Theoretical Evidence for Wave Nature of Micro Particle and New Theory of Its Collective Motion in Material

Since a material is composed of micro particles, investigating behavior of those particles is essentially dominant for materials science. The diffusivity of diffusion equation is relevant to not only a collective motion of micro particles but also a motion of single particle. An elementary process of diffusion was thus theoretically investigated in a local space and time. As a result, the investigation concluded that the wave nature of micro particle results from denying the mathematical density theorem of a real time in the Newton mechanics. In other words, the basic theory of quantum mechanics is established in accordance with the cause-and-effect relationship in the Newton mechanics, for the first time, regardless of the de Broglie hypothesis. In relation to the collective motion of micro particles, the new diffusion theory was also reasonably established using the universal expression of diffusivity obtained here. In the present paper, the new findings indispensable for the fundamental knowledge in physics are thus systematically discussed in accordance with the theoretical frame in physics.


Introduction
There is occasionally an important relation in natural phenomena where it is universally valid under a given condition. When we cannot reveal the theoretical evidence, it has been accepted as a law or a principle in physics. Further, the equation derived theoretically from its law or principle has been accepted as a basic The basic theory in physics lies in Newton's laws established under the condition of having the common time between arbitrary coordinate systems. Einstein's relativity [1], which is one of the modern physics, was established in 1905 by denying the absolute time in the Newton mechanics in accordance with the constant principle of light speed c. However, Newton's laws are still acceptable as dominant ones in physics under the condition of v c  , where v is a speed of a body with mass m.
On the other hand, the quantum theory, which is the other modern physics, was established by accepting the de Broglie hypothesis [2] of where , h λ are the Planck constant [3] and a wave length of matter-wave defined here. In the Newton mechanics, both the particle nature and wave one are not simultaneously accepted. The conception of matter-wave is thus extremely novel. We had not understood the cause-and-effect relationship between the Newton mechanics and the quantum mechanics until recently. In the following, it will be thus revealed that the quantum mechanics is theoretically established by denying the mathematical density theorem of a real time in the Newton mechanics.
In 1923, de Broglie assumed that the result 2 mc of Einstein's relativity and the photon energy hυ of the Planck theory relevant to a light of frequency υ are equivalent to each other. Further, he assumed that c υ corresponds to λ of a matter-wave, if we accept the replacement of c v → in the relation mc h c υ = obtained here. After that, the experimental results revealed that an electron has an intrinsic nature like a wave [4]. In relatively recent years, the experimental results revealed that an atom or a molecule has also an intrinsic nature like a wave [5]. However, we did not confirm whether such a micro particle satisfies Equation (1) or not, even if it had an intrinsic nature like a wave. Further, we did not understand the theoretical evidence that a micro particle in the Newton mechanics has a wave nature.
For a micro particle with mass m , the wave equation For a collective motion of micro particles in the space-time ( ) , , , t x y z , Fick's first law relevant to the diffusion flux F J given by was proposed in 1855, where ( ) , , , C t x y z and D are a concentration of micro particles and a diffusivity [7]. Fick's second law was accepted as a nonlinear partial differential equation of where † ∇ =− ∇  is used because of the Hermite conjugate of the Dirac bracket for a differential operator. In physics, Equation (3) shows that the diffusivity D is a proportional factor of a concentration gradient to the diffusion flux F J . In mathematics, Equation (4) shows that D is an operator in the operator Ω . As far as we accept Equations (3) and (4), therefore, we cannot understand the physical essence of D from their equations.
Here, the new findings are as follows.
1) The diffusion equation, having been accepted as the Fick second law since 1855, was theoretically derived from the mathematical theory of Markov's process [8] [9] [10]. It was then found that the elementary quantity of diffusivity D corresponds to the angular momentum and is expressed as 2 D m =  [11] [12] [13]. The universal expression of diffusivity, which is applicable to any diffusion problem, was also derived.
2) The diffusion equation using 2 D m =  for a micro particle in an isolated local space was transformed into the wave Equation (2) of Schrödinger by denying the mathematical density theorem of a real time in the Newton mechanics [11] [12] [13]. The wave nature of an arbitrary micro particle was, for the first time, theoretically revealed in accordance with the cause-and-effect relationship between the quantum mechanics and the Newton mechanics. Further, it was theoretically revealed that Equation (1) of matter-wave is not a hypothesis but a basic equation in physics [12] [13].
