85, 60]-=-. The system of 13-moment Grad equations linearized near equilibrium is (1.2) (1.3) ∂tρ = −∇ · u, ∂tu = −∇ρ − ∇T − ∇ · σ, ∂tT = − 2 (∇ · u + ∇ · q), 3 ∂tσ = −2∇u − 4 1 ∇q − 5 ǫ σ, ∂tq = − 5 2 ∇T − ∇ ·...

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...t equations of various complexity, which was introduced by Grad [69] and elaborated further by many authors. In particular, Levermore proved hyperbolicity of the properly constructed moment equations =-=[107]-=-. This staircase forms the basis of the Extended Irreversible Thermodynamics (EIT [85]). 1.3. Singular perturbation and separation of times in kinetics. The kinetic equations are singularly perturbed ...

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...m almost to its equilibrium is supported by monotonic entropy growth (Boltzmann’s H-theorem). This qualitative picture is illustrated in Fig. 1. Such a “nonrigorous picture of the Boltzmann dynamics” =-=[29]-=- which operates by the manifolds in the space of probability distributions is a seminal tool for production of qualitative hypotheses. The points (‘states’) in Fig. 1. correspond to the distributions ...

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...enominators affect the convergence of the Poincaré series in the classical many body problem and the theory of nearly integrable systems. They may even make the perturbation series approach senseless =-=[3]-=-. Many ideas have been proposed and implemented to deal with these singularities: use of the direct iteration method instead of power series in KAM [98, 2, 3], renormalization [39, 21, 117], summation...

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...okes equations by Korteweg [100] (for a review of some further results see [32]). Such terms appear naturally in theories of the phase transitions such as van der Waals liquids [132], Ginzburg–Landau =-=[1]-=- and Cahn–Hilliard equations [17, 16], and phase fields models [25]. Surprisingly, such terms are also found in the ideal gas dynamics as a consequence of the Chapman–Enskog expansion [134, 133]. In h...

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...description may be omitted. The transition from the microscopic to the macroscopic description without an intermediate kinetic equation is used in many physical theories like the Green–Kubo formalism =-=[101]-=-, the Zubarev method of a nonequilibrium statistical4 ALEXANDER N. GORBAN AND ILYA KARLIN operator [148], and the projection operator techniques [68]. This possibility is demonstrated rigorously for ...

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...y. The highest achievement here is the proof that rarefied gas of hard spheres will follow the Boltzmann equation during a fraction of the collision time, starting from a non-correlated initial state =-=[105, 43]-=-. The BBGKY hierarchy [13] provides the general framework for this problem. For the systems close to global thermodynamic equilibrium the global in time estimates are available and the validity of the...

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... make the perturbation series approach senseless [3]. Many ideas have been proposed and implemented to deal with these singularities: use of the direct iteration method instead of power series in KAM =-=[98, 2, 3]-=-, renormalization [39, 21, 117], summation and partial summation and rational approximation of the perturbation series [114, 34] and string theories [144, 28] in quantum field theory [146]. Various ad...

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... equation 44 7. Conclusion 48 Acknowledgements 53 About the authors 53 References 53 1. Introduction 1.1. Hilbert’s 6th Problem. The 6th Problem differs significantly from other 22 Hilbert’s problems =-=[77]-=-. The title of the problem itself is mysterious: “Mathematical treatment of the axioms of physics”. Physics, in its essence, is a special activity for the creation, validation and destruction of theor...

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... Some hints may be found in recent works about the universal asymptotics of solutions of PDEs with small dissipation [31] (which develop the ideas of Il’in proposed in the analysis of boundary layers =-=[83]-=-). We hope that problem #1 about the existence of hydrodynamic invariant manifolds for the linearized Boltzmann equation will be solved soon, problem #2 about the full nonlinear Boltzmann equation may...

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...old where the solutions stay after initial layer. The projection of the distributions onto the hydrodynamic fields and the parametrization of this manifold by the hydrodynamic fields are represented. =-=[69, 70]-=-) to ODEs. The recently developed geometric theory of singular perturbation [36, 37, 84] can be considered as a formalization of the Chapman–Enskog approach for the area where complete rigorous theory...

