As ligações entre a termodinâmica e a mecânica quântica mostram-se interessantes tópicos de pesquisa desde os anos 50 e tem atraído cada vez mais atenção nos últimos anos, tanto por suas possíveis aplicações tecnológicas, quanto pelo aspecto teórico - como, por exemplo, as relações de sistemas quânticos com a segunda lei da termodinâmica. Para sistemas quânticos mesoscópicos, restritos apenas a um número relativamente pequeno de estados energéticos, torna-se necessária uma generalização da termodinâmica usual. Neste trabalho mostramos como construir uma máquina de Stirling no contexto de íons aprisionados. Para isso, faz-se necessária a engenharia de frequências dependentes do tempo do modo vibracional do íon, além da engenharia de reservatórios térmicos com temperaturas controladas. Após a construção da máquina de Stirling e do cálculo do trabalho e da eficiência associados apresentamos um protocolo para a medida da função de distribuição do trabalho que recorre às medidas dos níveis de energia eletrônicos do íon para, a partir dessas, extrair-se informação sobre o seu estado vibracional. / The connections between quantum mechanics and thermodynamics have been an interesting research topic since the 1950´s and began attracting more and more attention recently, not only for the technological applications, but also from a theoretical point of view - as, for instance, when dealing with the relations between quantum systems and the second law of thermodynamics. For mesoscopic (or even macroscopic) quantum systems, restricted to relatively few energy states, a generalization of the usual thermodynamics becomes necessary. In the present work we show how to engeneer a Stirling engine in an ionic trap. To achieve this we have to engeneer an ionic vibrational mode with a time dependent frequency, and simutaneously engeneer a thermal reservoir with controled temperatures. After the construction of the Stirling machine and the calculation of the associated work and efficiency, we show a protocol that allows the measurement of the work distribution function which call on the measurement of the electronic energy levels of the ion and, from them, extract information about the vibrational state of the trap.

Nonequilibrium work theorems, such as the Jarzynski equality and the Crooks fluctuation theorem, allow one to use nonequilibrium measurements to determine equilibrium free energies. For example, it has been demonstrated that the Crooks fluctuation theorem can be used to determine RNA folding energies. We used single-molecule manipulation with an atomic force microscope to measure the work done on poly(dA) as it was stretched and relaxed. This single-stranded nucleic acid exhibits unique base-stacking transitions in its force-extension curve due to the strong interactions among A bases, as well as multiple pathways. Here we showed that free energy curves can be determined by using the Crooks fluctuation theorem. The nonequilibrium work theorem can be used to determine free energy curves even when there are multiple pathways.

Nonequilibrium work theorems, such as the Jarzynski equality and the Crooks fluctuation theorem, allow one to use nonequilibrium measurements to determine equilibrium free energies. For example, it has been demonstrated that the Crooks fluctuation theorem can be used to determine RNA folding energies. We used single-molecule manipulation with an atomic force microscope to measure the work done on poly(dA) as it was stretched and relaxed. This single-stranded nucleic acid exhibits unique base-stacking transitions in its force-extension curve due to the strong interactions among A bases, as well as multiple pathways. Here we showed that free energy curves can be determined by using the Crooks fluctuation theorem. The nonequilibrium work theorem can be used to determine free energy curves even when there are multiple pathways.

Nonequilibrium work theorems, such as the Jarzynski equality and the Crooks fluctuation theorem, allow one to use nonequilibrium measurements to determine equilibrium free energies. For example, it has been demonstrated that the Crooks fluctuation theorem can be used to determine RNA folding energies. We used single-molecule manipulation with an atomic force microscope to measure the work done on poly(dA) as it was stretched and relaxed. This single-stranded nucleic acid exhibits unique base-stacking transitions in its force-extension curve due to the strong interactions among A bases, as well as multiple pathways. Here we showed that free energy curves can be determined by using the Crooks fluctuation theorem. The nonequilibrium work theorem can be used to determine free energy curves even when there are multiple pathways.

Nonequilibrium work theorems, such as the Jarzynski equality and the Crooks fluctuation theorem, allow one to use nonequilibrium measurements to determine equilibrium free energies. For example, it has been demonstrated that the Crooks fluctuation theorem can be used to determine RNA folding energies. We used single-molecule manipulation with an atomic force microscope to measure the work done on poly(dA) as it was stretched and relaxed. This single-stranded nucleic acid exhibits unique base-stacking transitions in its force-extension curve due to the strong interactions among A bases, as well as multiple pathways. Here we showed that free energy curves can be determined by using the Crooks fluctuation theorem. The nonequilibrium work theorem can be used to determine free energy curves even when there are multiple pathways.

