A complete system of events of nonlinear processes in complex dynamical systems describes the evolution of the distribution. Different stages of evolution of the distribution declared stable, metastable and unstable systems. Variance of probabilistic distribution plays a crucial role in determining the state of the system. It was found that the system is metastable, when it carried Markovian processes in continuous time. Metastability is the original metallurgical phenomenon and actively exists in the structure of the materials. In a metastable state, the structural distances in material are exponentially distributed

B. Lalić

F. Sinani

I. Tanackov

V. Bogdanović

Đ. Ćosić

English

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Metastability – Markovian approach

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573 METALURGIJA 52 (2013) 4, 573-576I. TANACKOV, F. SINANI, V. BOGDANOVIĆ, Đ. ĆOSIĆ, B. LALIĆMETASTABILITY – MARKOVIAN APPROACHReceived – Prispjelo: 2013-01-21 Accepted – Prihvaćeno: 2013-03-30Review paper – Pregledni radISSN 0543-5846METABK 52(4) 573-576 (2013)UDC – UDK 531.39:531.3:539.5=111I. Tanackov, V. Bogdanović, Đ. Ćosić, B. Lalić, Faculty of Technical Sci-ences, University of Novi Sad, Novi Sad, SerbiaF. Sinani, Ministry of Transport and Communications, Skopje, Republic of MacedoniaA complete system of events of nonlinear processes in complex dynamical systems describes the evolution of the distribution. Diff  erent stages of evolution of the distribution declared stable, metastable and unstable systems. Variance of probabilistic distribution plays a crucial role in determining the state of the system. It was found that the system is metastable, when it carried Markovian processes in continuous time. Metastability is the original metal-lurgical phenomenon and actively exists in the structure of the materials. In a metastable state, the structural dis-tances in material are exponentially distributed.Key words: system stability, distributive evolution, material structure INTRODUCTIONStandard rating system stability is consistent with the principles of Aristotelian logic: the principle of identity, the principle of contradiction and the principle of non-existence of the third. Analogously, in theory de-clare the system steady state and unsteady state. The usual absence of the third principle is absent.Systems are declared as stable when the resources and activities of the system achieve goals of system. Complementary, if the system does not implement the goals, the situation is unstable. Control of systems al-ways focus on maintaining stability and preventing the destabilization of the system. In theory of the system, phenomenon of metastabil-ity is a temporally declared. In a qualitative sense, the metastable state is not the most stable state of the sys-tem. In the quantitative sense, the time that the system spends in the metastable state is always longer than the time that the system spends in a stable condition. According to available data, the phenomenon of metastability was ﬁ  rst used in the literature almost 100 years ago, just in metallurgy - in 1915 [1, 2].In accordance with the early ﬁ  ndings of the metal-lurgical industry, the phenomenon of metastability have a basic source in the system structure. This structure is variable, especially in chemical or thermal processing of materials. In analyzes of phase transitions in polymer [3], martensitic transformation [4], exponential decay as a function of surface separation in materials [5], cor-rosion process [6], crystal growth kinetics [7], etc. metastability phenomenon takes the form of an expo-nential expression. In continuous time, exponential pro-cesses are the basis of Markovian processes.   