The wave function of microparticles as a component of system reliability

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The article discusses the wavelength function of the microcosmos, introduced by Planck as a component of system reliability and its sustainability. In this aspect, we are investigating ensuring reliable sustainability at the different levels of the structural and informational organization of the phenomena in our material world.

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International Journal of Mechanical Engineering and Technology (IJMET) Volume 10, Issue 03, March 2019, pp. 1725–1734, Article ID: IJMET_10_03_174 Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=3 ISSN Print: 0976-6340 and ISSN Online: 0976-6359 © IAEME Publication Scopus Indexed THE WAVE FUNCTION OF MICROPARTICLES AS A COMPONENT OF SYSTEM RELIABILITY Nikolay Ivanov Petrov Institute of Metal Science, Equipment and Technologies with Hydro- and Aerodynamics Centre ,,Acad. A. Balevski” – Bulgarian Academy of Sciences, 67, Shipchenski prohod St. 1574 Sofia, Bulgaria ABSTRACT This article is based on the following scientific proposal, stated by the genius German physicist Max Planck: “No matter can exist on its own. All matter originates and exists only by virtue of a force which brings the particle of an atom to vibration and holds this most minute solar system of the atom together. We must assume behind this force the existence of a conscious and intelligent mind. This mind is the matrix of all matter“ [1]. The article discusses the wavelength function of the microcosmos, introduced by Planck as a component of system reliability and its sustainability. In this aspect, we are investigating ensuring reliable sustainability at the different levels of the structural and informational organization of the phenomena in our material world. The statement is maintained that the unified nature of the reliability of the various phenomena is based on the unity of the "microcosmos" - its material nature and the capability for duality and transformations. Key words: Reliability, microcosmos, sustainability, duality and trans-formations Cite this Article: Nikolay Ivanov Petrov, The Wave Function of Microparticles as a Component of System Reliability, International Journal of Mechanical Engineering and Technology 10(3), 2019, pp. 1725–1734. http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=3 1. INTRODUCTION The author of this paper proposes a study of the mechanism of the reliability of matter and its forms at the different levels of its structural organization. The notion of reliability is applicable in the world of the microcosm, as it reflects the physical nature of elementary particles that we are investigating. When observing the state of the microparticles (elementary particles), Planck's constant appears to be the absolute measure of uncertainty. In this sense, it should be pointed out that the uncertainty principle of Werner Heisenberg states the following: “Absolutely accurate measurement in the micro and macro world is impossible ! “[2, 3, 4]. Based on the principle of causality in physicochemistry, Max Planck poses the question in another way: "Are we obliged to seek a constant explanation of the universal unreliability and inaccuracy, associated with any physicochemical observation? It may be attributed to the http://www.iaeme.com/IJMET/index.asp 1725 editor@iaeme.com
The Wave Function of Microparticles as a Component of System Reliability peculiarities of the individual case, to the complexity of the material object under consideration, or to the imperfection of the measuring instrument, including even our visual organs. Therefore, the unreliability spreads in two directions - from the individual case to the general and vice versa, according to the laws of physicochemistry” [Planck M. The Unity of the physical world’s image. М., 1966] [1]. Max Planck's quantum theory and its basic equation for microparticle (MP) energy, even written on the memorial plaque, placed on his grave ( E  hv , where h is Planck’s constant, and v is the frequency of the physicochemical process) changes our notions in regard to the function of the error [2]. It testifies for the presence of a tendency for a dialectical "incarnation" of the error (equivalent to the failure in the physicochemical process) and its opposite – the reliability. Typical of this concept is the examination of the error (failure) in the physicochemical processes as a necessary moment in the "reliable" functioning of the systems. It (the concept) represents a gnoseological model of the objective reality [6]. 