3) The general solutions of nonlinear diffusion equation, which had never been solved since 1855, were reasonably obtained [14]. As a result, the new diffusion theory of a multi-components system was reasonably established in relation to the transformation from a diffusion equation of a moving coordinate system into that of a fixed one, and vice versa [9] [10].
As far as a material is composed of micro particles, investigating behavior of those particles is indispensable for research subjects in the materials science. In the following, it will be theoretically confirmed that the theory of diffusion plays an important role for fundamental problems in the materials science.

Theoretical Frame in Classical Quantum Theory
From a viewpoint of theoretical frame in physics, fundamental problems in the In the space-time ( ) , , , t x y z , the Markov process [8] in mathematics is applicable to such behavior as a collective motion of micro particles in an isolated physical system. As a result, the diffusion equation of was theoretically obtained as a nonlinear partial differential equation [9] [10]. In the isolated local space, the diffusivity D relevant to a micro particle with mass m is then obtained as ( ) where ( ) p mv = is a momentum of micro particle. Equation (6) shows that the diffusivity satisfies the relation of parabolic law. Further, a micro particle in the isolated local space has an angular momentum because of the term rp ∆ . In other words, the micro particle makes a circuit on the surface of isolated local space. This corresponds to phenomena known as a lattice vibration or a thermal vibration of atoms in a material.
The diffusion Equation (5) derived here is a moving coordinate system judging from the derivation process. It will be revealed that we can transform it into the relation (4) of a fixed coordinate system, and vice versa. It will be also revealed that the diffusivity expression obtained here plays a complementary role for incompletion of the theoretical frame in the quantum theory.
As seen from the atomic hypothesis of Dalton in 1803 and the law of Avogadro in 1811, the chemists in those days thought that a material is composed of atoms or molecules as fundamental particles. On the other hand, the velocity distribution function of Maxwell in 1859 or Boltzmann in 1968 was reported in physics as a problem of mechanical elastic collision between these fundamental particles. The theoretical frame developed here was relevant to a theory between the thermodynamics and the Newton mechanics, where the averaged impulse resulting from collisions between micro particles corresponds to a thermodynamic pressure of macro quantity in physics. Further, the equipartition of energy reported in 1876 was the theory that a mechanical energy of micro particle corresponds to an absolute temperature of macro quantity in the thermodynamic state [15].
By assuming a micro particle as a component of material, it seemed in those days that the equipartition of energy reveals what a material is composed of fundamental particles. However, it was found that the equipartition of energy cannot explain the theory of specific heat in a low temperature region. In the end of 19 century, therefore, there was no such firm theory in physics that a material is composed of atoms and/or molecules as fundamental particles.
In circumstances mentioned above, Planck in 1900 [3] [16] and Compton in 1923 [17] revealed that the light has both a wave nature and a particle one.
On the other hand, in relation to the Brown motion relevant to a random movement of pollen in water, Einstein in 1905 [18] revealed that the self-diffusion phenomena of water molecules are visualized by behavior of pollen. In other words, the relation of diffusivity obtained here shows that a material is composed of atoms and/or molecules as fundamental particles. As mentioned later in a head of Einstein's paradox, however, there is a problem in the theoretical frame developed then. In addition, Langevin in 1908 [19] also derived a similar relation to Einstein's theory from analyzing an equation of motion for a micro particle.
In accordance with the empirical equation of radiant light reported by Balmer in 1885 [20], Bohr in 1913 [21] proposed a model of atomic structure under the quantum condition and the frequency condition. In the model, an electron moves on a specific circular orbit n r ( 1, 2, n =  ) around the nucleus and it jumps from an orbit to an adjacent orbit through the radiation or absorption of energy E hυ ∆ = . Using a momentum p of electron, the notation the quantum condition is rewritten as After the Bohr model, the experimental results of Frank and Hertz in 1914 [22] suggested that an electron of the Bohr model moves on a specific circular orbit around the nucleus concerned. Further, it was experimentally revealed that the electronic beam has diffraction phenomena of a wave characteristic [4]. Based on the experimental results, it is considered that an electron of the Bohr model has a wave nature as a matter-wave. When an electron of the Bohr model satisfies the relation of 2 r it moves stably on a specific circular orbit as a stationary wave. Here, if we eliminate r ∆ from Equations (7) and (8), Equation (1) reasonably obtained.