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... [35, 24, 8]. The graph transformation approach was invented by Hadamard in 1901 [75] and developed further by many authors [80, 41, 74, 99]. The Newton-type direct iteration methods in various forms =-=[125, 51, 52, 53, 103]-=- proved their efficiency for model reduction and calculation of slow manifolds in kinetics. There is also a series of numerical methods based on the analysis of motion of an embedded manifold along th...

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...times; (2) not always; (3) possibly. Some hints about the problems with hydrodynamic asymptotics can be found in the series of works about the small dispersion limit of the Korteweg–de Vries equation =-=[106]-=-. Recently, analysis of the exact solution of the model reduction problem for a simple kinetic model [57, 136] has demonstrated that a hydrodynamic invariant manifold may exist and produce non-local h...

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...on method instead of power series in KAM [98, 2, 3], renormalization [39, 21, 117], summation and partial summation and rational approximation of the perturbation series [114, 34] and string theories =-=[144, 28]-=- in quantum field theory [146]. Various ad hoc analytical and numerical regularization tricks have been proposed too. Exactly solvable models give the possibility of exhaustive analysis of the solutio...

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... make the perturbation series approach senseless [3]. Many ideas have been proposed and implemented to deal with these singularities: use of the direct iteration method instead of power series in KAM =-=[98, 2, 3]-=-, renormalization [39, 21, 117], summation and partial summation and rational approximation of the perturbation series [114, 34] and string theories [144, 28] in quantum field theory [146]. Various ad...

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...then follow this manifold on the path to equilibrium? The answer to all three questions is essentially positive in the asymptotic regime when the Mach number Ma and the Knudsen number Kn tend to zero =-=[6, 46]-=- (see [127, 128]). This is a limit of very slow flows with very small gradients of all fields, i.e. almost no flow at all. Such a flow changes in time very slowly and a rescaling of time told = tnew/ε...

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...erms (∂xp) 2 in the energy density are standard in the thermodynamics of phase transitions. The bulk capillarity terms in fluid mechanics were introduced into the Navier– Stokes equations by Korteweg =-=[100]-=- (for a review of some further results see [32]). Such terms appear naturally in theories of the phase transitions such as van der Waals liquids [132], Ginzburg–Landau [1] and Cahn–Hilliard equations ...

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... physical theories like the Green–Kubo formalism [101], the Zubarev method of a nonequilibrium statistical4 ALEXANDER N. GORBAN AND ILYA KARLIN operator [148], and the projection operator techniques =-=[68]-=-. This possibility is demonstrated rigorously for a rarefied gas near global equilibrium [12]. The reduction from the Boltzmann kinetics to hydrodynamics may be split into three problems: existence of...

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...ery novel. There are at least three famous examples:18 ALEXANDER N. GORBAN AND ILYA KARLIN • The “ultraviolet catastrophe” in higher order terms because of physical phenomena at very short distances =-=[33]-=- and the perturbation series divergencies [147] are well known in quantum field theory, and many approaches have been developed to deal with these singularities [146]; • Singularities and divergence i...

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...ounded operator K with property (6.4): If −〈PTϕ, PTKPTϕ〉f ∗ ≥ 0 for all ϕ, ψ and all bounded K with property (6.4), then PT is an orthogonal projector with respect to the entropic inner product (6.3) =-=[58, 59]-=-. This projector acts on functions of v. For a local equilibrium f ∗ (x, v) the projector is constructed for each x and acts on functions ϕ(x, v) point-wise at each point x. Liu and Yu [110] also used...

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...then follow this manifold on the path to equilibrium? The answer to all three questions is essentially positive in the asymptotic regime when the Mach number Ma and the Knudsen number Kn tend to zero =-=[6, 46]-=- (see [127, 128]). This is a limit of very slow flows with very small gradients of all fields, i.e. almost no flow at all. Such a flow changes in time very slowly and a rescaling of time told = tnew/ε...

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...imits of the Boltzmann equation at the formal level”. The theory of singular perturbations was developed starting from complex systems, from the Boltzmann equation (Hilbert [78], Enskog [35], Chapman =-=[24]-=-, Grad8 ALEXANDER N. GORBAN AND ILYA KARLIN Initial conditions Initial layer Solution Projection to Macroscopic variables Slow manifold Parameterization by Macroscopic variables Macroscopic variables...

