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The system of inequalities I , II does not forbid the evolution shown in Figure 1.


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In this case, the model of the space homogeneous physical system is not correct. Hypothetical time evolution of the H -function. Now what happens to the fluctuations that develop in the system? Consequently, Eq. Relation 1. For multi-component gas, the analog of Eq. For ideal gas using the assumption about the statistical independence of molecules, the simplification of 1. To this end, one needs to answer the question of how it happened that inequality 1. This effect is a direct result of approximation 1. Thereby, the irreversibility of time is introduced as well.

In other words, one cannot speak of the principle of causality without using the concept of irreversibility of time. What may be the result of formal rejection of the principle of causality in this particular case? Inequality a 1. Therefore, the principle of entropy increase follows directly from the principle of irreversibility of time. The probability of reversible processes in closed dissipative physical systems is vanishingly small, and one must use GBE in the form of 1. However, as regards gigantic in comparison with human life time intervals, it is of interest to investigate dissipative and, nevertheless, reversible systems.

Evolution of the time parts of distribution function 1. It follows from Figure 1. Moreover, in the process of evolution the system may find itself in the state of thermodynamic equilibrium and, nevertheless, leave the latter state. What happens at these points? Consequently, the Boltzmann entropy may experience vibrations, shown in Figure 1. We use 1. One can see now that the reversibility of processes shows up, on the level of oneparticle description as well, as the reflection of reversible processes in a physical system, i. By analogy with the notation of difference schemas with weight, one can replace Eqs.

Of course, subsequent reasoning will no longer be rigorous, but it will be nevertheless useful for understanding the situation. Of course, the difference-differential operator in 1. In Eq. Is it possible to derive a differencedifferential approximation with the negative sign at the second substantial derivative?

No doubt, this is possible. We will now use the assumptions, which have brought us to the approximate notation of the GBE in the form of 1. Therefore, two mathematically equivalent difference approximations 1. Thereby, one of the main paradoxes, providing a subject for discussions in the Boltzmann physical kinetics, is resolved. We will now treat the theory of kinetically consistent difference schemes KCDS see, for example, Elizarova and Chetverushkin, , ; Elizarova, The ideas of this theory date back to the studies by Reitz in particular, Reitz, , who used the splitting method to solve problems of the theory of transfer to the kinetic and hydrodynamic stages.

As a result, we derive a system of integro-differential equations with additional terms on the right-hand sides, as distinct from the classical hydrodynamic equations , which present, by virtue of selected approximation, a combination of second space derivatives multiplied by the time step.

This approach does not lead to a new hydrodynamic description. Moreover, as it follows from derivation 1. In this case as well, the additional terms were introduced without adequate substantiation. This does not mean, naturally, that the return to differential The reason for this situation is quite obvious: it is impossible to obtain a qualitatively new physical description by using a formally higher difference approximation for the classical equation.

It is assumed that all the three coefficients are identical, and the difference between them may be taken into account by using some other, more complex, smoothing function. One can perceive an analogy between the continuity equations of Slezkin I. We see little point in discussing the remaining analogous HE.

Generalized Boltzmann Physical Kinetics

In closing this section we want to emphasize the fundamental point that the introduction of the third scale, which describes the distribution function variations on a time scale of the order of the collision time, leads to the single-order terms in the Boltzmann equation prior to Bogolyubov-chain-decoupling approximations, and to terms proportional to the mean time between collisions after these approximations.

It follows that the Boltzmann equation requires a radical modification, which, in our opinion, is exactly what the generalized Boltzmann equation provides. Generalized Boltzmann equation and iterative construction of higher-order equations in the Boltzmann kinetic theory Let us consider the relation between the generalized Boltzmann equation and the iterative construction of higher-order equations in the Boltzmann kinetic theory.

From Eq.


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  • To this end, we can write the second approximation in the form using the notation of approximation 1. Thus, with the notation of relation 1. This result can be easily proved by using the integration in 1. In the next section, these results will be discussed in the general theory of correlation functions. Return now to a spatially homogeneous system free from forces. We obtain from 1. The dots in this equation indicate that the procedure of constructing the series may be continued by this algorithm.

