Instytut Podstawowych Problemów Techniki

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The mixed three-moment hydrodynamic description of fermionic radiation transport based on the Boltzmann entropy optimization procedure is considered for the case of one-dimensional 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 non-negative 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 H-theorem 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 non-negative source term.

Słowa kluczowe:

fermionic radiation, mixed moments, moment realizability domain, entropy optimization problem, symmetric hyperbolicity

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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 closed-form analytic Kershaw-type and B-distribution closure procedures have been used. It is shown that the Kershaw-type closure procedure can also be applied to the spectra moment equations of fermionic radiation. First, a description of the Kershaw-type closure for the system consisting of an arbitrary number of one-dimensional moment equations is presented. Next, the Kershaw-type two-field and three-field 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 two-field case to three space dimensions is also presented. The fermionic Kershaw-type closures differ from those previously derived for classical and bosonic radiation. It is proved that the obtained one-dimensional systems of transport equations are strictly hyperbolic and causal. The fermionic Kershaw-type closure functions behave qualitatively in the same way as the fermionic maximum entropy closure functions, but attain different numerical values.

Słowa kluczowe:

Fermionic radiation, Moment equations, Moment realizability problem, Kershaw-type closure, Three-moment transport

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The two-field radiation hydrodynamics in n spatial dimensions is derived from the kinetic theory of radiation. Both the full-moment (frequency-independent) and spectral (frequency-dependent) formulations of radiation hydrodynamics are considered. The derivation is based on the entropy principle of extended thermodynamics of gases. In the case of the full-moment 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 full-moment 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 full-moment and spectral formulations.

Słowa kluczowe:

radiative transfer equation, H-theorem, radiation hydrodynamics, entropy principle, radiation pressure tensor

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This paper analyzes the propagation of the waves of weak discontinuity in a phonon gas described by the four-moment 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 dispersion-relation 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 wave-front speed on the sample’s temperature with the empirical relations of Coleman and Newman (1988) describing for NaF and Bi the variation of the second-sound speed with temperature. It is demonstrated that the calculated temperature dependence of the wave-front 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 second-sound 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 wave-type phonon heat transport and that the Dubey nonlinear isotropic dispersion-relation model can be very useful for this purpose.

Słowa kluczowe:

Phonon hydrodynamics, Waves in phonon gas, Genuine nonlinearity, Nonlinear phonon dispersion relation, Second sound

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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 high-frequency 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 λ-stability 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.

Słowa kluczowe:

Hyperbolic system, Multiple eigenvalue, Linearization, Harmonic wave, Dispersion relation, Weak discontinuity

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The spectral formulation of the nine-moment radiation hydrodynamics resulting from using the Boltzmann entropy maximization procedure is considered. The analysis is restricted to the one-dimensional flows of a gas of massless fermions. The objective of the paper is to demonstrate that, for such flows, the spectral nine-moment 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 one-to-one correspondence between the Lagrange multipliers and the moments. The maximum entropy closure of moment equations results in the symmetric conservative system of first-order 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.

Słowa kluczowe:

fermionic radiation, spectral moments, maximum entropy closure

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We analyse in detail the three-moment 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 one-dimensional flows. Thus, the independent or non-vanishing 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.

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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 first-order 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 well-known results of Minerbo are recovered. A similar single-variable 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 low-energy and high-energy 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 eighth-order 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.

Słowa kluczowe:

Spectral Eddington factor, Maximum entropy closure, Bosonic radiation, Fermionic radiation

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Various phenomenological theories of wave-type 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 four-moment 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 quasi-momentum 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 quasi-momentum 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 quasi-momentum 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.

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Under the assumption of Callaway’s model of the Boltzmann-Peierls equation, the Chapman-Enskog 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 second-order 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 Chapman-Enskog expansion gives the equations with linearly unstable rest solutions. In the context of Burnett and super-Burnett 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 Chapman-Enskog expansion, we combine the Chapman-Enskog procedure with the method of variable transformation and subsequently find a class of ε-dependent transformations through which it is possible to derive the second-order 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 third-order equations, it can be used to make the originally stable first-order equations asymptotically stable.

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A detailed treatment of the Chapman-Enskog method for a phonon gas is given within the framework of an infinite system of moment equations obtained from Callaway’s model of the Boltzmann-Peierls 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 one-dimensional 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 second-order 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 Chapman-Enskog 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 plane-wave solutions in the short-wavelength limit of the dispersion relations. This poses no problem if the equations are used in their proper range of validity.

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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 non-perturbative manner. The result is a system of nonlinear second-order 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 non-negative entropy production due to both resistive and normal processes.

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We consider a 3 × 3 hyperbolic symmetrisable phonon system of balance laws that describes the one-dimensional 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 Boltzmann-Peierls 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.

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The maximum-entropy 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 9-moment maximum-entropy system that involves the flux of the heat flux as an extra gas-state variable. After presenting a study of the one-dimensional, 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 non-equilibrium quantities about quasi-equilibrium states. This expansion is fundamentally a non-equilibrium expansion that includes the heat flux in a non-perturbative 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 non-equilibrium generalization of the method of Grad.

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The heat transport in non-metallic micro- and nanostructures is predominantly due to phonon processes, and therefore it can be analysed in terms of a flow of a phonon gas. Phonons are quantized lattice vibrations characterized by wave-vectors k from the first Brillouin zone and by the k dependent frequency ω. For simplicity, a single-branch phonon model is adopted, and consequently phonon polarizations are neglected. The dispersion relation ω(k) together with the relaxation times associated with normal and resistive phonon scattering processes determine the behaviour of a phonon gas. The commonly used linear isotropic approximation of the phonon dispersion relation ω(k) = c ǀkǀ employed in the phonon gas hydrodynamics leads to the constant speeds of thermal waves propagating into the region in thermal equilibrium. This contradicts the experimental results on the second sound propagation in solids.
The nonlinearity of the phonon dispersion relation significantly influences the thermal properties of micro- and nanostructures. Several forms of the nonlinear phonon dispersion relation ω(k) have been suggested for various structures and substrates in the literature, motivated either by empirical data or by the first-principle calculations. In micro- and nanostructures, the wave-type heat transport has been observed at low and elevated temperatures with temperature dependent propagation speeds. Hence, the phonon gas hydrodynamics employing the linear isotropic approximation of phonon dispersion relation seems to be inadequate in such cases. The governing conservation equations for energy and the quasi-momentum (understood as a vector internal state variable) are determined by the entropy function and by the additional scalar potential. Both, the entropy function and the additional potential are given by integral formulae involving the nonlinear isotropic phonon dispersion relation. Approximation of the finite domain of phonon wave-vectors by whole space R3 eliminates the additional potential and simplifies the form of conservation equations. For this approximation, the propagation of the waves of weak discontinuity into the region in thermal equilibrium has been analysed, and the dependence of the wave speeds on the temperature in a region ahead the wave front has been determined. In order to compare predictions of the theory derived with the second sound experimental data from the literature, the nonlinear isotropic phonon dispersion relation ǀkǀ = ω c-1 (1+b ω2) is adopted. For the values of the parameters c and b for NaF and Bi given in the literature, the dependence of the speed of weak discontinuity wave on the temperature ahead the wave front has been calculated and compared with the experimentally measured second sound velocity as a function of the sample temperature, and with the calculations based on the alternative second sound theory. Our results are in good agreement with the experimental data as well as with the known theoretical predictions, and show that the nonlinearity of the phonon dispersion relation plays the crucial role in the effect of temperature dependence of thermal wave speeds.

Słowa kluczowe:

phonon gas, phonon dispersion, phonon heat transport, second sound, nanostructures

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