The paper explained the Markovian basis of metastable state in the system structure.DISTRIBUTION EVOLUTIONThe system structure or ﬂ  ows structure in the sys-tem, have different forms in the time domain (Figure 1). Part of the complex system structure have a ﬂ  ow dy-namics analogy. This dynamics start from the equal dis-tance between successive identical parts or events (Fig-ure 1a), small and medium differences between succes-sive distances (Figure 1b), a large number of small dis-tances and small number of large distances (Figure 1c), initial and expression clusters (Figure 1d) and singulari-ties in one cluster (Figure 1e).At random, timed stochastic domain [0, T] derived bijection between intervals of homogeneous events n and axiomatic probability space [0, 1] (Figure 2). The basic random variable is equal to the quotient of your real length subintervals and the total time ti = Ti / T. Ap-plied bijective function is not bicontinuous function.Figure 1 System structure or fl  ows structure 574   METALURGIJA 52 (2013) 4, 573-576I. TANACKOV et al.: METASTABILITY – MARKOVIAN APPROACHThese system structure or ﬂ  ows structure in the sys-tem can be described by probability distributions and densities. Flow with identical distances between suc-cessive events are described with Dirac delta function (Figure 3a). Small and medium differences of succes-sive distances are described by a Normal or Erlang dis-tribution (Figure 3b). A large number of small distances and a small number of large distances are described the exponential distribution (Figure 3c). For a description of the initial and distinct clustering (Figure 3d) and sin-gularity (Figure 3e) is necessary to choose probability density function which in deﬁ  ned domain [0, 1] has minimum.The maximum probability density is most likely the state of the system. If the state of the system can be de-scribed by probability density function which has at least one maximum value, the system is stable. Fore-casting the state has the highest certainty.Minimum probability density is the least likely state of the system. If the state of the system can be described by probability density function which has at least one minimum, the system is unstable. Predicted state of the system has the largest uncertainty.If there is no probability density function of the ex-treme value (or minimum or maximum), his condition is forecasting a meta-case, and the system is metastable.Analog graphic interpretation of stability (Figure 4), the following declaration proposed for the assessment of stability, metastability and instability of the system.In line with the evolution of the continuum distribu-tion (Figure 3), a stable state of “a” to “b”, metastable states are “c”, and the unstable situation of the “d” and “e”. FUNCTION OF EVOLUTIONOne of the functions deﬁ  ned on the interval [0, 1] that can get normal forms, to move to the eccentricity and forms an Erlang distribution, than to exponential, and then take on a “U” shape is − β function dual sub-stituent parameter with deﬁ  ned functions of the proba-bility density and parameters (1):  1 1 1 − − −Γ Γ+ Γ = b a ) t ( t) b ( ) a () b a () t , b , a ( , 0 ≤ t ≤ 1,  b aa) t ( E+= , ) b a ( ) b a (ab) t ( V1 2 + + +=  (1)β function is deﬁ  ned in the domain [0, 1], according to the bijection of any stochastic ﬂ  ow [0, T] in axiomat-ic probabilystic domain [0, 1]. For n events deﬁ  ned on the interval [0, 1], the mathematical expectation of suc-cessive differe-nces is known and always is E (t) = 1 / n. On the basis of these facts, it is to have a functional re-lationship between the parameters a and b proposed functions (1), the value basic parametars (2) and one-parameter form of β function (3):Figure 2 Biection of real stochastic fl  owFigure 3 Distribution evolutionFigure 4   Graphical representation of the system stability, S-stable, M-metastable, N-unstable575 METALURGIJA 52 (2013) 4, 573-576I. TANACKOV et al.: METASTABILITY – MARKOVIAN APPROACH    ) an ( n) n () t ( V     ,nabn) t ( E11112 +−=−= ⇔ =  (2)  1 1 1 1 − − − −− Γ ΓΓ= ) n ( a a ) t ( t) a an ( ) a () an () t ( f    (3)Deterministic ﬂ  ow is described identical intervals ti = tj, ∀i, j∈[1, n] identical successive differences betwe-en events (Figure 3a). In the deterministic ﬂ  ow, the mean length of the interval is E(t) = n−1, ∀i∈[1, n] va-riance of the deterministic ﬂ  ow is zero, V(t) = 0. Deter-ministic ﬂ  ow can be described by the Dirac delta proba-bility function density form (4): 2210ctcc eclim ) t (−→ π= δ , ≠= ∞ += δ0 00t ,t ,) t ( c  (4)that by substitution c = σ 2 and translation for the period µ reduced down to a normal distribution N(µ,σ). Being known the mathematical expectation distance customers during the interval [0, 1] and is always equal to the reciprocal of clients 1/n, the density