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...ics. There is also a series of numerical methods based on the analysis of motion of an embedded manifold along the trajectories with subtraction of the motion of the manifold ‘parallel to themselves’ =-=[40, 62, 120, 60]-=-. The Chapman–Enskog method [35, 24] was proposed in 1916. This method aims to construct the invariant manifold for the Boltzmann equation in the form of a series in powers of a small parameter, the K...

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...the collision operator. For the Boltzmann equation, Q is a quadratic operator and, therefore, the notation Q(f, f) is often used. The second group of examples are the systems of Grad moment equations =-=[69, 9, 85, 60]-=-. The system of 13-moment Grad equations linearized near equilibrium is (1.2) (1.3) ∂tρ = −∇ · u, ∂tu = −∇ρ − ∇T − ∇ · σ, ∂tT = − 2 (∇ · u + ∇ · q), 3 ∂tσ = −2∇u − 4 1 ∇q − 5 ǫ σ, ∂tq = − 5 2 ∇T − ∇ ·...

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...apunov series [112], methods of geometric singular perturbation theory [36, 37, 84] and various power series expansions [35, 24, 8]. The graph transformation approach was invented by Hadamard in 1901 =-=[75]-=- and developed further by many authors [80, 41, 74, 99]. The Newton-type direct iteration methods in various forms [125, 51, 52, 53, 103] proved their efficiency for model reduction and calculation of...

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...s approach senseless [3]. Many ideas have been proposed and implemented to deal with these singularities: use of the direct iteration method instead of power series in KAM [98, 2, 3], renormalization =-=[39, 21, 117]-=-, summation and partial summation and rational approximation of the perturbation series [114, 34] and string theories [144, 28] in quantum field theory [146]. Various ad hoc analytical and numerical r...

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...siest of all hydrodynamic limits of the Boltzmann equation at the formal level”. The theory of singular perturbations was developed starting from complex systems, from the Boltzmann equation (Hilbert =-=[78]-=-, Enskog [35], Chapman [24], Grad8 ALEXANDER N. GORBAN AND ILYA KARLIN Initial conditions Initial layer Solution Projection to Macroscopic variables Slow manifold Parameterization by Macroscopic vari...

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...ics. There is also a series of numerical methods based on the analysis of motion of an embedded manifold along the trajectories with subtraction of the motion of the manifold ‘parallel to themselves’ =-=[40, 62, 120, 60]-=-. The Chapman–Enskog method [35, 24] was proposed in 1916. This method aims to construct the invariant manifold for the Boltzmann equation in the form of a series in powers of a small parameter, the K...

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...mic ones if the linearized collision operator has a spectral gap between the five times degenerated zero and the rest of the spectrum. The saturation of dissipation seems to be a universal phenomenon =-=[124, 52, 53, 102, 133, 91, 60]-=-. It appears in all exactly solved reduction problems for kinetic equations [91] and in the Bhatnagar–Gross–Krook [7] (BGK) kinetics [88] and is also proven for various regularizations of the Chapman–...

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... obtained are qualitatively the same as in the case of Maxwell molecules. Precisely the same result for the nonlinear elongational viscosity obtained first from the Grad equations [89] was derived in =-=[129, 44]-=- from the solution to the BGK kinetic equation in the regime of so-called homoenergetic extension flow. This remarkable fact gives more credit to the derivation of hydrodynamic manifolds from nonlinea...

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...del reduction problem for a simple kinetic model [57, 136] has demonstrated that a hydrodynamic invariant manifold may exist and produce non-local hydrodynamics. Analysis of more complicated kinetics =-=[91, 87, 88, 19, 20]-=- supports and extends this observations: the hydrodynamic invariant manifold may exist but sometimes does not exist, and the hydrodynamic equations when Ma ↛ 0 may differ essentially from the Euler an...

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...Several approaches to computation of invariant manifolds have been developed: Lyapunov series [112], methods of geometric singular perturbation theory [36, 37, 84] and various power series expansions =-=[35, 24, 8]-=-. The graph transformation approach was invented by Hadamard in 1901 [75] and developed further by many authors [80, 41, 74, 99]. The Newton-type direct iteration methods in various forms [125, 51, 52...

