The Annals of Frontier and Exploratory Science

Solid State Plasma Models

Have you noticed that in most if not in all the textbooks on atomic physics, statistical mechanics - there is no clear description:
- What is the subject of statistical averaging?
- How you do the statistical averaging ?

Do you count them (particles) one, two, three ... etc. ?

But they are moving all the time.

Do you count them as at the given point ?

But what is the point ?

If in the some physical theories (string theory, for example) the point could be taken as a one which is much less than the volume of 10(-105) m^3 =10(-78)nm^3 ??

There are theories recognizing the even smaller spatial scales.

Having said this, we are in no way saying some words of opinion or support in favor of this strangled theory, see more on that in -

" The Trouble With Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next"

From that piece - "The social constructivists claimed that the scientific community is no more rational or objective than any other community of human beings. ........
We tell our students that belief in a scientific theory must always be based on an objective evaluation of the evidence. Our opponents in the debate argued that our claims about how science works were mainly propaganda designed to intimidate people into giving us power, and that the whole scientific enterprise was driven by the same political and sociological forces that drove people in other fields."

Following our agenda, meanwhile, in the one scale mathematics the Point is the point (dot) - No Dimension in any direction!

At least - as for now on.

So, what to do with this? When the any almost (and might be everyone), and definitely all those we are talking about here particles are larger then the point? If we acknowledge the existence of different scales and treat physics on each scale REALLY - we need to address this issue seriously.


Do these particles come and leave the point if particle itself much larger then the point and we need to consider the centers of particles and their closest distance to the point?

So, we need to consider also the brim of the particle touching the Point, right ?

Well, please, don't remind me on the Quantum Mechanics some postulates, there should not be the sacred cows in physics, is this correct? We touch these subjects later on in this section.

Then we need to recognize that we are talking about the VOLUME for particle statistics, volume around of our point.

Even if we do not want to say this?

Well, this is not the physical rave, workers just don't see these "small" inconsistencies with definitions as a problem as long as they continue to do their way of living?

Now, let we drop the point definition off consideration for a while.

And so, let's do the volume to count for the statistics of the particle's behavior?

What is the volume to take for statistics?

In most, if not in any book on solid state and related software doing the MD modeling, statistics, they like to show the cube with the atoms, molecular movements -


And these objects are moving all the time - leaving and coming back into that selected volume -


So, they should cross the volume bounding surfaces. Right ?

What does this mean if taking these movements to the mathematics strictly as in Hierarchical Scaled Physics (HSP)?

Well, we come to the Volume-Surface relations in mathematics and physics, right?

Then, we need to go first through the introductory parts in all this new for most people in atomic physics approach to the mathematics of the Point (Dot) at the different spatial scale deliberation -


    Now we understand - this is not such a simple issue to build the conservative governing equations in this physics field - as the Plasma Physics.

    Each and every time when there are needs to talk, construct, erect the mathematical model for some specific plasma physical model - we need to find out

    - What is the single object (particle, anything) that we are dealing with ?


    - What is the Averaged subject we want to declare as directly related, tight to our single object when even it is not might be found at the definite point (point ?) in the given space at the sought time ?

    Many things discussed now in the plasma physics are already many times being concerned, debated and presented in the multiphase fluid mechanics, see our review on multiphase modeling in fluid mechanics in -

  • Are there any other Methods and Theories Available?

    Including the modeling, averaged governing equations derivation.

    Still, it is a big difference in physics and this should be mirrored in the mathematical modeling in plasma physics.

    We are presenting here just few selected points and mathematical outcomes for some modeling needs in solid state plasma (in other sub-sections, might be found various related issues).

    Again, we won't be involved into great number of plasma physics concerns, but few selected major points of mathematical modeling for multiphase collective subjects in plasmas are of interest for us as for the same reason - those are the multiphase, two- and more scale physical fields and their mathematics is the mathematics of HSP-VAT.

    We believe this will grow in time as this already happened with multiscale, heterogeneous, hierarchical media in other physical sciences through the last 40+ years.

    Any arguments, if not stupid would be considered openly here.

    When one starts to build the mathematical model for a solid state plasma physics problem, one of the main issues is - What is the Field's meaning? What does it mean - Local and Non-local, or Averaged, of the Upper (another) Scale value for this Field?