distribution deterministic ﬂ  ow becomes (5): ) a an ( ) a () t ( t ) an (lim )nt () n ( a aa − Γ Γ− Γ≅ − δ− − −∞ → σ1 1 121 1  (5)Distribution function of a deterministic ﬂ  ow, Dirac delta function is the Heaviside step function is 0 for t<1/n, and 1 for t≥1/n. This stage of evolution of the distribution is called Deterministic phase and is characterized by the value of the ﬂ  ow variance V(t) = 0.Deterministic phase is a special form of the normal phase. Minimal differences in the lengths of intervals ti under the inﬂ  uence of many other factors, leads to an increase in the standard deviation σ, inﬁ  nitesimal but ﬁ  nite value that is greater than zero, σ≥0. The intervals between events are no longer identical. Normal course of development begins. Mathematical expectation is universal and is always equal to the reciprocal value of the number of events, 1/n. (Figure 3b). The density dis-tribution has a maximum and is not eccentric. For a large number of events and ﬁ  nite n, the condition, which gives to the normal ﬂ  ow symmetric function probability density, must be fulﬁ   lled koji normalnom toku daje simetričnu funkciju gustinu verovatnoće: µ−3σ≥0. Ac-cording to (2), parameter „a” of distribution β becomes a≅9. Interval Variance of ﬂ  ow in Normal phase of distri-bution evolution is from 0 to 1/9n2, 0 <V(t)≤1/9n2.Violation of the conditions of Normal phase ﬂ  ow µ−3σ≤0 expresses eccentircity density functions of the distribution, but the probability density function is still a maximum. Increasing eccentricity follows the rise of the probability density function of the ordinate, increas-ing eccentricity (Figure 3b) and the ﬁ  nal position of the maximum of the asymptotic ordinate (Figure 3c). At this stage of evolution of the distribution, the key issue is the density function is lost when the maximum ordi-nate value in t=0 (6), which is for a ﬁ  nite number of events n possible only when a≅1 (7):   0 1 1 1 1 =′−− Γ ΓΓ − − − ) n ( a a ) t ( t) a an ( ) a () an (  (6)  () 110 1 1 1 1 ≅−= ⇔ = ′ − ⇔ − − −nna ) t ( t ) n ( a a  (7)One-parameter  β function gets the parameters of mathematical expectation, variance and standard devia-tion have a unique relationship characterized by an ex-ponential distribution (8) (note: a≅1): n) t ( E1= , 2 2111n ) an ( n) n () t ( V ≈+−=  (8)This interval the evolution of the distribution is called Erlang ﬂ  ow. Boundary condition at the maximum loss is exponential. From probability theory it is known that convolution exponential distribution gives Erlang, and that under certain conditions Erlang distribution can be approximated by a normal distribution. Also, de-pending on the parameters, the Erlang distribution has adequate ordinanty rise and eccentricity.Interval variance Erlang phase ﬂ  ow in the evolution of the distribution is from 1/9n2 to n−2.Deterministic phase, normal phase, and Erlang phas-es are phases in which the system is stable. Loss of maximum and minimum non-existence of the next phase of the evolution of the distribution system is the metastability period.With further reduction of the parameter „a” the ex-ponential evolution in the distribution function se inten-sively approaches the ordinate (Figure 3c). Exists a lar-ge number of small intervals and a small number of medium and large intervals. Such a distribution of the initial interval corresponds to the phenomenon of clus-tering (Figure 3d). The next crucial task in the course of evolution the distribution is a description of the proba-bility density of these random variables ti with a func-tion that receives a minimum. Differentiation is expres-sed in clusters and “disappearing” of middle distance. Number of small and large distances is greater than the number of “middle” range. If we consider the probabili-ty density (Figure 3e), in underexponential part we ex-pect convergence of the distribution of density values to the values of a distance and the ﬁ  rst appearance of a minimum close to this Singular ﬂ  ow (Figure 3d) have full concentration in n events to one arbitrary point on the interval [0, 1]. This means that the (n−1) distance equal to those of the random variable ti is zero, and the remaining distance is equal to 1. Minimum variance in which this is possible is obtained for the parameters a = (n−1)−1, and according to the conditions (2), za b=1. Minimum variance at which a sufﬁ   cient number of events and the ﬁ  nal n for the ﬁ  rst time in the evolution of the distribution occurs a minimum (9):()n n nn) b a ( ) b a (ab) t ( V21211 2 322 ≈−−=+ + +≈  (9)Interval Variance ﬂ  ow in the exponential phase of the evolution of the distribution is from 1/n2 to 1/2n. It 576   METALURGIJA 52 (2013) 4, 573-576I. TANACKOV et al.: METASTABILITY – MARKOVIAN APPROACHshould be noted that for large enough and ﬁ  nal n, the value of the parameter a≈1/n.The end of the exponential phase is the beginning stages of system instability. After the onset of minimum performance sub-exponential phase of evolution of the distribution. The phenomenon of concentration of all the events at a single point on the interval [0,1] is bor-derline phenomenon of singularity ﬂ  ow, with maximum value of the variance V(t)=1. Interval Variance ﬂ  ow in sub-exponential phase of evolution of the distribution is from 1/2n to 1.The approximate capacity distribution at this stage is not adequate and viable only up to the value of a param-eter a≥δ, δ→0. For condition (2) the parameter b→0. For parameters for the mathematical expectation E(t) = 1/n and variance V(t)=1, parametar value is a = −n−1<0, which is not in accordance to basic deﬁ  nition of β dis-tribution. Singular ﬂ  ow distribution in the phase of evo-lution may also be described by two Dirac delta func-tion. We know that (n−1) distance is equal to „0” and that one distance is equal to „1”. β distribution does not have this capability, and therefore its overall role is cat-alytic. At the same time, completely ambiguous posi-tion in which the interval [0, 1] forms singularity. There-fore, the state of the system is completely unstable. CONCLUSIONVariance plays a key role in assessing the stability of the system. General criteria for the evaluation of the system are given in Table 1:Table 1 System stability and varianceSystem states Variance, V(t)Stable 0 ≤ V (t) ≤ 1/n2Metastable 1/n2 ≤ V (t) ≤ 1/2nUnstable 1/2n ≤ V (t) ≤ 1Quantitatively, the variation interval in which the system is stable is less than the interval in which the system is metastable or unstable. From the above crite-ria indicate that the stability of the system decreases with the square of the elements. Exponential phase of the evolution of the distribution is system metastability phase. In the base of Markov continuous process is ex-ponential distribution. Metastable phase is Markov sys-tem [8, 9]. The ﬁ  nding in this study is not the ﬁ  rst and a unique relationship and metastability of Markovian processes [10-13].Generalization in a ﬂ  ow structure for one dimen-sional form, can be extended to the material structure for two-dimensional form (surface) or three-dimension-al form (volume).   According to the ﬁ  ndings, metastable states in the structures of materials are analogous to Markovian processes. In a metastable state, the struc-tural distance are exponentially distributed. Explicit metric system for metastability – based on the probabi-listic concept, is established.AcknowledgmentThe authors acknowledge the support of research project TR 36030, funded by the Ministry of Science and Technological Development of Serbia.REFERENCES [1]  E. Cohen, The inﬂ  uence of allotropy on the metastability of metals, and its bearing on chemistry, physics and tech-nology, Journal of the Franklin Institute, 179 (1915), 367.[2]  C. H. Desch, Martensite structure of steel, Journal of the Franklin Institute, 179 (1915), 367.[3]  A. Keller, S.Z.D. Cheng The role of metastability in poly-mer phase transitions, Polymer, 39 (1998), 4461-4487.[4]  Q. Meng, X. Chen, S. Guo, X. Zhao. The metastability and martensitic transformation of Co particles, J. Alloys Com-pd. (2012), doi:10.1016/j.jallcom.2012.03.03.[5]  F. Leal-Calderon, P. Poulin Progress in understanding emulsion metastability and surface forces, Current Opi-nion in Colloid & Interface Science, 4 (1999), 223-230.[6]  G. O. Ilevbare, G.T Burstein, The inhibition of pitting cor-rosion of stainless steels by chromate and molybdate ions, Corrosion Science, 45 (2003), 1545-1569.[7]  H. J. Fecht, W. L. Johnson, Thermodynamic properties and metastability of bulk metallic glasses, Materials Science and Engineering: A, 375–377 (2004), 2-8.