Sketching a probability theory of transition to turbulence

In this letter, we wish to initiate a probability theory to describe the (in)famous transition to turbulence phenomena. For normal systems, instability threshold can be predicted because the transition probability jumps from zero to P→1 when the control parameter exceeds the threshold. However, for nonnormal systems, transition is probabilistic, which depends on the control parameter and initial energy. A low-dimensional reduction idea is proposed to give a detailed description of the transition probability of nonnormal systems in future. We also wish that such an idea can be transplanted for understanding transition in more complex flows, e.g. atmospheric flows, mantle convection, ocean circulation.