Simple models for mesoscopic systems: from slender structures to stochastic resetting

The objective of this thesis is to advance the understanding of complex physical phenomena through the lens of statistical physics. Specifically, it addresses two fundamental questions: What types of interactions can induce buckling of slender structures when their temperature is increased? And, how can we devise an optimal strategy for locating a hidden target? The thesis is divided into two distinct parts, both employing mesoscopic descriptions -- neither fully microscopic nor fully macroscopic -- to capture the essential interactions and behaviours that qualitatively govern the phenomena under investigation. In the first part, we examine the buckling behavior of low-dimensional materials under thermal load. To this end, we develop a comprehensive model that characterises the system using a minimal setup for mimicking: (i) elastic and electronic degrees of freedom, and (ii) coupling between the elastic and the electronic modes. In the second part, we investigate stochastic resetting processes as a means to formulate efficient search strategies. We explore various resetting mechanisms to understand how to optimise the search performance in real scenarios, where: (i) resetting involves a finite cost, and (ii) the target location is only partially known.