Zbigniew Banach, Ph.D., Dr. Habil. 

Doctoral thesis
1980  Bezpośredni model ośrodka ciągłego współoddziaływującego z polem elektromagnetycznym
 354 
Habilitation thesis
1991  Momentowa procedura Grada a rozszerzona termodynamika procesów nieodwracalnych 
Recent publications
1.  Banach Z., Larecki W., Entropybased mixed threemoment description of fermionic radiation transport in slab and spherical geometries, Kinetic and Related Models, ISSN: 19375093, DOI: 10.3934/krm.2017035, Vol.10, No.4, pp.879900, 2017 Abstract: The mixed threemoment hydrodynamic description of fermionic radiation transport based on the Boltzmann entropy optimization procedure is considered for the case of onedimensional flows. The conditions for realizability of the mixed three moments chosen as the energy density and two partial heat fluxes are established. The domain of admissible values of those moments is determined and the existence of the solution to the optimization problem is proved. Here, the standard approaches related to either the truncated Hausdorff or Markov moment problems do not apply because the nonnegative fermionic distribution function, denoted f, must satisfy the inequality f _ 1 and, at the same time, there are three different intervals of integration in the integral formulae defining the mixed moments. The hydrodynamic equations are obtained in the form of the symmetric hyperbolic system for the Lagrange multipliers of the optimization problem with constraints. The potentials generating this system are explicitly determined as dilogarithm and trilogarithm functions of the Lagrange multipliers. The invertibility of the relation between moments and Lagrange multipliers is proved. However, the inverse relations cannot be determined in a closed analytic form. Using the Htheorem for the radiative transfer equation, it is shown that the derived system of hydrodynamic radiation equations has as a consequence an additional balance law with a nonnegative source term. Keywords:Fermionic radiation, mixed moments, moment realizability domain, entropy optimization problem, symmetric hyperbolicity Affiliations:
 
2.  Banach Z., Larecki W., Kershawtype transport equations for fermionic radiation, ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK ZAMP, ISSN: 00442275, DOI: 10.1007/s000330170847z, Vol.68, No.4, pp.100110024, 2017 Abstract: Besides the maximum entropy closure procedure, other procedures can be used to close the systems of spectral moment equations. In the case of classical and bosonic radiation, the closedform analytic Kershawtype and Bdistribution closure procedures have been used. It is shown that the Kershawtype closure procedure can also be applied to the spectra moment equations of fermionic radiation. First, a description of the Kershawtype closure for the system consisting of an arbitrary number of onedimensional moment equations is presented. Next, the Kershawtype twofield and threefield transport equations for fermionic radiation are analyzed. In the first case, the independent variables are the energy density and the heat flux. The second case includes additionally the flux of the heat flux as an independent variable. The generalization of the former twofield case to three space dimensions is also presented. The fermionic Kershawtype closures differ from those previously derived for classical and bosonic radiation. It is proved that the obtained onedimensional systems of transport equations are strictly hyperbolic and causal. The fermionic Kershawtype closure functions behave qualitatively in the same way as the fermionic maximum entropy closure functions, but attain different numerical values. Keywords:Fermionic radiation, Moment equations, Moment realizability problem, Kershawtype closure, Threemoment transport Affiliations:
 
3.  Larecki W., Banach Z., Twofield radiation hydrodynamics in n spatial dimensions, JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL, ISSN: 17518113, DOI: 10.1088/17518113/49/12/125501, Vol.49, No.12, pp.125501123, 2016 Abstract: The twofield radiation hydrodynamics in n spatial dimensions is derived from the kinetic theory of radiation. Both the fullmoment (frequencyindependent) and spectral (frequencydependent) formulations of radiation hydrodynamics are considered. The derivation is based on the entropy principle of extended thermodynamics of gases. In the case of the fullmoment hydrodynamics, the formulation of the entropy principle introduced by Boillat and Ruggeri (1997 Contin. Mech. Thermodyn. 9 205) is adapted and this suffices to determine the radiation pressure tensor. In the fullmoment formulation, the equations of radiation hydrodynamics take the same form for all possible types of radiation statistics. In the spectral formulation, the different radiation pressure tensors are assigned to Bose–Einstein, Fermi–Dirac and Maxwell–Boltzmann statistics, and consequently the different hydrodynamic equations are obtained for each of those statistics types. In order to derive the equations of the spectral radiation hydrodynamics, the relations for the radiation pressure tensor implied by the entropy principle must be supplemented by the additional conditions. Considering the limit of small heat flux, we arrive at the linearized equations of radiation hydrodynamics which assume the same form in both the fullmoment and spectral formulations. Keywords:radiative transfer equation, Htheorem, radiation hydrodynamics, entropy principle, radiation pressure tensor Affiliations:
 
