arXiv subjects

L. Pal

Publications and source records attributed to L. Pal.

11 recordsLinked to original sources

arXiv2005

Statistical considerations on safety analysis

Alerting experience with a well-acknowledged safety analysis code initiated the authors to pay attention to safety issues of complex systems. Their first concern was the statistical characteristics of such a code. We point out a remarkable weakness of the so called 0.95/0.95 methodology: when repeating the search for the tolerance limit, we get a higher value with non-negligible probability. We propose the sign test as an alternative method. We point out the correct form of Wilks' formula when the number of parameters subjected to limitation is two or more.

physics.data-an↗

arXiv2005

Transport on randomly evolving trees

The time process of transport on randomly evolving trees is investigated. By introducing the notions of living and dead nodes a model of random tree evolution is constructed which describes the spreading in time of objects corresponding to nodes. By using the method of the age-dependent branching processes we derive the joint distribution function of the number of living and dead nodes, and determine the correlation between these node numbers as a function of time. Also analyzed are the stochastic properties of the end-nodes; and the correlation between the numbers of living and dead end-nodes is shown to change its character suddenly at the very beginning of the evolution process. The survival probability of random trees is investigated and expressions are derived for this probability.

cond-mat.stat-mech↗

arXiv2004

Late comment on Astumian's paradox

In 2001 Astumian published a very simple game which can be described by a Markov chain with absorbing initial and final states. In August 2004 Piotrowski and Sladowski asserted that Astumian's analysis was flawed. However, as was shown by Astumian, this statement was wrong. In this comment the properties of Markov chains corresponding to games that are more general than that studied by Astumian, are investigated.

physics.data-an↗

arXiv2004

Lifetime of small systems controlled by autocatalytic reactions

By using the point model of reaction kinetics we have studied the stochastic properties of the lifetime of small systems controlled by autocatalytic reaction A+X -> X+X -> A+X, X -> B. Assuming that a system is living only when the number of autocatalytic particles is larger than zero but smaller than a positive integer N, we have calculated the probability of the lifetime provided that the number of substrate particles A is kept constant by a suitable reservoir, and the end-products B do not take part in the reaction. We have shown that the density function of the lifetime is strongly asymmetric and in certain cases it has a well-defined minimum at the beginning of the process. It has been also proven that the extinction probability of systems of this type is exactly 1.

cond-mat.stat-mech↗

arXiv2004

Remarks on statistical aspects of safety analysis of complex systems

We analyze safety problems of complex systems using the methods of mathematical statistics for testing the output variables of a code simulating the operation of the system under consideration when the input variables are uncertain. We have defined a black box model of the code and derived formulas to calculate the number of runs needed for a given confidence level to achieve a preassigned measure of safety. In order to show the capabilities of different statistical methods, firstly we have investigated one output variable with unknown and known distribution functions. The general conclusion has been that the different methods do not bring about large differences in the number of runs needed to ensure a given level of safety. Analyzing the case of several statistically dependent output variables we have arrived at the conclusion that the testing of the variables separately may lead to false, safety related decisions with unforseen consequences. We have advised two methods: the sign test and the tolerance interval methods for testing more than one mutually dependent output variables.

physics.data-an↗

arXiv2004

Stochastic properties of systems controlled by autocatalytic reactions II

We analyzed the stochastic behavior of systems controlled by autocatalytic reaction A+X -> X+X, X+X -> A+X, X -> B provided that the distribution of reacting particles in the system volume is uniform, i.e. the point model of reaction kinetics introduced in arXiv:cond-mat/0404402 can be applied. Assuming the number of substrate particles A to be kept constant by a suitable reservoir, we derived the forward Kolmogorov equation for the probability of finding n=0,1,... autocatalytic particles X in the system at a given time moment. We have shown that the stochastic model results in an equation for the mean value of autocatalytic particles X which differs strongly from the kinetic rate equation. It has been found that not only the law of the mass action is violated but also the bifurcation point is disappeared in the well-known diagram of X particle- vs. A particle-concentration. Therefore, speculations about the role of autocatalytic reactions in processes of the "natural selection" can be hardly supported.