    The representation of the distribution function in a series form, Eqs. Note also that the appearance of the minus sign on the right-hand sides of Eqs. Generalized Boltzmann equation 83 1. Generalized Boltzmann equation and the theory of non-local kinetic equations with time delay It is of interest to examine the relation between the Boltzmann equation and the theory of kinetic equations accounting for time delay effects. We resort to the Bogolyubov equation 1. Consequently, in this approach, it is the integral term containing W2 which must lead to the Boltzmann or a more general collision integral.

    The correlation function W3 may be neglected. Using Eq. We resort to the Bogolyubov condition of the weakening of initial correlations corresponding to a certain initial instant of time t0 see Eq. Because of Assumption 3, Eq. The collision of the probe particles, 1 and 2, is dominated by the forces of their internal interaction, so that see Eqs. Delay is sufficiently small, so that the linearization in the delay time can be used. The sum of the first two terms in Eq. If we now substitute f2 1, 2 from Eq. The second collision integral accounts for the time delay effect and is amenable to a differential approximation analogous to Eq.

    To obtain this approximation, the following assumption is made. Generally speaking, integration with respect to time in Eq. However, to the linear approximation in the delay time this contribution is negligible. Thus, the appearance of the second substantial derivative with respect to time in the generalized Boltzmann equation may be considered as a differential approximation to the time delay integral that emerges in the theory of correlation functions for kinetic equations.

    However, this is not the case. Each of the approaches outlined above actually complements one another and is interrelated with one another. The generalized Boltzmann equation can be treated both from the point of view of a higher-order Boltzmann theory and as a result of differential approximations to the collision integral accounting for time delay effects. There is another point to be made. Dt Dt 1. However, in the transition to the hydrodynamic limit after multiplying the kinetic equation by the collision invariants and subsequently integrating over velocities , the Boltzmann integral term vanishes, while the second term on the right-hand side of Eq.

    A well-known example of a local approximation to a non-local integral term in the kinetic theory is the Enskog theory of transport processes in a dense gas composed of hard spheres. Finally we can state that introduction of control volume by the reduced description for ensemble of particles of finite diameters leads to fluctuations of velocity moments in the volume. This fact can be considered in a definite sense as a classical analog of Heisenberg indeterminacy principle of quantum mechanics.

    Johnson by the measurement of current fluctuations of thermo-electron emission. Let n be an alternate direction coinciding with positive directions of coordinate axes in physical space. Then relation 2. Then from 2. But apart from energy from translation movement, molecules have, generally speaking, vibration and rotation energy. Given chemical reactions, the transition of potential energy of molecules into kinetic energy of particles should be taken into account.

    All these kinds of energy including energy of chaotic movement identify as internal energy of gas. Hydrodynamic Enskog equations Recall derivation of the Enskog hydrodynamic equations. Therefore, GBKT does not create additional difficulties connected with collision integrals.

    Transformations of the generalized Boltzmann equation The generalized Boltzmann equation GBE inevitably leads to formulation of new hydrodynamic equations, which are called generalized hydrodynamic equations GHE. Classical hydrodynamic equations of Enskog, Euler and Navier—Stokes are particular cases of these equations.

    For example, in 2. The last term on the right side of relation 2. As a result, the transition to generalized hydrodynamic equations GHE requires more effort. The result of this procedure leads to the generalized Enskog equations of continuity, momentum and energy. Generalized continuity equation 1 Multiply GBE 2. Formally 22 terms should be transformed, here and in the following we of necessity demonstrate only character and complicated elements of these transformations and restrict our consideration to some comments in more simple cases. Consider integrals the transformation of which is realized by integration in parts.

    Nevertheless, we repeat the main points of this consideration because of their importance. Recall the phenomenological derivation of continuity equation. Control volume is defined in an area occupied by gas. The characteristic length of this control volume is much bigger than mean free path between collisions but much smaller than hydrodynamic scale.