It was thus found that Equation (1) is valid for an electron of the Bohr model.
Here, we think the function ϕ of progressive wave in the space-time ( ) , t r given by where the notations , A k and ω are an amplitude of vibration of a frequency υ , a wave number vector of   (1) is valid also for a motion of free electron. At this point, as far as we discuss the motion of an electron, Equation (1) is not a hypothesis but a basic equation in accordance with the theoretical frame in physics.
In circumstances mentioned above, de Broglie in 1923 [2] proposed Equation (1) as a hypothesis applicable to an arbitrary micro particle. After that, Schrödinger in 1926 [2] derived a wave Equation (2) from using Equation (1). It was found that the Schrödinger equation is applicable to behavior of an arbitrary micro particle. The Schrödinger equation is directly derived from the relation of matter-wave then. The theoretical evidence that a micro particle in the Newton mechanics has a wave nature has never been revealed since 1926. The theoretical frame of the quantum mechanics is thus still incomplete without revealing the causality for the Newton mechanics.

Consistency of Quantum Theory with Basic Theory in Physics
The basic idea in physics lies in the Newton mechanics. For example, Einstein's relativity was established by denying the absolute time in the Newton mechanics.
In the following, it will be revealed that we can establish the quantum theory by denying the mathematical density theorem of a real time in the Newton mechanics.
As a formal problem between Equations (2) and (5) Here, the Fick second law shows that the diffusivity is related to a micro particle in the diffusion region. Nevertheless, we cannot grasp the physical essence of diffusivity relevant to a micro particle from the Fick laws. About that matter, as mentioned above, Okino began by deriving the diffusion equation from the mathematical theory of Markov process in order to grasp the physical essence of diffusivity. In the following, the fundamental theory in physics will be developed using Equation (6) obtained then.
Applying the equipartition of energy to a free electron in a material gives the relation of ( ) where B k , ε and α are the Boltzmann constant, a correction term for the uncertain principle at T = 0 and a degree of freedom of micro particle, for example, 3 α = in case of a mono-atomic molecule. Since the free electron satisfies depends only on an absolute temperature. Here, note that there is no characteristic quantity of a free electron in Equation (10) in spite of the discussion about the free electron itself. This means that Equation (10) is valid also for an arbitrary micro particle in a material [13] when the equipartition of energy is applied to it. It is, therefore, revealed that Equation (7) of L rp ∆ = ∆ =  becomes also valid for an arbitrary micro particle in a material [11] [12] [13].
By substituting Equation (7) into Equation (6), the elementary quantity of diffusivity yielding is obtained in the isolated local space. In the following, the wave nature of an arbitrary micro particle will be revealed using Equation (11).
When d is a distance between two micro particles A and B of the same kind in the isolated local space, it is necessary to observe a reflected light of wave length d λ < for the discrimination of A from B. If d is very small, a high energy of c λ  is necessary for the discrimination. As a result, we cannot then discriminate them because of a turbulence caused by the high energy. In that situation, we consider an elastic collision problem between the above micro particle A and B in the following, where the particle A moves with a velocity A 0 v v = and the particle B is in the rest state of the velocity B 0 v = .
If we can identify the difference between the micro particles A and B in the space and time given by ( ) On the other hand, if we cannot identify the difference between them, it seems For the behavior of the particle A in the collision time resulting from the impossibility of the discrimination of A from B between B 0 t t < ∆ < ∆ [11] [12] [13].
The matter mentioned here is equivalent to denying the mathematical density theorem of a real time in the Newton mechanics. In other words, there is a minimum unit time t ε as a real time and the relation of t i t ∆ → ∆ is reasonably acceptable in the region of t t ε ∆ < [11]. Generally, as can be seem from rewriting a partial differential equation into the difference equation, it means a re- → is valid, the general solution of partial differential equation becomes not a complex value function but a complex function then.
