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

Abstract

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.

Figures (39)

• Figure 1: Schematic representation of a thin plate of thickness $\Delta$ that lies on the $xy$-plane. The displacement field $\mathbf{u}=(u_x,u_y,\eta)$ has two in-plane components $u_x$ and $u_y$, and an out-of-plane component $\eta$.
• Figure 2: Height of the STM tip as a function of the applied voltage for several constant tunneling currents. Rippled profiles, represented by red curves, are observed for low voltages: the height $Z$ has a smooth and reversible dependence with the voltage. The black curve indicates the critical value, which depends on the local heating---product of the voltage and the current. At this critical voltage, the system undergoes an abrupt transition to a buckled state where the height $Z$ shows an irreversible behaviour: over the blue curve, the system remains buckled even if the voltage is decreased. Curves are slightly offset from each other for the sake of clarity. Image taken from journalarticle:Schoelz.etal_GrapheneRipplesRealization_Phys.Rev.B15.
• Figure 3: Schematic representation of the current-constant operating mode in scanning tunneling microscopy. The height of the tip $z$ varies in order to keep $I_{\text{tunnel}}$ constant while the tip maps the surface. Image taken from thesis:Pandelov_InvestigationStructureReactivity_07.
• Figure 4: Left panel: Phase diagram of the one-dimensional spin-string model in the plane $(J,T)$---($\kappa,\theta$) in the nomenclature of journalarticle:Ruiz-Garcia.etal_BifurcationAnalysisPhase_Phys.Rev.E17. The blue solid line corresponds to the phase transition lines of the system: a second-order one for temperatures above that of the critical point $K\equiv(J_K,T_K)$, and a first-order for temperatures below it. In region I, the only phase is the flat or rippled one, whereas a stable buckled phase appears in region II, where the rippled one becomes unstable. Region III is a coexistence zone where both phases are (locally) stable. Image taken from journalarticle:Ruiz-Garcia.etal_BifurcationAnalysisPhase_Phys.Rev.E17. Right panel: Numerical simulation of a simple model to qualitatively reproduce the Schoelz experiment. The height of the central atom is represented as a function of the temperature $\theta$, for $\kappa=0.1$. The system, prepared at a rippled phase at low temperature, is heated up until the phase transition occurs, leaving the system in an irreversible buckled state. Image taken from journalarticle:Ruiz-Garcia.etal_STMdrivenTransitionRippled_Phys.Rev.B16.
• Figure 5: Illustrative trajectory of Brownian dynamics with stochastic resetting to the initial position $x_0$ with rate $r$. Dashed blue lines stand for the resetting events occurring at times $t_i$, labelled by order of occurrence. The dotted black line indicates the position of the target $x_T$, whereas the dashed black line corresponds to the first-passage time $\mathcal{T}$ to reach $x_T$.