论文标题
通过重置对数电势扩散
Diffusion with resetting in a logarithmic potential
论文作者
论文摘要
我们研究重置对对数潜力中扩散的影响。在此模型中，粒子在潜在的$ u（x）= u_0 \ log | x | $中的扩散是重置的，即带回其初始位置，并以恒定的速率$ r $ $。我们表明，这种可分析的可处理模型系统表现出一系列的相变为单个参数的函数，$βU_0$，电势强度与热能的强度之比。对于$βU_0<-1 $，电势具有强烈的排斥性，从而阻止了粒子达到原点。然后重置产生一个非平衡稳态，该稳态被精确而彻底地分析。相反，对于$βU_0> -1 $，电势要么是弱排斥或吸引力，而且扩散的粒子最终达到了起源。在这种情况下，我们为后续的第一-邮编时间分布提供了封闭式表达式，并表明重置过渡发生在$βU_0= 5 $。也就是说，我们发现当$ -1 <βU_0<5 $ $ -1 <βU_0> 5 $时，重置可以加快到达来源。本文提供的结果将结果推广到简单扩散与重置的结果 - 一种广泛适用的模型，是通过设置$ u_0 = 0 $从我们的获得的。为了扩展一般潜在的优势，我们的工作为大量问题的理论和实验研究打开了大门，这些问题将对数潜力的重置和扩散融合在一起。
We study the effect of resetting on diffusion in a logarithmic potential. In this model, a particle diffusing in a potential $U(x) = U_0\log|x|$ is reset, i.e., taken back to its initial position, with a constant rate $r$. We show that this analytically tractable model system exhibits a series of phase transitions as a function of a single parameter, $βU_0$, the ratio of the strength of the potential to the thermal energy. For $βU_0<-1$ the potential is strongly repulsive, preventing the particle from reaching the origin. Resetting then generates a non-equilibrium steady state which is characterized exactly and thoroughly analyzed. In contrast, for $βU_0>-1$ the potential is either weakly repulsive or attractive and the diffusing particle eventually reaches the origin. In this case, we provide a closed form expression for the subsequent first-passage time distribution and show that a resetting transition occurs at $βU_0=5$. Namely, we find that resetting can expedite arrival to the origin when $-1<βU_0<5$, but not when $βU_0>5$. The results presented herein generalize results for simple diffusion with resetting -- a widely applicable model that is obtained from ours by setting $U_0=0$. Extending to general potential strengths, our work opens the door to theoretical and experimental investigation of a plethora of problems that bring together resetting and diffusion in logarithmic potential.