In accordance with the discussion mentioned above, accepting the impossibility of discrimination between two particles of the same kind in a local space corresponds to accepting the relation of differential operator given by in the present theory. It was revealed that there is no conception of acceleration for a motion of micro particle in a local space [13]. As shown in the relation of matter-wave, however, the conception of velocity is still valid in a local space. Therefore, judging from the correlation between differential operators expressed should be consequently valid. It will be found that the imaginary operator shown here corresponds to a real eigenvalue of the Hermite operator. In addition, as seen from the above discussion, note that a local space is clearly real and r i r ∆ → ± ∆ is not valid then.
A plus or a minus sign of the above imaginary operators is determined from eigenvalues of Equation (9). When i ± ∇ operates on Equation (9), we determine i ∇ → − ∇ like the obtained eigenvalue corresponds to the direction of movement of progressive wave. In a similar manner, we determine Thus, the differential operators in the Newton mechanics corresponds to in the quantum mechanics in accordance with the causality [11] [12] [13].
Substituting Equations (11) and (12) into Equation (5)  is valid not only for a free electron but also for an arbitrary micro particle. Hereinbefore, since it was theoretically revealed that Equations (7) and (8) are valid for an arbitrary micro particle, eliminating r ∆ from their equations yields the relation of matter-wave expressed by Equation (1). In accordance with the theoretical frame in physics, at this point, Equation (1) proposed by de Broglie as a hypothesis is now not a hypothesis but a basic equation in physics. Hereafter, the relation of matter-wave should not be thus named a hypothesis except the historical description.
In the above theoretical development, for a micro particle in a material, the relation of ( ) was obtained as a new expression of matter-wave [13]. It is considered that Equation (13) is applicable to the diffusion theory based on the matter-wave. For example, when the averaged distance between micro particles in a material is expressed as 2a, a micro particle under the condition of ( ) m a k T α ε < +  cannot exist in a local space and it moves through interstices between micro particles while repeating collisions with other micro particles. As can be seen from the definition of diffusivity given by Equation (6), the behavior of such micro particles as satisfying should be expressed not by diffusion Equation (5) but by the wave equation expanded into the many-body problem in the quantum mechanics. As an example, the tunnel effect is known as such a case if the number of collision times is too few in a thin film.
In the past, the correlation between the quantum mechanics and the Newton mechanics has been discussed as an afterthought in accordance with the correspondence principle between an operator in the quantum mechanics and the corresponding physical quantity in the Newton mechanics. For example, the relations of a momentum p and energy E in the Newton mechanics have been accepted as in the quantum mechanics. In accordance with the causality for the Newton mechanics discussed above, Equation (12) shows that Equation (14) is reasonably derived as

Importance of Derivation of Diffusion Equation
Hereinbefore, it was found that the theory of elementary process of diffusion And the Fick second law of diffusion equation in a broad sense, corresponding to the thermal conduction equation of Fourier, was defined as Equation (4) of It seems that the reason for the proposal of laws in those days results from the realization of parabolic law shown in experimental profiles of distribution relevant to a concentration as well as a temperature in a material.
When Fick's laws were proposed, Gauss's diversion theorem was already reported in 1813. As shown in the following, therefore, judging from the theoretical frame in physics, it is inadequate that we accept each of equations (3) and (4) as an independent law.
For a differentiable spatial vector ( ) , , A x y z in a region V within a single closed surface S, Gauss's diversion theorem shows that the correlation between a volume integral and a surface integral yielding is valid, where n is a unit vector perpendicular to a surface element dS. When Gauss's diversion theorem is applied to a flux vector is physically valid in relation to the law of material conservation. Equations (15) and (16) yield the well-known continuous equation given by Here, we should mathematically consider a degree of freedom for a diffusion flux because of In the following, it will be revealed that ( ) J t and eq J are indispensable for understanding diffusion phenomena [9] [10].
If we substitute Equations (3) or (18) into Equation (17), Equation (4)  2) Theoretical expression of diffusivity The diffusion Equation (5) was theoretically derived as a basic equation of moving coordinate system from a behavior of micro particle in an isolated local space. The diffusivity obtained here was expressed as Equation (6) then, and further it was also expressed as Equation (11) resulting from applying the equipartition of energy to a free electron in a material. In general, each of diffusion particles in local spaces is physically different conditions from each other. In a case where Equations (4) or (5) is applied to a whole diffusion region, we must consider a dependence of the space-time ( ) , , , t x y z on the diffusivity. However, a thermal effect on a diffusion particle in a material and a mechanical interaction between a diffusion particle and the surrounding other micro particles have not been incorporated into Equations (6) or (11) yet.