4.  Banach Z., Dispersion relation governing the propagation of wave pulses in a mixture of interacting longitudinal and transverse phonon gases, WAVE MOTION, ISSN: 01652125, DOI: 10.1016/j.wavemoti.2015.05.008, Vol.58, pp.203221, 2015 Abstract: The investigations by Koreeda et al. (2009) and Ma (2013) of the effect of second sound in dielectric crystals assume the representation of the phonon system as a mixture composed of the gas of longitudinal phonons and the gas of transverse phonons. The theoretical background of those investigations can be related either to linear phonon gas hydrodynamics based on the moment equations of Dreyer and Struchtrup (1993) or to the heuristic model of energy transport in a phonon gas proposed by Rogers (1971). The relationship between these two approaches is analyzed. Specifically, restricting attention to the onedimensional flows, the system of linear hyperbolic equations of sixmoment phonon gas hydrodynamics is discussed together with the ‘equivalent’ system of two coupled thirdorder in time partial differential equations for the energy densities of longitudinal and transverse phonon branches. Given the sixmoment system and its equivalent, the dispersion relation governing the propagation of complex plane harmonic waves is studied in detail. It is demonstrated that the interaction between the longitudinal phonon gas and the transverse phonon gas plays an important role when determining how the complex wave number depends on the real angular frequency of plane harmonic waves. The derived dispersion relation is systematically compared with that postulated by Ma (2013). Using the principle of superposition and the real plane harmonic waves as a basis, the manifestly real pulselike solution to the equations of phonon hydrodynamics is explicitly constructed. It is shown that the computed energypulse profiles have the features that agree well with the main features of the energy or temperature profiles observed in the heatpulse experiments. Keywords:Phonongas mixture, Longitudinal mode, Transversal mode, Sixmoment model, Plane harmonic wave, Pulselike solution Affiliations:
 
5.  Larecki W., Banach Z., Influence of nonlinearity of the phonon dispersion relation on wave velocities in the fourmoment maximum entropy phonon hydrodynamics, PHYSICA DNONLINEAR PHENOMENA, ISSN: 01672789, DOI: 10.1016/j.physd.2013.10.006, Vol.266, No.1, pp.6579, 2014 Abstract: This paper analyzes the propagation of the waves of weak discontinuity in a phonon gas described by the fourmoment maximum entropy phonon hydrodynamics involving a nonlinear isotropic phonon dispersion relation. For the considered hyperbolic equations of phonon gas hydrodynamics, the eigenvalue problem is analyzed and the condition of genuine nonlinearity is discussed. The speed of the wave front propagating into the region in thermal equilibrium is first determined in terms of the integral formula dependent on the phonon dispersion relation and subsequently explicitly calculated for the Dubey dispersionrelation model:k=ωc−1(1+bω2). The specification of the parameters cc and bb for sodium fluoride (NaF) and semimetallic bismuth (Bi) then makes it possible to compare the calculated dependence of the wavefront speed on the sample’s temperature with the empirical relations of Coleman and Newman (1988) describing for NaF and Bi the variation of the secondsound speed with temperature. It is demonstrated that the calculated temperature dependence of the wavefront speed resembles the empirical relation and that the parameters cc and bb obtained from fitting respectively the empirical relation and the original material parameters of Dubey (1973) are of the same order of magnitude, the difference being in the values of the numerical factors. It is also shown that the calculated temperature dependence is in good agreement with the predictions of Hardy and Jaswal’s theory (Hardy and Jaswal, 1971) on secondsound propagation. This suggests that the nonlinearity of a phonon dispersion relation should be taken into account in the theories aiming at the description of the wavetype phonon heat transport and that the Dubey nonlinear isotropic dispersionrelation model can be very useful for this purpose. Keywords:Phonon hydrodynamics, Waves in phonon gas, Genuine nonlinearity, Nonlinear phonon dispersion relation, Second sound Affiliations:
 
