AI 中文总结
本文针对重置率依赖位置的自适应扩散，确定了λ=-2为临界阈值，发现平方反比重置对应平衡态对数势，存在温度依赖幂律尾部的离域转变，明确了空间依赖型重置的定域条件。
AI 中文摘要
随机重置可使扩散定域并产生非平衡稳态，但空间依赖型重置产生定域的条件仍不明确。本文建立了自适应重置下扩散的通用分类方法，其中重置率r(x)依赖于位置。对于渐近标度r(x)~|x|^λ的重置率，确定了λ=-2为临界阈值：当λ>-2时，稳态为定域态且呈现 stretched-exponential（拉伸指数）尾部；当λ<-2时，重置渐近过弱无法使粒子定域。在临界标度r(x)~|x|^{-2}时，出现全新的定性区域：稳态呈现幂律尾部，其指数依赖于温度，并表现出有限温度下的离域转变。因此，平方反比重置在非平衡体系中起到了平衡态下对数势的作用，建立了与平衡态中温度驱动离域相对应的非平衡对应关系。
英文摘要
Stochastic resetting can localize diffusion and generate nonequilibrium steady states, but the conditions under which spatially dependent resetting produces localization remain unclear. Here, we establish a general classification for diffusion under adaptive resetting, where the resetting rate $r(x)$ depends on position. For rates with the asymptotic scaling $r(x)\sim |x|^λ$, we identify a sharp threshold at $λ=-2$. For $λ>-2$, the steady state is localized and exhibits stretched-exponential tails, whereas for $λ<-2$, resetting is asymptotically too weak to localize the particle. Precisely at the marginal scaling $r(x)\sim |x|^{-2}$, a qualitatively new regime emerges: the steady state develops power-law tails with a temperature-dependent exponent and exhibits a finite-temperature delocalization transition. Thus, inverse-square resetting plays the role of the logarithmic potential in equilibrium, establishing a nonequilibrium counterpart of the temperature-driven delocalization seen there.