    Solid State and Complex Plasmas Multiscale Governing Equations


    A solid state is a medium which fairly can be recognized as a heterogeneous medium at any chosen scale when we judge the phenomena in each of two- (or more) "phases" and looking for these phenomena to be modeled with proper mathematical concepts.

    The great number of facts evidence in a favor of interplay of different physical processes in each "phase" (consistent component of the physical picture) at the same scale while they are being the important elements of a common process or transport.

    There is the need to understand the physics and to model multiscale characteristics in heterogeneous media. Also, at present, the common majority of experimental methods are designed for homogeneous processes or like homogeneous, making it difficult to assess the impact of various features of heterogeneity.

    The governing hydrodynamics, electrodynamics, and energy transport equations used in most past work in plasma physics had treated particulate transport and wave propagation, whether they are of differential or integro-differential type, as if they were in a homogeneous medium.

    This type of idealization significantly reduces the robustness of the physical description loosing the relationship between each scale from atomic to microscale parameters on the macroscopic behavior.

    The present work continues our suggested first in 90th the mathematical language for scaled formulation of heterogeneous problems in electrodynamics and plasma physics - the tools of heterogeneous media scaled description - Heterogeneous Scaled Physics-Volume Averaged Theory (HSP-VAT):


  • Electrodynamics

    and Optics (including scattering issues)

  • Optics

    and Acoustics

  • Acoustics

    At the present time it is the only consistent and reliable theory available for multiscale either 1D or 3D, linear or nonlinear, cross-connected between scales mathematical statement tasks. The VAT approach has been successfully applied in the last twenty plus years to a large number of difficult scaled problems in fluid mechanics, thermal physics, meteorology, electrodynamics, acoustics and other disciplines and technologies for heterogeneous media treatment.

    Some Basic Postulates Used in the Conventional One Scale Plasma Physics

    Here we intentionally will try to give references to the old publications, that are in a direct connection to the main theories of sub-atomic world and in current time COHP books, manuscripts, textbooks, articles, etc.

    In the book by Steel and Vural (1969) we can find -

    We can read in the page 43:

    "In summary, the basic equations of electron plasma considered as a conducting fluid are





    where the equation of "averaged" velocity $\QTR{bf}{u}$ conservation is



    where $v_{e}$ is the effective collision frequency coefficient.

    The current density $\QTR{bf}{j}_{m}$ and the charge density $\rho _{m}$ obey the equation of continuity


    which follows from Eqs. (4-20) and (4-22). In equation (4-23) we assumed that the pressure is isotropic. "

    "We now must relate the source terms to the dynamical motion of the electrons:






    This text above, the equations ground and the definitions for the charge density $\rho _{m}$ and the current density $\QTR{bf}{j}_{m} $ are clearly speak on the volumetric description of the physical fields (functions).

    We see - that in this excerpt in the conventional one scale homogeneous physics the subject of the point in the Dirac's function and of the electron's location point is taken as the same by the definition. That means - that either the electron is of no dimensions and the size or the delta function MATH acts over the volumetric space occupied by the electron. Which is not a volumeless.

    Both assumptions are contradict mathematically.

    Still, more of the physical ground is that - the delta function has spread it's qualities over the volume occupied by electron. O.K., but this is the volumetrical integration even for a single electron! And should be of volumetrical nature for the cloud of electrons in the solid state volume taken as the characteristical one.

    In both situations for the one scale homogeneous physics of the first half of the 20-th century it was convenient to have not (or to ignore) the description of the volume for integration.

    It has no clear description - just the "Volume" and this sum ! When everybody knows that to perform an integration one needs to depict exactly - What is the space for integration ? Any feature of it is needed for integration.

    Electrodynamics Basic Governing Equations in One Scale Plasmas

    Consider as the already like "averaged" homogeneously conventional Maxwell governing equations in semiconductor plasma




    In these above GE the variable's fields are already considered as the "averaged" variables - which is completely wrong, as soon as the fields are acting in Heterogeneous media, but Maxwell equation are used to be developed (at least justified and explained) on the base of Homogeneous GO theorem.

