Bounds on Performance in Continuous and Discrete Thermal Machines: Insights from Quantum and Thermal Fluctuations

Abstract

Fluctuations play a crucial role in the operation of small-scale devices, where thermal and quantum effects significantly influence the efficiency, power output, and stability of machines. This thesis investigates performance constraints of small-scale thermodynamic machines, focusing on the impact of fluctuations in both autonomous and non-autonomous setups, as well as in continuous and discrete cycles. For autonomous absorption refrigerators operating in the linear response regime, we uncover a hierarchy among the relative fluctuations in the currents associated with the cold, hot, and work terminals. This hierarchy, established using Onsager reciprocity and refrigeration conditions, reveals that the tightest bound on cooling power is governed by the fluctuation of the work current. These universal bounds can be stricter than those predicted by standard thermodynamic uncertainty relations (TURs) and converge in the tight-coupling limit. The results are demonstrated using two models: a four-level system in the weak coupling regime and a two-level system with strong system-bath interactions. We then extend our analysis to non-autonomous continuous machines, where time-reversal symmetry is broken. By symmetrizing the operational regime, we derive a universal relationship between the relative fluctuations of input and output currents, even in the absence of Onsager reciprocity. These results are illustrated with a periodically driven classical Brownian heat engine. For discrete machines, specifically the quantum Otto cycle, we examine work and heat fluctuations for asymmetrically driven engines. First, we analyze Otto cycles with specific working fluids, such as a single qubit and a harmonic oscillator, noting that unlike continuous machines, these cycles lack a well-established linear response formalism. We establish bounds on non-equilibrium fluctuations for both engine and refrigerator regimes, revealing distinct relationships between the fluctuations of work and heat in different reservoirs compared to continuous machines. Additionally, we derive generalized thermodynamic uncertainty relations (GTURs) for the qubit-Otto cycle, which remain valid even under far-from-equilibrium driving. Finally, we develop a rigorous linear response framework for generic Otto cycles using the Schwinger-Keldysh nonequilibrium Green’s function (NEGF) technique. This approach uncovers that the fluctuation-dissipation relation (FDR) for the work current breaks down in the quantum domain due to external driving, while it remains valid for heat. This leads to important consequences for fluctuation bounds and thermodynamic performance in different operational regimes (engine and refrigerator). The violation of the work-FDR in the Otto cycle explains the observed differences in fluctuation bounds between discrete and continuous machines.