For a micro particle in a state of activation energy Q in a material at a temperature T, we incorporate the Boltzmann factor [ ] B exp Q k T − relevant to an existence probability of the diffusion particle into Equation (11) [25]. When a micro particle interacts with the surrounding other particles, we also incorporate the potential energy U of an external force F operating on the micro particle into the Boltzmann factor. The universal expression of diffusivity is thus expressed as Judging from Equation (6) in a local space, a jumping velocity p v of a micro particle from a local space to another one is obtained as a diffusivity gradient of where the normalized condition of For a k element among N elements in the diffusion region, the relation of ( ) is valid because of the normalized condition of in the parabolic space x t ξ = [26]. The general solutions of Equation (26) had never been also reported until recently [14] [27] [28]. In accordance with the usual analytical method of differential equation, the general solutions of equation (26) were reasonably obtained as ( ) in the previous work [14]. Here, the notation ± corresponds to ± of the parabolic coordinate ξ ( )  (27) and (28) In addition, when the physical field around a diffusion particle is considered to be uniform, the diffusivity D is accepted as a physical constant D 0 . In that case, Equation (26) is rewritten as On the other hand, the diffusivity expression of Equation (27) Here, Equation (30) itself is the general solution of Equation (29). Therefore, the general solution of Equation (29) is included in those of Equations (27) and (28) as an especial case.  (27) and (28).
Nevertheless, the importance has not yet been universally known to researchers.
As mentioned later, there has been such a situation that misunderstanding theory is widely accepted for a long time in the existing diffusion field [24]. In the following, therefore, the application of Equations (27) and (28) to actual problems is briefly explained here.
In the diffusion experiments, we can obtain not a diffusivity profile but an only concentration profile. When the diffusivity is physically considered to be such a constant value as a case of a self-diffusion or an impurity-diffusion, we can determine the diffusivity unknown quantity by fitting Equation (30) to the concentration profile of j element obtained from experiments, using the given initial concentration values for Equation (30).
In general, a diffusivity of j element for a diffusion system of L elements in a material A depends on ( ) , t x in case of Equation (25). There had been no mathematical methods to determine a diffusivity of j element in a diffusion system of L elements. In accordance with the present method, however, A j D of j element in the material A is reasonably obtained by using Equations (27), (28) and (30).
Here, we suppose experiments of a diffusion couple, where it is smoothly jointed at an initial interface between the material A mentioned above and a pure material B composed of k element among L elements in the material A. As a matter of course, the initial value of concentration A j C of j element in the material A is known. In that case, the diffusivity values applicable to initial and/or boundary ones in the material B are determined as (30) to concentration profiles obtained from usual experiments for an impurity diffusion of ( )  (27) and (28).
For the analysis of interdiffusion problems, it is thus extremely important that the general solutions of Equations (27) and (28) are theoretically obtained.

4) Coordinate system of diffusion equation
Generally, there is no such a conception as a migration or a concentration for the space in physics. However, if we consider a thermodynamic influence on a material in the region V within a single closed surface S, the expanding or the shrinking S is physically conceived. In that case, the observer on the surface S seems that the space migrates relatively against the observer itself and the original region V changes.
is valid under the condition of t t ′ = and 0 r′ = , Using Equation (31), the relations between differential operators in their coordinate systems are expressed as Substituting Equation (32) into Equation (4) is valid.
Since the diffusion region space itself is continuous and has no mass, the relation of is physically valid between v of Equation (31) and p v of Equation (20).
Thus, the second term in Equation (33) in the moving coordinate system. As a matter of course, the inverse transformation is also possible [9] [10]. At this point, Equation (4) is now not a law but a basic equation in physics because of the theoretical derivation of Equation (5).
Rewriting Equation (33) into and if we compare it with Equation (17), a diffusion flux J ′ of is obtained as a moving coordinate system, using a diffusion flux in the fixed coordinate system.
The Brown motion reveals that micro particles move randomly also under the condition of concentration gradient zero like a pure material. As far as Equation (3) is accepted as a law, the theoretical equation of diffusion flux is impossible then, because of 0 C ∇ = . In the existing diffusion theory, therefore, the self-diffusion has been understood from diffusion phenomena of isotope elements of extremely small quantities introduced into a pure material concerned.
