Thermodynamics typically concerns the physical behaviour of macroscopic systems comprised of many particles. However, recent theoretical progress has extended the theory to both the classical-stochastic and quantum regimes, where systems are comprised of just a small number of particles. In this thesis I investigate a range of situations in which new thermodynamic phenomena emerge due to the reduced size of the systems involved. One central assumption in macroscopic thermodynamics is the weak coupling approximation, which posits that the equilibrium properties of a system are not influenced by the interactions with its surrounding environment. However, for microscopic systems this assumption can break down, and I derive new fluctuation relations that provide a refined form of the second law of thermodynamics in this strong-coupling regime, taking into account corrections stemming from these interactions. In this work I provide a characterisation of stochastic heat and entropy production for small scale classical systems that are defined regardless of the strength of interaction. I then show that these definitions lead to a consistent thermodynamic framework valid beyond the usual weak-coupling regime. The thesis also concerns the effect of interactions on the equilibrium properties of strongly-coupled quantum systems, and I investigate how these interactions can influence the resulting temperature fluctuations in this regime. Using tools from quantum estimation theory, I derive an uncertainty relation between energy and temperature valid at all coupling strengths and system sizes. The relation reveals how quantum energy coherences contribute to statistical fluctuations in the estimated temperature of small- scale systems. Finally, I investigate how quantum fluctuations influence the statistics of work extracted from slowly-driven quantum systems. I prove that, unlike in classical systems, the work dissipated by a general quantum system is no longer proportional to its statistical fluctuations. This result reveals new subtleties involved in designing optimal quantum thermodynamic processes.
Thermodynamics typically concerns the physical behaviour of macroscopic systems comprised of many particles. However, recent theoretical progress has extended the theory to both the classical-stochastic and quantum regimes, where systems are comprised of just a small number of particles. In this thesis I investigate a range of situations in which new thermodynamic phenomena emerge due to the reduced size of the systems involved. One central assumption in macroscopic thermodynamics is the weak coupling approximation, which posits that the equilibrium properties of a system are not influenced by the interactions with its surrounding environment. However, for microscopic systems this assumption can break down, and I derive new fluctuation relations that provide a refined form of the second law of thermodynamics in this strong-coupling regime, taking into account corrections stemming from these interactions. In this work I provide a characterisation of stochastic heat and entropy production for small scale classical systems that are defined regardless of the strength of interaction. I then show that these definitions lead to a consistent thermodynamic framework valid beyond the usual weak-coupling regime. The thesis also concerns the effect of interactions on the equilibrium properties of strongly-coupled quantum systems, and I investigate how these interactions can influence the resulting temperature fluctuations in this regime. Using tools from quantum estimation theory, I derive an uncertainty relation between energy and temperature valid at all coupling strengths and system sizes. The relation reveals how quantum energy coherences contribute to statistical fluctuations in the estimated temperature of small- scale systems. Finally, I investigate how quantum fluctuations influence the statistics of work extracted from slowly-driven quantum systems. I prove that, unlike in classical systems, the work dissipated by a general quantum system is no longer proportional to its statistical fluctuations. This result reveals new subtleties involved in designing optimal quantum thermodynamic processes.
To understand in detail the relation between unitary quantum theory that describes our world at the microscopic scale and thermodynamics, which was long believed only to apply to macroscopic objects, is one of the most interesting and long-standing problems in physics. Recently, this problem has received renewed attention, in particular from the community of quantum information theory, but also from the field of statistical mechanics, inspired from stochastic thermodynamics. These results suggest that thermodynamics is also relevant for individual quantum systems, provided that they can be brought into contact with thermal baths. In this thesis, I use recently developed tools to provide new results both on fundamental questions, but also on questions which are of practical relevance for potential miniaturized thermal machines. In terms of fundamental questions, the results in this thesis contribute to understanding how the basic laws of thermodynamics and statistical mechanics can be understood directly from unitary quantum mechanics. In particular, I discuss and answer the following questions: i) How can we quantify the third law of thermodynamics using information theoretic methods? ii) How can we quantify the "thermodynamic value" of a state-transition in quantum systems? iii) How can we axiomatically characterize the non-equilibrium free energy and relative entropy? iv) How can we justify statistical ensembles from an operational perspective, without having to introduce either some probability measures or an information theoretic entropy measure? v) How can we understand the equilibration of closed quantum systems, how long does it take and how difficult is it to avoid? In the second part of the thesis I discuss in detail how experimental restrictions, which become important at the quantum scale, influence the ultimate thermodynamic bounds for thermal machines. In particular, the results in this thesis provide thermodynamic bounds on work-extraction and efficiencies of thermal machines in situations where 1.) an experimenter only has limited field strengths available, 2.) an experimenter cannot control the interactions between particles, but external fields arbitrarily well, and 3.) situations in which a small quantum system can only be strongly coupled to heat baths. These bounds are tight and I provide explicit examples illustrating the different behaviours. Finally, I come back to a classic problem in statistical physics: The emergence of spontaneous symmetry breaking. Here, I provide general and rigorous new results that show how symmetry-breaking stationary states emerge from fluctuations in order parameters in dissipative lattice models.