2. REGARDING THE WAVELENGTH FUNCTION OF MICROPARTICLES The idea of the author of this study is to consider the wave function based on the following experiment with the light - the most important phenomenon of nature on Earth and in the Universe. A phenomenon that ensures the existence of humanity and of all forms of a reliable and real biological world. Since light possesses corpuscular-wave properties, then the following experiment is possible. Let a beam of light rays fall on the surface of a transparent plate. According to the wave notion regarding the nature of these beams, they are reflected from the upper surface of the plate, and another portion is refracted and passes through it. If the intensity of the falling, reflected and refracted beam is designated with I , I r and I l , the following inequality will be valid: I  I r  Il . (1) The following equation is known I  A , where A is the amplitude of the falling wave of 2 light [3]. As a result of this ratio, the following equation can be recorded: A  Ar  Al , 2 2 2 (2) where A, A r and A l are the amplitudes of the falling, reflected and refracted wave. If the light is manifested by its corpuscular properties, the intensity of the falling beam of light will be proportional to the number of photons that are contained therein. If the number of falling, reflected, and refracted photons is designated with N , N r and N l , then follows that N  N r  Nl . (3) In order to combine the two concepts regarding the nature of the light (wave and quantum), the assumption is made that the number of the photons in the beam of light is proportional to the square of the module of the respective wave. From this assumption, follow the following formulas for the squares of modules of the reflected and refracted wave [3]: Ar  Al  1. 2 2 (4) http://www.iaeme.com/IJMET/index.asp 1726 editor@iaeme.com
Nikolay Ivanov Petrov Nr N Ar  ; Al  l . 2 2 (5) N N The relationship N r N determines what portion of the total number of falling photons is reflected or the probability that a single photon is being reflected from the surface of the considered plate. The relationship N l N determines what portion of the total number of falling photons passes through the surface of the plate, i.e. the probability that a single photon can pass through the surface of the plate. We do not know which of the falling photons will be reflected, nor which one will pass through. Their behavior is described by virtue of a probability function, by using the squares of the amplitudes of their respective waves. Originating from similar theoretical considerations, the German physician and chemist Max Planck has suggested that the behavior of each microparticle can be described by single function   x, y, z, t  , which he named wave function. The probability of finding the microparticle in a small volume dV in the space W is determined by using the wave function in the following way [1, 4]: dW dW   dV ;      , 2 2 (6) dV where  is a function, which is a complex conjugate function of  . From (6) follows  that the square of the module of the wave function actually determines the probability density, i. e. the probability that a particle may be located in a particular unit volume of the space W. Therefore, the physical meaning of Planck's wave function   x, y, z, t  is probabilistic in nature. This leads to the conclusion, that is more appropriate for it to be called wave function of the microparticle reliability (WFMPR). This is so, because the reliability of a given microparticle is a probability of its presence at a certain time, precisely at the supposed place in the object (space), in the presence of the respective uncertainty of measurement (even when observation is performed by using the most exceptional microscope) Of course, it must be taken into consideration that Max Planck introduces the term wave function during the time period from 1900 to 1905, when the term “reliability” had not been formally introduced to science and only the quality of this or that physico-chemical process or object was discussed. In order to determine the probability of a given microparticle to be located at a particular W , then integration of the WFMPR must be performed into the time at any point in the space elementary volume dV within the range from  to  , as a result of which follows:     x, y, z, t  .dV  1 2 W (7)  . Since, the probability is a magnitude that varies within the range of 0 to 1, the condition (7) is called normalization condition of the WFMPR. The physical significance of (7) is related to the fact that, under certain conditions, the microparticle must surely be located at some point in the space W . Therefore, the cumulative probability of the particle being located somewhere in the space shall be equal to one. As a result of this follows that the normalization condition, expressed in (7) confirms the objective existence of the microparticle in space and time. http://www.iaeme.com/IJMET/index.asp 1727 editor@iaeme.com