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...ies: use of the direct iteration method instead of power series in KAM [98, 2, 3], renormalization [39, 21, 117], summation and partial summation and rational approximation of the perturbation series =-=[114, 34]-=- and string theories [144, 28] in quantum field theory [146]. Various ad hoc analytical and numerical regularization tricks have been proposed too. Exactly solvable models give the possibility of exha...

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...the preservation of entropy production for hydrodynamics produced by projection of kinetics onto an approximate invariant manifold even for rough approximations10 ALEXANDER N. GORBAN AND ILYA KARLIN =-=[59]-=-. This construction is presented in Sec. 6. In Conclusion, we discuss solved and unsolved problems and formulate several hypotheses. 2. Invariance equation and Chapman–Enskog expansion 2.1. The idea o...

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...ingular perturbation theory [36, 37, 84] and various power series expansions [35, 24, 8]. The graph transformation approach was invented by Hadamard in 1901 [75] and developed further by many authors =-=[80, 41, 74, 99]-=-. The Newton-type direct iteration methods in various forms [125, 51, 52, 53, 103] proved their efficiency for model reduction and calculation of slow manifolds in kinetics. There is also a series of ...

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...tion of the inverse operator Q −1 M . (Nevertheless these calculations may be very bulky and one of the creators of the method, S. Chapman, compared reading his book [24] to “chewing glass”, cited by =-=[15]-=-). Differentiability is needed also because the transport operator A should be bounded to provide strong sense to the manipulations (see the discussion in [128]). The second order in ǫ hydrodynamic eq...

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...ures in Wonderland.) The small parameter disappears (we take ǫ = 1) but the effect of its presence persists in the analytic invariant manifold. There are some other effects of such a grin in kinetics =-=[67]-=-. The use of the term “slow manifold” for the case ǫ = 1 seems to be an abuse of language. Nevertheless, this manifold has some offprints of slowness, at least for smooth solutions bounded by small nu...

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...rs are more suitable from thermodynamic point of view. In addition, the flexible choice of projectors allows us to treat various nonlinear functionals (like scattering rates) as macroscopic variables =-=[56, 65]-=-.12 ALEXANDER N. GORBAN AND ILYA KARLIN If f M is a solution to the invariance equation (2.2) then the reduced model (2.3) has two important properties: • Preservation of conservation laws. If a diff...

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...y. The highest achievement here is the proof that rarefied gas of hard spheres will follow the Boltzmann equation during a fraction of the collision time, starting from a non-correlated initial state =-=[105, 43]-=-. The BBGKY hierarchy [13] provides the general framework for this problem. For the systems close to global thermodynamic equilibrium the global in time estimates are available and the validity of the...

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...dissipation inequalities may occur when one uses the approximate solutions of the invariance equation. For these situations, two techniques are invented: modification of the projection operation (see =-=[51, 59]-=- and Sec. 6 below) and modification of the entropy functional [72, 71]. They allow to retain the dissipation inequality for the approximate equations. It is obvious that the invariance equation (2.2) ...

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...icles based on the central limit theorems was proposed [96, 30]. The modern development of the limit theorems in high dimensions is based on the geometrical ideas of the measure concentration effects =-=[73, 138]-=- and gives new insights into the foundation of statistical physics (see, for example, [47, 139]). Despite many open questions, this part of the Hilbert programme is essentially fulfilled – the probabi...

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... to global thermodynamic equilibrium the global in time estimates are available and the validity of the linearized Boltzmann equation is proven recently in this limit for rarefied gas of hard spheres =-=[12]-=-. The second part, model reduction in dissipative systems, from kinetics to macroscopic dynamics, is ready for a mathematical treatment. Some limit theorems about this model reduction are already prov...

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... Solomonoff and Kolmogorov (see the review [149] and the textbook [108]). The rigorous foundation of equilibrium statistical physics of many particles based on the central limit theorems was proposed =-=[96, 30]-=-. The modern development of the limit theorems in high dimensions is based on the geometrical ideas of the measure concentration effects [73, 138] and gives new insights into the foundation of statist...