6.  Banach Z., Larecki W., Ruggeri T.^{♦}, Dispersion relation in the limit of high frequency for a hyperbolic system with multiple eigenvalues, WAVE MOTION, ISSN: 01652125, DOI: 10.1016/j.wavemoti.2014.03.008, Vol.51, pp.955966, 2014 Abstract: The results of a previous paper (Muracchini et al., 1992) are generalized by considering a hyperbolic system in one space dimension with multiple eigenvalues. The dispersion relation for linear plane waves in the highfrequency limit is analyzed and the recurrence formulas for the phase velocity and the attenuation factor are derived in terms of the coefficients of a formal series expansion in powers of the reciprocal of frequency. In the case of multiple eigenvalues, it is also verified that linear stability implies λstability for the waves of weak discontinuity. Moreover, for the linearized system, the relationship between entropy and stability is studied. When the nonzero eigenvalue is simple, the results of the paper mentioned above are recovered. In order to illustrate the procedure, an example of the linear hyperbolic system is presented in which, depending on the values of parameters, the multiplicity of nonzero eigenvalues is either one or two. This example describes the dynamics of a mixture of two interacting phonon gases. Keywords:Hyperbolic system, Multiple eigenvalue, Linearization, Harmonic wave, Dispersion relation, Weak discontinuity Affiliations:
 
7.  Banach Z., Larecki W., Spectral maximum entropy hydrodynamics of fermionic radiation: a threemoment system for onedimensional flows, NONLINEARITY, ISSN: 09517715, DOI: 10.1088/09517715/26/6/1667, Vol.26, No.6, pp.16671701, 2013 Abstract: The spectral formulation of the ninemoment radiation hydrodynamics resulting from using the Boltzmann entropy maximization procedure is considered. The analysis is restricted to the onedimensional flows of a gas of massless fermions. The objective of the paper is to demonstrate that, for such flows, the spectral ninemoment maximum entropy hydrodynamics of fermionic radiation is not a purely formal theory. We first determine the domains of admissible values of the spectral moments and of the Lagrange multipliers corresponding to them. We then prove the existence of a solution to the constrained entropy optimization problem. Due to the strict concavity of the entropy functional defined on the space of distribution functions, there exists a onetoone correspondence between the Lagrange multipliers and the moments. The maximum entropy closure of moment equations results in the symmetric conservative system of firstorder partial differential equations for the Lagrange multipliers. However, this system can be transformed into the equivalent system of conservation equations for the moments. These two systems are consistent with the additional conservation equation interpreted as the balance of entropy. Exploiting the above facts, we arrive at the differential relations satisfied by the entropy function and the additional function required to close the system of moment equations. We refer to this additional function as the moment closure function. In general, the moment closure and entropy–entropy flux functions cannot be explicitly calculated in terms of the moments determining the state of a gas. Therefore, we develop a perturbation method of calculating these functions. Some additional analytical (and also numerical) results are obtained, assuming that the maximum entropy distribution function tends to the Maxwell–Boltzmann limit. Keywords:fermionic radiation, spectral moments, maximum entropy closure Affiliations:
 
8.  Banach Z., Larecki W., Onedimensional maximum entropy radiation hydrodynamics: threemoment theory, JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL, ISSN: 17518113, DOI: 10.1088/17518113/45/38/385501, Vol.45, No.38, pp.385501(124), 2012 Abstract: We analyse in detail the threemoment maximum entropy radiation hydrodynamics which describes the motion of a gas of massless particles obeying Maxwell–Boltzmann, Bose–Einstein or Fermi–Dirac statistics. The primitive fields of this hydrodynamics are the energy density, the heat flux and the flux of the heat flux. We focus on the study of onedimensional flows. Thus, the independent or nonvanishing variables defining the state of the gas are 3 in number. Two formulations are used: the formulation in terms of the Lagrange multipliers of the constrained optimization problem and the formulation in terms of the primitive fields. We prove that there exists a diffeomorphism between the open convex domain of Lagrange multipliers and the whole natural domain of primitive fields. Affiliations:
 
