O princípio da mínima ação estocástico: um estudo no contexto dos movimentos randômicos e sua aplicação em eventos tipo cisne negro

Abstract

The least action principle is one of the most important physical principles and is applicable to many distinct areas of knowledge. This principle states that the trajectory in which a system evolves is the one with the minimal action. Therefore, within the scenery of randomic movements, due to the fact of not being a deterministic movement, it is not directly applicable. That means that to approach this kind of situation it was needed to develop the stochastic least action principle (SAP). This stochastic least action principle must reproduce analog results to those obtained by the trajectory integrals. To do so, it is considered that the mean of the variation of the classical action of the possible trajectories must be minimized, obtaining then, the constraints which the distribution of the trajectory probabilities of a movement must obey. Using the principle of Jaynes, the trajectory probability’s distribution is found, being proportional to the inverse of the action’s exponential. Taking this movement as a markovian system, it is possible to associate the probability of each mark of a trajectory to the immediately previous mark, obtaining the trajectory integral. Considering a gausian noise, it is possible to obtain a specific shape for such trajectory integral and also to observe that the integral is proportional to the inverse of the classical action. To verify the relation between the action and the trajectory’s probability, it has been built a computational simulation for the unidimensional random motion particles: i) within free regimen; ii) under the influence of a constant force and iii) under the influence of a restorative force, in which the results corroborate the relation predicted by the SAP. Once the results of the simulation with gaussian noise were verified, this formalism was applied to the description of black swan events. Such events are rare and extreme, therefore, for a system where it can occur, it is needed that events on the extremities of the distributions to be probabilistically relevant. To be so, heavy tailed distributions must be used, we elect stable distributions that have such characteristics and describe a process of anomalous diffusion. This kind of process is obtained by the equation of Fokker-Planck with a fractionary spatial- differential operator, a non-local operator which results on long-range correlations. Consequently it is needed to modify the entropy that describes an anomalous diffusion process, once the Shannon’s entropy is only valid to systems with short range interactions. Analogously, the generalized entropy of Tsallis was used to obtain the randomic movements’ trajectory probability with black swan events and it was verified, throughout computational simulations, the validity of this result.