Le fluttuazioni su scala statistica - dalla teoria fisica statistica alle applicazioni biologiche

Fluctuations at different scales arise naturally as a result of coarse-graining dynamics at smaller spatial levels. In this thesis, we will see how tools from statistical physics can help analyse fluctuations in different systems and the resulting effects, especially in a biological context. We move from theoretical computations in non equilibrium statistical mechanics and numerical simulations of biological examples to brief experimental verifications through microbial communities to understand such stochasticity at different spatiotemporal scales. We begin initially with techniques to understand fluctuations of non equilibrium currents in a stochastic process. We focus on entropy production which is a significant marker of out-of-equilibrium regime, and propose a simple and graphical method to compute its various moments, not just in a discrete example of interest, but also in general spatially continuous systems. From non-equilibrium produced by heat baths at different temperatures, we pivot to present another kind of out-of-equilibrium processes due to sink-driven boundary dynamics. Such processes, found in systems with absorbing boundaries invalidate standard relations between fluctuations and dissipations. We propose a generalization of powerful fluctuation-dissipation theorems to include the effect of such boundary driven effects, and apply the results to strongly biologically motivated examples of birth-death forest dynamics and DNA target search on proteins. Here, we compute responses of previously infeasible quantities which point to strong ecological considerations. Boundary effects due to sinks in a local setting is significantly different from a spatially structured system. This necessitates an explicit incorporation of information of the landscape structure into biological models, offered through the lens of metapopulation theory. We present a first principles derivation of classical theoretical models and demonstrate additional benefits that this statistical mechanics route offers, including the ability to incorporate heterogeneous landscape and dispersal network information. In such systems, close to extinction, demographic fluctuations become significant. Contrarily, at the opposite spatial scale, in a laboratory environment, this regime is hardly reached, with minimum population of microbes being in tens to hundreds of thousands of individuals. Nevertheless, a natural setting involves constant environmental variability, captured through stochasticity in parameters in theoretical models. Specifically, serial dilution, a common experimental technique, gets represented as a periodic fluctuation in the dynamics of species and resources. Incorporating the different aspects of this technique through theoretical methods demonstrates a relation between different associated parameters which enable a sensible comparison across different dilution frequencies, opening the doors to further analysis of effects of environmental variability in microscale biological systems. Techniques developed for specific processes yield specific results whose scope remain limited. However, generalizations open up new paths of investigation into extended systems offering novel findings. Through investigations of systems at different levels using generalizing methods, we not only understand the specific process, but also observe the potential patterns of non-equilibrium across scales. Such results serve to underscore the importance of fluctuations, either thermal or externally driven, in determining different observed behaviours. This not only aids in the search for general non-equilibrium principles, which is inspired by biological systems, but also motivates further interdisciplinary research.