Active noise-driven particles under space-dependent friction in one dimension

We study a Langevin equation describing the stochastic motion of a particle in one dimension with coordinate $x$, which is simultaneously exposed to a space-dependent friction coefficient $\gamma(x)$, a confining potential $U(x)$ and non-equilibrium (i.e., active) noise. Specifically, we consider frictions $\gamma(x)=\gamma_0 + \gamma_1 | x |^p$ and potentials $U(x) \propto | x |^n$ with exponents $p = 1,2$ and $n = 0, 1, 2$. We provide analytical and numerical results for the particle dynamics for short times and the stationary probability density functions (PDFs) for long times. The short-time behaviour displays diffusive and ballistic regimes while the stationary PDFs display unique characteristic features depending on the exponent values $(p, n)$. The PDFs interpolate between Laplacian, Gaussian and bimodal distributions, whereby a change between these different behaviours can be achieved by a tuning of the friction strengths ratio $\gamma_0/\gamma_1$. Our model is relevant for molecular motors moving on a one-dimensional track and can also be realized for confined self-propelled colloidal particles.Comment: 10 page, 9 figure

Torus spectroscopy of the chiral Heisenberg quantum phase transition with quantum Monte Carlo simulations

The study of the universality classes of phase transitions is one of the most important topics in solid state physics, as it allows us to draw comparisons between the behaviour of systems that vastly differ microscopically. Usually, the universal numbers that define a phase transition are found measuring the various critical exponents of the system. In this thesis we aim to characterize the chiral Heisenberg quantum phase transition, represented in our case by the transition of a Fermi-Hubbard model on the honeycomb lattice between a semi-metal and an anti-ferromagnet, by studying the properties of its energy spectrum instead. We carry out this analysis employing a Determinant Quantum Monte Carlo (DQMC) algorithm, written specifically to deal with fermionic systems in an efficient and stable way. In this work we determine the fingerprint of this phase transition, while also finding evidence for a location of the critical point different from the one present in literature.Arbeit an der Bibliothek noch nicht eingelangt - Daten nicht geprüftInnsbruck, Univ., Masterarb., 2019(VLID)458213

Brownian particles driven by spatially periodic noise

We discuss the dynamics of a Brownian particle under the influence of a spatially periodic noise strength in one dimension using analytical theory and computer simulations. In the absence of a deterministic force, the Langevin equation can be integrated formally exactly. We determine the short- and long-time behaviour of the mean displacement (MD) and mean-squared displacement (MSD). In particular, we find a very slow dynamics for the mean displacement, scaling as $$t^{-1/2}$$ with time t. Placed under an additional external periodic force near the critical tilt value we compute the stationary current obtained from the corresponding Fokker–Planck equation and identify an essential singularity if the minimum of the noise strength is zero. Finally, in order to further elucidate the effect of the random periodic driving on the diffusion process, we introduce a phase factor in the spatial noise with respect to the external periodic force and identify the value of the phase shift for which the random force exerts its strongest effect on the long-time drift velocity and diffusion coefficien

A one-dimensional three-state run-and-tumble model with a ‘cell cycle’

We study a one-dimensional three-state run-and-tumble model motivated by the bacterium Caulobacter crescentus which displays a cell cycle between two non-proliferating mobile phases and a proliferating sedentary phase. Our model implements kinetic transitions between the two mobile and one sedentary states described in terms of their number densities, where mobility is allowed with different running speeds in forward and backward direction. We start by analyzing the stationary states of the system and compute the mean and squared-displacements for the distribution of all cells, as well as for the number density of settled cells. The latter displays a surprising super-ballistic scaling $$\sim t^3$$ at early times. Including repulsive and attractive interactions between the mobile cell populations and the settled cells, we explore the stability of the system and employ numerical methods to study structure formation in the fully nonlinear system. We find traveling waves of bacteria, whose occurrence is quantified in a non-equilibrium state diagram