The Wave Function of Microparticles as a Component of System Reliability From everything that has been analyzed above, follows the conclusion: “The wave function of the reliability of microparticles combines their wave and quantum properties and serves in order to describe their behavior in the space of states considered. Planck's constant appears to be the absolute measure of the uncertainty in measuring the reliability of microparticles (elementary particles)”. 3. THEOREM REGARDING THE RELIABLE SUSTAINABILITY OF THE MICROCOSMOS Already scholars during the antiquity started to discuss the idea of reliability of the microcosmos. The main idea in Democritus's teachings is : “The being is reality – the atoms, while non-being is the empty space”. The study of the probability-stochastic reliability phenomena at the level of the microparticles and at the atomic levels throughout the entire twentieth century (including the CERN experiments that have so far given the world only the “World Wide Web” and some conjectures regarding the existence of the X-boson particle) opens up new opportunities for the resolution of this problem and ensures the solving of problems the in man-machine systems [13]. Philosophical form of the theorem: "In order for different systems (obje-cts - atoms, molecules, chemical compounds, etc.) to be generated and formed from microparticles (microvesicles), it is necessary to fulfil the following criterion of sustainability: ,, The energy of the internal connections between the elements of the physicochemical systems must exceed the sum of the kinetic energy of the microparticles and the energy of the external influences (continuous or impulse) under the respective conditions of existence” Mathematically, this is expressed by the inequality: 𝐸𝐶𝐸𝑆 ≥ 𝐸𝐾,Σ + 𝐸𝐸𝐼 , (8) where 𝐸𝐶𝐸𝑆 is energy of the connections between the elements of the system (CES), 𝐸𝐾,Σ - the cumulative kinetic energy of microparticles, 𝐸𝐸𝐼 - energy of the external influences (EI). Mathematical form of the theorem - equation for the sustainability of the system of microparticles (SMP). We are looking at an ever-evolving system of microparticles (SMP). Its functioning (interactions) is represented by a system of ordinary differential equations, written in their generic form: xi  t   fi  t , x1 ,..., xn  , i  1,..., n; xi t0   xi 0 , (9) x  t  ,..., xn  t  are the functions where t is an independent variable of the ongoing time; 1 sought that determine the basic parameters of the RTS; t 0 and xi 0 are the initial set operating x  t  dxi  t  dt denotes a conditions of the system under consideration, and the symbol i   derivates of the function xi  t  . In (9) it is presumed that the functions i 1 f  t , x ,..., x n  are real. Under these conditions, it is possible to record the system of differential equations (9) in the following vector form: xi  t   fi  t , x  , x  colon  x1 ,..., xn   Rn , x t0   x0 (10) http://www.iaeme.com/IJMET/index.asp 1728 editor@iaeme.com
Nikolay Ivanov Petrov We can denote with S H a sphere with a radius H and center at the beginning of the x   i 1 xi2 n coordinates of the Euclidean space R n with norm . With T is designated an interval from the real axis of the space, as the following condition is fulfilled T  a  t   , where a is  or some number. It shall be considered that the function f  t , x  , f : T  S H  Rn is continuous with respect to two arguments and fulfills the conditions of Lipschitz (L) with respect to the second argument, i.e. the following inequality is valid f  t , x   f  t , y   L  x  y  , L  const  0 (11) Under these conditions, the theorem for local existence is valid, as well as the uniqueness and continuity of the solution x  t , t0 , x0  of the task from (9) regarding the sustainability of functioning of the SMP under the conditions t0  T and 0 x  Sn [6]. From the famous axiom of Andrey Lyapunov follows, that the solution x  t , t0 , x0  is called undisturbed (stable) solution, and x t  - disturbed solution of the system of ordinary differential equations (9) in the presence of impulse interferences [7, 8, 9, 11]. In some of the earliest scientific papers [3, 5], it is proposed to use impulse differential equations in order to describe aggregate disturbances on the system of microparticles (SMP). In the following works [7], it is considered that the investigation of the process of studying of the functioning of a SMP, which is influenced by aggregate impulse disturbances, is determined by a simplified version of the system of ordinary differential equations (9), which are expressed as follows: x  t   f  t , x(t )  , x  t0   x0 ,  x, f  R n  , (12) and the jumps of the argument 𝑥 of the first order i x  i  N  for which are set only their points in time ti , where t0  t1  ...  