    A soon as we regard the scales in solids the ones that can be characterized as the meso- nano- and atomic scales we have to admit that the variables $\QTR{bf}{D}_{m},$ $\QTR{bf}{B}_{m},$ $\QTR{bf}{E}_{m},$ $\QTR{bf}{H}_{m},$ in this set and the equations itself can not be accepted as averaged, unless we formalize at least the two scale physics and their related mathematical models more strictly than just saying that these variables are "averaged" somehow. We would shortly feature below the few moments and arguments of the HSP-VAT applied to the formulation of basic governing equations in solid state and complex plasmas theories.

    The Representative Point Mathematical Definition and its Physical Meaning for Different (and Neighboring) Scales.

    We would like to say some notes regarding the definitions of these concepts, those taken as being granted. When the problem, the physics of the problem demanding and has the features of different scale physics, then we need to do something about it.

    The easiest way is just to ignore this and say for a particular problem - that "these are the respected scale features."

    As it is done with, for example, the Molecular Dynamics (MD) simulations. After simulation of the huge amount of atoms or molecules in the box, then the whole box properties are declared as the specified material's properties.

    The trick is that in MD also used numerous atomic scale physics approximations and assumptions, and still the results can be driven to a reasonable proximity to experiment, or vice versa.

    The reason to distinguish the point defined local and the non-local field's values at the same coordinate system is following the same famous theorems we are discussing here in this website - the Ostrogradsky-Gauss (OG) (or Gauss-Ostrogradsky - GO) theorem

    or the Heterogeneous Whitaker-Slattery-Anderson-Marle (WSAM) kinds of theorem.

    Thus, we start with the definition of a point, a dot used in the mathematical formulation of physical problems (not a strict one, which definition we leave for a more appropriate situation with the mathematics discussed).

    Definition 1:

    At the known and assigned previously system of coordinate the point is the object with no dimension in any of the three spatial (cartesian) coordinates, and this object has the descriptive features, which determine the location of that point in the assigned system of coordinates. The point located physical field's property has this spatial point determined value.

    Following the OG theorem we now know that - if at any point with the coordinates inside of the problem's domain is known the functional dependency for a physical field, which in the most of physical sciences right now is the partial differential or just differential equation(s), then we imply that the domain which served for the derivation of this equation via the OG theorem was the domain of the Lower subspace - because in that theorem we start doing an integration over the Homogeneous finite volume and the finite surface(s).

    Now - why not apply the same concept to the heterogeneous media, which means - that after the averaging provided according to the one of the WSAM theorems - we get the mathematical equation (dependency) of the higher (another) space. With the corresponding spatial dependencies and the topology of the physical spatial fields.


    Figure 4: Representative elementary averaging volumes (REVs) with the fixed points of representation.

    With the one substantial different feature - we can not infinitely reduce the size (well, if we not considering the sizes of kind of the string theory when the physical point could be taken as a one which is much less than the volume of MATH the volume of this domain - because we want to keep the most of descriptive features of the both (or more) phases inside of our spatial domain(s).

    Definition 2:

    The Upper Space point when connected to the Lower Scale Domain might reflect, determine, and establish the features of communications between the both space physics'. These features can be of different physical description in accordance to their respected space physics definitions. And vise versa - the Higher Space point physics might control, govern, reflect, and determine in some ways the properties of the Lower Scale physical Domain.

    The brightest example for viewing the situation as the said above is the transition from the atomic (atomistic) spatial scale dependencies - dependencies described by atomic physics discipline, to the continuum physics laws and dependencies.

    Solid State Plasma REV

    Meanwhile, no one seems can argue that the media in solid state of most of origin are heterogeneous media. If we would like to except the notion that the Dot, the Point on the Lower Scale in these physical fields are related, assumed as, to the volume approximately with MATH $m$ the side of the cube, then we can say that the Representative Elementary Volume (REV) or Point (Dot) in Plasma Physics fields, for governing equations for these kinds of fields - the commonly explained and excepted fields are $\Delta h_{U}=$ MATHor MATH

    And this is the three orders of linear one dimension magnitude separating the Point in the Lower Scale field and the Point of the Upper Scale physical field. Reasonably enough for most of theoretical schemes.

    With meaning of the physical representative (REV) Upper Scale point (REV) variable of the size of


    We do not discuss here in this work the lower than MATH $m$ spatial scales of physical fields as well as their corresponding physical notions, objects (as, for example, the string theory and strings size of MATH $m,$when a point could be taken as a one which is much less than the volume of MATH, their properties, peculiarities, relation to the atomic scale physics etc., etc.

