Les solitons dissipatifs (SD) sont des paquets d'onde auto-localisés qui apparaissent dans les systèmes dissipatifs spatialement étendus. En optique, tous les SD observés jusqu'à présent dans des systèmes propagatifs peuvent être classés en deux catégories, suivant la présence ou non d'un forçage externe, i.e. si la symétrie de phase est brisée ou non. Dans les systèmes forcés, les DS sont accrochés en phase au forçage alors que sans forçage, leur phase est libre et peu dériver en fonction du temps. Dans cette thèse nous étudions la formation d'états localisés propagatifs dans deux systèmes expérimentaux qui diffèrent fondamentalement par la présence ou l'absence d'un forçage externe. Le premier système est un laser à semiconducteur à cavité verticale (VCSEL) soumis à une boucle de rétro-action sélective en fréquence, qui accueille des DS se formant dans la dimension transverse à la propagation (2D). Nous analysons comment la synchronisation des fréquences longitudinales du système (verrouillage modal) peut mener à la formation d'un état localisé dans les trois dimensions : balles de lumière. Le deuxième système est un laser en anneau à semi-conducteur fortement multimode le long de la propagation, et forcé par une injection externe. Lorsque le forçage est légèrement désaccordé de la fréquence naturelle du système, il est possible d'observer des états localisés constitués par un tour de phase de 2 pi, immergés dans l'état homogène (synchronisé). Nous reportons ainsi la première observation de SD qui se forment dans la phase de l'onde optique : solitons de phase dissipatifs / Dissipative solitons (DS) are self-localized wave-packets appearing in spatially extended dissipative systems. In optics, all the DS that have been observed in propagative systems can be cast in two categories, depending on the presence or absence of an external forcing, i.e. the phase symmetry is broken or not. In forced systems, DS are locked in phase to forcing whereas without forcing, their phase is free an can wander in the course of time. In this thesis, we study the formation of propagative DS in two different experimental systems that fundamentally differ from the presence or lack of an external forcing. The first one is a Vertical Cavity Surface Emitting Laser (VCSEL) submitted to a frequency selective feedback, in which DS form in the transverse plane of the system (2D). We analyze how the synchronization of the longitudinal frequencies (mode-locking) can give rise to tri-dimensionnal localization of light: light bullets. The second system is a highly multimode semiconductor ring laser with external forcing, whose spatial extension takes place along the propagation dimension. When the forcing frequency is slightly detuned from the natural frequency of the system, we can see the appearance of self-confined 2 pi phase rotations embedded in a homogeneous (synchronized) state. We then report on the first observation of DS that form in the phase of the optical wave : dissipative phase solitons

Quantum systems that are in weak contact with a thermal heat bath will ultimately relax to an equilibrium state which is characterized by the temperature of the environment only. This state is independent of the specific properties of the bath and of how it is coupled to the system. This changes completely, when the system is additionally driven. Such a driven-dissipative situation can emerge, for example, due to an additional time-periodic modulation of the system, or when it is brought into contact with a second bath of different temperature. Then, the system will run into a well-defined nonequilibrium steady state. This state, however, will depend on the very details of the environment and its coupling to the system. We study whether this freedom can be used to engineer interesting properties of quantum systems, which are not found in their equilibrium states, i.e. in the absence of a drive. We focus on bosonic quantum many-body systems. We investigate when far-from-equilibrium ideal gases feature Bose condensation in a group of single-particle states, as opposed to situations where Bose condensation is completely absent in the nonequilibrium steady state. We show that Bose condensation can be induced in a finite one-dimensional ideal gas by the competition of two heat baths whose temperatures both lie well above the equilibrium condensation temperature. This setup also allows to engineer condensation in excited single-particle states. We discuss first ideas to study