As ligações entre a termodinâmica e a mecânica quântica mostram-se interessantes tópicos de pesquisa desde os anos 50 e tem atraído cada vez mais atenção nos últimos anos, tanto por suas possíveis aplicações tecnológicas, quanto pelo aspecto teórico - como, por exemplo, as relações de sistemas quânticos com a segunda lei da termodinâmica. Para sistemas quânticos mesoscópicos, restritos apenas a um número relativamente pequeno de estados energéticos, torna-se necessária uma generalização da termodinâmica usual. Neste trabalho mostramos como construir uma máquina de Stirling no contexto de íons aprisionados. Para isso, faz-se necessária a engenharia de frequências dependentes do tempo do modo vibracional do íon, além da engenharia de reservatórios térmicos com temperaturas controladas. Após a construção da máquina de Stirling e do cálculo do trabalho e da eficiência associados apresentamos um protocolo para a medida da função de distribuição do trabalho que recorre às medidas dos níveis de energia eletrônicos do íon para, a partir dessas, extrair-se informação sobre o seu estado vibracional. / The connections between quantum mechanics and thermodynamics have been an interesting research topic since the 1950´s and began attracting more and more attention recently, not only for the technological applications, but also from a theoretical point of view - as, for instance, when dealing with the relations between quantum systems and the second law of thermodynamics. For mesoscopic (or even macroscopic) quantum systems, restricted to relatively few energy states, a generalization of the usual thermodynamics becomes necessary. In the present work we show how to engeneer a Stirling engine in an ionic trap. To achieve this we have to engeneer an ionic vibrational mode with a time dependent frequency, and simutaneously engeneer a thermal reservoir with controled temperatures. After the construction of the Stirling machine and the calculation of the associated work and efficiency, we show a protocol that allows the measurement of the work distribution function which call on the measurement of the electronic energy levels of the ion and, from them, extract information about the vibrational state of the trap.

Indiana University-Purdue University Indianapolis (IUPUI) / Open quantum systems have become a rapidly developing sector for research. Such systems present novel physical phenomena, such as topological chirality, enhanced sensitivity, and unidirectional invisibility resulting from both their non-equilibrium dynamics and the presence of exceptional points. We begin by introducing the core features of open systems governed by non-Hermitian Hamiltonians, providing the PT -dimer as an illustrative example. Proceeding, we introduce the Lindblad master equation which provides a working description of decoherence in quantum systems, and investigate its properties through the Decohering Dimer and periodic potentials. We then detail our preferred experimental apparatus governed by the Lindbladian. Finally, we introduce the Liouvillian, its relation to non-Hermitian Hamiltonians and Lindbladians, and through it investigate multiple properties of open quantum systems.

This thesis explores the second law of thermodynamics at scales where fluctuations in thermodynamic quantities become significant, revealing the inherent stochastic characteristics of thermodynamic principles. The results are structured into two sections, each dedicated to addressing particular facets and scenarios. The first part of the thesis focuses on the behavior of two-level systems undergoing instantaneous changes in the energy level difference. Specifically, it examines the probability of observing microscopic realizations where the work is smaller than the Helmholtz free energy difference, which somewhat overstated can be referred to as microscopic second law 'violations'. As the number of two-level systems increases and the thermodynamic limit is approached, a non-monotonic behavior of the probability of second law 'violations' is observed. Surprisingly, the addition of just one additional two-level system can significantly increase the probability of second law 'violations', which at first sight seems counterintuitive but can be attributed to the discreteness of the system. In the second part of the thesis we derive a work fluctuation theorem similar to the Jarzynski equality, but applicable to a Brownian particle confined in a potential well with finite depth that is changed in time by an external protocol. Due to the weak confining effect of the potential well such a system is unable to relax to an equilibrium state, resulting in a mean squared displacement of the Brownian particle that diverges with time. This divergence leads to an additional term in the fluctuation theorem that differs from the Jarzynski equality. The inequality resulting from this theorem places a fundamental lower bound on the work required to change the potential over time.:Contents 1 Introduction 1 2 Equilibrium thermodynamics 5 2.1 Macroscopic vs. microscopic descriptions . . . . . . . . . . . . . . . . 5 2.2 First law of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Thermodynamic equilibrium . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Thermodynamic state variables and properties . . . . . . . . . . . . . 7 2.5 Work and heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.6 Second law of thermodynamics for an isothermal process . . . . . . . 11 2.7 Quasi-static and adiabatic processes . . . . . . . . . . . . . . . . . . . 12 3 A non-equilibrium description: The Langevin equation 15 3.1 The Langevin equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 From the Langevin to the Fokker-Planck equation . . . . . . . . . . . 17 3.3 The overdamped limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4 Diffusion equation for non-Gaussian noise . . . . . . . . . . . . . . . 22 3.5 A probability measure for a single trajectory . . . . . . . . . . . . . . 25 4 The Jarzynski equality 31 4.1 Definition of thermodynamic work for small systems . . . . . . . . . 31 4.2 Introducing the Jarzynski equality . . . . . . . . . . . . . . . . . . . . 33 4.3 What the Jarzynski equality does not say . . . . . . . . . . . . . . . . 35 4.4 Unzipping the Jarzynski equality . . . . . . . . . . . . . . . . . . . . . 37 4.5 The underlying symmetry of the Jarzynski equality: the Crooks theorem 40 4.6 Mean second law 'violations' . . . . . . . . . . . . . . . . . . . . . . . 40 4.7 Derivation of the Jarzynski equality based on the Feynman-Kac for- mula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.8 The Feynman-Kac formula . . . . . . . . . . . . . . . . . . . . . . . . 46 VI Contents 5 Stochastic Thermodynamics 51 5.1 The laws of stochastic thermodynamics . . . . . . . . . . . . . . . . . 51 6 Second law 'violations' in two level systems 57 6.1 The toy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.2 Verification of JE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.3 Apparent second law 'violations' . . . . . . . . . . . . . . . . . . . . 59 6.4 Work distribution in the thermodynamic limit . . . . . . . . . . . . . 60 6.5 Second law 'violations' in the thermodynamic limit . . . . . . . . . . 63 6.6 Quasi-static limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.7 Second law 'violations' in the quasistatic limit . . . . . . . . . . . . . 68 6.8 Outlook: Towards coupled two level systems . . . . . . . . . . . . . . 69 6.9 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7 An extension of the Jarzynski equality 71 7.1 Setting the stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 7.2 A motivating special case: The infinitely fast protocol . . . . . . . . . 73 7.3 Derivation of the work fluctuation theorem . . . . . . . . . . . . . . . 75 7.4 A possible physical Interpretation . . . . . . . . . . . . . . . . . . . . 77 7.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.6 A comparison with Seifert’s fluctuation theorem . . . . . . . . . . . . 90 7.7 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8 Conclusion 93

