We study run and tumble particles on the one-dimensional lattice $\mathbb{Z}$. We explicitly compute the Fourier-Laplace transform of the position of the particle and as a consequence obtain explicit expressions for the diffusion constant and the large deviation free energy function. We also do the same computations in a corresponding continuum model. In the latter, when adding an external field, we can explicitly compute the large deviation free energy, and the deviation from the Einstein relation due to activity. Finally, we generalize the model to the $d$-dimensional lattice $\mathbb{Z}^d$, with an arbitrary finite set of velocities, and show that the large deviation free energy for the position of the particle can be computed via the largest eigenvalue of a matrix of Schrödinger operator form, for which we can derive an explicit variational formula via occupation time large deviations of the velocity flip process.
First-order condensation transition in the position distribution of a run-and-tumble particle in one dimension
我们考虑一个单一的运行和翻滚粒子（RTP）在一维运动。我们假设粒子的速度是独立绘制在每个翻滚从零均值高斯分布和运行时间是指数分布。利用$N\gg 1$研究了粒子在$N$运行后的位置$X$的概率分布$P(X,N)$。我们表明，在政权$ X \sim N^{3/4}$的分布$P(X,N)$具有大的偏差形式，其特征在于由一个不连续的衍生物在临界值$X=X_c>0$的速率函数。由于$P(X,N)$的对称性，$X=-X_c$也是如此。我们发现，这种奇异性对应于一阶凝聚过渡：对于$X>X_c$一个单一的大跳跃占主导地位的RTP轨迹。我们考虑的参与率的单次运行的位移作为系统的序参量，表明这个量是不连续的$X=X_c$。我们的研究结果支持与约束马尔可夫链蒙特卡罗算法进行数值模拟。
Activity-driven clustering and many-body steady state of jamming run-and-tumble particles
Université Clermont-Auvergne（克莱蒙奥弗涅大学）；CNRS（法国国家科学研究中心）；Laboratoire de Mathématiques Blaise Pascal UMR 6620（布莱斯·帕斯卡数学实验室）；Institut de Mathématiques, Université de Neuchâtel（纳沙泰尔大学数学研究所）
