Probability density evolution method: Background, significance and recent developments

Investigation of a stochastic system from a viewpoint of studying the randomness propagation process in a physical system starts to play an important role in understanding the complete performance or behaviour of engineering systems. Following this path, the probability density evolution method (PDEM) has been developed by Li and Chen at the beginning of this century on the basis of the principle of probability preservation and its random event description. A family of generalized probability density evolution equation (GPDEE) was then derived. The systematic analysis indicates that the new family of equation actually reveals the logical fundamentals of randomness propagation in a physical system: the transition of the probabilistic structure in a stochastic system definitely relies on the change of physical state of the system. This paper devotes to a summary on the background and basic theoretical developments of the PDEM. Considering the limitation of probability conservative systems for the GPDEE, a novel concept of probability dissipation is introduced and a completely uncoupled partial differential equation is derived as well with respect to the evolutionary probability density function, which holds for any physical quantity of a probability dissipative system. For illustrative purposes, several engineering applications, including the dynamic reliability assessment of controlled structures with fractional derivative viscoelastic dampers and the stability analysis of structures under dynamic loading, are investigated, respectively.