<div>In the 1970's, Arthur Ashkin developed a remarkable system called the ``optical tweezer'' which utilizes the radiation pressure of light to manipulate particles. Because of its non-invasive nature and controllability, optical tweezers have been widely adopted in biology, chemistry and physics. In this dissertation, two applications related to optical tweezers will be discussed. The first application is about the demonstration of multiple feedback controlled optical tweezers which let us conduct novel experiments which have not been performed before. For the second application, levitation of a silica nanodumbbell and cooling its motion in five degrees of freedom is executed.</div><div><br></div><div>To be more specific, the first chapter of the thesis focuses on an experiment using the feedback controlled optical tweezers in water. A well-known thought experiment called ``Feynman's ratchet and pawl'' is experimentally demonstrated. Feynman’s ratchet is a microscopic heat engine which can rectify the random thermal fluctuation of molecules to harness useful work. After Feynman proposed this system in the 1960’s, it has drawn a lot of interest. In this dissertation, we demonstrate a solvable model of Feynman’s ratchet using a silica nanoparticle inside a feedback controlled one dimensional optical trap. The idea and techniques to realize two separate thermal reservoirs and to keep them in contact with the ratchet is discussed in detail. Also, both experiment and simulation about the characteristics of our system as a heat engine are fully explored.</div><div><br></div><div>In the latter part of the dissertation, trapping silica nanodumbbell in vacuum and cooling its motion in five degrees of freedom is discussed. A levitated nanoparticle in vacuum is an extraordinary optomechanical system with an exceptionally high mechanical quality factor. Therefore, levitated particles are often utilized as a sensor in various research. Different from a levitated single nanosphere, which is only sensitive to force, a levitated nanodumbbell is sensitive to both force and torque. This is due to the asymmetry of the particle resulting it to have three rotational degrees of freedoms as well as three translational degrees of freedoms. In this dissertation, creating and levitating a silica nanodumbbell will be demonstrated. Active feedback cooling also known as cold damping will be employed to stabilize and cool the two torsional degrees of freedom of the particle along with the three center of mass DOF in vacuum. Additionally, both computational and experimental analysis is conducted on a levitated nanodumbbell which we call rotation-coupled torsional motion. The complex torsional motion can be fully explained with the effects of both thermal nonlinearity and rotational coupling. The new findings and knowledge of a levitated non-spherical particles leads us one step further towards levitated optomechanics with more complex particles.</div>

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This thesis is devoted to the theoretical study of slow thermodynamic processes in non-equilibrium stochastic systems. Its main result is a physically and mathematically consistent construction of relevant thermodynamic quantities in the quasistatic limit for a large class of non-equilibrium models. As an application of general methods a natural non-equilibrium generalization of heat capacity is introduced and its properties are analyzed in detail, including an anomalous far-from-equilibrium behavior. The developed methods are further applied to the related problem of time-scale separation where they enable to describe the effective dynamics of both slow and fast degrees of freedom in a more precise way. Powered by TCPDF (www.tcpdf.org)