Multifunctional proteins that undergo major structural changes to perform different functions are known as “Transformer Proteins”, which is a recently identified class of proteins. One such protein that shows a remarkable structural plasticity and has two distinct functions is the transcription antiterminator, RfaH. Depending on the interactions between its N-terminal domain and its C-terminal domain, the RfaH CTD exists as either an all-α-helix bundle or all-β-barrel structure. Another example of a transformer protein is the Ebola virus protein VP40 (eVP40), which exists in different conformations and oligomeric states (dimer, hexamer, and octamer), depending on the required function.I performed Molecular Dynamics (MD) computations to investigate the structural conversion of RfaH-CTD from its all-a to all-b form. I used various structural and statistical mechanics tools to identify important residues involved in controlling the conformational changes. In the full-length RfaH, the interdomain interactions were found to present the major barrier in the structural conversion of RfaH-CTD from all-a to all-b form. I mapped the energy landscape for the conformational changes by calculating the potential of mean force using the Adaptive Biasing Force and Jarzynski Equality methods. Similarly, the interdomain salt-bridges in the eVP40 protomer were found to play a critical role in domain association and plasma membrane (PM) assembly. This molecular dynamic simulation study is supported by virus like particle budding assays investigated by using live cell imaging that highlighted the important role of these saltbridges. I also investigated the plasma membrane association of the eVP40 dimer in various PM compositions and found that the eVP40 dimer readily associates with the PM containing POPS and PIP2 lipids. Also, the CTD helices were observed to be important in stabilizing the dimer-membrane complex. Coarse-grained MD simulations of the eVP40 hexamer and PM system revealed that the hexamer enhances the PIP2 lipid clustering at the lower leaflet of the PM. These results provide insight on the critical steps in the Ebola virus life cycle.

[pt] A Igualdade de Jarzynski (IJ) é um tipo especial de Teorema de Flutuação, de trabalho, que caracteriza sistemas termodinâmicos microscópicos fora do equilíbrio. A IJ pode ser usada como uma calibração de experimentos e simulações, o que nos permite estudar comportamentos não triviais da dinâmica desses sistemas. Um desses comportamentos é a troca de entropia e informação que o sistema realiza junto a um banho térmico de contato. Neste ensejo, modelamos via uma dinâmica não-Markoviana, i.e., uma dinâmica com memória, que leva a fluxos reversos de informação do reservatório para o sistema. / [en] The Jarzynski Equality (JE) is a special kind of Fluctuation Theorem, of work, which characterizes non-equilibrium small thermodynamics systems. The JE can be used as gauge of experiments and simulations allowing us to study the non-trivial behaviours of these systems dynamics. One of these behaviours is the entropy and information flow the system makes in contact with a thermal bath. In this framework, we modelled through a non-Markovian dynamic, i.e., with a memory effect, leading to reverse flows of information from the reservoir to the system.