    But we do this in other studies related.

    For fields of dusty plasmas with the $\mu m$ size "dust" particles the REV size would be reasonably taken as - MATH with the Lower scale point of the same size MATH $m$

    Electrodynamics Basic Governing Equations in Two Scale Solid State Plasmas

    The full description of derivation of VAT non-local Upper scale electrodynamics governing equations is given in publications by Travkin et al. (1999, 2001) and some earlier works.

    We use the notion that the Point (dot) in our averaged (Upper Scale) governing equations of electrodynamics in solid state are considered on the scale of MATH.

    Here we will write down only few governing equations, as for the electric field those after averaging over the phase $\left( m\right) $ using MATH become




    The phase averaged magnetic field equations are






    These equations and some of their variations as, for example,

    the electric field wave equation


    becomes (assuming the fixed (unmovable) interface surfaces)




    An analogous form has the often used averaged equation for the time-harmonic electric field




    with MATH

    The VAT based transient charge conservation equation in heterogeneous media obtains the form



    As it can be observed the most advantages feature of the heterogeneous media electrodynamics equations is the inclusion of terms reflecting phenomena on the interface surface $\partial S_{ms},$ and that fact can be used to incorporate morphologically precisely multiple effects occuring at the interfaces.

    Now we will add here for the completeness of the picture the averaged nonlinear equation for the electric potential



    Dusty (Complex) Plasmas - Local and Non-local Presentation. Does Local Magnitudes Meaning That?

    We consider here the following scales to make hierarchical scaled physical processes cross-connection of the different scales physics in the dusty plasmas evident. The models for reflecting each scale processes and dynamics are based on the recognition that the point determined and the averaged over the some domain physical fields are the different fields, cross-connected, directly tight but still the different physical fields and require the different mathematical models for each one.

    For physical fields of dusty plasmas with the $\mu m$ size "dust" particles the REV size would be reasonably taken as the Upper scale 1D span of the size - MATH with the Lower scale point of the same size MATH $m$ MATH $\mu m$

    This multiscale physical object does need the fields with meaning of physical representative (REV) point variable of the size of MATH $?$

    In the plasma chamber of the size

    MATH $10^{15}\mu m^{3}?$

    Probably - yes, inspite the huge amount of calculation this size REV domain needs for.

    For some statements this span of scales MATH is most likely too large, and depending on the dusty particles size, distribution features, and physics of that plasma the third intermediate scale might be introduced as of MATH $m$ MATH $\mu m.$

    As it is commonly accepted in the one scale plasma physics and usually can be taken as this kind (with, well, not important anyway variations) of governing equations for dusty (complex) plasmas:

    as the conservation of electrons equation (assumed as with the averaged number $n_{e}$ )


    and the equation of electron momentum conservation


    with the effective rate of electron collision $v_{e}^{eff},$

    which we may comparing to the electrons momentum conservation equation of Fushinobu et al. (1995)


    where included the terms - the last term MATH symbolizing the momentum change due to electron collisions with particles in the medium.

    Here one would explain that $v_{e}^{eff}$ could be taken as MATH

    In all other governing equations in this kind of the one scale studies can be seen the similar particles interaction reflecting the phenomenological terms. Interesting enough, that all the source terms as, for example, $S_{e}$ which is the overall rate of production-destruction of electrons are given through the coefficients


    where $v_{ion}$ is the rate of electron impact ionization of the neutral particles,

    $v_{ed}$ is the rate of collection of plasma species by the dust grains,

    $v_{att}$ is the rate of electron attachment to the neutral atoms resulting in negative ion production,

    $v_{wall}^{e}$ is the particle (electrons) loss at the walls of the discharge" chamber.

    In other conservation equations the same kind of production-destruction rates are explained as the nessessary parts of the balance in each process.

    So - all these production-destruction, as well as collision rates terms - are the result of knowledge about real events in the atomic scale plasma, and at the same time - the helplessness in terms of really strict explanation and derivation of models for these existing effects.

    We need to say that these are not the averaged equations as well.