similar setups in weakly interacting Bose gases. Describing the microscopic dynamics of interacting many-body systems coupled to thermal baths is extremely challenging, due to the fact that generally the full many-body spectrum is inaccessible. Using ideas from semiclassics, we develop an approximation to the dynamics that yields good results at high and intermediate bath temperatures. We also investigate the transient dynamics of driven-dissipative quantum systems. Our studies are motivated by a result that is well known for isolated quantum systems: for a system whose dynamics is generated by a time-periodic Hamiltonian, the stroboscopic dynamics (observed at integer multiples of the driving period) can always be understood as if it would stem from a time-independent Hamiltonian, the Floquet Hamiltonian. For open quantum systems in contact with an environment, we ask if a similar mapping to an effective generator, the Floquet Lindbladian, is always possible. For a simple qubit model we show that there are two extended parameter regions, one in which the Floquet Lindbladian exists, and one in which it does not. We discuss problems of analytical expansions that can give rise to this Floquet Lindbladian and discuss how we can interpret the region where it does not exist. These results are important for dissipative Floquet engineering and open up new perspectives for the control of open quantum systems via time-periodic driving.:1. Introduction 2. Master equation for open quantum systems 3. Existence of the Floquet Lindbladian 4. Number of Bose-selected modes in driven-dissipative ideal Bose gases 5. High-temperature nonequilibrium Bose condensation induced by a hot needle 6. Weakly interacting Bose gases far from thermal equilibrium 7. Summary and outlook / Quantensysteme, die in schwacher Wechselwirkung mit einem thermischen Wärmebad stehen, relaxieren stets in einen Gleichgewichtszustand, welcher allein durch die Temperatur der Umgebung beschrieben ist. Dieser Zustand ist unabhängig von den spezifischen Eigenschaften des Bades, und davon wie dieses an das System gekoppelt ist. Dies ändert sich, wenn das System zusätzlich angetrieben wird. Ein solches getrieben-dissipatives Szenario kann beispielsweise durch einen zusätzlichen zeitperiodischen Antrieb entstehen, oder wenn das System mit einem zweiten Bad unterschiedlicher Temperatur in Kontakt gebracht wird. In diesem Fall läuft das System in einen wohldefinierten stationären Nichtgleichgewichtszustand. Dieser Zustand hängt jedoch von den Details der Umgebung, und davon wie diese an das System gekoppelt ist, ab. Es wird untersucht ob diese Freiheit genutzt werden kann um interessante Eigenschaften von Quantensystemen zu konstruieren, die in deren Gleichgewichtszuständen, d.h. in Abwesenheit des Antriebs, nicht zu finden sind. Der Fokus der Arbeit liegt auf bosonischen Quantenvielteilchensystemen. Es wird ergründet unter welchen Bedingungen ideale Gase fernab des thermischen Gleichgewichts Bose Kondensation in einer Gruppe von Einteilchenzuständen aufweisen, im Gegensatz zu Szenarien in denen überhaupt keine Bose Kondensation im stationären Nichtgleichgewichtszustand auftritt. Weiterhin wird gezeigt, dass Bose Kondensation in einem eindimensionalen idealen Gas durch das Wechselspiel zweier Wärmebäder induziert werden kann. Die Temperatur beider Bäder liegt dabei weit über der Kondensationstemperatur des Gleichgewichts. Diese Anordnung erlaubt außerdem kontrollierte Kondensation in angeregten Einteilchenzuständen. Erste Ideen für das theoretische Studium ähnlicher Anordnungen für schwach wechselwirkende Bosegase werden diskutiert. Eine Beschreibung der mikroskopischen Dynamik wechselwirkender Vielteilchensysteme ist extrem anspruchsvoll, da typischerweise das volle Vielteilchenspektrum unzugänglich ist. Unter Zurhilfenahme semiklassischer Ideen wird eine Näherung der Dynamik entwickelt, welche eine gute Beschreibung für hohe und intermediäre Temperaturen liefert. Weiterhin wird die transiente Dynamik getrieben-dissipativer Quantensysteme untersucht. Die Motivation bietet ein bekanntes Resultat für abgeschlossene Quantensysteme: Für ein System, dessen Dynamik durch einen zeitperiodischen Hamiltonoperator bestimmt ist, kann die stroboskopische Dynamik (unter Beobachtung zu Zeiten, die Vielfache der Antriebsperiode sind) immer so verstanden werden als würde sie von einem zeitunabhängigen Hamiltonoperator, dem Floquet Hamiltonian, induziert. Für offene Quantensysteme im Kontakt mit einer Umgebung