The research work I have carried out since my master's degree (2014) focuses on out-of-equilibrium statistical physics in biological, hydrodynamic and electromagnetic contexts. My approach mainly consists in analytically solving physical problems by developing asymptotic expressions, while performing numerical solutions of equations and stochastic simulations, and confronting these results with experiments. My contributions focused on six major themes:

**1. Reduced search time for NK lymphocytes in the presence of bystanders [P1].** NK lymphocytes search to eliminate infected cells, and encounter bystander cells favouring their migration during their search. We have developed a model to understand in-vitro experiments. Numerical simulations show that the search time is reduced in the presence of these bystander cells, as much by their number as their radius of action.

**2. Quantitative approximation schemes for the glass transition [P2].** The glass transition is a first order transition depending on the kinetics of cooling, which does not occur at a given temperature (at fixed pressure), unlike the liquid-solid transition. The phase diagram is then extremely complex to draw. We carried out a systematic development of the equation of state around its exact solution in infinite dimension, allowing to obtain the thermodynamic properties of the glass transition in dimension $d\ge3$ and to construct the phase diagram.

**3. Dispersion in complex media [P3,P4,P5,P6].** The dispersion allows to understand the spread of a cloud of particles initially close in complex media. In these media, the Boltzmann distribution is not reached in the long time regime and this spreading is characterized by the effective diffusivity. We have studied the dispersion in microchannels, looking only at its link with the containment geometry. Three dispersion regimes were found and an expression of the effective diffusivity was proposed for each of them. Then, we analyzed the dispersion in a lattice of obstacles, identifiable with the dispersion in the microchannels. Finally, we looked at this dispersion by adding an attractive potential on the obstacle surface, which is strangely enhanced compared to reflective obstacles despite the addition of an energy trap.

**4. Brownian vortices created by optical traps [P7,P8,P9].** Optical traps are often assumed to be harmonic, but they are anharmonic and the radiation pressure from the laser creates a non-conservative force. Stationary currents are then present forming Brownian vortices. We are interested in the underdamped regime reached in experiments carried out in parallel for low pressures. We have derived an analytical expression for the stationary current and power spectrum density in agreement with numerical simulations and experimental observations.

**5. Escape time in cells and cytoskeleton organization [P10,P11].** The intracellular transport of various cargo-particles to the immunological synapse is crucial for the correct functioning of cells. The time of this transport can be minimized depending on the cytoskeleton organization, and in particular on the size of the actin cortex. We have calculated the first passage time within the narrow escape limit (small synapse) for a simple cytoskeleton organization with heterogeneous diffusivity. This allowed us to show that it was necessary to have a mechanism forcing the particles to go towards the cell membrane to minimize this search time.

**6. Collective motion of Potts spins [P12,P13].** Collective motions of large clusters of active particles are widely observed in nature and studied in various man-made systems. This is an out-of-equilibrium phenomenon described by a phase transition. We have analyzed the active Potts model for which the internal states of the particles correspond to their directions of motion and locally align like Potts spins. The flocking transition is similar to a liquid-gas phase transition without the supercritical region. We have also observed a reorientation of the coexistence phase and analytically explained its origin.

In the following paragraphs, I present a more detailed summary for each of the previous research topics, observing the same outline. Note that some important results of my thesis [P12] have not been published yet and are nevertheless discussed in this summary.

*This work was carried out during my 5-month internship of the fisrt year of my master's degree under the supervision of H. Rieger and K. Schwarz in Saarbrücken (Germany), during the second semester of 2014. It gave rise to one publication [P1].*

During this internship, I modeled the search process for natural killer cells (NK lymphocytes) in the presence of spectator cells, in collaboration with the experimental physics team of B. Qu in Hombourg (Germany). NKs search to eliminate cells infected with a pathogen (virus) or swollen, hereinafter called target cells, to prevent the spread of infection. During their search, they encounter other types of non-target (healthy) cells which are bystanders but which increase the efficiency and migration of these NKs. This is due to the production of hydrogen peroxide (${\rm H}_2 {\rm O}_2$) by these bystander cells. In-vitro experiments have shown that the rate and persistence of NKs were increased in their presence, without destroying their lethal capacity.

