聚集行为是常见的自然现象。动物的迁徙、细菌的运动以及心脏和脉搏的跳动都表现出一定的聚集行为。2007年, Cucker和Smale提出了可有效地刻画聚集行为的Cucker-Smale模型。2017年, Ha等人考虑了温度效应对聚集现象的影响, 对Cucker-Smale模型进行了改进。改进后的模型一般称为热力学Cucker-Smale模型 (简称TCS模型) 。具体而言: 本文探讨了一类特定作用核的热力学Cucker-Smale模型的渐近稳定性。当TCS模型的两个作用核函数ϕ (r) 和ζ (r) 均有界且存在严格大于0的下界时, 我们得到了TCS模型Cauchy问题的解会在一类特定初值下, 呈现指数稳定性的一个充分条件。特别的, 与以往的研究通常会假设ϕ (r) 和ζ (r) 有界且单调递减不同, 本文中我们对ϕ (r) 和ζ (r) 的单调性不做要求。此外, 我们还利用Matlab程序对本文结论的一维情形进行数值模拟。模拟结果显示, 在一组符合结论条件的初值下, 粒子的速度和温度会趋于同步, 而且它们的间距也保持有界, 从而进一步地验证了结论的正确性。

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.