P. Tartaglia

Stochastic dynamic approach to the decay of an unstable state

We show that the problem of the decay from an unstable state can be approached by means of a systematic expansion which directly approximates the trajectories associated with the stochastic process. The first step of the expansion is a quasi-deterministic theory (QDT) which gives good results when far from the critical point and is valid only for the early times when approaching it. In the latter case the successive terms allow the description of the behavior for intermediate times and close to the steady state. The method is illustrated in the case of a symmetric double well potential and compared with the results of a direct computer simulation of the stochastic process.

Transient laser radiation as a stochastic process near an instability point

A new approximate solution of the stochastic process associated to the laser transient radiation is proposed. It is shown that, by means of the classical equation of motion, it is possible to introduce a non-linear time-dependent mapping between the original stochastic process and a new process which, near an unstable state, has a limited range for short times. Detailed comparisons with numerical results and experimental data are also performed.

Transient behavior of nonequilibrium phase transitions

Abstract The transient behavior of a simple Malthus-Verhulst like nonlinear chemical system is analyzed. This model exhibits a nonequilibrium phase transition analogous to the laser instability. We first perform a qualitative analysis of the model in the space of the rate parameters. We show, both with a nonperturbative calculation and a direct computer simulation, that the fluctuations in the transient regime are enhanced above the steady state value. This anomalous behavior occurs in a given range for the rate parameters.

Analysis of the coexistence curve of a liquid system with a global equation of state

The data of Green (1976) on the difference of volume fractions in the coexisting phases of the isobutyric acid-water system are analysed in terms of an equation of state, which is a model for the crossover from the critical region to a van der Waals-like behaviour. The results provide evidence of the importance of non-scaling correction terms even in a small range close to the critical point. Correspondingly the effective critical index describing the coexistence curve shows a significant departure from its asymptotic value.

Transient behaviour near an instability point: Vectorial stochastic representation of the Malthus-Verhulst model

SummaryWe study the fluctuations in the Malthus-Verhulst model of population dynamics in the vicinity of the instability point. We introduce a representation in terms of an-dimensional simple stochastic process, wheren is an integer related to the rate parameters of the model. We are thus able to obtain the transient behaviour for the decay from an unstable state by means of the quasi-deterministic approximation. Forn=1 and 2 we recover the well-known case of a double-well potential and the single-mode laser, respectively. We show that it is also possible to perform an analytic continuation of the results for any noninteger positiven.RiassuntoIl modello di Malthus-Verhulst per la dinamica della popolazione è analizzato, in prossimità del punto d’instabilità, introducendo un processo stocastico vettoriale adn componenti, doven è un intero legato ai parametri del modello. Con un’approssimazione quasi deterministica si ricava l’evoluzione temporale durante il transiente del decadimento dello stato instabile. Pern=1 e 2 si ritrovano i casi ben noti del potenziale a doppia buca e del laser unimodo. Si mostra inoltre che il risultato è analiticamente continuabile a valori din non interi.РезюмеМы исследуем флуктуации в модели Малтуса-Верхулста динамики заселения в окрестности точки неустойчивости. Мы вводим представление в терминахn-мерного простого стохастического процесса, гдеn является целым числом, связанным со скоростными параметрами модели. Таким образом, мы можем получить переходное поведение для распада из неустойчивого состояния с помощяю квазидетерминистического приближения. Дляn=1 иn=2 мы получаем хорошо известный случай потенциала двойной ямы и одномодового лазера, соответственно. Мы показываем, что также вокже возможно получить аналитическое продолжение полученных результатов для любого нецелого положительногоn.