Fluctuations are of great importance in systems of small length and energy scales. Measuring the pulling of single molecules or the stationary fiow of mesospheres dragged through a viscous media enables the direct analysis of work and entropy distributions. These probability distributions are the result of a large number of repetitions of the same experiment. Due to the small scale of these experiments, the outcome can vary significantly from one realization to the next. Strong theoretical predictions exist, collectively called Fluctuation Theorems, that restrict the shape of these distributions due to an underlying time reversal symmetry of the microscopic dynamics. Fluctuation Theorems are the strongest existing statements on the entropy production of systems that are out of equilibrium. Being the most important ingredient for the Fluctuation Theorems, the probability distribution of the entropy change is itself of great interest. Using numerically exact methods we characterize entropy distributions for various stochastic reaction-diffusion systems that present different properties in their underlying dynamics. We investigate these systems in their steady states and in cases where time dependent forces act on them. This study allows us to clarify the connection between the microscopic rules and the resulting entropy production. The present work also adds to the discussion of the steady state properties of stationary probabilities and discusses a non-equilibrium current amplitude that allows us to quantify the distance from equilibrium. The presented results are part of a greater endeavor to find common rules that will eventually lead to a general understanding of non-equilibrium systems. / Ph. D.

Landauer's principle states that it costs at least k_BT ln 2 of work to reset one bit in the presence of a heat bath at temperature T. The bound of k_BT ln 2 is achieved in the unphysical infinite-time limit. Here we consider two different finite-time protocols - one with discretised time and the second in the continuous limit. We prove analytically that the discrete time protocol enables one to reset a bit with a work cost close to k_BT ln 2 in a finite time. We construct an explicit protocol that achieves this, which involves thermalising and changing the system's Hamiltonian so as to avoid quantum coherences. Using concepts and techniques pertaining to single-shot statistical mechanics, we furthermore prove that the heat dissipated is exponentially close to the minimal amount possible not just on average, but guaranteed with high confidence in every run. Moreover we exploit the protocol to design a quantum heat engine that works near the Carnot efficiency in finite time. We further contrast this to a continuous time version of the protocol which is substantially less energy sufficient. We also consider the fluctuations in the work cost, and calculate how their magnitude is suppressed by a factor depending on the length of the protocol. We demonstrate with an experiment how molecules are a natural test-bed for probing fundamental quantum thermodynamics. Single-molecule spectroscopy has undergone transformative change in the past decade with the advent of techniques permitting individual molecules to be distinguished and probed. By considering the time-resolved emission spectrum of organic molecules as arising from quantum jumps between states, we demonstrate that the quantum Jarzynski equality is satisfied in this set-up. This relates the heat dissipated into the environment to the free energy difference between the initial and final state. We demonstrate also how utilizing the quantum Jarzynski equality allows for the detection of energy shifts within a molecule, beyond the relative shift.