[8]  S. Vukadinović, Sistemi masovnog opsluživa-nja, Naučna knjiga, Beograd, 1989.[9]  E. B. Davies, Dynamical stability of metastable states, Journal of Functional Analysis, 46 (1982), 373-386.[10]  F. den Hollander, Metastability under stochastic dynamics, Stochastic Processes and their Applications, 114 (2004), 1-26.[11]  R. Mahnke, J. Kaupužs, I. Lubashevsky, Probabilistic de-scription of trafﬁ  c  ﬂ   ow, Physics Reports, 408 (2005), 1-130.[12]  W. Hachicha, A Simulation Metamodelling Based Neural Networks for Lot-Sizing Problem in MTO Sector, Interna-tional Journal of Simulation Modelling, 10 (2011) 4, 191-203.[13]  J. Beltrán, C. Landim, Metastability of reversible ﬁ  nite sta-te Markov processes, Stochastic Processes and their Appli-cations, 121 (2011), 1633-1677.Note:   The responsible translator for English language is N. Kozul, Novi Sad, Serbia

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573METALURGIJA 52 (2013) 4, 573-576I. TANACKOV, F. SINANI, V. BOGDANOVIĆ, Đ. ĆOSIĆ, B. LALIĆMETASTABILITY – MARKOVIAN APPROACHReceived – Prispjelo: 2013-01-21 Accepted – Prihvaćeno: 2013-03-30Review paper – Pregledni radISSN 0543-5846METABK 52(4) 573-576 (2013)UDC – UDK 531.39:531.3:539.5=111I. Tanackov, V. Bogdanović, Đ. Ćosić, B. Lalić, Faculty of Technical Sci-ences, University of Novi Sad, Novi Sad, SerbiaF. Sinani, Ministry of Transport and Communications, Skopje, Republic of MacedoniaA complete system of events of nonlinear processes in complex dynamical systems describes the evolution of the distribution. Diff erent stages of evolution of the distribution declared stable, metastable and unstable systems. Variance of probabilistic distribution plays a crucial role in determining the state of the system. It was found that the system is metastable, when it carried Markovian processes in continuous time. Metastability is the original metal-lurgical phenomenon and actively exists in the structure of the materials. In a metastable state, the structural dis-tances in material are exponentially distributed.Key words: system stability, distributive evolution, material structure INTRODUCTIONStandard rating system stability is consistent with the principles of Aristotelian logic: the principle of identity, the principle of contradiction and the principle of non-existence of the third. Analogously, in theory de-clare the system steady state and unsteady state. The usual absence of the third principle is absent.Systems are declared as stable when the resources and activities of the system achieve goals of system. Complementary, if the system does not implement the goals, the situation is unstable. Control of systems al-ways focus on maintaining stability and preventing the destabilization of the system. In theory of the system, phenomenon of metastabil-ity is a temporally declared. In a qualitative sense, the metastable state is not the most stable state of the sys-tem. In the quantitative sense, the time that the system spends in the metastable state is always longer than the time that the system spends in a stable condition. According to available data, the phenomenon of metastability was fi rst used in the literature almost 100 years ago, just in metallurgy - in 1915 [1, 2].In accordance with the early fi ndings of the metal-lurgical industry, the phenomenon of metastability have a basic source in the system structure. This structure is variable, especially in chemical or thermal processing of materials. In analyzes of phase transitions in polymer [3], martensitic transformation [4], exponential decay as a function of surface separation in materials [5], cor-rosion process [6], crystal growth kinetics [7], etc. metastability phenomenon takes the form of an expo-nential expression. In continuous time, exponential pro-cesses are the basis of Markovian processes.  