标题： 自驱动粒子在幂律势阱中空间分布的连续和不连续形状转变 显示英文标题

标题： Continuous and discontinuous shape transitions in the spatial distribution of self-propelling particles in power-law potential wells

摘要： 我们研究了在二维幂律势阱中活性布朗粒子和跑跳粒子的定态，在平动扩散可以忽略的极限情况下。 粒子的势能取形式$U(r)\propto r^{n}$，其中$n\geq 2$且为偶数。 我们推导出二维情况下的位置概率分布$\phi({\bf r})$的精确方程，并在假设粒子的方向角是高斯变量的情况下求解该方程。 我们表明$\phi({\bf r})$具有紧支集，并且随着活性速度的增加，其形状会发生类似相变的变化。 对于活性布朗粒子，我们的理论预测在$n=2$时形状发生连续转变，在$n>2$时发生不连续转变，两者均与模拟结果一致。 在强活性区域，方向概率分布靠近外边界时为单峰，但向内部则变为双峰，表明存在环绕运动。 跑跳粒子的角分布中的单峰-双峰转变几乎不存在。 显示英文摘要

摘要： We study the stationary states of an active Brownian particle and run-and-tumble particle in a two dimensional power-law potential well, in the limit where translational diffusion is negligible. The potential energy of the particle is taken to have the form $U(r)\propto r^{n}$, where $n\geq 2$ and even. We derive an exact equation for the positional probability distribution $\phi({\bf r})$ in two dimensions, and solve for the same, under the assumption that the particle's orientation angle is a Gaussian variable. We show that $\phi({\bf r})$ has compact support and undergoes a phase transition-like change in shape as the active velocity increases. For active Brownian particle, our theory predicts a continuous transition in shape for $n=2$ and a discontinuous transition for $n>2$, both of which agree with simulation results. In the strongly active regime, the orientational probability distribution is unimodal near the outer boundary but becomes bimodal towards the interior, signifying orbiting motion. The unimodal-bimodal transition in the angular distribution is nearly absent for run-and-tumble particle.