One- and two-dimensional active transport

Coordinator : Estelle Pitard

Transport phenomena are ubiquitous in physics, involving simple or complex fluids, as well as in biophysics. Such phenomena can be modelled by various paradigmatic models.

In one dimension, the TASEP model (Totally asymetric simple exclusion model) is used in various forms and can be applyed to traffic, waiting files, directed transport in channels and many more. It can be modified and studied according to various pratical cases, for example by the inclusion of disorder or the addition of diffusion and reaction rules.

In a first project, we asked ourselves the question of the time it takes for particles to leave a network on which they are transported by a TASEP model. More precisely, if N particles are initially located on the first sites of a network of L sites, we investigate the time it takes for all of them to leave this network. This study was carried out in collaboration with J. Dorignac and F. Geniet from the SCPN team at L2C. We have shown that if N << L, the average particle exit time is asymptotically given by TN (L) ∼ L + βN √L when L is large. Using exact results obtained for 2 particles, we constructed an approximate time- and space-continuous model of random particle motion that enabled us to obtain a recurrence relation. These results are in excellent agreement with the numerical values obtained by simulation, but also with the exact results obtained for N=2 and N=3 via a link between this problem and the determination of the expectation of the maximum eigenvalue of certain GUE-type random matrices [J. Dorignac, F. Geniet and E. Pitard, 2024].

In a second project, we have studied the motion of active oscillating particules in a viscous fluid, either in one or two dimensions. This model was generated by the interest for the motion of active cilia grafted on a biological substrate such as the bronchial epithelium, surrounded by the bronchious viscous fluid. This interest was motivated by the role played by these cilia in the cleaning process at stake in respiratory diseases.

We have shown that the spontaneous active oscillation of the cilia induces a collective motion in the presence of noise, which takes the form of a metachronal wave. Spatial properties of this wave (such as the wavelength) are independent of viscosity but depend on the activity of the cilia. This study was carried out in collaboration with G. Massiera from the team Matière Molle at L2C and with S. Dey from University SRM Andra Pradesh (India). [S.Dey, G. Massiera, E. Pitard, 2024].