    Physical principles of the generalized Boltzmann kinetic theo..|INIS

    Write down the mass balance for this volume with transparent boundary Generalized Boltzmann Physical Kinetics surface. In another way, variation of mass in control volume could be connected only with fluxes of particles directed inside or outside of reference surface. It means that only point-like particles are taken into consideration. This fact — as it was in Chapter 1 in the course of GBE derivation from the Bogolyubov chain — is principal restriction of Boltzmann kinetic theory. In the generalized Boltzmann kinetic theory GBKT , considered particles have finite sizes and then at some instant of time can be placed partly inside, partly outside of reference surface.

    This fact, along with the use of the DF form oriented for describing the point structureless particles, leads to appearance of fluctuation terms, in particular in 2. In dimensionless form 2. Therefore, indicated fluctuation terms are proportional to Knudsen number and are really small in the case of small Knudsen numbers in the regime of continuum media. Qualitative pictures of these regimes are shown for small Kn Figures 2. Theory of generalized hydrodynamic equations Fig. Small Kn, particles of finite size.

    Small Kn, point-like particles. Large Kn, particles of finite size. Large Kn, point-like particles. But no reason to think that for Kn 1 fluctuation terms could be omitted. As will be shown in the next chapter, the consideration of these terms is of principal significance for turbulence description on the Kolmogorov micro or sub-grid scale. Also should be noticed that for the case of Kn 1 hydrodynamic equations including the generalized continuity equation belong to the class of differential equations with small parameter in view of senior derivatives.

    Consider now transformations of the second integral in 2. Below are given examples of mentioned transformations. Generalized hydrodynamic Euler equations Formulate now the summary of the generalized Enskog hydrodynamic equations for components. In particular, GEnE are of higher order than classical Enskog equations. This brings up immediately two problems: 1 is it possible to simplify these equations? Let us begin by looking at the first question.

    The use in BKT instead of the first approximation, the convergence series in Sonine polynomials for hard spheres leads to slightly changing of mentioned coefficient. One obtains 0. Theory of generalized hydrodynamic equations In the theory of generalized hydrodynamic equations, relations 2. If a mixture of gases contains particles of which the masses are not too different, it is possible to set in Eqs. The difficulties of principal character leads to the problem calculation of averaged values in 2. But for local Maxwellian DF all averaged values in 2.

    Summary of results follows. In calculations realized in 2. We have the following generalized Euler continuity equation compare with 2. Boundary conditions in the theory of the generalized hydrodynamic equations Let us consider now the problem of the additional boundary conditions in the theory of GHE.

    With this aim write down once more Eq. The explicit form of coefficient Av is not significant for us here, notice only that the thermal velocity vT is used as scale for molecular velocity near the surface. The length L can be significantly different and should be introduced in correspondence with concrete solving problem.

    Tend now L to zero using an arbitrary law of this tendency. It means that in 2. After dividing Eq. This very remarkable fact will be discussed in the next chapter from turbulence positions. Relations 2. Domain in the vicinity of the streamed wall. These conditions are the direct consequence of the Boltzmann equation and allow to expand the area of the formal appliance of the Navier— Stokes description. Further we intend to show that GHE contain known kinetic boundary conditions as asymptotic solutions near the wall. It means on the one hand that GHE can be applied for description of the rarefied gas without additional kinetic conditions serving for adjusting kinetic and hydrodynamic solutions in boundary kinetic layer.

    With the aim of analytical investigation we introduce reasonable assumptions often used in the theory of kinetic Knudsen layer. By application of 2. In kinetic layer the process of relaxation is being realized by adjusting the dynamic parameters of particles after desorption to the hydrodynamic parameters of flow.

    For us of interest is the process on the bottom of Knudsen layer and corresponding asymptotic GHE solutions. In this case we omit the second logarithmic in square brackets in 2. From 2. After substitution of 2. In physical chemistry cases are known Glasstone, Laidler and Eyring, , when absorbed particles move with a velocity along the surface while being bound by mentioned surface. In classical hydrodynamics usl cannot be introduced without taking into account the Boltzmann equation.