cond-mat.stat-mech↗

arXiv2004

Stochastic properties of systems controlled by autocatalytic reactions I

We analyzed the stochastic behavior of systems controlled by autocatalytic reaction A+X -> X+X. Assuming the distribution of reacting particles in the system volume to be uniform, we introduced the notion of the point model of reaction kinetics, and derived a system of differential equations for probabilities of finding n=0,1,... autocatalytic particles at a given time moment. It has been found that the kinetic law of the mass action cannot be supported by stochastic model.

cond-mat.stat-mech↗

arXiv2003

Randomly evolving trees III

The properties of randomly evolving special trees having defined and analyzed already in two earlier papers (arXiv:cond-mat/0205650 and arXiv:cond-mat/0211092) have been investigated in the case when the continuous time parameter converges to infinity. Equations for generating functions of the number of nodes and end-nodes in a stationary (i.e. infinitely old) tree have been derived. In order to solve exactly these equations we have chosen three simple distributions for the number of new nodes produced by one dying node. By using appropriate method we have calculated step-by-step the probabilities of finding n=1,2,... nodes as well as end-nodes in a stationary random tree. The conclusion to be correct that in the evolution process the formation of a rod-like stationary tree is much more probable than the formation of a tree with many branches.

cond-mat.stat-mech↗

arXiv2002

Randomly evolving trees II

Generating function equation has been derived for the probability distribution of the number of nodes with $k \ge 0$ outgoing lines in randomly evolving special trees. The stochastic properties of end-nodes (k=0) have been analyzed, and it was shown that the relative variance of the number of end-nodes vs. time has a maximum when the evolution is either subcritical or supercritical. On the contrary, the time dependence of the relative dispersion of the number of dead end-nodes shows a minimum at the beginning of the evolution independently of its type. For the sake of better understanding of the evolution dynamics the survival probability of random trees has been investigated, and asymptotic expressions have been derived for this probability in the cases of subcritical, critical and supercritical evolutions. In critical evolution it was shown that the probability to find the tree lifetime larger than x, is decreasing to zero as 1/x, if x tends to infinity. Approaching the critical state it has been found the fluctuations of the tree lifetime to become extremely large, and so near the critical state the average lifetime could be hardly used for the characterization of the process.

cond-mat.stat-mech↗

arXiv2002

Randomly evolving trees I

By introducing the notions of living and dead nodes a new model of random tree evolution with continuous time parameter has been constructed. It is assumed that two random variables, the lifetime and the offspring number of living nodes control the evolution process. It has been shown that the generating function of the probability to find the evolving tree in a given state satisfies a non-linear integral equation. Analyzing the time dependence of the average number of living nodes three different types of evolution (subcritical, critical and supercritical) can be observed. It has been proved that in the case of subcritical evolution there is a well-defined time point at which the variance of the number of living nodes has a maximum. The joint distribution function of the numbers of living and dead nodes has been derived, and the time dependence of the correlation between these node numbers has been calculated. It is found that in the critical evolution with increasing time the correlation converges to a fixed number which is independent of the distributions of the lifetime and the offspring number of living nodes.

cond-mat.stat-mech↗

arXiv1999

Stochastic model of hysteresis

The methods of the probability theory have been used in order to build up a new model of hysteresis. It turns out that the reversal points of the control parameter (e. g., the magnetic field) are Markov points which determine the stochastic evolution of the process. It has been shown that the branches of the hysteresis loop are converging to fixed limit curves when the number of cyclic back-and-forth variations of the control parameter between two consecutive reversal points is large enough. This convergence to limit curves gives a clear explanation of the accommodation process. The accommodated minor loops show the return-point memory property but this property is obviously absent in the case of non-accommodated minor loops which are not congruent and generally not closed. In contrast to the traditional Preisach model the reversal point susceptibilities are non-zero finite values. The stochastic model can provide a good approximation of the Raylaigh quadratic law when the external parameter varies between two sufficiently small values.

cond-mat.stat-mech↗