    For example, the conservation of electrons really "phase" averaged variables equation will have the form



    where MATH means averaging over the major phase which is the interatomic and inter-particle subvolume of the REV volume in the dusty plasma,

    where the r.h.s. term might be annihilated because of the multiphase presense in the subvolume and of the correct averaging of the gradient term in the left hand side of this equation,

    where $\partial S_{md}$ is the "interface" (real or imaginable) of impact with the dust "phase" and scatterers,

    $\partial S_{mion}$ is the "interface" (real or imaginable) of impact with the neutral and ion "phase", while

    $\partial S_{m-}$ is the "interface" (real or imaginable) of impact with the negative ion "phase."

    It will be assumed here that not only immobile scatterers produce phase separation as in the case of solid state mostly.

    We skip here the more strict developments, saving the space for other features, etc.


    Laser Physics EM Wave Incident Upon the One, Two and Many Atoms (Nuclei) Matter

    It is one of the areas where the clear necessity for multi-body consideration appears via attempts to model the interference of the power laser radiation with the one, or two, and more atoms, nuclei. Which is the known problem not only in the laser physics. We would like to stress our attention onto the more exact expression of this kind of interaction of electromagnetic fields with the particles using more exact formulation and the solution, that still has been done so far for the single electron and a single nucleus.

    When intense EM wave (laser radiation) acts upon the piece of a matter, it turned out that under certain circumstances there is a possibility of amplification of the electromagnetic waves when they are going through the matter. To be more specific, laser radiation, going through the plasma, where the processes of the dispersion of the electrons on the nuclei are occuring, may intensify!

    The theory of strong electromagnetic radiation is dealing with this problem has been developed in some recent years.

    Using the Volkov's solution for wave functions, describing the state of real particles, it's possible to calculate the probabilities of different processes in the field of the EM wave. The mentioned method of calculation of interaction with the intensive field of the wave is used in many researches, as it allows to get general solution's formulae correctly, when there are arbitrary intensions of the wave. This mehod of examination of processes in the intensive electromagnetic field is a half-classic - the falling wave is considered as a classic field, and all the rest of the particles (electron, absorbed and radiated quantum) are considered as via the quantum-mechanics.

    We write down the Dirac's equation in the field of the plane EM wave for finding differential cross-sections of radiation and absorption:


    where MATH, MATH MATH где $\widehat{p}_{\mu }$ is the impulse operator component,$\ \psi (x)$ is electron wave function, $m$ is the electron mass, $\gamma ^{\mu }$ is the Dirac matrix, see more detail in -

  • Laser Optics and Problems of One (Many) Atom(s) (Nuclei) in Streaming EM Fields


    This equation was first solved by D. Volkov in 1937 and was widely adopted at the beginning of 60-th due to the creation of the laser technology. One of us (V.A.Tsibulnik) used this Volkov's solution models to develop the model for interference of the electron particle colliding with the nucleus resulting fields and the laser EM radiation.

    Computing the our task while choosing the 4-potential of the external electromagnetic field as a sum of two plane waves propagating along the axis $z$:


    we find to expand the Volkov's solution in this field in a double Fourier series:


    where partial wave functions MATH, corresponding to various processes of forced radiation and absorption by the electron the $n$ photons of the first wave and $n^{\prime }$ photons of the second wave.

    The so-called interference and non-interference areas were studied separately. Interference area is considered as an area where the dispersion occurs in the plane formed by the impulse of the initial electron and the wave vector.

    We would like to point out that the given here dependence for the average total differential cross-section dependent on the initial speed of the electron in the interference area as seen in this figure


    Among results of this analysis we had found that comparison of absolute quantities of the total cross-section in the interference and non-interference areas show that amplification of the electromagnetic waves usually occurs in the interference area and probability of that event almost 3 times higher than the corresponding probability in the non-interference area. In this way it has been shown that for certain conditions the laser rays going through the matter can amplify.

    The reason to continue study in this direction is obviously the same - as soon as we have to study and find out the interference picture for not a single nucleus and a single electron, but for the matter of collective interaction of the intense EM radiation with the cloud of nuclei and electrons in the same volume. The correct procedure for development in this field is also using the HSP-VAT.

    Two Scale HSP-VAT Solutions of Classical Problems in Many Body, Heterogeneous Media (Hydrodynamics, Wave Mechanics, Electrodynamics etc.).