We have then developed two-dimensional diffusive models, in periodic domains, allowing us to understand the experimental observations. NKs have been represented as discoid Brownian cells searching immobile target cells instantly destroyed when found. The bystander cells, also immobile, increase the NKs diffusivity locally in a finite radius. We have used the kinetic Monte Carlo method [1,2] on the one hand, as well as a lattice model on the other hand to reproduce numerically this stochastic dynamic. These numerical simulations show that the average time to locate the target cells, defined as the half-life of the targets, is reduced in the presence of spectator cells, both in terms of their number and their radius of action. This impact is greater than the presence of simple obstacles, even if they already favor the search by reducing the accessible surface. Our numerical study therefore shows that the NKs search is more efficient in the presence of bystander cells, perfectly reproducing the observations of in-vitro experiments.

*This work was carried out during my 2-month internship of the second year of my master's degree under the supervision of F. Zamponi at LPTENS in Paris, during the winter of 2015. It gave rise to my first publication [P2].*

During this internship, I studied the thermodynamic properties of the glass transition by developing an approximation scheme starting from their exact solutions in infinite dimension. The glass transition is a random first order transition, belonging to the universality class of spin glasses [3,4]. The glass phase is therefore not unique, unlike the crystal for example. The phase diagram is then extremely complex, exhibiting several glassy states characterized by different thermodynamic properties (for example the equation of state) which generally depend on the kinetics of the glass transition.

Like many statistical models, the thermodynamic properties of the glass transition are solvable in infinite dimension with a mean field structure [5]. We have then carried out a systematic development around this solution in infinite dimension, similar to a high temperature or low density development. Using the replica method [6], for $m$ copies of the original system, we have then derived analytical expressions for the entropy and the equation of state of the glass phase in all dimensions $d$. These equations depend only on the thermodynamics and the structure of the liquid, and in particular on the pair correlation $g_{\rm liq}$ and the entropy $S_{\rm liq}$, and have a structure comparable to those obtained with the mode coupling theory (MCT) [7] valid only for $d=3$. It should be noted that there are only approximate solutions of these latter quantities for the liquid, the most known are called Hyper-Netted-Chain (HNC) and Percus-Yevick (PY), derived from the virial equation [8].

From the equation of state and the entropy (also called complexity $\Sigma_{\rm eq}$) of the glass, as well as the numerical values of the pair correlation of the liquid, we have calculated numerically, in dimension $d \ge 3$, the non-ergodicity factor and the densities: (i) of the dynamic transition $\varphi_d$ for which the liquid becomes infinitely viscous and several glassy states appear, and from which the equation of state for $m=1$ admits solutions; (ii) the Kauzmann transition $\varphi_K$ for which the number of glass states becomes subexponential with an entropy crisis, $\Sigma_{\rm eq}=0$ for $m=1$, corresponding to the ideal glass transition of the second order; (iii) the jamming transition ($m=0$) going from the threshold density $\varphi_{\rm th}$ to the close packing density of the glass $\varphi_{\rm GCP}$, corresponding respectively to the dynamic and Kauzmann transitions. For the dimension $d=3$, we obtained numerical values ($\varphi_d \simeq 0.53$, $\varphi_K \simeq 0.62$, $\varphi_{\rm th}\simeq 0.45$ and $\varphi_{\rm GCP}\simeq 0.68$) quasi-similar to the previous studies [9,10], based in particular on the MCT. We have also obtained values for all dimensions $d \ge 3$. This work therefore allowed to relate the thermodynamic properties of the glass transition in dimension $d=3$ and in infinite dimension.