记忆稳定自发粒子聚集：线性化Vlasov-Fokker-Planck分析
Memory Stabilizes Spontaneous Particle Aggregation: A Linearized Vlasov--Fokker--Planck Analysis
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中文总结 AI 辅助
该研究通过线性化Vlasov-Fokker-Planck分析，证明记忆可稳定自发粒子聚集的不稳定模式，且不会破坏无记忆时的稳定均匀态，还推导了相关定态解分支。
中文摘要 AI 辅助
我们对Vlasov-Fokker-Planck方程进行线性化稳定性分析，该方程是带记忆的随机自发聚集模型的平均场描述。记忆由K个内部变量链表示。我们表征空间均匀平衡态，并推导空间非均匀扰动的标量色散关系。当所有弛豫率相等时，我们证明无记忆模型中不稳定的每个傅里叶模式都具有唯一的临界弛豫参数：足够长的记忆会使该模式稳定，而短记忆时该模式仍不稳定。尽管记忆可能使与感知核负傅里叶系数相关的单个模式不稳定，但我们证明，对于径向对称的距离递减核，记忆无法使无记忆模型中稳定的均匀平衡态不稳定。因此，记忆无法在原本稳定的均匀状态中产生整体不稳定性，尽管它可能改变不稳定模式的集合。我们给出归一化顶帽核的数值例子，证明增加有效记忆长度会依次稳定傅里叶模式，更高频率模式比更低频率模式先稳定，这与近期粒子模拟中观察到的粗化效应一致。最后，在适当的非退化和正则性假设下，我们使用Crandall-Rabinowitz定理获得从均匀态分岔出的空间非均匀定态解分支。
英文摘要
We perform a linearized stability analysis of the Vlasov--Fokker--Planck equation obtained as the mean-field description of a stochastic spontaneous aggregation model with memory. Memory is represented by a chain of $K$ internal variables. We characterize the spatially homogeneous equilibria and derive a scalar dispersion relation for spatially inhomogeneous perturbations. When all relaxation rates are equal, we show that every Fourier mode that is unstable in the memoryless model possesses a unique critical relaxation parameter: sufficiently long memory stabilizes the mode, whereas it remains unstable for short memory. Although memory may destabilize individual modes associated with negative Fourier coefficients of the sensing kernel, we prove that, for radially symmetric distance-decreasing kernels, it cannot destabilize a homogeneous equilibrium that is stable in the memoryless model. Thus, memory cannot create an overall instability of an otherwise stable homogeneous state, although it may change the set of the unstable modes. We present a numerical example for the normalized top-hat kernel, demonstrating that increasing the effective memory length successively stabilizes the Fourier modes, with higher frequencies being stabilized before lower ones. This is consistent with the coarsening effect observed in recent particle simulations. Finally, under suitable nondegeneracy and regularity assumptions, we use the Crandall--Rabinowitz theorem to obtain branches of spatially inhomogeneous stationary solutions bifurcating from the homogeneous state.