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    • Formula 2. As a result, formula 2. Let us neglect density jump in Knudsen layer. Notice that indicated by Cercignani four ciphers after point in numerical factors in 2. As a conclusion we state that GHE incorporate the kinetic effect of slip as an element of adjusting of hydrodynamic and kinetic regimes of flow and allow — avoiding artificial fashions — to describe flows by intermediate flow numbers.

      This peculiar feature of GHE will be demonstrated next, particularly in calculations of sound propagation in rarefied gas and shock wave structure for arbitrary Mach numbers. The classical Enskog, Euler, and Navier—Stokes equations of fluid dynamics are special cases of these new equations. The derivation of the generalized hydrodynamical equations is given in the previous chapter.

      But the area of applications of GHE and corresponding principles used for their derivation is much vaster. We intend to discuss these new possibilities of generalized hydrodynamic equations in the theory of turbulent flows, electrodynamics of continuum media and quantum mechanics. About principles of classical theory of turbulent flows The turbulent fluid motion has been the subject of intense research for over a hundred years because it has numerous applications in aerodynamics, hydraulics, combustion and explosion processes, and hence is of direct relevance to processes occurring in turbines, engines, compressors, and other modern-day machines.

      The scientific literature on this subject is enormous, and a detailed analysis of all the existing models is beyond the scope of this book.

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      Here the object is to discuss the currently available turbulence concept in the context of generalized equations of fluid dynamics. We will also see how this picture corresponds to the generalized Boltzmann kinetics and will try to find out which of the known approaches may be used and which should be abandoned. It is commonly held that a fully developed turbulence may be characterized by the irregular variation of velocity with time at each point in the flow, and that hydrodynamic quantities undergo fluctuations turbulent ones or pulsations.

      Their scale varies over the wide range from the external using the terminology of Landau and Lifshitz, scale comparable to the characteristic flow size, to a small scale on which the dynamic fluid viscosity begins to dominate. From the large-scale fluctuations, the energy goes practically undissipated to the small-scale ones, where viscous dissipation takes place Richardson model of For want of a better model, it is assumed that turbulent motion is described by the same equations of fluid mechanics Navier—Stokes equations used for laminar flows, with a consequence that turbulence emerges as a flow instability or, in this particular case, as an instability in the Navier—Stokes flow model.

      This gives rise to many inconsistencies, however. This contradicts experimental data. In , W. Heisenberg published a study on the instability of laminar flows Heisenberg, A year later, E. What about the rigorous mathematics then? The notions of averaged and fluctuating motions prompted Reynolds to explicitly isolate the fluctuation terms in the Navier—Stokes equations and to subsequently average them over a certain time interval. But neither this approach nor the later technique of averaging over the masses of liquid volumes sometimes called Favre averaging Favre, provides close solutions, and indeed neither of them is adequate when it comes to physics because, as we will see below, the Navier—Stokes equations are not written for true physical quantities.

      Strict theory of turbulence One further approach to the problem involves the evaluation of velocity correlation functions with the aim of establishing the relation between the velocities at two neighboring flow field points within the theory of local turbulence. For example, the simplest correlation function is the second-rank tensor!

      The theory of correlation functions attracted a great deal of attention after Keller and Fridman first introduced them into the hydrodynamics of turbulent motions back in In , Landau gave a comprehensive assessment of this line of research. In reality, however, such a formula cannot exist at all as the following argument shows. One can but agree with this view. To put it another way, if the Kolmogorov scale admits an explicit universal formulation for turbulent fluctuations as we will show later on , then large-scale fluctuations are determined by solving a specific boundary-value problem.

      Theory of turbulence and generalized Euler equations We now apply the generalized hydrodynamical equations to the theory of turbulence and demonstrate that they enable one to write explicitly the fluctuations of all hydrodynamic quantities on the Kolmogorov turbulence scale lK. We start by writing down the generalized hydrodynamical equations and, for the sake of simplicity, employ the generalized Euler equations for the special case of a one-component gas flow in a gravitational field. Lg Thus, the continuity equation 3.