9.  Larecki W., Banach Z., Entropic derivation of the spectral Eddington factors, JOURNAL OF QUANTITATIVE SPECTROSCOPY AND RADIATIVE TRANSFER, ISSN: 00224073, DOI: 10.1016/j.jqsrt.2011.06.011, Vol.112, No.15, pp.24862506, 2011 Abstract: In general, the closure of the finite system of moment equations by the corresponding maximum entropy distribution function results in the symmetric conservative system of firstorder partial differential equations for the Lagrange multipliers of the constrained Boltzmann entropy maximization problem. Then the transformation of dependent variables yields the system of conservation equations for the moments which is consistent with the additional conservation equation identified with the balance of entropy. The objective of this paper is to employ these facts for the analysis of the spectral Eddington factors obtained from the maximum entropy distribution functions. The supposition that the spectral Eddington factors should depend on the energy density and the heat flux only through the single variable representing the heat flux normalized in some way by the energy density predominates in the literature on the subject. Here, it is demonstrated that this is true only for classical Maxwell–Boltzmann radiation and, in this case, the wellknown results of Minerbo are recovered. A similar singlevariable dependence postulated by Cernohorsky and Bludman for fermionic radiation cannot be justified since it leads to the contradiction with the consistency conditions between the moment evolution equations and the entropy balance. For Bose–Einstein radiation, we rederive and analyze the results given in the literature for lowenergy and highenergy limits. We also show that, except for those limiting cases, the Eddington factor for bosonic radiation cannot be represented as a function of a single normalized variable. In the present approach, the entropy function plays a crucial role in determining the system of evolution equations for the energy density and the heat flux. In this system, the flux of the heat flux, and hence the Eddington factor, is determined by the additional scalar potential uniquely related to the entropy function for each type of statistics. Since the Eddington factor cannot be expressed in terms of elementary functions, we propose to use the polynomial approximation. Namely, for Maxwell–Boltzmann, Fermi–Dirac, and Bose–Einstein statistics, we expand the entropy function in powers of the square of the heat flux and also calculate the corresponding power series expansion of the additional potential. By truncating the latter, we obtain the Eddington factor represented as the eighthorder polynomial in the heat flux with coefficients being the elementary functions of the energy density and the parameter which determines statistics. Finally, we analyze the behavior of the scalar Eddington factors in the limiting case when the normalized heat flux tends to one. Keywords:Spectral Eddington factor, Maximum entropy closure, Bosonic radiation, Fermionic radiation Affiliations:
 
10.  Larecki W., Banach Z., Consistency of the phenomenological theories of wavetype heat transport with the hydrodynamics of a phonon gas, JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL, ISSN: 17518113, DOI: 10.1088/17518113/43/38/385501, Vol.43, No.38, pp.385501124, 2010 Abstract: Various phenomenological theories of wavetype heat transport, which can be interpreted as the models of an isotropic rigid heat conductor with an internal vector state variable, have been proposed in the literature with the objective to describe the second sound propagation in dielectric crystals. The aim of this paper is to analyze the relation between these phenomenological approaches and the phonon gas hydrodynamics. The fourmoment phonon gas hydrodynamics based on the maximum entropy closure of the moment equations with nonlinear isotropic phonon dispersion relation is considered for this purpose. We reformulate the equations of this hydrodynamics in terms of energy and quasimomentum as the primitive fields and subsequently demonstrate that, from the macroscopic point of view, they can be understood as describing the reference model of an isotropic rigid heat conductor with quasimomentum playing the role of the internal vector state variable. This model is determined by the entropy function and the additional scalar potential, but if the finite domain of phonon wave vectors is approximated by the whole space, the additional potential can be expressed in terms of the entropy function and its first derivatives. Then the transformation of primitive fields and the expansion of thermodynamic potentials in powers of the square of quasimomentum enable us to compare the reference model with the models proposed earlier in the literature. It is shown that the previous models require some subtle modifications in order to achieve full consistency with phonon gas hydrodynamics. Affiliations:
 
11.  Banach Z., Larecki W., Zajączkowski W.^{♦}, Stability analysis of phonon transport equations derived via the ChapmanEnskog method and transformation of variables, PHYSICAL REVIEW E, ISSN: 15393755, DOI: 10.1103/PhysRevE.80.041114, Vol.80, No.4, pp.41114114, 2009 Abstract: Under the assumption of Callaway’s model of the BoltzmannPeierls equation, the ChapmanEnskog method for a phonon gas forms the basis to derive various hydrodynamic equations for the energy density and the drift velocity of interest when normal processes dominate over resistive ones. The first three levels of the expansion (i.e., the zeroth, first, and secondorder approximations) are satisfactory in that they are entropy consistent and ensure linear stability of the rest state. However, the entropy density contains a weakly nonlocal term, the entropy production is a degenerate function of variables, and the next order in the ChapmanEnskog expansion gives the equations with linearly unstable rest solutions. In the context of Burnett and superBurnett equations, a similar type of problem was recognized by several authors who proposed different ways to deal with it. Here we report on yet another possible device for obtaining more satisfactory equations. Namely, inspired by the fact that there exists no unique way to truncate the ChapmanEnskog expansion, we combine the ChapmanEnskog procedure with the method of variable transformation and subsequently find a class of εdependent transformations through which it is possible to derive the secondorder equations possessing a local entropy density and nondegenerate expression for the entropy production. Regardless of this result, we also show that although the method cannot be used to construct linearly stable thirdorder equations, it can be used to make the originally stable firstorder equations asymptotically stable. Affiliations:
 