    These classical problems - on straight capillary, single diameter globular morphologies, and 1D layered systems, are already present the examples of pure statements of the test-problem for more then hundred years. Each new generation of researchers trying to solve these problems based on the new tools, but did not obtain a complete solution. Further, the conjugate problem of heat transport in capillary systems has neither been stated fully nor correctly. The studies referred to in this website are the solutions for both scales obtained with the HSP-VAT and provide for the direct and strict communication of physical properties between the both scales.

    In all three classical categories of medium morphologies and for many types of physical processes we have achieved the solutions -

  • Classical Problems - in Fluid Mechanics

  • Classical Problems - in Thermal Physics

  • Two Scale EM Wave Propagation in Superlattices - 1D Photonic Crystals

  • Two Scale Solution for Acoustic Wave Propagation Through the Multilayer Two-Phase Medium

  • Globular Morphology Two Scale Electrostatic Exact Solutions

  • When the 2x2 is not going to be 4 - What to do ?

  • Effective Coefficients in Electrodynamics

    These solutions on the Lower Scale are just the same that known to physicists and presented in each university textbook on fundamental disciplines. Meanwhile, the Upper scale solutions could not be obtained with the COHP methods in a previous more than 100 years. What is given for some cases in the university textbooks is the profanation of solutions, are not the correct Upper scale solutions.

    The Upper Scale solutions in HSP have been obtained in some cases as exact. Also is important that the Upper Scale classical media solutions for few problems attained in an analytical form, which is unusual so far in HSP-VAT mathematics.

    The resolution of these questions on the results, meanings, connection of the Lower and Upper scale physical fields, on the balance and conservation of the Lower and Upper scale fields give the sufficient ground to seek the same kind features in the physical problems and their modeling in particle, atomic, nucler, and plasma physics.


    $\QTR{bf}{B}$ - magnetic flux density [Wb/m$^{2}$]

    $c$ - speed of light in vacuum MATH $[m/s]$

    $c_{p}$ - specific heat $[J/(kg\cdot K)]$

    $C$ - heat capacity per unit volume $[J/(m^{3}\cdot K)]$

    $ds$ - interface differential area in porous medium [$m^{2}$]

    $\partial S_{12}$ - internal surface in the REV [$m^{2}$]

    $\QTR{bf}{D\ }$- electric flux density [C/m$^{2}$]

    $e$ - charge of electron, MATH $[C]$

    $\QTR{bf}{E\ }$- electric field [V/m]

    $\widetilde{f_{i}}$ $\equiv $ MATH- VAT intrinsic phase averaged over $\Delta \Omega _{i}$ value $f$

    $<f>_{f}$ - VAT phase averaged value $f$, averaged over $\Delta \Omega _{i}$ in a REV

    MATH - VAT morpho-fluctuation value of $f$ in a $\Omega _{i}$

    $\hbar $ - reduced Planck constant MATH $[J$ $s]$

    $I_{\nu }$ - radiation intensity [W/(m$^{2}$ sr Hz)]

    $I_{\nu b}$ - spectral blackbody intensity [W/(m$^{2}$ sr Hz)]

    $I_{b}$ - total blackbody radiation intensity [W/(m$^{2}$ sr)]

    $I_{b21}$ - blackbody specific surface radiation intensity [W/(m$^{2}$)]

    $I$ - current $[A]$

    $\QTR{bf}{j}$ - current density $[A/m^{2}]$

    $<f>_{t}$ - time averaged value $f$

    $k_{B}$ - Boltzmann constant, MATH $[J/K]$

    $k_{1}=k_{f}$ - fluid phase thermal conductivity $[W/(mK)]$

    $k_{2}=k_{s}$ - homogeneous effective thermal conductivity of solid phase [$W/(mK)$]

    $\QTR{bf}{H}$ - magnetic field [A/m]

    $m$ - porosity [-]

    MATH - averaged porosity [-]

    $m_{e}$ - electron rest mass MATH $[kg]$

    $m^{\ast }$ - electron effective mass $=0.066\times m_{e}$ $\ [kg]$

    $n$ - is the electron number density $[1/m^{3}]$

    $N_{D}$ - doping concentration of the active layer $[1/m^{3}]$

    $n$ - refraction index [-]

    $\QTR{bf}{q}^{r}$ - radiation flux [$W/m^{2}$]

    $p$ - phase function [-]

    MATH - solid phase fraction [-]

    $S_{12}$ - specific surface of a porous medium MATH [$1/m$]