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Avec les progrès de la technologie, il est désormais devenu possible de manipuler des faibles quantités d’objets nanométriques, voire des objets uniques. Observer une réaction chimique de quelques centaines de molécules sur des catalyseurs, étudier le travail exercé lors du déploiement d’un brin d’ADN unique ou mesurer la chaleur émise par un unique électron dans un circuit électrique constituent aujourd’hui des actes expérimentaux courants. Cependant, à cette échelle, le caractère aléatoire des processus physiques étudiés se fait plus fortement ressentir. Développer une théorie thermodynamique à ces échelles nécessite d'y inclure de manière exhaustive ces fluctuations.Ces préoccupations et les résultats expérimentaux et théoriques associés ont mené à l’émergence de ce que l’on appelle aujourd’hui la thermodynamique stochastique. Cette thèse se propose de développer une approche originale à la thermodynamique stochastique, basée sur une extension de l'hypothèse d'équilibre local aux variables fluctuantes d'un système. Cette théorie offre de nouvelles définitions des grandeurs thermodynamiques stochastiques, dont l'évolution est donnée par des équations différentielles stochastiques (EDS).Nous avons choisi d'étudier cette théorie à travers des modèles simplifiés de phénomènes physiques variés; transport (diffusif) de chaleur ou de masse, transport couplé (comme la thermodiffusion), ainsi que des modèles de réactions chimiques linéaires et non-linéaires. A travers ces exemples, nous avons proposé des versions stochastiques de plusieurs grandeurs thermodynamiques d'intérêt. Une large part de cette thèse est dévolue à l'entropie et aux différents termes apparaissant dans son bilan (flux d'entropie, production d'entropie ou dissipation). D'autres exemples incluent l'énergie libre d'Helmholtz, la production d'entropie d'excès, ou encore les efficacités thermodynamiques dans le transport couplé.A l'aide de cette théorie, nous avons étudié les propriétés statistiques de ces différentes grandeurs, et plus particulièrement l'effet des contraintes thermodynamiques ainsi que les propriétés cinétiques du modèle sur celles-là. Dans un premier temps, nous montrons comment l'état thermodynamique d'un système (à l' équilibre ou hors d'équilibre) contraint la forme de la distribution de la production d'entropie. Au-delà de la production d'entropie, cette contrainte apparaît également pour d'autres quantités, comme l'énergie libre d'Helmholtz ou la production d'entropie d'excès. Nous montrons ensuite comment des paramètres de contrôle extérieurs peuvent induire des bimodalités dans les distributions d'efficacités stochastiques.Les non-linéarités de la cinétique peuvent également se répercuter sur la thermodynamique stochastique. En utilisant un modèle non-linéaire de réaction chimique, le modèle de Schlögl, nous avons calculé la dissipation moyenne, non-nulle, engendrée par les fluctuations du système. Les non-linéarités offrent aussi la possibilité de produire des bifurcations dans le système. Les différentes propriétés statistiques (moments et distributions) de la production d'entropie ont été étudiées à différents points avant, pendant et après la bifurcation dans le modèle de Schlögl.Ces nombreuses propriétés ont été étudiées via des développements analytiques supportés par des simulations numériques des EDS du système. Nous avons ainsi pu montrer la fine connexion existant entre les équations cinétiques du système, les contraintes thermodynamiques et les propriétés statistiques des fluctuations de différentes grandeurs thermodynamiques stochastiques. / Over the last decades, nanotechnology has experienced great steps forwards, opening new ways to manipulate micro- and nanosystems. These advances motivated the development of a thermodynamic theory for such systems, taking fully into account the unavoidable fluctuations appearing at that scale. This ultimately leads to an ensemble of experimental and theoretical results forming the emergent field of stochastic thermodynamics. In this thesis, we propose an original theoretical approach to stochastic thermodynamics, based on the extension of the local equilibrium hypothesis (LEH) to fluctuating variables in small systems. The approach provides new definitions of stochastic thermodynamic quantities, whose evolution is given by stochastic differential equations (SDEs).We applied this new formalism to a diverse range of systems: heat or mass diffusive transport, coupled transport phenomena (thermodiffusion), and linear or non-linear chemical systems. In each model, we used our theory to define key stochastic thermodynamic quantities. A great emphasis has been put on entropy and the different contributions to its evolution (entropy flux and entropy production) throughout this thesis. Other examples include also the stochastic Helmholtz energy, stochastic excess entropy production and stochastic efficiencies in coupled transport. We investigated how the statistical properties of these quantities are affected by external thermodynamic constraints and by the kinetics of the system. We first studied how the thermodynamic state of the system (equilibrium \textit{vs.} non-equilibrium) strongly impacts the distribution of entropy production. We then extended those findings to other related quantities, such as the Helmholtz free energy and excess entropy production. We also analysed how some external control parameters could lead to bimodality in stochastic efficiencies distributions.In addition, non-linearities affect stochastic thermodynamics quantities in different ways. Using the example of the Schlögl chemical model, we computed the average dissipation of the fluctuations in a non-linear system. Such systems can also undergo a bifurcation, and we studied how the moments and the distribution of entropy production change while crossing the critical point.All these properties were investigated with theoretical analyses and supported by numerical simulations of the SDEs describing the system. It allows us to show that properties of the evolution equations and external constraints could strongly reflect in the statistical properties of stochastic thermodynamic quantities. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

We characterize finite-time thermodynamic processes of multidimensional quadratic overdamped systems. Analytic expressions are provided for heat, work, and dissipation for any evolution of the system covariance matrix. The Bures-Wasserstein metric between covariance matrices naturally emerges as the local quantifier of dissipation. General principles of how to apply these geometric tools to identify optimal protocols are discussed. Focusing on the relevant slow-driving limit, we show how these results can be used to analyze cases in which the experimental control over the system is partial.