12.  Banach Z., Larecki W., Modified ChapmanEnskog moment approach to diffusive phonon heat transport, PHYSICAL REVIEW E, ISSN: 15393755, DOI: 10.1103/PhysRevE.78.061137, Vol.78, No.6, pp.61137118, 2008 Abstract: A detailed treatment of the ChapmanEnskog method for a phonon gas is given within the framework of an infinite system of moment equations obtained from Callaway’s model of the BoltzmannPeierls equation. Introducing no limitations on the magnitudes of the individual components of the drift velocity or the heat flux, this method is used to derive various systems of hydrodynamic equations for the energy density and the drift velocity. For onedimensional flow problems, assuming that normal processes dominate over resistive ones, it is found that the first three levels of the expansion (i.e., the zeroth, first, and secondorder approximations) yield the equations of hydrodynamics which are linearly stable at all wavelengths. This result can be achieved either by examining the dispersion relations for linear plane waves or by constructing the explicit quadratic Lyapunov entropy functionals for the linear perturbation equations. The next order in the ChapmanEnskog expansion leads to equations which are unstable to some perturbations. Precisely speaking, the linearized equations of motion that describe the propagation of small disturbances in the flow have unstable planewave solutions in the shortwavelength limit of the dispersion relations. This poses no problem if the equations are used in their proper range of validity. Affiliations:
 
13.  Banach Z., Larecki W., ChapmanEnskog method for a phonon gas with finite heat flux, JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL, ISSN: 17518113, DOI: 10.1088/17518113/41/37/375502, Vol.41, No.37, pp.375502118, 2008 Abstract: The Chapman–Enskog perturbation method for a phonon gas is investigated with the use of Callaway's model for the Boltzmann–Peierls equation. Assuming that the effective relaxation time for normal processes is small and the effective relaxation time for resistive processes is large, this perturbation method proposes to expand the phase density about a displaced Planck distribution and to include the above two relaxation times in the expansion. The main advantage of using the displaced Planck distribution is that the drift velocity of a phonon gas is incorporated into the model in a nonperturbative manner. The result is a system of nonlinear secondorder parabolic equations for the energy density and the drift velocity which, unlike the usual set of hydrodynamic equations, does not restrict the magnitude of the individual components of the drift velocity and the heat flux in any way. This system is linearly stable at all wavelengths and is also fully consistent with the second law of thermodynamics in the sense that there exists a macroscopic entropy density which depends locally on the hydrodynamic variables and satisfies the balance equation with a nonnegative entropy production due to both resistive and normal processes. Affiliations:
 
14.  Banach Z., Larecki W., Kawashima condition for a hyperbolic moment model of phonon hydrodynamics, International Journal of Dynamical Systems and Differential Equations, DOI: 10.1504/IJDSDE.2008.023003, Vol.1, No.4, pp.263275, 2008 Abstract: We consider a 3 × 3 hyperbolic symmetrisable phonon system of balance laws that describes the onedimensional flow evolution of the energy density, the heat flux and the flux of the heat flux. This system possesses a strictly concave homogeneous entropy function and is derived by taking moments of the reduced BoltzmannPeierls equation with Callaway's collisional term and subsequently truncating and closing the resulting moment equations by means of the entropic approximation. Employing the entropy dissipation condition and the Kawashima condition, we verify the existence of global smooth solutions for initial data close enough to a constant equilibrium state. The two formulations are used: the formulation in terms of the entropy variables and the formulation in terms of the primitive variables. Affiliations:
 