wird untersucht ob eine ähnliche Abbildung auf einen effektiven Generator, den Floquet Lindbladian, existiert. Für ein einfaches Qubit Modell wird gezeigt, dass es zwei ausgedehnte Parameterregionen gibt, eine in welcher der Floquet Lindbladian existiert und eine weitere in der dieser nicht existiert. Es werden Probleme von analytischen Entwicklungen des Floquet Lindbladian diskutiert. Auch wird eine Interpretation der Region gegeben, in der dieser nicht existiert. Diese Resultate sind maßgeblich für dissipatives Floquetengineering und eröffnen neue Blickwinkel auf die zeitperiodische Kontrolle offener Quantensysteme.:1. Introduction 2. Master equation for open quantum systems 3. Existence of the Floquet Lindbladian 4. Number of Bose-selected modes in driven-dissipative ideal Bose gases 5. High-temperature nonequilibrium Bose condensation induced by a hot needle 6. Weakly interacting Bose gases far from thermal equilibrium 7. Summary and outlook

Bridge substructures are generally constructed using cast-in-place concrete and designed to undergo inelastic deformation in earthquake events. Although this construction approach has proven to be economical and provides adequate seismic performance through the formation of ductile plastic hinges, there are downsides relating to construction speed and quality, and post-earthquake repairability. This thesis explores two categories of Accelerated Bridge Construction (ABC) connection types, which use precast concrete instead of cast-in-place concrete to offer advantages including increased construction speed and quality. High Damage (HD) ABC connection types emulate the seismic behaviour of cast-in-place construction through the formation of ductile plastic hinges. Controlled Damage (CD) ABC connection types use unbonded post-tensioned precast connections to offer additional advantages including reduced residual drifts, limited and controlled damage and simple repair options. Novel buckling-restrained, fused mild steel energy dissipators suitable for use in CD connections are also developed and tested. These designs utilise 'dry' fabrication to simplify the fabrication process and minimise cost. Half-scale experimental testing is carried out to demonstrate both the assembly processes and behaviour under reversed cyclic uniaxial and biaxial loading representing an earthquake event. Following benchmark testing, repair strategies are applied to the CD connection types and the columns are tested again, representing a subsequent earthquake event. Good results are obtained from all cases with relatively straightforward construction and repair processes. With further developments and testing, the connection types proposed can provide competitive alternatives to conventional bridge pier design with regard to seismic performance and life cycle costs, with the additional benefits associated with precast construction.

Waves, which arise when restoring forces act on small perturbations, are ubiquitous in fluids. Their counterpart, mean flows, capture the remainder of the motion and are often characterised by a slower evolution and larger scale patterns. Waves and mean flows, which are typically separated by time- or space-averaging, interact, and this interaction is central to many fluid-dynamical phenomena. Wave-mean flow interactions can be classified into dissipative interactions and non-dissipative interactions. The former is important for small-scale flows, the latter for large-scale flows. In this thesis these two kinds of interactions are studied in the context of microfluidics and geophysical applications. Viscous wave-mean flow interactions are studied in two microfluidic problems. Both are motivated by the rapidly increasing number of microfluidic devices that rely on the mean-flow generated by dissipating acoustic waves - acoustic streaming - to drive small-scale flows. The first problem concerns the effect of boundary slip on steady acoustic streaming, which we argue is important because of the high frequencies employed. By applying matched asympototics, we obtain the form of the mean flow as a function of a new non-dimensional parameter measuring the importance of the boundary slip. The second problem examined is the development of a theory applicable to experiments and devices in which rigid particles are manipulated or used as passive tracers in an acoustic wave field. Previous work obtained dynamical equations governing the mean motion of such particles in a largely heuristic way. To obtain a reliable