      All the generalized equations of Euler fluid dynamics contain the Reynolds, Euler, and Frud numbers similarity criteria. Naturally, the inclusion of forces of electromagnetic origin would lead to additional similarity criteria. For small-scale fluctuations i. The fluctuation terms thus determine turbulent Kolmogorov-scale fluctuations small-scale fluctuations Strict theory of turbulence or, using the computational hydrodynamics term, sub-grid turbulence which are of a universal nature and not problem specific.

      To fully understand the situation, however, the following questions remain to be answered: 1 Are there no contradictions in the system of fluctuations introduced in this way? In other words, is the set of fluctuations self-consistent? In answering the above questions, the generalized Euler equations for one-component gas will be employed for the sake of clarity.

      Implicit in the following analysis will be the fact, already noted above, that we are dealing with small-scale fluctuations. Ignoring the fluctuation terms squared and keeping only first-order small quantities in relations of type 3. Dependent fluctuations for example, pf should be calculated using independent fluctuations of hydrodynamic values.

      Let us find the fluctuation of this moment. Use now 3. Are they coincided, Mv 3. However, a rigorous approach based on the generalized Euler equations leads to a residual with respect to true quantities on the right of the equation of energy 3. This problem does not exist for generalized Enskog equations written for genuine DF. But situation is more complicated. Even if relation 3. The theory presented here overcomes this problem by simply reverting to the formulation of the hydrodynamic equations in terms of the true quantities.

      And it is only in the case when turbulent fluctuations are completely absent or, equivalently, when the average product of hydrodynamic quantities is equal to the product of their averages, that we arrive at the classical form of the Euler and, of course, Navier—Stokes equations. In this case the following Kovalevskaya theorem can be formulated for Cauchy problem: 0 If all functions Fi are analytical in a vicinity of point t0 , x10 ,.

      The system of generalized hydrodynamic equations cf. System of fluctuations of hydrodynamic values can be used for investigation of the flow stability. Basic guidelines of stability can be obtained on the basement of theory of differential equations with small parameter in front of derivative Elsholz, The formal possibility is existing of applications of these qualitative considerations for investigation of the problems of stability in the frame of GHE.

      Notice really that in one-dimensional non-stationary case the velocity fluctuation has the form see Table 3. But in all cases we can confirm the Heiseberg affirmation — stated in on the basement of the stability investigation of the Navier—Stokes equations Heisenberg, — that small but finite liquid viscosity leads in the definite sense to the destabilizing influence on the flow in comparison with the case of ideal liquids.

      Let us consider the procedure of calculations of dependent fluctuations with help of independent fluctuations coming from the minor tensor velocity moments containing in continuity equation 2. The general character of procedure corresponds to obtaining of related values for generalized Euler equations and, as in previous case, all found fluctuations are summarized in Table 3. As this takes place, the fluctuations of tensor moments of the zeroorder and the first-order appearing in momentum equation coincide in other words do not contradict with corresponding fluctuations found from continuity equation.

      Now we state that the most general boundary conditions for GHE should we written as follows compare with 2. In the past many attempts were made at phenomenological refining of the Boltzmann equation see, for example, Belotserkovski and Oparin, , p. In conclusion several remarks are necessary: 1 By calculation of turbulent fluctuations contained in Table 3.

      It means that calculation of terms in curly brackets of Table 3. These fluctuations of hydrodynamic quantities are universal and tabulated. Recall that this problem does not exist in the generalized Boltzmann physical kinetics; one needs only to return to equations written for genuine variables.

      It is interesting to discuss this fact from position of the GBKT. Notice immediately, that existence of relation 3. Transform now Eq. With this aim divide left and right sides of 3. Obviously, in classical limit Uqu turns to zero.

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      One of these problems is considered as an example below. Consider quantum mechanics analogue of stabilized flow. Then from Eqs. Appearance of the additional electro-kinetic momentum leads to the change of probability density; quantum hydrodynamics reflects this fact by the rate of fictitious particles formation. Difficulties, arising by numerical investigations of quantum mechanics problems, are well known.