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

Les petites machines, comme les moteurs moléculaires ou les particules actives, fonctionnent dans un environnement fortement fluctuant qui affecte leur efficacité ou leur puissance. L'objectif de cette thèse est de décrire les petites machines à l'aide de la thermodynamique stochastique et de la théorie des grandes déviations. En reliant localement puis globalement les courants aux forces thermodynamiques, on introduit une matrice de conductance hors d'équilibre, qui généralise la matrice d'Onsager pour un système stationnaire hors d'équilibre. Cela permet de majorer l'efficacité des machines par une fonction universelle qui ne dépend que du degré de couplage entre les courants d'entrée et de sortie. On obtient aussi de nouvelles relations générales entre puissance et efficacité. Du point de vue des fluctuations, la matrice de conductance hors d'équilibre est reliée à une borne quadratique pour les fonctions de grande déviation des courants. Cette borne permet d'obtenir des bornes pour les fonctions de grande déviation de l'efficacité, mais aussi de revisiter le théorème de fluctuation-dissipation comme une inégalité dans le cas des systèmes loin de l'équilibre. Pour terminer, on étudie l'effet d'une brisure d'ergodicité sur les fluctuations d'observables comme l'activité, les courants ou l'efficacité. En particulier, on calcule la fonction de grande déviation de l'efficacité pour un ensemble de nanomachines en interaction pour lesquelles un couplage fort et une brisure d'ergodicité apparaissent à la limite thermodynamique. / Small machines -- like molecular motors or active particles -- operate in highly fluctuating environments that affect their efficiency and power. This thesis aims at describing small machines using stochastic thermodynamics and large deviation theory. By relating mean currents to thermodynamic forces, locally first and then at the global level, we introduce the non-equilibrium conductance matrix that generalizes the Onsager matrix for stationary non-equilibrium systems. We use it to bound machine efficiency by a universal function depending only on the degree of coupling between input and output currents and to find new general power-efficiency trade-offs. On the fluctuations side, the non-equilibrium conductance matrix can be used to find a quadratic bound on the large deviation function of currents. This enables to revisit the fluctuation-dissipation theorem as an inequality when dealing with far-from-equilibrium systems, but also to derive bounds on the efficiency large deviation function. Finally, we study the effects of ergodicity breaking on the fluctuations of observables like activity, currents or efficiency. In particular, we derive the efficiency large deviation function for a model of interacting nanomachines, for which tight coupling and ergodicity breaking emerge in the thermodynamic limit.

Stochastic processes with temporal delay play an important role in science and engineering whenever finite speeds of signal transmission and processing occur. However, an exact mathematical analysis of their dynamics and thermodynamics is available for linear models only. We introduce a class of stochastic delay processes with nonlinear time-local forces and linear time-delayed forces that obey fluctuation theorems and converge to a Boltzmann equilibrium at long times. From the point of view of control theory, such ‘equilibrium stochastic delay processes’ are stable and energetically passive, by construction. Computationally, they provide diverse exact constraints on general nonlinear stochastic delay problems and can, in various situations, serve as a starting point for their perturbative analysis. Physically, they admit an interpretation in terms of an underdamped Brownian particle that is either subjected to a time-local force in a non-Markovian thermal bath or to a delayed feedback force in a Markovian thermal bath. We illustrate these properties numerically for a setup familiar from feedback cooling and point out experimental implications.

Title: Stochastic dynamics and energetics of biomolecular systems Author: Artem Ryabov Department: Department of Macromolecular Physics Supervisor: prof. RNDr. Petr Chvosta, CSc., Department of Macromolecular Physics Abstract: The thesis comprises exactly solvable models from non-equilibrium statistical physics. First, we focus on a single-file diffusion, the diffusion of particles in narrow channel where particles cannot pass each other. After a brief review, we discuss open single-file systems with absorbing boundaries. Emphasis is put on an interplay of absorption process at the boundaries and inter-particle entropic repulsion and how these two aspects affect the dynam- ics of a given tagged particle. A starting point of the discussions is the exact distribution for the particle displacement derived by order-statistics argu- ments. The second part of the thesis is devoted to stochastic thermodynam- ics. In particular, we present an exactly solvable model, which describes a Brownian particle diffusing in a time-dependent anharmonic potential. The potential has a harmonic component with a time-dependent force constant and a time-independent repulsive logarithmic barrier at the origin. For a particular choice of the driving protocol, the exact work characteristic func- tion is obtained. An asymptotic analysis of...