Evolution paradigm introduced in biology [1] became popular in social and natural sciences, economy and liberal arts [2-6]. Although thousands of papers and books are published about evolution of different systems and artifacts, most are not more than collection of facts and speculations. All attempts to construct a forecasting theory or method for predicting future of technique were nonsuccess. Some TRIZ experts claimed recently the creation of simple (qualitative) forecasts for technical systems (TS) and technological processes (TP) that is based in regularities of technique evolution [7, 8]. The target of this work is to check a possibility for quantitative forecast of TS.
Any forecast must provide the value of probability of achieving a given level of TS performance by a certain time [4,5]. To provide the right description for forecasting of technique we should choose the appropriate mathematical apparatus for its description. It seems that such apparatus has been developed in the statistical physics that describe sets of objects regardless to their nature [9-11]. The statistical physics works with micro-states of objects (analog of the technique states), time, space and different averages that also applicable to TS and TP [12]. There is an extensive number of states Gi scaled like eN for a TS consists of N elements or subsystems in the phase space Q of a technique. All states Gi are metastable because technique changes due to development.
First of all, the question about the ergodicity is raised for the objects that should be forecasted [13, 5]. Let me remind that due to ergodicity the time <Q>t and ensemble <Q>E averages of microstates for a set (or ensemble in the statistical physics language) of the ergodic objects are equal and, therefore, the behavior of such object can be predicted if their current state is known [9-11]. The following important necessary conditions for the ergodicity must occur in a system [9-11, 14-16]:
i). Conservation of the Hausdorff dimensions d for the invariant ergodic informational measure of a dynamically system.
ii). Accessibility of almost all phase space with the equal probability for the Lebesgue measure.
iii). Completely nondegeneration of full system's Hamiltonian and the energy equipartition.
iv). Positive values of the maximal Lyapunov's exponent Lmax, i.e. the exponential divergence of initially close trajectories in the phase space.
Let us discuss the ergodicity of TS based on the modern knowledge about technique. Unfortunately, it is impossible to check any of the necessary conditions of ergodicity in direct experiments with real technical systems, because we cannot measure the ergodicity metrics like the maximal Lyapunov's exponent. On the other hand, the numerical experiments at modern superfast CRAY computers for analysis of a set that consists of non-identical elements require more time that a design of a new technique takes usually, so the numerical experiments are non-effective [17]. Hence, for the analysis of TS ergodicity we shall use the indirect information that can be extracted from the knowledge about technique. The summary of the necessary conditions analyses is following.
i). It is well known that the technique evolution leads to change of number of elements and links between subsystems, i.e. the Hausdorff dimensions d is not conservative metrics of the TS or TP.
iii). Any technique does not develop itself yet. Various factors, such as availability of raw materials, quality of labor forces, necessary knowledge for the development, etc. play important role in the evolution of technique. Due to complexity of a real technique various barriers can separate the transitions between any neighboring states N of the phase spaces. Then the probabilities of the transition Gi-1à Già Gi+1 and the transition Gi+1à Già Gi-1 are different, even if the equal “weight” corresponds to the states Gi+1, Gi, and Gi-1, i.e. the equal accessible of all states Gi of is violated. Each such transition can be considered as a step in a technique development.
iv). If the long-term development observed in some techniques meets the change of the phase space ( |dQ/dt|> 0 ), then gij = Gi - Gj decreases with time t and the maximal Lyapunov's exponent
Lmax = lim {ln[gij(t)/gij(0)]}/2t
t ->oo.
is negative. This slow down or saturation in evolution [2-7] begins just in times when the technique needs additional sources for its development which usually lie out of main knowledge domain of the system resources.
Therefore, the necessary conditions for the ergodicity are violated for the technical systems. What is why the application of the statistical methods to the design of TS [12] is under a question.
Besides the technique phase space is non-compact and is divided into some fragments. Any technique has a finite time-of-life, hence the average on time can be performed only up to TS life-time. Therefore, some states Gi of the phase space Q are not probed in a finite period because not all possibilities of the system development are tried by its designers. It is possible to speculate that these fragments allow existence of competition between various technical systems with the same primary function.