tk  ...   , and the respective estimate of the jumps  i x , which are caused by the impulse disturbances. The system of differential equations (12) including taking into consideration the impulse disturbances, causing jumps ∆𝑖 𝑥, is recorded as follows: 𝑑𝑥(𝑡)⁄𝑑𝑡 = 𝑓(𝑡, 𝑥(𝑡)) + ∑+∞ 𝑖=1(∆𝑖 𝑥) 𝛿Д (𝑡 − 𝑡𝑖 ) (13) where 𝛿Д is the delta function of Dirac during the time interval (𝑡 − 𝑡𝑖 ). A natural transformation is performed, by replacing the second member on the right side of (13) with the function g   t  , where 𝑔(𝑡): [0, +∞) → 𝑅𝑛 is a defined continuous left-hand function, locally limited by variation. The function g  t  is assumed to be an impulse disturbance of the equations (12). The result is a new look of the system of differential equations (13): 𝑑𝑥(𝑡)⁄𝑑𝑡 = 𝑓(𝑡, 𝑥(𝑡)) + g   t  , (14) http://www.iaeme.com/IJMET/index.asp 1729 editor@iaeme.com
The Wave Function of Microparticles as a Component of System Reliability The transformation thus performed, preserves the simplicity and clarity that are characteristic of the "classical" impulse differential equations. It allows to be encompassed some "non-classical" cases of resolving the problem of impulse disturbances. Such is the case, where during the observed interval with fixed duration is being limited not the number of the impulses, but their cumulative impulse. When stating the problem under investigation, it should be taken into consideration that: there is no general definition of sustainability of the functioning of the SMP, as related to the reliability of this system [9]. In ad-dition, it is necessary to know the properties of functions with locally limited variations, including the Lebesgue theorem for decomposition of these functions into a sum of continuous set functions and functions of the jumps [6, 7]. On the other hand, in every reliability standard of a system of elements (particles), which form objects (tangible and / or intangible) in the countries in the EU and globally, the following is emphasized: “Reliability is a complex property of a system of objects which, depending on the purpose of the object and the conditions of its operation, includes faultlessness, repairability, durability and storage during transportation, either individually or in combination with these properties, in the presence of external disturbances” [15, 16, 20, 22, 23]. 4. DEFINITION OF SUSTAINABILITY OF SMP Let us examine Cauchy’s problem in regard to SMP, which is represented by a system of equations and fulfillment of the condition 𝑥(𝑡0 ) = 𝑥0 : 𝑑𝑥(𝑡)⁄𝑑𝑡 = 𝑓{𝑥(𝑡), 𝑡} + 𝑑𝑔(𝑡)⁄𝑑𝑡,𝑥(𝑡0 ) = 𝑥0 , (15) In (15) the functions x, f , g assume values in the field of the real numbers 𝑅 𝑛 (𝑡0 ≤ 𝑡 ≤ +∞), as the function f satisfies the conditions of Caratheodory [8]. The g functions 𝑔(𝑡) and 𝑥(𝑡) have a locally limited variation and continuity to the left. Thus, the accepted conditions provide for the resolution of the problem (15) in the interval  0 0 t , t  h) , where ℎ(𝜏) is the function of the jumps of the impulse disturbances. Uniqueness of the solutions of (15) is not supposed. It is not difficult to formulate a problem (11) in such a way, so that the equality in the basic equation can be understood in the ordinary sense. For this, it is sufficient for the system of equations (15) to be integrated in an interval (𝑡0 , 𝑡) for each t t 0 . The assumption is accepted, that in this interval the solution of the problem (15) exists and this results in the following integral equation: t x  t   x0   f x   ,  d  g  t   g  t0  t0 , (16) equivalent in the event of t t 0 as in problem (12). In (16) the time interval 𝜏 is determined by 𝜏 = 𝑡 − 𝑡0 . By ℎ(𝜏) is denoted the function of the jumps (i.e. the discontinuous summand in the decomposition of Lebesgue) of the main function of the risk g t  . The function h is determined with accuracy up to a random constant addend; therefore, as certainty is considered that h t   x 0 0 . It is noticed that the second addend on the right side of the formula (13) is continuously dependent on the argument t . An “impulse transformation” is performed by carrying out the following substitution http://www.iaeme.com/IJMET/index.asp 1730 editor@iaeme.com
