Entropy analysis of nonequilibrium systems

Abstract

Entropy is among the most enigmatic notions in modem science. It has firstbeen introduced as the non-compensated quantity of beat exchanged in an isothermalexpansion of an ideal gas in a steam engine. Later on its deep probabilisticfoundations have been recognized. In information theory the entropy has been introduced naturally in the study of probability distributions as a measure of uncertainty. The effort for generalizationof the entropy of an event in a State space to the case of a succession ofStates, led to the notion of the block entropy. Here, the measurable quantitieswould be the frequencies of occurrence of certain blocks of events, instead ofindividual events. An obvions question that corne immediately in one’s mindin such a situation is: ‘what is the appropriate way of reading the sequence’?Because there are many different possible ways of reading.However, one should observe that, the block entropies obey to a strong monotonicity property (reminiscent of the H-theorem familiar from statisticalphysics) only when one measures the blocks having a sliding window on anarbitrary symbolic sequence. This type of reading the sequence enjoys also abetter numerical stability because it improves statistics. Thus, the use of blockentropies when one reads the sequence in different ways has been abandoned fora long time. Specialists thought that the entropie study of symbolic sequencesby the usual prescription of gliding suffices to give a correct characterization oftheir complexity. However, as we shall see in Chapters 2 and 4, for arbitrary symbolic sequencesone can readily show that the estimation of the block entropies dependsactually on the way of reading, that is on the observer [91, 92, 93, 95, 96]. This isessentially a strong uncertainty argument, related to old philosophical problems.Recently also, interesting new links appeared in the literature between thenotion of the entropy as a measure of complexity. These studies leave wide roomto future research and almost promise that these notions will be important inthe upcoming years