arXiv subjects

Mathis Guéneau

Publications and source records attributed to Mathis Guéneau.

8 recordsLinked to original sources

arXiv2026

Active Brownian Dynamics in Channels: First-Passage and Spatiotemporal Properties via Siegmund Duality

Accumulation at boundaries represents a widely observed phenomenon in active systems with implications for microbial ecology and engineering applications. To rationalize the underlying physics, we study the first-passage properties and spatial distributions of an active Brownian particle (ABP) in a channel. Leveraging Siegmund duality, we establish a direct mapping between the propagators of ABPs with absorbing and hard-wall boundary conditions, yielding analytical results in both problems. We analyze the system across low and high activity regimes -- quantifying persistent motion relative to diffusion -- and show that active motion, together with a favorable initial orientation, typically lowers the mean first-passage time relative to passive diffusion. Notably, the full time-dependent propagator between hard walls approaches a wall-accumulated stationary state, given by the derivative of the splitting probability as a consequence of Siegmund duality.

cond-mat.stat-mech↗

arXiv2026

Exact Stationary State of a $d$-dimensional Run-and-Tumble Particle in a Harmonic Potential

We derive the exact nonequilibrium steady state of a run-and-tumble particle (RTP) in $d$ dimensions confined in an isotropic harmonic trap $V(\mathbf r)=\mu r^{2}/2$, with $r=\|\mathbf r\|$. Rotational invariance reduces the problem to the stationary single-coordinate marginal $p_X(x)$, from which the radial distribution $p_R(r)$ and the full joint stationary density follow by explicit integral transforms. We first focus on a generalized trapped RTP in one dimension, where post-tumble velocities are drawn from an arbitrary distribution $W(v)$. Using a Kesten-type recursion, we represent its stationary position in terms of a stick-breaking (or Dirichlet) process, yielding closed-form expressions for its distribution and its moments. Specializing $W(v)$ to the projected velocity law of an isotropic RTP, we reconstruct $p_R(r)$ and the full joint distribution of all the coordinates in $d=1,2,3$. In $d=1$ and $d=2$, the radial law simplifies to a beta distribution, while in $d=3$, we derive closed-form expressions for $p_R(r)$ and the stationary joint distribution $P(x,y,z)$, which differ from a beta distribution. In all cases, we characterize a persistence-controlled shape transition at the turning surface $r=v_0/\mu$, where $v_0$ is the self-propulsion speed. We further include thermal noise characterized by a diffusion coefficient $D>0$, showing that the stationary law is a Gaussian convolution of the $D=0$ result, which regularizes turning-point singularities and controls the crossover between persistence- and diffusion-dominated regimes as $D \to 0$ and $D \to \infty$ respectively. All analytical predictions are systematically validated against numerical simulations.

cond-mat.stat-mech↗

arXiv2025

Non-Equilibrium Dynamics and First-Passage Properties of Stochastic Processes: From Brownian Motion to Active Particles

In this thesis, we develop analytical methods to study out-of-equilibrium stochastic processes driven by colored noise, i.e., noise with temporal correlations. These non-Markovian processes pose significant analytical challenges compared to processes driven by white noise, such as Brownian motion. A primary focus is on active particle systems, specifically the run-and-tumble particle subjected to an arbitrary force. We derive exact expressions for its mean first-passage time (MFPT) and exit probability from an interval using the backward Fokker-Planck equation. Remarkably, we find that the MFPT can be optimized as a function of the tumbling rate. Additionally, we investigate stochastic resetting and switching diffusion models. For switching diffusion models which are examples of "Brownian yet non-Gaussian diffusions", we use a renewal approach and large deviation theory to derive exact results for various observables. These include the distribution of the position of the particle and its moments, but also its cumulants which are key observables to characterize non-Gaussian fluctuations. Notably, we uncover an unexpected connection between this model and free cumulants. We also examine these models in the presence of a harmonic potential by using Kesten variables. This approach enables us to write an integral equation for the steady-state distribution, which we solve in specific cases. Furthermore, we extend Siegmund duality - a concept that is not widely known in the physics literature - to active particles, random diffusion models, stochastic resetting, and continuous-time random walks. This duality establishes a direct relation between first passage observables and the spatial properties of a dual process, which we explicitly construct.

cond-mat.stat-mech↗

arXiv2025

Large Deviations in Switching Diffusion: from Free Cumulants to Dynamical Transitions

We study the diffusion of a particle with a time-dependent diffusion constant $D(t)$ that switches between random values drawn from a distribution $W(D)$ at a fixed rate $r$. Using a renewal approach, we compute exactly the moments of the position of the particle $\langle x^{2n}(t) \rangle$ at any finite time $t$, and for any $W(D)$ with finite moments $\langle D^n \rangle$. For $t \gg 1$, we demonstrate that the cumulants $\langle x^{2n}(t) \rangle_c$ grow linearly with $t$ and are proportional to the free cumulants of a random variable distributed according to $W(D)$. For specific forms of $W(D)$, we compute the large deviations of the position of the particle, uncovering rich behaviors and dynamical transitions of the rate function $I(y=x/t)$. Our analytical predictions are validated numerically with high precision, achieving accuracy up to $10^{-2000}$.

