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.

<p>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.</p> <p><br></p> <p>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.</p>

The modelling of single-molecule experiments is of vital interest to gain new insights into processes which were hitherto not accessible by measurements performed on bulk systems. In the first part of this thesis, the kinetics of a triple-branch DNA molecule with four conformational states is investigated by employing pulling experiments with optical tweezers and theoretical modelling. Probability distributions of first rupture forces, which are calculated by applying transition rate theory to a free energy model, show good agreement with experimental findings. Permanently frayed molecules could be identified by analysing the number of opening base pairs in force jumps. In the second part of the thesis, DNA hairpin molecules with periodic base sequences are studied. Their unfolding kinetics allows an analytical treatment, because they exhibit a regular coarse-grained free energy landscape as a function of the number of opened base pairs. A procedure is developed for determining all relevant parameters of the landscape, which relies on probabilities that can be easily sampled from the unfolding trajectories. By means of Monte Carlo simulations it is shown that already 300 trajectories, as typically measured in single-molecule experiments, provide faithful results for the energetic parameters. The approach in particular opens a new access to improve loop contributions in the free energy landscape. In the third part of the thesis, a simulation method is developed for modelling the unfolding kinetics of DNA molecules with arbitrary base sequences. The method is validated against experimental data for five DNA hairpin molecules with different length of the end-loop. Applications of the method enable one, among others, to improve the parameter determination in functional forms suggested for the tail behaviour of work distributions. Such work distributions enter detailed and integral fluctuation theorems, which are useful for estimating free energy differences between folded and unfolded states from nonequilibrium measurements.

In this thesis we examine methodologies for determining free energy differences (FEDs) of phases via Monte Carlo simulation. We identify and address three generic issues that arise in FED calculations; the choice of representation, the choice of estimator, and the choice of sampling strategy. In addition we discuss how the classical framework may be extended to take into account quantum effects. Key words: Phase Mapping, Phase Switch, Lattice Switch, Simulated Tempering, Multi-stage, Weighted Histogram Analysis Method, Fast Growth, Jarzynski method, Umbrella, Multicanonical, Path Integral Monte Carlo, Path Sampling, Multihamiltonian, fluctuation theorem.