The paper explained the Markovian basis of metastable state in the system structure.DISTRIBUTION EVOLUTIONThe system structure or fl ows structure in the sys-tem, have different forms in the time domain (Figure 1). Part of the complex system structure have a fl ow dy-namics analogy. This dynamics start from the equal dis-tance between successive identical parts or events (Fig-ure 1a), small and medium differences between succes-sive distances (Figure 1b), a large number of small dis-tances and small number of large distances (Figure 1c), initial and expression clusters (Figure 1d) and singulari-ties in one cluster (Figure 1e).At random, timed stochastic domain [0, T] derived bijection between intervals of homogeneous events n and axiomatic probability space [0, 1] (Figure 2). The basic random variable is equal to the quotient of your real length subintervals and the total time ti = Ti / T. Ap-plied bijective function is not bicontinuous function.Figure 1 System structure or fl ows structure 574  METALURGIJA 52 (2013) 4, 573-576I. TANACKOV et al.: METASTABILITY – MARKOVIAN APPROACHThese system structure or fl ows structure in the sys-tem can be described by probability distributions and densities. Flow with identical distances between suc-cessive events are described with Dirac delta function (Figure 3a). Small and medium differences of succes-sive distances are described by a Normal or Erlang dis-tribution (Figure 3b). A large number of small distances and a small number of large distances are described the exponential distribution (Figure 3c). For a description of the initial and distinct clustering (Figure 3d) and sin-gularity (Figure 3e) is necessary to choose probability density function which in defi ned domain [0, 1] has minimum.The maximum probability density is most likely the state of the system. If the state of the system can be de-scribed by probability density function which has at least one maximum value, the system is stable. Fore-casting the state has the highest certainty.Minimum probability density is the least likely state of the system. If the state of the system can be described by probability density function which has at least one minimum, the system is unstable. Predicted state of the system has the largest uncertainty.If there is no probability density function of the ex-treme value (or minimum or maximum), his condition is forecasting a meta-case, and the system is metastable.Analog graphic interpretation of stability (Figure 4), the following declaration proposed for the assessment of stability, metastability and instability of the system.In line with the evolution of the continuum distribu-tion (Figure 3), a stable state of “a” to “b”, metastable states are “c”, and the unstable situation of the “d” and “e”. FUNCTION OF EVOLUTIONOne of the functions defi ned on the interval [0, 1] that can get normal forms, to move to the eccentricity and forms an Erlang distribution, than to exponential, and then take on a “U” shape is − β function dual sub-stituent parameter with defi ned functions of the proba-bility density and parameters (1): 11 1 −− −ΓΓ+Γ = ba )t(t)b()a()ba()t,b,a( , 0 ≤ t ≤ 1,  baa)t(E+= , )ba()ba(ab)t(V12 +++=  (1)β function is defi ned in the domain [0, 1], according to the bijection of any stochastic fl ow [0, T] in axiomat-ic probabilystic domain [0, 1]. For n events defi ned on the interval [0, 1], the mathematical expectation of suc-cessive differe-nces is known and always is E (t) = 1 / n. On the basis of these facts, it is to have a functional re-lationship between the parameters a and b proposed functions (1), the value basic parametars (2) and one-parameter form of β function (3):Figure 2 Biection of real stochastic fl owFigure 3 Distribution evolutionFigure 4  Graphical representation of the system stability, S-stable, M-metastable, N-unstable575METALURGIJA 52 (2013) 4, 573-576I. TANACKOV et al.: METASTABILITY – MARKOVIAN APPROACH   )an(n)n()t(V   ,nabn)t(E11112 +−=−=⇔=  (2) 111 1 −−− −−ΓΓΓ= )n(aa )t(t)aan()a()an()t(f   (3)Deterministic fl ow is described identical intervals ti = tj, ∀i, j∈[1, n] identical successive differences betwe-en events (Figure 3a). In the deterministic fl ow, the mean length of the interval is E(t) = n−1, ∀i∈[1, n] va-riance of the deterministic fl ow is zero, V(t) = 0. Deter-ministic fl ow can be described by the Dirac delta proba-bility function density form (4): 2210ctcc eclim)t(−→ π=δ , ≠=∞+=δ000t,t,)t(c  (4)that by substitution c = σ 2 and translation for the period µ reduced down to a normal distribution N(µ,σ). Being known the mathematical expectation distance customers during the interval [0, 1] and is always equal to the reciprocal