cond-mat.stat-mech↗

arXiv2024

Run-and-tumble particle in one-dimensional potentials: mean first-passage time and applications

We study a one-dimensional run-and-tumble particle (RTP), which is a prototypical model for active system, moving within an arbitrary external potential. Using backward Fokker-Planck equations, we derive the differential equation satisfied by its mean first-passage time (MFPT) to an absorbing target, which, without any loss of generality, is placed at the origin. Depending on the shape of the potential, we identify four distinct ``phases'', with a corresponding expression for the MFPT in every case, which we derive explicitly. To illustrate these general expressions, we derive explicit formulae for two specific cases which we study in detail: a double-well potential and a logarithmic potential. We then present different applications of these general formulae to (i) the generalization of the Kramer's escape law for an RTP in the presence of a potential barrier, (ii) the ``trapping'' time of an RTP moving in a harmonic well and (iii) characterizing the efficiency of the optimal search strategy of an RTP subjected to stochastic resetting. Our results reveal that the MFPT of an RTP in an external potential exhibits a far more complex and, at times, counter-intuitive behavior compared to that of a passive particle (e.g., Brownian) in the same potential.

cond-mat.stat-mech↗

arXiv2024

Relating absorbing and hard wall boundary conditions for a one-dimensional run-and-tumble particle

The connection between absorbing boundary conditions and hard walls is well established in the mathematical literature for a variety of stochastic models, including for instance the Brownian motion. In this paper we explore this duality for a different type of process which is of particular interest in physics and biology, namely the run-tumble-particle, a toy model of active particle. For a one-dimensional run-and-tumble particle subjected to an arbitrary external force, we provide a duality relation between the exit probability, i.e. the probability that the particle exits an interval from a given boundary before a certain time $t$, and the cumulative distribution of its position in the presence of hard walls at the same time $t$. We show this relation for a run-and-tumble particle in the stationary state by explicitly computing both quantities. At finite time, we provide a derivation using the Fokker-Planck equation. All the results are confirmed by numerical simulations.

cond-mat.stat-mech↗

arXiv2024

Siegmund duality for physicists: a bridge between spatial and first-passage properties of continuous and discrete time stochastic processes

We consider a generic one-dimensional stochastic process $x(t)$, or a random walk $X_n$, which describes the position of a particle evolving inside an interval $[a,b]$, with absorbing walls located at $a$ and $b$. In continuous time, $x(t)$ is driven by some equilibrium process $\mathbf{\theta}(t)$, while in discrete time, the jumps of $X_n$ follow a stationary process that obeys a time reversal property. An important observable to characterize its behaviour is the exit probability $E_b(x,t)$, which is the probability for the particle to be absorbed first at the wall $b$, before or at time $t$, given its initial position $x$. In this paper we show that the derivation of this quantity can be tackled by studying a dual process $y(t)$ very similar to $x(t)$ but with hard walls at $a$ and $b$. More precisely, we show that the quantity $E_b(x,t)$ for the process $x(t)$ is equal to the probability $\tilde \Phi(x,t|b)$ of finding the dual process inside the interval $[a,x]$ at time $t$, with $y(0) =b$. This is known as Siegmund duality in mathematics. Here we show that this duality applies to various processes which are of interest in physics, including models of active particles, diffusing diffusivity models, a large class of discrete and continuous time random walks, and even processes subjected to stochastic resetting. For all these cases, we provide an explicit construction of the dual process. We also give simple derivations of this identity both in the continuous and in the discrete time setting, as well as numerical tests for a large number of models of interest. Finally, we use simulations to show that the duality is also likely to hold for more complex processes such as fractional Brownian motion.

cond-mat.stat-mech↗

arXiv2024

Optimal mean first-passage time of a run-and-tumble particle in a class of one-dimensional confining potentials

We consider a run-and-tumble particle (RTP) in one dimension, subjected to a telegraphic noise with a constant rate $γ$, and in the presence of an external confining potential $V(x) = α|x|^p$ with $p \geq 1$. We compute the mean first-passage time (MFPT) at the origin $τ_γ(x_0)$ for an RTP starting at $x_0$. We obtain a closed form expression for $τ_γ(x_0)$ for all $p \geq 1$, which becomes fully explicit in the case $p=1$, $p=2$ and in the limit $p \to \infty$. For generic $p>1$ we find that there exists an optimal rate $γ_{\rm opt}$ that minimizes the MFPT and we characterize in detail its dependence on $x_0$. We find that $γ_{\rm opt} \propto 1/x_0$ as $x_0 \to 0$, while $γ_{\rm opt}$ converges to a nontrivial constant as $x_0 \to \infty$. In contrast, for $p=1$, there is no finite optimum and $γ_{\rm opt} \to \infty$ in this case. These analytical results are confirmed by our numerical simulations.

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