In the strict sense of the word, however, that corresponds to a diffusion problem relevant to the impurity diffusion.

T. Okino Journal of Modern Physics
In the present diffusion theory, the theoretical equation of self-diffusion is given by , , , J t x y z [9] [23]. Therefore, judging from the discussion mentioned here, the diffusion flux eq J is thus indispensable for understanding the diffusion theory in case of The well-known Kirkendall effect (K-effect eff x ∆ ) reveals that the jointed interface of diffusion couple shifts from the initial position to the diffusion direction in the interdiffusion problems [29]. At the same time, the phenomena indicate that the diffusion region space migrates in a diffusion region. Further, the matter also indicates that the moving and fixed coordinate systems of diffusion equation are indispensable for understanding diffusion phenomena. In the following, the formative mechanism of the K-effect is discussed.
In case of the one-dimensional space for Equation (31), using the relation of Vacancies in the vacancy rich region diffuse into the vacancy poor region during the temperature fall of E R T T → like the diffusion region space reaches a thermal equilibrium state. In that case, it is also considered that a quantity Q of vacancies flows from a vacancy rich region to the specimen surface, because of the specimen surface of a sink of vacancies. At the same time, a quantity Q of vacancies flows from the specimen surface to a vacancy poor region, because of the specimen surface of a source of vacancies then.
The formative mechanism of the K-effect depends on a material characteristic T. Okino Journal of Modern Physics of a specimen used as a diffusion couple. In the above discussion, the K-effect is expressed by the relation of where S is a cross section of specimen used as a diffusion couple [30] [31] [32].
In addition, the theoretical equation of the K-effect yielding is valid in accordance with the parabolic law then, where the suffix γ means and eff α is a parameter dependent on a material characteristic used for the diffusion couple [9].

5) Einstein's paradox
Using the van't Hoff law relevant to an osmostic pressure, the Stocks law in a fluid, and the Fick first law relevant to a diffusion flux, Einstein theoretically investigated behavior of the well-known Brown motion of pollen in water and he obtained the expression of diffusivity yielding where R, A N and k are the gas constant, the Avogadro constant and a proportional constant of an external force F used as for a micro particle moving with a velocity p v . As a result, it was revealed that a self-diffusion of water molecules is visualized by the behavior of pollen. In other words, it was revealed that a material is composed of such a fundamental particle as an atom and/or a molecule.
Einstein conceived then that the diffusion flux J becomes under the condition of a mechanical equilibrium state for diffusion particles, where the diffusion particle moves with a velocity p v during an interaction of . Here, substituting Equation (42) into the continuous Equation (17) yields not the diffusion equation but the well-known Euler's equation in a liquid given by If we compare the present theory to a carved statue by single knife, Einstein's theory corresponds to one by assembling complicated pieces of wood. The matter discussed here gives evidence that the new diffusion theory is meaningful for a fundamental physics.

6) Historical misunderstanding problems in diffusion theory
In relation to having been no conception of a moving or a fixed coordinate system for diffusion equation, misunderstanding problems have been widely accepted in the existing diffusion theory. Further, in relation to analyzing a diffusion equation, mathematically wrong methods for solving a differential equation have been also widely accepted for a long time.
In the following, interdiffusion problems in case of N = 2 for Equation (24) are discussed in order to reveal misunderstanding problems.
As mentioned above, it is apparent that the discussion about a moving or a fixed coordinate system for diffusion equation is indispensable for understanding diffusion theory. The K-effect affords an experimental evidence for the correlation between those coordinate systems. Nevertheless, a relation of diffusion flux, which is similar to Equation (34) of a moving coordinate system, has been widely accepted as a fixed coordinate system in the existing diffusion theory [33].
In the history of diffusion, the relation of  I  II  II  I  rin rin where D  was accepted as not an operator but a physical diffusivity [34]. The so-called Darken equation has been widely used for numerical analyses of interdiffusion problems [35] [36] [37]. It was, however, reported that Equation (44) is not mathematically valid because of mathematical errors in the derivation process [24]. Before that, it is believed that such conception of an intrinsic diffusion is an illusion conceived in those days, judging from the conception of operator D  and further from the physical essence of diffusivity resulting from the derivation of diffusion equation.
