Dynamical phase transition in the first-passage probability of a Brownian motion

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

Keywords:

• Phase transition
• Brownian motion
• Probability distribution
• Particle
• Monotonic function
• Mathematical physics
• Critical value
• Langevin dynamics
• Sigma
• Physics

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