Dynamical phase transition in the first-passage probability of a Brownian motion – Archive ouverte HAL

Benjamin Besga ¹ Felix Faisant ¹ Artyom Petrosyan ¹ Sergio Ciliberto ¹ Satya N. Majumdar ² Satya Majumdar ²

Benjamin Besga, Felix Faisant, Artyom Petrosyan, Sergio Ciliberto, Satya N. Majumdar, et al.. Dynamical phase transition in the first-passage probability of a Brownian motion. Physical Review E , American Physical Society (APS), 2021, 104 (1), ⟨10.1103/PhysRevE.104.L012102⟩. ⟨hal-03301450⟩

We study theoretically, experimentally and numerically the probability distribution $F(t_f|x_0,L)$ of the first passage times $t_f$ needed by a freely diffusing Brownian particle to reach a target at a distance $L$ from the initial position $x_0$, taken from a normalized distribution $(1/\sigma)\, g(x_0/\sigma)$ of finite width $\sigma$. We show the existence of a critical value $b_c$ of the parameter $b=L/\sigma$, which determines the shape of $F(t_f|x_0,L)$. For $b>b_c$ the distribution $F(t_f|x_0,L)$ has a maximum and a minimum whereas for $b

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