of clients 1/n, the density distribution deterministic fl ow becomes (5): )aan()a()t(t)an(lim)nt()n(aaa −ΓΓ−Γ≅−δ−−−∞→σ111211  (5)Distribution function of a deterministic fl ow, Dirac delta function is the Heaviside step function is 0 for t<1/n, and 1 for t≥1/n. This stage of evolution of the distribution is called Deterministic phase and is characterized by the value of the fl ow variance V(t) = 0.Deterministic phase is a special form of the normal phase. Minimal differences in the lengths of intervals ti under the infl uence of many other factors, leads to an increase in the standard deviation σ, infi nitesimal but fi nite value that is greater than zero, σ≥0. The intervals between events are no longer identical. Normal course of development begins. Mathematical expectation is universal and is always equal to the reciprocal value of the number of events, 1/n. (Figure 3b). The density dis-tribution has a maximum and is not eccentric. For a large number of events and fi nite n, the condition, which gives to the normal fl ow symmetric function probability density, must be fulfi lled koji normalnom toku daje simetričnu funkciju gustinu verovatnoće: µ−3σ≥0. Ac-cording to (2), parameter „a” of distribution β becomes a≅9. Interval Variance of fl ow in Normal phase of distri-bution evolution is from 0 to 1/9n2, 0 <V(t)≤1/9n2.Violation of the conditions of Normal phase fl ow µ−3σ≤0 expresses eccentircity density functions of the distribution, but the probability density function is still a maximum. Increasing eccentricity follows the rise of the probability density function of the ordinate, increas-ing eccentricity (Figure 3b) and the fi nal position of the maximum of the asymptotic ordinate (Figure 3c). At this stage of evolution of the distribution, the key issue is the density function is lost when the maximum ordi-nate value in t=0 (6), which is for a fi nite number of events n possible only when a≅1 (7):  01 111 =′−−ΓΓΓ −−− )n(aa )t(t)aan()a()an(  (6) ( ) 1101 111 ≅−=⇔=′−⇔ −−−nna)t(t )n(aa  (7)One-parameter β function gets the parameters of mathematical expectation, variance and standard devia-tion have a unique relationship characterized by an ex-ponential distribution (8) (note: a≅1): n)t(E 1= , 22111n)an(n)n()t(V ≈+−=  (8)This interval the evolution of the distribution is called Erlang fl ow. Boundary condition at the maximum loss is exponential. From probability theory it is known that convolution exponential distribution gives Erlang, and that under certain conditions Erlang distribution can be approximated by a normal distribution. Also, de-pending on the parameters, the Erlang distribution has adequate ordinanty rise and eccentricity.Interval variance Erlang phase fl ow in the evolution of the distribution is from 1/9n2 to n−2.Deterministic phase, normal phase, and Erlang phas-es are phases in which the system is stable. Loss of maximum and minimum non-existence of the next phase of the evolution of the distribution system is the metastability period.With further reduction of the parameter „a” the ex-ponential evolution in the distribution function se inten-sively approaches the ordinate (Figure 3c). Exists a lar-ge number of small intervals and a small number of medium and large intervals. Such a distribution of the initial interval corresponds to the phenomenon of clus-tering (Figure 3d). The next crucial task in the course of evolution the distribution is a description of the proba-bility density of these random variables ti with a func-tion that receives a minimum. Differentiation is expres-sed in clusters and “disappearing” of middle distance. Number of small and large distances is greater than the number of “middle” range. If we consider the probabili-ty density (Figure 3e), in underexponential part we ex-pect convergence of the distribution of density values to the values of a distance and the fi rst appearance of a minimum close to this Singular fl ow (Figure 3d) have full concentration in n events to one arbitrary point on the interval [0, 1]. This means that the (n−1) distance equal to those of the random variable ti is zero, and the remaining distance is equal to 1. Minimum variance in which this is possible is obtained for the parameters a = (n−1)−1, and according to the conditions (2), za