A New Approach To Determination
A NEW APPROACH TO DETERMINATION
OF THE EMERGENCY SITUATION PROBABILITY:
THE RISK LEVELS AND THE ERGODIC THEORY
A.L. Tsykalo
State Academy of Refrigeration of Odessa, Ukraine
The determination of the risk level (we mean ecological and industrial risks, i. e. probability of potentially possible emergencies, accidents and catastrophes of an elemental, technogenic or social origin) is a very sophisticated complex problem as it requires the account of very many factors (both “external” ones related to surrounding conditions, for example, the impact of extraterrestrial objects and radiation’s and “internal” - for instance, the faults of the personnel at enterprises and objects, sabotage, potential errors, etc.). Even the use of especially powerful up-to-date computers and soft-ware (e.g. modifications of Monte-Carlo method) is connected with considerable difficulties, especially if it is necessary to take into account the influence of many factors of different nature (physical, chemical, biological, geological, factors connected with the social, psychological and criminal sphere, etc.).
Nevertheless, the solution of this problem is of extreme importance, especially for regions characterized by high density of population combined with a great number of potentially harmful objects (enterprises of atomic, chemical, mining industries, storage’s of harmful substances and materials, ports and terminals, main pipelines, etc.). and with the availability of recreation zones and resources, objects of the natural reserve found, etc. It is such conditions that characterize the near-sea zones of the Black Sea; and first of all it is true for the coastal line of the North-West part of the Black Sea (Odessa, Nikolaev, Kherson regions) and the Crimean shores. At the same time the solution of this problem will take it possible to find motivated answers for a number of important questions, related to the location of new objects and development of existing ones, the determination of the safety degree and resulting from it the cost of land and other local resources, the
development of programs to withstand emergencies, the defining of necessary means of protection and their number for personnel of enterprises and objects, the population and etc.
The objective of this work is to show a new in principle approach to the solution of this problem based on the use of up-to-date results of the ergodic theory.
As it is known, the ergodic theory which was developed mainly in theoretical (mathematical) physics and is closely connected with the solution of the fundamental problem of matter and thermodynamic statistical theory, takes its origin in the works of L. Boltzmann (see, for example, 1 developed by I. Prigogine, N.S. Krylov, N.N. Bogoliubov, G.M. Zaslavsky, V.I. Arnold, Ya.G. Sinay et al. 2-13. Though the first most important results obtained in this field (including the famous Boltzmann’s formula imprinted on the monument which was erected on his grave) and modern results reffered mainly to objects of atomic-molecular physics and physical chemistry, the essence of the ergodic theory is much wider as it is based on the theotetical-probabilistic mathematical concepts. Many other objects, which are characterized first of all by a great many interacting (or independent) factors influencing to some degree the behaviour of the object on the whole, in principle can be objects for the application of these results.
As it is known, the essence of the ergodic theory (sometimes called “ergodic hypothesis”) can be expressed in the following simplest form: evolution of the object, whose behaviour is influenced by a great number of factors (in statistical-molecular theory this great number is a great number of atoms or molecules composing some macroscopic object), is such that as time goes on the isolated object transits to the most probable state (or to “equilibrium state” in terms of statistical physics). This corresponds to the known second law of thermodynamics. The idea of 6N-dimentional “ergodic surface” used in the ergodic statistical theory enables the use of the graphic illustration given in Figure 1. The number N corresponds to a great number of influencing factors (in the case of molecular object it corresponds to the number of particles composing the macroscopic object - there are atoms or molecules, of the order of Avogadro number, N А), multiplier 6 corresponds to the sum of the each atomic (or molecular) mass center coordinates x, y, z (3) and number of coordinate projections of the vector of the each atom (or molecule) impulse ( mvx, mvy, mvz ) (3).
The ergodic surface region having the largest area (zone 1) corresponds to the most probable (“equilibrium”) macrostate of the object; zone 2 correspons to the less probable (“nonequilibrium”) macrostates. Each point of this ergodic 6N-dimentional surface corresponds to one microstate. If the initial state of the object is characterized by point A, in the process of evolution (trajectory A-B, for example) the system gets, earlier or later, to region 1 of the largest area by all means and then for ifinitely long time stays in this region which corresponds to its most probable (“equilibrium”) state (trajectory B-C, for example). Though there can be going out of the equilibrium state (for instance segment C-D), but hardly probable and short duration. The ergodic theory and statistical-probabilistic methods make it possible to assess the probability of the object being in the equilibrium state as well as the probability of the system transit to other states (by the way, the probability of the object transit to the nonequilibrium state is the higher, the less dimension of the system N or the less number of influencing microfactors).
The idea of using the ergodic theory to solve our problem is the following. Suppose we have some object at some “macrostate” characterized by a relatively small number of factors (let us call them “macrofactors” similarly with the parameters of “macrostate” in the molecular statistical theory) and its behaviour is influenced by a rather large number of factors (let us call them “microfactors” similarly with parameters of “microstate” used in the molecular statistical theory).
Figure 1. The ergodic 6N-dimension surface:
1 - zone of the most probable ("equilibrium") state;
2 - zone of the less probable ("nonequilibrium") states;
A-B-C-D - trajectories of the representation point.
A living organism, a population of some region, natural ecological system or industrial object can be examples of such objects. The macrofactors are body temperature, blood pressure and some other functional parameters of the living organism; for ecological system or for region population the macrofactors are population density, birth- sickness- and death-rates, etc.; for an industrial object there are productivity, the number of employees, profitability, etc. Microfactors here are numerous characteristics (for a living organism there are characteristics of outer and inner spheres including heredity factors, many characteristics of all systems and organs; for region population there are individual peculiarities, local home features; for all of these examples there are numerous factors of outer surroundings, psychological and professional features of people and of enterprises personnel, hydrogeological and climatic factors of the locality and many others).
The application of the ergodic theory makes it possible not only to comprehend from a new position an important and pressing problem of determining the probability of emergencies and connected with it the risk levels. there appears the possibility of quantitative assessment and detailed calculations of risk factors using all the resources of the ergodic hypothesis. For example, one can use the famous Boltzmann’s formula which connects the entropy of the system S with the “thermodynamic probability” W (the latter is meant as the number of microstates by means of which it is possible to realize the given macrostate):
S=k lnW,
where k is Boltzmann’s constant. The modern results based on the ergodic theory opens the many new possibilities in the interested field, however we shall discuss these possibilities on the basis of some concrete examples and objects in the following papers.
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