A Symptotic Stability of Thermodynamic Cucker-Smale Model with a Specific Interacting Kerne

Gathering behavior. are commonly observed in nature. Animal migration, bacterial movement, and the beating of the heart and pulse, exhibit collective behaviors all exhibit certain aggregation behaviors. In 2007, Cucker and Smale proposed the Cucker-Smale model, which can effectively characterize aggregation behavior. In 2017, Ha et al improved the Cucker-Smale model, which considered the effect of temperature on aggregation phenomena. This improved model is generally referred to as the thermodynamic Cucker- Smale model (TCS model for short) . Specifically, this article explores the asymptotic stability of thermodynamic Cucker-Smale model with a specific interacting kernel. When the two action functions ϕ (r) and ζ (r) of the TCS model are both bounded and have a lower bound which is strictly greater than 0, we obtain a sufficient condition for the exponential stability of the solution to the Cauchy problem of the TCS model. Specifically, previous studies have typically assumed that ϕ (r) and ζ (r) are bounded and monotonically decreasing, but in this article, we do not require monotonicity for ϕ (r) and ζ (r) . In addition, we also used Matlab programs to numerically simulate the one-dimensional case of the conclusions presented in this paper. The simulation results show that under a set of initial values that meet the conclusion conditions, the velocity and temperature of particles tend to synchronize, and their spacing remains bounded, further verifying the correctness of the conclusion.