      From this point of view it is interesting to apply well-developed methods of numerical hydrodynamics see, for example, Richtmyer and Morton, ; Doering and Gibbon, ; Fletcher, for solution of quantum mechanics tasks. Numerical calculation of system of hydrodynamic equations 3. Relation 3. The results are in good coincidence with analogous results obtained by the method of partial waves Abakumov and Vinogradov, a dashed curve.

      This method of quantum hydrodynamics was applied later to calculations with taking into account external force fields and non-spherical atom potentials. Strict theory of turbulence Fig. From this point of view no surprise comes in appearance of discrete structures in flow investigations like strange attractors Doering and Gibbon, Quantum mechanics technique of quantization can be propagated in physics of continuum on the whole, and in particular in hydrodynamics. Chapter 8 contains examples of corresponding approach. But for open systems, for example for interaction of atom with radiation, the situation is radically different.

      For construction of quantum mechanics of open systems the principles of the generalized hydrodynamics can be used. This situation reflects the state of total thermodynamic equilibrium. In general case of open quantum system, the energy equation should be used with consideration.

      As we see the generalized hydrodynamic equations could be the theoretical basement of quantum mechanics of open systems. Investigations of transport processes in gas discharge, quantum generators, magnetohydrodynamic generators and so on are based on physics of weakly ionized gases and, in particular, on the Boltzmann kinetic theory BKT.

      Methods of physics of weakly ionized gas are significant not only for technical applications but also for fundamental physics, for example for physics of ionosphere and astrophysics. The traditional area of application of the Boltzmann kinetic theory is the physics of a weakly ionized gas. It is interesting to see what the GBE yields in this case and how its results differ from those of the classical theory.

      To answer this fundamental question, let us consider the classical Lorentz formulation of the problem. We assume that the magnetic field is either absent or has a static component Bz , and that the electric field is along the x-axis; all inelastic interactions are neglected. Topology of the external electromagnetic field and models of particle interactions will be discussed in the following for every considered case separately.

      Dt Dt Dt 4. As the GBE theory suggests, the collision integral can be taken in the Boltzmann form. Multiplying Eqs. The continuity equations obtained from Eq. The solution of Eq. Then, from Eqs. From Eqs. For a weakly ionized Lorentz gas, the effect of the self-consistent forces of electromagnetic origin can be neglected. CBE for multicomponent reacting mixture of gases can be written in the form 1.

      Then, multiplying the GBE 4. The right-hand sides of Eqs. The generalized Boltzmann equation and the system of GHEs can be used to study plasma in an electric field, in particular to understand the electron energy runaway ef- Generalized Boltzmann Physical Kinetics fect Golant, Zhilinski and Sakharov, We now proceed to apply the generalized Boltzmann kinetic theory to the classical problems of plasma physics.

      Later on, Druyvesteyn a, b and Davydov obtained analytical expressions for the distribution function and transport coefficients for the special case of elastic collisions. More recent work note, in particular, the monograph of Ivanov, Lebedev and Polak has been aimed principally at investigating the effect of inelastic collisions on the DF and transport processes within the BKT framework.

      It is important to note that the calculation of the DF depends heavily on what model of particle interaction is adopted — and hence ultimately on the collision cross sections involved. The corresponding system of linked equations was obtained elsewhere Ginzburg and Gurevich, a, b. The solution to the GBE 4. For our further calculations in this section we assume that the force Fe is along the positive direction of a certain chosen coordinate axis. We Physics of a weakly ionized gas now substitute expansion 4.

      It is relations 4. We take up the former case first. Multiplying Eq. Substituting Eq. These are in fact quite obvious. Thus, Eq. There is no need to do this, however. Indeed, multiplying Eq. To accomplish this, we multiply Eq. In concluding the discussion of this limiting case, we present the corresponding form of Eq. In this case, the analogue of Eq. The boundary conditions for Eq. In order to numerically integrate Eq. Although Eq. Let us consider here some numerical results for the distribution function of charged particles in an external electric field, produced when employing the generalized Boltzmann equation.