b=1. Minimum variance at which a suffi cient number of events and the fi nal n for the fi rst time in the evolution of the distribution occurs a minimum (9):( )nnnn)ba()ba(ab)t(V21211 2322 ≈−−=+++≈  (9)Interval Variance fl ow in the exponential phase of the evolution of the distribution is from 1/n2 to 1/2n. It 576  METALURGIJA 52 (2013) 4, 573-576I. TANACKOV et al.: METASTABILITY – MARKOVIAN APPROACHshould be noted that for large enough and fi nal n, the value of the parameter a≈1/n.The end of the exponential phase is the beginning stages of system instability. After the onset of minimum performance sub-exponential phase of evolution of the distribution. The phenomenon of concentration of all the events at a single point on the interval [0,1] is bor-derline phenomenon of singularity fl ow, with maximum value of the variance V(t)=1. Interval Variance fl ow in sub-exponential phase of evolution of the distribution is from 1/2n to 1.The approximate capacity distribution at this stage is not adequate and viable only up to the value of a param-eter a≥δ, δ→0. For condition (2) the parameter b→0. For parameters for the mathematical expectation E(t) = 1/n and variance V(t)=1, parametar value is a = −n−1<0, which is not in accordance to basic defi nition of β dis-tribution. Singular fl ow distribution in the phase of evo-lution may also be described by two Dirac delta func-tion. We know that (n−1) distance is equal to „0” and that one distance is equal to „1”. β distribution does not have this capability, and therefore its overall role is cat-alytic. At the same time, completely ambiguous posi-tion in which the interval [0, 1] forms singularity. There-fore, the state of the system is completely unstable. CONCLUSIONVariance plays a key role in assessing the stability of the system. General criteria for the evaluation of the system are given in Table 1:Table 1 System stability and varianceSystem states Variance, V(t)Stable 0 ≤ V (t) ≤ 1/n2Metastable 1/n2 ≤ V (t) ≤ 1/2nUnstable 1/2n ≤ V (t) ≤ 1Quantitatively, the variation interval in which the system is stable is less than the interval in which the system is metastable or unstable. From the above crite-ria indicate that the stability of the system decreases with the square of the elements. Exponential phase of the evolution of the distribution is system metastability phase. In the base of Markov continuous process is ex-ponential distribution. Metastable phase is Markov sys-tem [8, 9]. The fi nding in this study is not the fi rst and a unique relationship and metastability of Markovian processes [10-13].Generalization in a fl ow structure for one dimen-sional form, can be extended to the material structure for two-dimensional form (surface) or three-dimension-al form (volume).  According to the fi ndings, metastable states in the structures of materials are analogous to Markovian processes. In a metastable state, the struc-tural distance are exponentially distributed. Explicit metric system for metastability – based on the probabi-listic concept, is established.AcknowledgmentThe authors acknowledge the support of research project TR 36030, funded by the Ministry of Science and Technological Development of Serbia.REFERENCES [1] E. Cohen, The infl uence of allotropy on the metastability of metals, and its bearing on chemistry, physics and tech-nology, Journal of the Franklin Institute, 179 (1915), 367.[2] C. H. Desch, Martensite structure of steel, Journal of the Franklin Institute, 179 (1915), 367.[3] A. Keller, S.Z.D. Cheng The role of metastability in poly-mer phase transitions, Polymer, 39 (1998), 4461-4487.[4] Q. Meng, X. Chen, S. Guo, X. Zhao. The metastability and martensitic transformation of Co particles, J. Alloys Com-pd. 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Hachicha, A Simulation Metamodelling Based Neural Networks for Lot-Sizing Problem in MTO Sector, Interna-tional Journal of Simulation Modelling, 10 (2011) 4, 191-203.[13] J. Beltrán, C. Landim, Metastability of reversible fi nite sta-te Markov processes, Stochastic Processes and their Appli-cations, 121 (2011), 1633-1677.Note:  The responsible translator for English language is N. Kozul, Novi Sad, Serbia

Metastability – Markovian approach

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