It is known that temperature estimates of macroscopic systems in equilibrium are most precise when their energy fluctuations are large. However, for nanoscale systems deviations from standard thermodynamics arise due to their interactions with the environment. Here we include such interactions and, using quantum estimation theory, derive a generalised thermodynamic uncertainty relation valid for classical and quantum systems at all coupling strengths. We show that the non-commutativity between the system's state and its effective energy operator gives rise to quantum fluctuations that increase the temperature uncertainty. Surprisingly, these additional fluctuations are described by the average Wigner-Yanase-Dyson skew information. We demonstrate that the temperature's signal-to-noise ratio is constrained by the heat capacity plus a dissipative term arising from the non-negligible interactions. These findings shed light on the interplay between classical and non-classical fluctuations in quantum thermodynamics and will inform the design of optimal nanoscale thermometers.
Abstract It is known that temperature estimates of macroscopic systems in equilibrium are most precise when their energy fluctuations are large. However, for nanoscale systems deviations from standard thermodynamics arise due to their interactions with the environment. Here we include such interactions and, using quantum estimation theory, derive a generalised thermodynamic uncertainty relation valid for classical and quantum systems at all coupling strengths. We show that the non-commutativity between the system’s state and its effective energy operator gives rise to quantum fluctuations that increase the temperature uncertainty. Surprisingly, these additional fluctuations are described by the average Wigner-Yanase-Dyson skew information. We demonstrate that the temperature’s signal-to-noise ratio is constrained by the heat capacity plus a dissipative term arising from the non-negligible interactions. These findings shed light on the interplay between classical and non-classical fluctuations in quantum thermodynamics and will inform the design of optimal nanoscale thermometers.
Relatively small many-body quantum systems are often used as hardware for quantum devices. Most of these devices will operate below the thermal limit where thermodynamics must be treated differently to account for quantum behaviours. It is therefore imperative that the thermodynamic properties of these systems are well understood, especially as they can limit the technologies but also help the fabrication and running of efficient quantum devices. \n \nIn this thesis we study quantum work and entropy production in closed many-body quantum systems out-of-equilibrium. We find that, for the systems studied, the largest average quantum work can be extracted in adiabatic weakly correlated regimes. These regimes are also seen to minimise the entropy produced, making them efficient regimes in which to operate devices based on these systems. \n \nAdiabatic evolutions are important for many quantum devices, and so it is important that they can be accurately characterised. The validity of current methods has been questioned recently, so in this thesis we propose the use of metrics as a good quantitative measure to characterise adiabaticity. We found that the density distance (a more accessible quantity than the wavefunction and its distance measures) alone can determine adiabaticity in a range of quantum systems, even at finite temperature. \n \nHowever, when calculating properties of many-body systems, there are many challenges often resulting in the need to approximate. In this thesis we propose a new style of approximation for quantum thermodynamic properties, taking inspiration from density functional theory (DFT). This new style uses the exact initial state of the system but approximates the dynamics and is seen to be computationally cheap but largely accurate. We test this with non-interacting and DFT approximated dynamics, finding that, surprisingly, the non-interacting dynamics give the most accurate results in most regimes, with the cheapest cost.
We build a framework for the thermodynamics of macroscopic quantum systems. In contrast with approaches requiring access to the full density matrix, our framework relies on a coarse-grained description, based on measurement statistics of a few observables. When these observables commute, the outcomes define classical macrostates whose entropy is quantified by observational entropy, accounting for uncertainty about both the macrostate and the microstate within it. We extend this notion to non-commuting observables forming a subalgebra of the operator space, and use Jaynes' principle to define an algebra-dependent entropy interpolating between von Neumann and observational entropies. Given initial and final measurement sets, connected by internal and/or environment-induced dynamics, we derive a second law for the coarse-grained dynamics. Unlike formulations based on von Neumann entropy, our inequality captures irreversibility from both non-unitary environment-induced dynamics and internal equilibration. It takes the usual form of a positive entropy production when the system is initially at internal equilibrium, while correction terms capture nonequilibrium resources ignored by the coarse-graining. We also derive fluctuation theorems for coarse-grained thermodynamic quantities. Along a quasi-static path of measurement schemes, we identify quantum macroscopic notions of work and heat fulfilling the first and second laws, including an additional work contribution from manipulating the algebra to which the system is confined, through external constraints or quantum measurement backaction. Finally, we apply our framework to examples illustrating the impact of varying the coarse-graining scheme. Our approach unifies macroscopic and stochastic thermodynamics in a genuinely quantum framework, laying the basis for a versatile, experimentally friendly toolbox to analyze complex quantum dynamics.
Rapid progress in the fabrication and manipulation of micro and nanoscale devices has made it necessary to extend the concepts of thermodynamics to small systems and finally to quantum systems. In such systems the extensive thermodynamic quantities, such as entropy, heat, and work, are not described by their average values alone but by distributions. \n \nRemarkably, it has been shown that the stochastic thermodynamic variables often obey fluctuation relations which usually appear in the form of relations between exponential averages. While the two-measurement protocol of thermodynamic variables, especially work, is now well studied in closed quantum systems, there have been conceptual and experimental problems in open quantum systems. \n \nIn this thesis, we theoretically study work statistics and fluctuation relations in open quantum systems. To calculate the work statistics, we use the quantum jump method and direct master equation calculations. We show that work definitions equivalent in closed systems can lead to mutually different results in the case of open quantum systems due to a different order of approximations. Moreover, we show that the fluctuation relations can be extended to nearly adiabatically driven systems by using the adiabatic renormalization procedure. \n \nLast, we focus on how a finite size of the environment affects the dynamics and work statistics of open quantum systems. This is important from the experimental point of view because a finite-size environment allows the detection of heat exchange between the system and the environment. We derive a master equation that takes into account these effects and develop a stochastic model that unravels the master equation. We explicitly show that the finite size of the environment influences the dynamics and the work statistics. We also show that the common fluctuation relations are still valid.