Nikolay Ivanov Petrov 𝑦(𝑡) = 𝑥(𝑡) − ℎ(𝜏), 𝛼(𝑡) = 𝑔(𝑡) − ℎ(𝜏) (17) From (17) follows that 𝑥(𝑡) = 𝑦(𝑡) + ℎ(𝜏) and 𝑔(𝑡) = 𝛼(𝑡) + ℎ(𝜏), (18) The so performed impulse transformations (17) and (18) result in: ,,The equation of function continuity y  t  in the event of locally limited variation”. It looks the following t y  t    f  y    h   ,   d    t     t0  (19) t0 It is not difficult to carry out the reverse transition from equation (19) to equation (16), i. e. these two equations are equivalent. However, unlike (16), in equation (19) all summands are continuous functions. This means that equation (19) has been defined in an "elementary sense". In the assumptions stated above, equations (19) and (16) have one solution that is located within the interval  0 t ,T   t  T    0 . An example of a local variation in the equation of function continuity (19) is shown in [15,21], where by using the method of nonlinear mathematical programming, have been solved in parallel two multifactor iterative one- parameter technogenic optimization problems, related to the reservation of technical and economic systems. Let us investigate the question regarding the sustainability (stability) of the solution of the equation (15), defining Cauchy’s problem for a system of microparticles (SMP). Above all, it is clear that the problem is reduced to the study of the stability of the null solution x0  t   0 , i.e. pertaining to the case f  0, t   0 at x0  0 . We shall consider that, in equation (15), the external influences appear to be a function of g   t  , i.e. this concerns researching the stability of SMP in the event of permanently acting external disturbances. The null solution of equation (15) is called stable in the event of aggregate impulse external influences, if for each   0 exists such   0 , whereby the integral inequality is satisfied k  g   t  .dt   , k  N , (20) k 1 the inequality is valid, that looks like this: x  t    , t 0,   (21) In the event that (20) and (21) are not fulfilled, the null solution of (9) is unstable due to the aggregate impulse external influences. As a result of this stability follows the analogous sustainability, in the event of which the initial conditions are set for each t0  0 . If  does not depend on t 0 , then the null solution of (9) is referred to as uniformly stable in the event of aggregate impulse influences on the SMP. The null solution of (9) is asymptotically stable in the event of aggregate impulse disturbances[12], sufficiently small variation of the function g  t  in the research (study) interval  k  1, k   k  N  and limitation of this variation in the interval 0,  , resulting from x  t   0 at t   . 5. INVESTIGATION OF CAUCHY’S PROBLEM IN REGARD TO SMP The normal functioning of the SMP is represented by the differential equation of the type http://www.iaeme.com/IJMET/index.asp 1731 editor@iaeme.com
The Wave Function of Microparticles as a Component of System Reliability 𝑑𝑥(𝑡)⁄𝑑𝑡 = 𝐴𝑥(𝑡) + 𝑑𝑔(𝑡)⁄𝑑𝑡, 𝑥(0) = 0, (22) where   and   satisfy the assumption of locally limited variation and A is a square x t g t matrix of the n-th order with real constant coefficients. The investigation of Cauchy’s problem in regard to SMP is related to postulating the "Theorem on the stability of the functioning of SMP (the reliability of the microparticle system).” It is represented by the following definition: If all values of the matrix A from (22) have a negative real part, then the solution of the equation will be uniform and asymptotically stable. However, if the condition so formulated is violated, then this solution is unstable, which results as well in the instability in the functioning of the SMP, i.e. its unreliability. Proof. The solution of Cauchy’s problem, represented by (22) exists only in the following form t x  t   g  t   e g  0   A et   g   d . tA (23) 0 In equation (23) all summands appear as common functions and their sum is understood in the “elementary” sense. If it is possible to choose a number c  0 such that esA  cebs for all s  0,   , then formula (23) through integration by parts is represented in the form of: t x  t    et   g    d , (24) 0 where in the right-hand side of (24) the expression g    represents a generalized function. When implementing the inequality (21) from equation (24), it follows: t 1 mink ,t x t     e( t  ) A  g    d  M , (25) k 1 k 1 where the designation:  t  k t  M  ce  bt    g    d  ebt  g    d  (26)  k 1 k 1 t   After entering intermediate designation, the result is:  1  M  c   1 , (27) 1 e b  whereby  denotes the expression   k  g    d  0 at k   (28) k 1 In inequalities (25) and (26), the designation   (located above the sign for the t mathematical sum) represents the whole part of the number t . As a result of the so performed mathematical analysis in order to investigate the stability (reliability) of the system of microparticles (SMP), it becomes evident that if in (20) and (21) is assigned a number   0, then such a value of   0 can be found, for which the right-hand side of (25) can turn out to be less than  . This demonstrates the stability of the solution to Cauchy’s problem for SMP as represented by the differential equation (22). The uniformity of this sustainability follows as a result of the autonomy of the assigned problem [12]. http://www.iaeme.com/IJMET/index.asp 1732 editor@iaeme.com
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