Information Entropy Principle Research Articles

Rate statistics and thermodynamic analogies for relaxation processes in systems with static disorder: Application to stretched exponential

The paper deals with the relationships between the total rate of a relaxation process occurring in a system with static disorder and the decay rates attached to the different individual reaction channels. It is proven that the models of relaxation constructed on the basis of these two types of rates are equivalent to each other. From an experimentally observed relaxation curve it is possible to evaluate only the density of channels characterized by different relaxation rates and the overall probability distribution of the total relaxation rate. For evaluating the probability density of the individual relaxation rates attached to different channels an approach based on the maximum information entropy principle is suggested. A statistical thermodynamic formalism is developed for the relaxation time of a given channel, i.e., for the reciprocal value of the individual relaxation rate. The probability density of the relaxation time is proportional to the product of the density of channels to an exponentially decreasing function similar to the Boltzmann’s factor in equilibrium statistical mechanics. The theory is applied to the particular case of stretched exponential relaxation for which the density of channels diverges to infinity in the limit of large relaxation times according to a power law. The extremal entropy of the system as well as the moments and the cumulants of the relaxation times and of the relaxation rates are evaluated analytically. The probability of fluctuations can be expressed by a relationship similar to the Greene–Callen generalization of Einstein’s fluctuation formula. In the limit of large rates the density of channels and the probability density of individual rates have the same behavior; both functions have long tails of the negative power law type characterized by the same fractal exponent. For small rates, however, their behavior is different; the probability density tends to zero in the limit of very small rates whereas the density of channels displays an infrared divergence in the same region and tends to infinity. Although in the limit of small rates the density of channels is very large the probability of occurrence of these channels is very small; the compensation between these two opposite factors leads to the self-similar features displayed by the stretched exponential relaxation. The thermodynamic approach is compared with a model calculation for the problem of direct energy transfer in finite systems. The connections between stretched exponential relaxation and the thermal activation of the channels are also investigated. It is shown that stretched exponential relaxation corresponds to a distribution of negative and positive activation energies of the Gompertz-type.

Self-organization in the nuclear system. II. Formation of a new order

For pt. I see ibid., vol. 19, p. 2045 (1993). The information entropy of the atomic nucleus is calculated from the wavefunctions of the resonance states. The transition from low- to high-level density is traced as a function of the coupling strength between the discrete nuclear states and the environment of decay channels. In the critical region of the coupling strength where a redistribution inside the nucleus takes place, information entropy in relation to the discrete states of the closed system is created. Beyond the critical value, a few relevant short-lived modes exist: together with long-lived noise. This result is in full correspondence to the maximum information entropy principle of synergetics formulated by Haken (1992). The noise is characterized by disorder expressed by a large information entropy while the relevant modes have a high order and take, correspondingly, a small part of the information entropy of the whole system. The entropy excess accompanying the formation of the new order is used inside the system for creation of noise. Further, the noise is not structureless and the corresponding information entropy is smaller than its maximal value.

Information and Self-Organization: A Macroscopic Approach to Complex Systems

First Page

Self-Organization and Information

This paper is concerned with processes of self-organization which can take place both in the inanimate and animate world. In particular we study the question what physics can contribute to the understanding of these processes. Its traditional disciplines, namely thermodynamics and statistical mechanics which are concerned with the behavior of multi-component systems, require new ideas and concepts in order to cope with self-organizing systems. These concepts were elaborated in the new field of synergetics from the microscopic point of view. The present paper is mainly concerned with a macroscopic approach.In Section 1 we briefly remind the reader of various concepts of entropy and information and we give brief definitions of structure and self-organization. At present there seems to be no satisfactory definition of complexity available. Section 2 provides two examples of self-organizing systems, namely the laser and slime mold. In Section 3 we briefly remind the reader of the microscopic approach used in synergetics. As a new result it is shown that the information of the total system close to instability points is essentially contained in the information in the order parameters for which the specific example of a single order parameter is then treated explicitly. Finally we show how adequate constraints can be found to formulate the maximum information entropy principle for self-organizing systems. Our approach allows one to deduce the order parameters and dominant spatial patterns of a system which undergoes a non-equilibrium phase transition by means of an algorithm rather than by guessing.

Application of the maximum information entropy principle to selforganizing systems

Self-organizing systems are systems which can acquire macroscopic spatial, temporal, or spatio-temporal structures by means of internal processes. Hitherto the distribution functions of the order parameters governing the macroscopic structures could be calculated by microscopic theories only. In the present paper we derive them from macroscopic quantities, where we demonstrate the procedure explicitly by means of the single and multimode laser close to the lasing threshold.