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...in the thermodynamics of phase transitions. The bulk capillarity terms in fluid mechanics were introduced into the Navier– Stokes equations by Korteweg [100] (for a review of some further results see =-=[32]-=-). Such terms appear naturally in theories of the phase transitions such as van der Waals liquids [132], Ginzburg–Landau [1] and Cahn–Hilliard equations [17, 16], and phase fields models [25]. Surpris...

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...rs are more suitable from thermodynamic point of view. In addition, the flexible choice of projectors allows us to treat various nonlinear functionals (like scattering rates) as macroscopic variables =-=[56, 65]-=-.12 ALEXANDER N. GORBAN AND ILYA KARLIN If f M is a solution to the invariance equation (2.2) then the reduced model (2.3) has two important properties: • Preservation of conservation laws. If a diff...

Citation Context

...mic ones if the linearized collision operator has a spectral gap between the five times degenerated zero and the rest of the spectrum. The saturation of dissipation seems to be a universal phenomenon =-=[124, 52, 53, 102, 133, 91, 60]-=-. It appears in all exactly solved reduction problems for kinetic equations [91] and in the Bhatnagar–Gross–Krook [7] (BGK) kinetics [88] and is also proven for various regularizations of the Chapman–...

Citation Context

...ingular perturbation theory [36, 37, 84] and various power series expansions [35, 24, 8]. The graph transformation approach was invented by Hadamard in 1901 [75] and developed further by many authors =-=[80, 41, 74, 99]-=-. The Newton-type direct iteration methods in various forms [125, 51, 52, 53, 103] proved their efficiency for model reduction and calculation of slow manifolds in kinetics. There is also a series of ...

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... 1500) , 153k 2 + 35 867k 2 k 2 + 2105k 2 + 300 . The corresponding hydrodynamics are non-local but stable and were first obtained by a partial summation (regularization) of the Chapman–Enskog series =-=[49]-=-. A numerical solution of the invariance equation (4.13) is also straightforward and does not produce any serious problem. Select of the proper (Chapman–Enskog) branch of the solution, is set by the a...

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...und in the series of works about the small dispersion limit of the Korteweg–de Vries equation [106]. Recently, analysis of the exact solution of the model reduction problem for a simple kinetic model =-=[57, 136]-=- has demonstrated that a hydrodynamic invariant manifold may exist and produce non-local hydrodynamics. Analysis of more complicated kinetics [91, 87, 88, 19, 20] supports and extends this observation...

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...of the approximation obtained are qualitatively the same as in the case of Maxwell molecules. Precisely the same result for the nonlinear elongational viscosity obtained first from the Grad equations =-=[89]-=- was derived in [129, 44] from the solution to the BGK kinetic equation in the regime of so-called homoenergetic extension flow. This remarkable fact gives more credit to the derivation of hydrodynami...

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... is the proof that rarefied gas of hard spheres will follow the Boltzmann equation during a fraction of the collision time, starting from a non-correlated initial state [105, 43]. The BBGKY hierarchy =-=[13]-=- provides the general framework for this problem. For the systems close to global thermodynamic equilibrium the global in time estimates are available and the validity of the linearized Boltzmann equa...

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...r, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics”. This is definitely “a programmatic call” =-=[23]-=- for the axiomatization of the formal parts of existent physical theories and no new universal logical framework for the representation ofHILBERT’S 6TH PROBLEM: EXACT HYDRODYNAMICS 3 reality is neces...

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...details of the thermodynamic projector construction we refer to [59, 60]. Some examples with construction of the thermodynamic projector with preservation of linear conservation laws are presented in =-=[48]-=-. Another possible modification is a modification of the entropy functional. Recently, Grmela [71, 72] proposed to modify the entropy functional after each step of the Chapman–Enskog expansion in orde...

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...ions of the invariance equation. For these situations, two techniques are invented: modification of the projection operation (see [51, 59] and Sec. 6 below) and modification of the entropy functional =-=[72, 71]-=-. They allow to retain the dissipation inequality for the approximate equations. It is obvious that the invariance equation (2.2) for dynamical systems usually has too many solutions, at least locally...