    $T$ - temperature $[K]$

    $T_{2}=T_{s}$ - solid phase temperature $[K]$

    MATH - interface surface temperature when $i$ is in upward direction $[K]$

    $W$ - energy density per unit volume $[J/m^{3}]$


    $e$ - electron

    $f$ $\equiv 1$ - fluid phase

    $s$ $\equiv 2$ - solid phase


    $\thicksim $- value in fluid phase averaged over the $\Delta \Omega _{1}$

    $\ast $ - complex conjugate variable

    Greek letters

    MATH - averaged heat transfer coefficient over $\partial S_{12}$ $[W/(m^{2}K)]$

    $\beta $ - total extinction coefficient

    $\beta _{\nu }$ - extinction coefficient [1/m]

    $\varepsilon _{d}$ $-$ dielectric permittivity [Fr/m]

    $\varepsilon _{s}$ $-$ dielectric permittivity of GaAs MATH $[Fr/m]$

    $\varepsilon _{r}$ $-$ low-frequency relative dielectric permittivity of GaAs $=12.8$ $[$ -$]$

    MATH $-$ high-frequency relative dielectric permittivity of GaAs $=10.9$ $[$ -$]$

    $\varepsilon _{0}$ $-$ dielectric permittivity of vacuum MATH $[Fr/m]$

    $\varepsilon _{ij}$ - radiative hemispherical emissivity from phase $i$ to phase $j$ with phase $i$ being up into the direction $\QTR{bf}{\Omega }$

    MATH - total radiative hemispherical emissivity from phase $2$ to phase $1$ in the REV

    $\lambda _{j}$ - prescribed Markovian transition length in medium $j$ [m]

    $\mu _{m}$ - magnetic permeability $[H/m]$

    $\mu _{e}$ - electron mobility MATH $]$

    $\nu $ - frequency $[Hz]$

    $\rho _{ch}$ - electric charge density $[C/m^{3}]$

    $\rho $ - mass density $[kg/m^{3}]$

    $\sigma =k_{B}$ - Stephan-Boltzmann constant [W/(m$^{2}$ K$^{4}$)] $[J/K]$

    $\sigma _{e}$ - medium specific electric conductivity $[A/V/m]$

    $\Phi $ - electric scalar potential $[V]$

    $\psi (x)$ - electron wave function (atomic physics) $[$ $\ ]$

    $\psi $ - particle intensity per unit energy (frequency)

    MATH - interface ensemble-averaged value of $\psi ,$ with phase $j$ being to the left

    MATH - ensemble-averaged value of $\psi $

    $\omega $ - angular frequency $[rad/s]$

    $\varkappa _{\nu a}$ = $\varkappa _{a}$ - absorption coefficient $[1/m]$

    $\varkappa _{\nu s}$ = $\varkappa _{s}$ - scattering coefficient $[1/m]$

    $\Delta \Omega $- representative elementary volume (REV) $[m^{3}]$

    MATH $\Delta \Omega _{f}$ - pore volume in a REV $[m_{3}]$

    MATH- solid phase volume in a REV $[m_{3}]$

    $\tau $ - relaxation time $[s]$

    $\tau _{m}$ - electron momentum relaxation time $[s]$

    References in Solid State Plasmas, Atomic and Sub-atomic Scales Physics:

    Steel, M.C. and Vural, B., Wave Interaction in Solid State Plasmas, McGraw-Hill, New York, 1969.

    Vorotyntsev, M.A. and Kornyshev, A.A., Electrostatics of a Medium with the Spatial Dispersion, Nauka, Moscow (in Russian). 240 p. 1993.

    Travkin, V.S., Catton, I., Hu, K., Ponomarenko, A.T., and Shevchenko, V.G., "Transport Phenomena in Heterogeneous Media: Experimental Data Reduction and Analysis," in Proc. ASME, AMD 233, 1999, Vol. 233, pp. 21-31.

    Travkin, V.S. and Catton, I., "Radiation Heat Transport in Porous Media," in Proc. ASME, HTD-364-3, 1999, Vol. 3, pp. 31-40.

    Travkin, V.S. and Catton, I., "Tranport Phenomena in Heterogeneous Media Based on Volume Averaging Theory", in Advances in Heat Transfer, 2001, Vol. 34, pp. 1-144.