The non-uniform probability P of the existence in various states can be used for definition of the non-ergodicity parameter F [18] that is the width of the state distribution probability due to the violations of the equality
"Gi = "Gj;Gi,GjÌ N.
This parameter can be applied to various systems including technique [18]. In the ergodic system F = 0 and the dependence P(N) is characterized by the delta-function. On the other hand, in the absolutely non-ergodic system F = 1 and the uniform distribution is realized for P(N) in interval from 0 up to 1. In non-ergodic the majority of the non-ergodic systems the values of F that are far from the lower and upper limits [18] and P(N) is characterized by non-uniform distribution. Such distribution leads to the well-known S - shape evolution curve [2-8]. This S-curve can be represented by the equation of the so-called inverse Kohlrausch law
M ~ exp [(-t/t0)-d].
Here M is the measure of TS performance, t0 is the characteristic mean time of the system development, and d is the Kohlrausch exponent that depends on a value of the driven force for some types of the movement of the representative point through the phase space [19]. The comparison of the Kohlrausch law with the evolution curves of various TS (airplanes, semiconductor chips, light sources) leads to the values 0.4-0.8 for the exponent d. The uncertainty for forecast is in order of the Kohlrausch exponent d that reflects the non-ergodicity parameter F. Unfortunately, d or F are too big for any reasonable practical predictions about the evolution of a technique.
Therefore the lack of time dependence of evolution in non-ergodic systems (including TS and TP) does forbid a quantitative forecasting for such objects.
REFERENCES
1. C. Darwin, On the Origin of Species. Harvard University Press, 1964.
2. K. Binder and A. P. Young. Rev. Mod. Phys. 58(1986) 801.
3. Evolution and Culture. Eds. S. Marshall and E. R. Service. University of Michigan Press, 1960
4. The Economy as a Complex Evolving System. Eds. K. Arrow, P. W. Anderson and D. Pines. Addison Wesley, New York, 1988
5. Masanao Aoki, New Approaches to Macroeconomic Modeling. Oxford University Press, 1996.
6. T. Kuhn, The Structure of Scientific Revolutions, University of Chicago Press, 1962
7. G.S. Altshuller, To Find an Idea, Novosibirsk, Nauka, 1991.
8. G.S. Altshuller, B.L. Zlotin, A.V. Zusman, and V.I. Filatov, Search of New Ideas, Kartya Moldovenyaske, Kishinev, 1989.
9. M. Toda, R. Kubo and N. Saitf, Statistical Physics I. Equilibrium Statistical Mechanics. Berlin, Springer, 1983.
10. R. Balescu, Equilibrum and Nonequilibrum Statistical Mechanics. New York, John Willey & Sons, 1975.
11. D ter Haar , Rev. Mod. Phys., 27 (1955) 288.
12a. Yoshikawa, H., “Extended General Design Theory”, in Proceedings of IFIP W.G. 5.2: Design Theory for CAD, North-Holland, Amsterdam. 1985.
12b. Y. Reych, Research in Engineering Design 7 (1995) 1
13. S. N. Durlauf, "Nonergodic Economic Grown" Review of Economics Studies 60 (1993) 349.
14. V. M. Shurenkov, Markov's Ergodic Processes. /In Russian/. Moscow, Nauka, 1989.
15. K. Petersen, Ergodic Theory. Cambridge University Press, 1983.
16. I. P. Kornfel'd, Ya. G. Sinai and S. V. Fomin, Ergodic Theory. /In Russian/. Moscow, Nauka, 1980.
17. D. Thirumalai and R. D. Mountain, Phys. Rev. E47 (1993) 479.
18. S.D. Savransky, "Quasielastic Neutron Scattering QENS'93" Singapore, World Scientific, (1994) 194.
19. R. G. Palmer, D. L. Stein, E. Abrahams and P. W. Anderson, Phys. Rev. Lett. 53(1984) 958.