      Numerical integration of corresponding differential equations was realized by the three-diagonal method of Gauss elimination techniques for the differential second-order equation see Appendix 4. In Figure 4. By the end of his life Boltzmann went over to fluctuation theory, in which the decrease of the H -function in time is only treated as the process the system is most likely to follow. This interpretation, however, is not substantiated by his kinetic theory since the origin of the primary fluctuation remains unclear the galactic scale of such fluctuation included.

      One of the first physicists to see that Boltzmann equation must be modified in order to remove the existing contradictions was J Maxwell. Maxwell thought highly of the results of Boltzmann, who in his turn did much to promote Maxwell electrodynamics and its experimental verification. We may summarize Maxwell's ideas as follows. The equations of fluid dynamics are a consequence of the Boltzmann equation. From the energy equation, limiting ourselves to one dimension for the sake of simplicity and neglecting some energy transfer mechanisms in particular, convective heat transfer , we obtain the well-known heat conduction equation.

      The temperature distribution given by this equation is unsatisfactory physically. On the other hand, at any arbitrarily distant point x the temperature produced by an instantaneous heat source will be different from zero for arbitrarily small times. While this difference may be small, it is a point of principal importance that it has a finite value. This implies an infinitely fast propagation of heat, which is absurd from the point of view of molecular-kinetic theory.

      In the courses of mathematical physics this result is usually attributed to the fact that the heat conduction equation is derived phenomenologically, neglecting the molecular-kinetic mechanism of heat propagation. However, as has been already noted, the parabolic equation I. Some of Maxwell's ideas, phenomenological in nature and aimed at the generalization of the Boltzmann equation, are discussed in Woods' monograph [10].

      Work on the hyperbolic equation of heat conduction was no longer related directly to the Boltzmann equation but rather was of a phenomenological nature. Without expanding the details of this approach, we only point out that the idea of the improvement of Eq. For the first time in modern physics this idea was formulated by B Davydov [ 11] see also interesting discussion about priority between C Cattaneo and P Vernotte [12—15]. The wave equation I.

      We do not present the technical details in Introduction but refer the reader to the classical works cited above or, for example, to Ref. Some fundamental points of the problem are worth mentioning here, however.

      In other words, the set of integro-differential equations turns out to be a linked one, so that in the lowest-order approximation the distribution function f 1 depends on f 2. This means formally that, strictly speaking, the solution procedure for such a set should be as follows. First find the distribution function fN and then solve the set of BBGKY equations subsequently for decreasingly lower-order distributions. But if we know the function fN , there is no need at all to solve the equations for fs and it actually suffices to employ the definition of the function.

      We thus conclude that the rigorous solution to the set of BBGKY equations is again equivalent to solving Liouville equations. On the other hand, the seemingly illogical solution procedure involving a search for the distribution function f 1 is of great significance in kinetic theory and in non-equilibrium statistical mechanics. This approach involves breaking the BBGKY chain by introducing certain additional assumptions which have a clear physical meaning, though.

      These assumptions are discussed in detail below. The Boltzmann equation is invalid for time lengths of the order of the collision times. Notice that a change from the time scale to the length scale can of course be made if desired. For Boltzmann particles the distribution function is automatically normalized to an integer because a point-like particle may only be either inside or outside a trial contour in a gas — unlike finite-diameter particles which of course may overlap the boundary of the contour at some instant of time. Another noteworthy point is that the mean free path in Boltzmann kinetic theory is only meaningful for particles modeled by hard elastic spheres.

      Other models face difficulties related, though, to the level of one-particle description employed. The requirement for the transition to a one-particle model is that molecular chaos should exist prior to a particle collision. The advent of the BBGKY chain led to the recognition that whatever generalization of Boltzmann kinetic theory is to be made, the logic to be followed should involve all the elements of the chain, i. This logical construction was not generally adhered to.

      In , N Slezkin published two papers [ 22, 23] on the derivation of alternative equations for describing the motion of gas. The assumption of a variable-mass particle implies that at each point a liquid particle, close to this point and moving with a velocity v , adds or loses a certain mass, whose absolute velocity vector U differs, as Slezkin puts it, by a certain appreciable amount from the velocity vector v of the particle itself.