    Vinogradov, A.P., Electrodynamics of Composite Materials, Moscow, Aditorial URSS, (2001).

    Vinogradov, A.P., Physics Uspekhi, 172, No. 3, 363-370 (2002).

    Landau, L.D. and Lifshits, E.M., Electrodynamics of Continuous Media. Oxford, Pergamon Press, (1960).

    Primak, A.V., Shcherban, A.N. and Travkin, V.S., "Turbulent Transfer in Urban Agglomerations on the Basis of Experimental Statistical Models of Roughness Layer Morphological Properties," in Trans. World Meteorological Organization Conference on Air Pollution Modelling and its Application, Geneva, 1986, Vol. 2, pp.259-266.

    Travkin, V.S., Catton, I., Ponomarenko, A.T., and Kalinin, Yu.E., "Bottom Up and Top Down, from Nano-Scale to Micro-Scale, Hierarchical Descriptions of Electrodynamic, Thermal and Magnetic Fields in Ferromagnets and HTSCs", in Proc. DOE 20th Symposium on Energy Engineering Sciences, Argonne National Laboratory, 2002, pp. 296-304.

    Travkin, V.S. and I.Catton, Advances in Colloid and Interface Science, 76-77, 389-443 (1998).

    Travkin, V. S. and Ponomarenko, A. T., Inorganic Materials, 40, Suppl. 2, S128 - S144 (2004).

    Travkin, V. S. and Ponomarenko, A. T., Journal of Alternative Energy and Ecology, No. 3, 9-19 (2005).

    Travkin, V. S. and Ponomarenko, A. T., Journal of Alternative Energy and Ecology, No. 4, 9-22 (2005).

    Travkin, V. S. and Ponomarenko, A. T., Journal of Alternative Energy and Ecology, No. 5, 34-44 (2005).

    Travkin, V.S., Nanotechnologies - General Concept for Pretty Large Amount of Pretty Small Gadgets Embedded Into Something and Consequences for Design and Manufacturing,, (2006)

    Travkin, V.S., Electrodynamics 2 - Elements 3P (Polyphase-Polyscale-Polyphysics),, (2013)

    Travkin, V.S. and Bolotina, N.N., "The Classical and Sub-Atomic Physics are the Same Physics,", (2013)

    Travkin, V.S., Particle Physics - Heterogeneous Polyscale Collectively Interactive,, (2011)

    Travkin, V.S., Particle Physics (Particle Physics 2). Fundamentals,, (2013)

    Travkin, V.S., Nuclear Physics Structured. Introduction,, (2006-2013)

    Travkin, V.S., What's Wrong with the Pseudo-Averaging Used in Textbooks on Atomic Physics and Electrodynamics for Maxwell-Heaviside-Lorentz Electromagnetism Equations,, (2009)

    Travkin, V.S., Incompatibility of Maxwell-Lorentz Electrodynamics Equations at Atomic and Continuum Scales,, (2009)

    Travkin, V.S., Experimental Science in Heterogeneous Media,, (2005)

    Travkin, V.S., Statistical Mechanics Homogeneous for Point Particles. What Objects it Articulates?, (2014)

    Travkin, V.S., Solid State Polyscale Physics. Fundamentals,, (2014)

    Travkin, V.S., "Two-Scale Three-Phase Regular and Irregular Shape Charged Particles (Electrons, Photons) Movement in MHL Electromagnetic Fields in a Vacuum0 (Aether),"

    Travkin, V.S. and Bolotina, N.N., "Two-Scale Two-Phase Formation of Charged 3D Continuum Particles - Sphere and Cube From Electrons in a Vacuum0 (Aether). An Example of Scaleportation of Charge from the Sub-Atomic to Continuum Charged Particles, Conventional MD Cannot be Applied,"

    Travkin, V.S. and Bolotina, N.N., "One Structured Electron in an Aether (Vacuum0) Electrodynamics, Many Electrons in an Aether Fixed in Space - the Upper Scale Galilean Electrodynamics ,"

    Travkin, V.S. and Bolotina, N.N., "Electrons and CMBR (Cosmic Microwave Background Radiation) Flux of Photons in a Vacuum0 (Aether) - Two-Scale Galilean Theory ,"

    and in publications mentioned above.

    With time, more topics will be uploaded into this and neighboring sub-sections for education and for advertisement.

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