Magnetic fields are ubiquitous in the Universe and observed in various astrophysical systems, with different amplitudes. In galaxy clusters, they are typically on the order of a few $\mu\mathrm{G}$. Comparing this value with the amplitude of magnetic seed fields predicted by several theoretical scenarios (between $10^{-20}$ to $10^{-9}~\mathrm{G}$) suggests that a dynamo mechanism might have amplified them. Their study is of paramount interest because they are thought to influence the intracluster medium (ICM) dynamics, affecting processes like gas cooling and the propagation of cosmic rays. Understanding their origin and evolution in the ICM is a necessity to deepen our knowledge of the large-scale structure formation within the framework of the $\Lambda$CDM model.

In this thesis, I investigate the magnetic field amplification in the ICM, taking into account its weakly collisional nature that prevents the application of ideal magnetohydrodynamics (MHD) equations. I also explore the evolution and properties of the Faraday rotation measure (RM), a common observable of cosmic magnetism. To do so, I developed a semi-analytical model based on merger trees generated by the modified GALFORM model, which probes the evolution of various plasma parameters during a typical galaxy cluster formation through dark matter halos mergers. The regulation of pressure anisotropies by kinetic instabilities is implemented, along with a model of the small-scale dynamo (SSD), which amplifies the magnetic fields at scales smaller than the biggest eddies produced by the turbulence. My work reveals that the SSD is self-accelerating as the magnetic field transitions into the magnetized kinetic regime, suggesting that magnetic fields could be amplified from seed fields to equipartition with the turbulent velocity field (a few $\mu\mathrm{G}$) within a few hundred million years. Consequently, my results indicate that $\mu\mathrm{G}$ magnetic fields should be present in high redshift clusters. Yet, the implications of such strong fields in the $\Lambda$CDM model are still to be explored. The RM radial distributions from my models at $z=0$ were compared to radio observations of a selection of low-redshift clusters. Overall, my model is in good agreement with observational data, and the most significant discrepancies are due to a high level of gas mixing in some clusters of the sample. My work highlights the importance of considering the weakly collisional nature of the ICM in studying its dynamics over cosmological timescales and suggests exploring alternative formulations to classical MHD.