15.  Banach Z., On the general relativistic symmetric hyperbolic moment system for matter and radiation: local rotational symmetry, CLASSICAL AND QUANTUM GRAVITY, ISSN: 02649381, DOI: 10.1088/02649381/24/16/007, Vol.24, No.16, pp.40794106, 2007 Abstract: For the onedimensional, locally rotationally symmetric case, we employ the relativistic Boltzmann equations and the moment methods to derive a 6D causal evolution system of firstorder symmetric hyperbolic form for a fluidradiation mixture, i.e., a relativistic gas composed of material particles and photons. The material medium is assumed to be a nonbarotropic perfect gas and the evolution equations for this medium are obtained by using the local Jüttner equilibrium distribution function as seen by an observer with arbitrary 4velocity. In order to derive the evolution equations for the photon gas, we develop the modified Gradtype method which consists of expanding the phase density (i.e., the number density of photons) about a local Planck equilibrium distribution function and including the tracefree anisotropic pressure in the expansion. In our approach, it is significant that the matter fluid's peculiar velocity and the radiative heat flux (relative to an arbitrary 4velocity vector field) are incorporated into the model in a nonperturbative manner, thereby allowing arbitrarily large values for the two nonvanishing components of these dynamical variables. The additional difference about the treatment here as opposed to other treatments is that we do not neglect terms in the evolution equations which involve small firstorder quantities multiplying the spatial frame derivatives of hydrodynamical variables. Affiliations:
 
16.  Banach Z., A symmetric hyperbolic formulation of the general relativistic moment equations for matter and radiation, JOURNAL OF HYPERBOLIC DIFFERENTIAL EQUATIONS, ISSN: 02198916, DOI: 10.1142/S0219891607001021, Vol.4, No.1, pp.65104, 2007 Abstract: For a fluidradiation mixture, i.e. a relativistic gas consisting of material particles and photons, we derive a 14D causal evolution system of firstorder symmetric hyperbolic form for a set of basic physical gasstate variables, which is defined relative to a network of observers with arbitrary 4velocities. This system is obtained by taking moments of the relativistic Boltzmann equation (the socalled central moments) and by truncating and closing the resulting system of moment equations by means of an appropriate (5+9)moment closure prescription. The material medium under consideration is assumed to be a nonbarotropic perfect gas described by a local Jüttner equilibrium distribution function. With regards to the photon gas, we employ the modified Gradtype approach, which begins by expanding the phase density (i.e. the number density of photons) about a local Planck equilibrium distribution and including the tracefree anisotropic pressure in the expansion. The main advantage of using the above (5+9)moment closure prescription is that the matter fluid's peculiar velocity and the radiative heat flux are incorporated into the model in a nonperturbative manner, thereby allowing virtually arbitrarily large values for the individual components of these dynamical variables. Within the framework of a reduced 3 + 1 orthonormal frame formalism, we also explain how to choose the initially arbitrary 4velocity vector field, as this is crucial for the determination of the full 38D causal matterradiationgravity system. Keywords:Generalrelativistic kinetic theory, central moments, fluidradiation mixture, modified Gradtype perturbative expansion approach, (5+9)moment closure, reduced 3+1 orthonormal frame formalism, causal symmetric hyperbolic systems Affiliations:
 
17.  Banach Z., Larecki W., Ninemoment phonon hydrodynamics based on the maximumentropy closure: one dimensional flow, JOURNAL OF PHYSICS AMATHEMATICAL AND GENERAL, ISSN: 03054470, DOI: 10.1088/03054470/38/40/018, Vol.38, No.40, pp.87818802, 2005 Abstract: The maximumentropy distribution functions applied to Callaway's model for phonon gas dynamics lead to a hierarchy of closed systems of moment equations. The system of equations for the energy density and the heat flux is the first member of this hierarchy of closures. Here emphasis is placed on analysing the next member, the 9moment maximumentropy system that involves the flux of the heat flux as an extra gasstate variable. After presenting a study of the onedimensional, rotationally symmetric reduction of this system, we explicitly calculate a single generating function of three Lagrange multipliers in terms of which the reduced system of three evolution equations for these multipliers can be cast into a symmetric hyperbolic form. In the context of determining the Lagrange multipliers as explicit functions of the moment densities, we discuss new aspects of the expansion of various nonequilibrium quantities about quasiequilibrium states. This expansion is fundamentally a nonequilibrium expansion that includes the heat flux in a nonperturbative manner, i.e., there are no unphysical limitations on the magnitude of the nonvanishing component of the heat flux to maintain a theory. Results are presented both at the first order in the expansion and at the second order. This enables us to verify the internal consistency of our approach and to justify the nonequilibrium generalization of the method of Grad. Affiliations:
 
18.  Banach Z., Grad`s Moment Truncation Procedure Versus Extended Irreversible Thermodynamics (Praca habilitacyjna), Prace IPPT  IFTR Reports, ISSN: 22993657, No.2, pp.1174, 1990 