      Since there are different directions for this mass to come or go off, the associated mass flux density vector Q is introduced. By applying the laws of conservation of mass, momentum, and energy in the usual way, Slezkin then proceeds to formulate a set of hydrodynamical equations, of which we will here rewrite the continuity equation for a one-component nonreacting gas:.

      The mass flux density Q is written phenomenologically in terms of the density and temperature gradients. Thus, we now have fluctuation terms on the right-hand side of Eq. At very nearly the time of the publication of Slezkin's first paper [ 23], Vallander [25] argued that the standard equations of motion are ill grounded physically and should therefore be replaced by other equations based on the introduction of additional mass Qi fluxes i where, to quote, " D 1 is the density self-diffusion coefficient, D 2 is the thermal self-diffusion coefficient, k 1 is the density heat conductivity, and k 2, the temperature heat conductivity.

      Heuristic and inconsistent with Boltzmann's theory, the work of Siezkin and Vallander came under sufficiently severe criticism. Note that Siezkin and Vallander also modified the equations of motion and energy for a one-component gas in a similar way by including self-diffusion effects. Possible consequences of additional mass transfer mechanisms for the Boltzmann kinetic theory were not analyzed by these authors. Boltzmann's fluctuation hypothesis was repeatedly addressed by Ya Terletskii see, for instance, Refs.

      To secure that fluctuations in statistical equilibrium be noticeable, Terletskii modifies the equation of perfect gas state by introducing a gravitational term, which immediately extends his analysis beyond the Boltzmann kinetic theory leaving the question about the irreversible change of the Boltzmann H -function unanswered. Some comments concerning terminology should be done.

      In recent years, possible generalizations of the Boltzmann equation have been discussed widely in the scientific literature. Since the term generalized Boltzmann equation GBE has usually been given to any new modification published, we will only apply this term to the particular kinetic equation derived by me for example in Refs. The corresponding equation is known also in the literature as Alexeev equation. Obviously it is not convenient for me to apply this term.

      Moreover in the following this kinetic equation will be transformed in the basic equations of the unified theory of transport processes BEUT valid in the tremendous diapason of scales—from the internal structures of so called elementary particles to the Universe expansion. L Woods see, e. The phenomenological equation I. A weak point of the classical Boltzmann kinetic theory is the way it treats the dynamic properties of interacting particles.

      On the one hand, as the so-called physical derivation of the BE suggests [ 1, 6, 35, 36], Boltzmann particles are treated as material points; on the other hand, the collision integral in the BE brings into existence the cross sections for collisions between particles. A rigorous approach to the derivation of the kinetic equation for f from the BBGKY equations, then a passage to the BE implies the neglect of non-local effects it time and space. Given the above difficulties of the Boltzmann kinetic theory BKT , the following clearly interrelated questions arise.

      First, what is a physically infinitesimal volume and how does its introduction and, as a consequence, the unavoidable smoothing out of the DF affect the kinetic equation [ 30]? Conversely, a change in the macroscopic description will inevitably affect the kinetic level of description. Because of the complexity of the problem, this interrelation is not always easy to trace when solving a particular TTP problem.

      The important point to emphasize is that at issue here is not how to modify the classical equations of physical kinetics and hydrodynamics to include additional transport mechanisms in reacting media, for example ; rather we face a situation in which, those involved believe, we must go beyond the classical picture if we wish the revised theory to describe experiment adequately. The alternative TTPs can be grouped conventionally into the following categories:. One of the pioneering efforts in the first line of research was a paper by Davydov [ 11], which stimulated a variety of studies see, for instance, [37] on the hyperbolic equation of thermal conductivity.

      Introducing the second derivative of temperature with respect to time permitted a passage from the parabolic to the hyperbolic heat conduction equation, thus allowing for a finite heat propagation velocity. However, already in his paper B I Davydov points out that his method cannot be extended to the three-dimensional case ' and that here the assumption that all the particles move at the same velocity would separate out a five-dimensional manifold from the six-dimensional phase space, suggesting that the problem cannot be limited to the coordinate space alone.

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