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We consider a general system of two run-and-tumble particles interacting by hardcore jamming on the unidimensional torus. RTP are a paradigmatic active matter model, typically modeling the evolution of bacteria. By directly modeling the system at the continuous-space and -time level thanks to piecewise deterministic Markov processes (PDMP), we derive the conservation conditions which sets the invariant distribution and explicitly construct the two universality classes for the steady state, the detailed-jamming and the global-jamming classes. They respectively identify with the preservation or not of a detailed symmetry at the level of the dynamical internal states between probability flows entering and exiting jamming configurations. Thanks to a spectral analysis of the tumble kernel, we give explicit expressions for the invariant measure in the general case. The non-equilibrium features exhibited by the steady state includes positive mass for the jammed configurations and, for the global-jamming class, exponential decay and growth terms, potentially modulated by polynomial terms. Interestingly, we find that the invariant measure follows, away from jamming configurations, a catenary-like constraint, which results from the interplay between probability conservation and the dynamical skewness introduced by the jamming interactions, seen now as a boundary constraint. This work shows the powerful analytical approach PDMP provide for the study of the stationary behaviors of RTP sytems and motivates their future applications to larger systems, with the goal to derive microscopic conditions for motility-induced phase transitions.

The persistent character of the motion of active particles gives rise to accumulation at boundaries. I investigate the problem of run-and-tumble swimmers confined in a 1D box with hard walls, reporting expressions for the particles probability distribution and wall pressure. A crossover box length value is found below which the initial value of the pressure turns out to be higher than the asymptotic one, indicating a bounce effect of the active "wave" of swimmers. The case of attracting and repelling boundaries are also investigated using two different tumble rates for particles in the bulk and at walls. Escape problems are finally analyzed by considering partially permeable walls through which particles can leave the box., Comment: 16 pages

In this thesis, we study active particles with focus on statistical properties of trapping time near a boundary. Active particles have a self-propelled force that drives the system out of equilibrium. The trapping time is here the time during which a particle remains in contact with a boundary. Two types of self-propelled active particles models are considered in our study, namely, active Brownian particles and run-and-tumble particles. These models are applied in a two-dimensional space. To study the trapping statistics we use the framework of first-passage times. The trapping problems can be mapped onto a first passage problem associated with the angular dynamics of the particles. We calculate the angular first-passage time distribution and the mean first-passage time. The trapping time depends on the random incoming collision angle relative to the boundary, and we calculate both conditional and averaged first passage time distributions with respect to this random variable. For the active Brownian particle, we obtain the angular distribution from the solution of the Fokker-Planck equation with absorbing boundary conditions. From this the survival probability, first passage time distribution and mean trapping time are calculated. For run-and-tumble particles, the survival probability function can be deduced from its angular tumbling dynamics directly. This allows us to find the first-passage time distribution easily from the survival probability. The incoming angle distribution is obtained from simulations. We verify the theory through stochastic dynamic simulations that are constructed from the Langevin equation. The result of this study is that statistical features of trapping time have certain dependence on both the particle dynamics and the incoming angle distribution. For the active Brownian particle, the averaged first-passage time distribution is influenced significantly from the incoming angle distribution, giving rise to a non-monotonic trapping time distribution. However, the incoming angle does not influence the trapping of run-and-tumble particles. The theoretical predictions fit well with the numerical data. When particles are confined in a channel with flat boundaries, the run-and-tumble particles are trapped almost twice as long as the active Brownian particles in units of the persistence time.

Absorption problems of run-and-tumble particles, described by the telegrapher's equation, are analyzed in one space dimension considering partially reflecting boundaries. Exact expressions for the probability distribution function in the Laplace domain and for the mean time to absorption are given, discussing some interesting limits (Brownian and wave limit, large volume limit) and different case studies (semi-infinite segment, equal and symmetric boundaries, totally/partially reflecting boundaries)., 13 pages