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...kerm) ′ at point f 0 . Recently, the approach to invariant manifolds based on the invariance equation in combination with the Lyapunov auxiliary theorem were used for the reduction of kinetic systems =-=[93, 94, 95]-=-. 2.2. The Chapman–Enskog expansion. The Chapman–Enskog and geometric singular perturbation approach assume the special singularly perturbed structure of the equations and look for the invariant manif...

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... recognized in mechanics by Lyapunov in his thesis (1892) [112]. The problem of persistence and bifurcations of invariant manifolds under perturbations is one of the most seminal problems in dynamics =-=[98, 2, 3, 4, 80, 140]-=-. Several approaches to computation of invariant manifolds have been developed: Lyapunov series [112], methods of geometric singular perturbation theory [36, 37, 84] and various power series expansion...

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...limit theorems in high dimensions is based on the geometrical ideas of the measure concentration effects [73, 138] and gives new insights into the foundation of statistical physics (see, for example, =-=[47, 139]-=-). Despite many open questions, this part of the Hilbert programme is essentially fulfilled – the probability theory and the foundations of equilibrium statistical physics are now well-established cha...

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...kerm) ′ at point f 0 . Recently, the approach to invariant manifolds based on the invariance equation in combination with the Lyapunov auxiliary theorem were used for the reduction of kinetic systems =-=[93, 94, 95]-=-. 2.2. The Chapman–Enskog expansion. The Chapman–Enskog and geometric singular perturbation approach assume the special singularly perturbed structure of the equations and look for the invariant manif...

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...uage. Nevertheless, this manifold has some offprints of slowness, at least for smooth solutions bounded by small number. The definition of slow manifolds for a single system may be a non-trivial task =-=[27, 60]-=-. There is a problem with a local definition because for a given vector field the “slowness” of a submanifold cannot be invariant with respect to diffeomorphisms in a vicinity of a regular point. Ther...

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...ions of the invariance equation. For these situations, two techniques are invented: modification of the projection operation (see [51, 59] and Sec. 6 below) and modification of the entropy functional =-=[72, 71]-=-. They allow to retain the dissipation inequality for the approximate equations. It is obvious that the invariance equation (2.2) for dynamical systems usually has too many solutions, at least locally...

Citation Context

...del reduction problem for a simple kinetic model [57, 136] has demonstrated that a hydrodynamic invariant manifold may exist and produce non-local hydrodynamics. Analysis of more complicated kinetics =-=[91, 87, 88, 19, 20]-=- supports and extends this observations: the hydrodynamic invariant manifold may exist but sometimes does not exist, and the hydrodynamic equations when Ma ↛ 0 may differ essentially from the Euler an...

Citation Context

... Solomonoff and Kolmogorov (see the review [149] and the textbook [108]). The rigorous foundation of equilibrium statistical physics of many particles based on the central limit theorems was proposed =-=[96, 30]-=-. The modern development of the limit theorems in high dimensions is based on the geometrical ideas of the measure concentration effects [73, 138] and gives new insights into the foundation of statist...

Citation Context

...del reduction problem for a simple kinetic model [57, 136] has demonstrated that a hydrodynamic invariant manifold may exist and produce non-local hydrodynamics. Analysis of more complicated kinetics =-=[91, 87, 88, 19, 20]-=- supports and extends this observations: the hydrodynamic invariant manifold may exist but sometimes does not exist, and the hydrodynamic equations when Ma ↛ 0 may differ essentially from the Euler an...

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...hydrodynamic limits of the Boltzmann equation at the formal level”. The theory of singular perturbations was developed starting from complex systems, from the Boltzmann equation (Hilbert [78], Enskog =-=[35]-=-, Chapman [24], Grad8 ALEXANDER N. GORBAN AND ILYA KARLIN Initial conditions Initial layer Solution Projection to Macroscopic variables Slow manifold Parameterization by Macroscopic variables Macrosc...

Citation Context

...del reduction problem for a simple kinetic model [57, 136] has demonstrated that a hydrodynamic invariant manifold may exist and produce non-local hydrodynamics. Analysis of more complicated kinetics =-=[91, 87, 88, 19, 20]-=- supports and extends this observations: the hydrodynamic invariant manifold may exist but sometimes does not exist, and the hydrodynamic equations when Ma ↛ 0 may differ essentially from the Euler an...