The misunderstood matters relevant to the K-effect, the intrinsic diffusion, the Darken [39]. Misunderstanding theory in the existing diffusion field causes thus serious problems for not only researchers but also students. It is, therefore, required that the existing fundamental textbooks are suitably revised as soon as possible, also taking account of problems of coordinate systems of the diffusion equation.

Discussion and Conclusions
As far as a material is composed of micro particles, investigating behavior of those particles is indispensable for research subjects in the materials science. In that case, the Schrödinger Equation (2)  Judging from the theoretical frame in physics, it is considered that the quantum theory is still incomplete without revealing the causality for the Newton mechanics. From a viewpoint of fundamental physics, it is necessary to reveal theoretical evidence for the wave nature of an arbitrary micro particle in accordance with the cause-and-effect relationship in the Newton mechanics.
The diffusion equation having been accepted as a law for a long time since 1855 is formally transformed into the Schrödinger equation as mentioned in the text. Since the diffusion equation shows that the diffusivity depends on behavior of a micro particle in an isolated local space in a material, the transformation of the diffusion equation into the Schrödinger equation is thus reasonably accepted.
It is, however, apparent that the theoretical transformation is impossible as far as we accept the Fick laws as it is. Before investigating the theoretical transformation between them, therefore, we must first grasp the physical essence of diffusivity itself.
Recently, the diffusion Equation (5) was reasonably derived from the mathematical theory of Markov process. As a result, it was first theoretically revealed that the diffusivity D correlates to the angular momentum as expressed by 2 D m =  for a micro particle with mass m in an isolated local space. In other words, it was found that a diffusion particle makes a circuit on the surface of a local space in a material. On the other hand, it was also revealed that the impossibility of discrimination between two micro particles of the same kind in close vicinity to each other is equivalent to denying the mathematical density theorem of a real time in the Newton mechanics. It was thus revealed that the time t in the Newton mechanics has a time t ε of minimum unit as a real time. In future, the conception of the time t ε may be accepted as a dominant conception in the fundamental physics.
As a result, it was also revealed that the differential operators t ∂ ∂ and ∇ in the Newton mechanics becomes i t ∂ ∂ and i − ∇ in the quantum mechanics. By rewriting t i t ∂ ∂ → ∂ ∂ , i ∇ → − ∇ and 2 D m →  in the diffusion Equation (5), the Schrödinger Equation (2) is reasonably obtained. At this point, for the first time in physics, the wave nature of an arbitrary micro particle was theoretically revealed through the transformation from the diffusion equation relevant to a picture of micro particle into the wave equation of Schrödinger relevant to a wave picture. We could thus reasonably understand the necessity of quantum theory for behavior of a micro particle in accordance with the cause-and-effect relationship in the Newton mechanics.
In addition, it was also revealed that the well-known relation of matter-wave is valid as not a hypothesis but a basic equation in physics. Further, the validity of the matter-wave Equation (1) was not only reasonably revealed but also the new Equation (13) was theoretically obtained. In future, Equation (13) will be useful for understanding behavior of micro particles in a material.
Further, the derivation of diffusion equation first revealed that a moving coordinate system as well as a fixed coordinate system for the diffusion equation is essentially indispensable for understanding diffusion phenomena. The discussion about coordinate system of diffusion equation indicated that the wrong theory of diffusion has been accepted for a long time in the existing field. Concretely, the conception of intrinsic diffusion coefficient supposed to understand the Kirkendall effect is not essentially accepted judging from the basic theory of mathematical physics. In accordance with the transformation theory between those coordinate systems and the general solutions of a nonlinear diffusion equation, the new diffusion theory was reasonably established.
In history, most of laws, principles, and basic equations in physics were yielded in Europe and they have been widely accepted in the world. In such circumstances, it was revealed in the Asia country that the equations having been accepted as Fick's laws and de Broglie's hypothesis for a long time are now basic ones in physics, and further that the wave nature of an arbitrary micro particle  From a viewpoint of the theoretical frame in physics, the matters discussed in the present paper are extremely fundamental ones as shown in the physical textbooks for students. We must have a responsibility to develop the physical truth in the textbooks. From a viewpoint of the physical education for younger people, therefore, we thus hope that researchers planning to write a fundamental textbook on physics would publish it taking account of the matters discussed above.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.