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...ehaviour of the hydrodynamic solutions becomes dependent only on the extreme cases. Some hints may be found in recent works about the universal asymptotics of solutions of PDEs with small dissipation =-=[31]-=- (which develop the ideas of Il’in proposed in the analysis of boundary layers [83]). We hope that problem #1 about the existence of hydrodynamic invariant manifolds for the linearized Boltzmann equat...

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...iteria either but some important parts of them have been already claimed as solved. Several axiomatic approaches to probability have been developed and the equivalence of some of them has been proven =-=[45]-=-. Kolmogorov’s axiomatics (1933) [97] is now accepted as standard. Thirty years later, the complexity approach to randomness was invented by Solomonoff and Kolmogorov (see the review [149] and the tex...

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...d the Tamm–Mott-Smith lifting operation provides the approximate equations on the Tamm–Mott-Smith manifold with positive entropy production. Several other projections have been tested computationally =-=[81]-=-. All of them violate second law of thermodynamics because for some initial conditions the entropy production for them becomes negative at some points. Indeed, introduction of the entropy density as a...

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... how to project the Boltzmann equation onto the linear manifold (6.1) and create the macroscopic equation. To respect second law of thermodynamics and provide positivity of entropy production, Lampis =-=[104]-=- used the entropy density s as a new variable. The entropy density is defined as a functional of f(v), s(x) = − ∫ f(x, v)ln f(x, v)d3v. For each distribution f the time derivative of s is defined by t...

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...del reduction problem for a simple kinetic model [57, 136] has demonstrated that a hydrodynamic invariant manifold may exist and produce non-local hydrodynamics. Analysis of more complicated kinetics =-=[91, 87, 88, 19, 20]-=- supports and extends this observations: the hydrodynamic invariant manifold may exist but sometimes does not exist, and the hydrodynamic equations when Ma ↛ 0 may differ essentially from the Euler an...

Citation Context

...kerm) ′ at point f 0 . Recently, the approach to invariant manifolds based on the invariance equation in combination with the Lyapunov auxiliary theorem were used for the reduction of kinetic systems =-=[93, 94, 95]-=-. 2.2. The Chapman–Enskog expansion. The Chapman–Enskog and geometric singular perturbation approach assume the special singularly perturbed structure of the equations and look for the invariant manif...

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...s of them have been already claimed as solved. Several axiomatic approaches to probability have been developed and the equivalence of some of them has been proven [45]. Kolmogorov’s axiomatics (1933) =-=[97]-=- is now accepted as standard. Thirty years later, the complexity approach to randomness was invented by Solomonoff and Kolmogorov (see the review [149] and the textbook [108]). The rigorous foundation...

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...s onto the hydrodynamic fields and the parametrization of this manifold by the hydrodynamic fields are represented. [69, 70]) to ODEs. The recently developed geometric theory of singular perturbation =-=[36, 37, 84]-=- can be considered as a formalization of the Chapman–Enskog approach for the area where complete rigorous theory is achievable. A program of the derivation of (weak) solutions of the Navier–Stokes equ...

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...finite dissipation is quite typical for the application of the Newton method to solution of the invariance equation. The ‘solid jet’ limit for the extremely fast compressions gives us another example =-=[55]-=- (see also Sec. 5.2). 4.4. Invariant manifolds, entanglement of hydrodynamic and non-hydrodynamic modes and saturation of dissipation for the 3D 13 moments Grad system . The thirteen moments linear Gr...

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...n in quantum field theory, and many approaches have been developed to deal with these singularities [146]; • Singularities and divergence in the semiclassical Wentzel-Kramers-Brillouin (WKB) approach =-=[42, 145, 82]-=-; • The small denominators affect the convergence of the Poincaré series in the classical many body problem and the theory of nearly integrable systems. They may even make the perturbation series appr...

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...ingular perturbation theory [36, 37, 84] and various power series expansions [35, 24, 8]. The graph transformation approach was invented by Hadamard in 1901 [75] and developed further by many authors =-=[80, 41, 74, 99]-=-. The Newton-type direct iteration methods in various forms [125, 51, 52, 53, 103] proved their efficiency for model reduction and calculation of slow manifolds in kinetics. There is also a series of ...