mean dynamical equation for particles, we apply a systematic multiscale approach that captures a broad range of parameter space. Our results clarify the limits of validity of previous work and identify a new parameter regime where the motion of particles and of the surrounding fluid are coupled nonlinearly. Non-dissipative wave-mean flow interactions are studied in two geophysical fluid problems. (i) Motivated by the open question of mesoscale energy transfer in the ocean, we study the interaction between a mesoscale mean flow and near-inertial waves. By applying generalized Lagrangian mean theory, Whitham averaging and variational calculus, we obtain a Hamiltonian wave-mean flow model which combines the familiar quasi-geostrophic model with the Young & Ben Jelloul model of near-inertial waves. This research unveils a new mechanism of mesoscale energy dissipation: near-inertial waves extract energy from the mesoscale ow as their horizontal scale is reduced by differential advection and refraction so that their potential energy increases. (ii) We study the interaction between topographic waves and an unidirectional mean flow at an inertial level, that is, at the altitude where the Doppler-shifted frequency of the waves match the Coriolis parameter. This interaction can be described using linear theory, using a combination of WKB and saddle-point methods, leading to explicit expressions for the mean-flow response. These demonstrate, in particular, that this response is switched on asymptotically far downstream from the topography, in contrast to what is often assumed in parameterisation.

Este trabalho tem por objetivo avaliar a emissão de metano em dois reservatórios amazônicos, Tucuruí e Samuel. A análise leva em consideração flutuações da área/cota e medidas isotópicas e de fluxos de d13CH4 obtidas em campo, as quais são influenciadas por diversos fatores simultaneamente, como ventos, pressão atmosférica, oxidação na coluna d´água e outros. A composição isotópica do metano em bolhas do sedimento indica que a principal via metanogênica é a redução de CO2. Os resultados também mostram que as emissões em reservatórios rasos são bem mais elevadas que nos mais profundos. Usando conceitos e técnicas da teoria fractal, baseada fundamentalmente em leis de potência, um modelo entre nível de água, área do reservatório e fluxo de metano permitiu estimar a emissão do reservatório de Tucuruí, num horizonte de 100 anos de vida útil, em 2,5 ± 2,7 x 10^6 toneladas de CH4. As emissões locais são amplificadas com a diminuição da cota, porém a emissão total do reservatório diminui, em função da contração da área do reservatório. Num cenário mais otimista, as emissões de Tucuruí em CO2 equivalentes devem ser 0,13 ± 0,14 x 10^6 toneladas de C/ano. / The purpose of this work is evaluating methane emissions from two Amazonian reservoirs, Tucuruí and Samuel. The analysis considers fluctuations of the reservoir area-height and isotopic measurements and fluxes of d13CH4 acquired in the field, which are influenced by many factors simultaneously, such as winds, atmospheric pressure, oxidation in the water column, and others. Isotopic composition of methane in bubbles indicates that CO2 reduction is a major pathway for methane production in the sediments. The results also show that fluxes are higher in shallow reservoirs than in deeper ones. Using concepts and techniques of fractal theory, fundamentally based on power law functions, and considering a time horizon of 100 years, a model between water level, flux and reservoir area of Tucuruí allowed to estimate methane emission from Tucuruí at about 2,5 ± 2,7 x 10^6 tons of CH4. When water levels drop local methane emissions are enhanced, although, total reservoir emission decreases, due to the shrinking of the reservoir area. Over an optimistic scenario, the emissions of Tucuruí reservoir in CO2 equivalents would be around 0,13 ± 0,14 x 10^6 tons C/year.

Realizamos nesse trabalho um tratamento abrangente da interação entre um sistema quântico e o meio ambiente modelado como um conjunto de osciladores harmônicos. Partimos para isso de um tratamento prévio de redes de osciladores harmônicos quânticos dissipativos. Utilizando a função característica, transformamos a equação de von Neumann em uma equação diferencial, e explorando a sua linearidade, essa é transformada em uma equação vetorial, cuja resolução é computacionalmente eficiente. Nosso formalismo, que parte de uma rede de osciladores harmônicos, não necessariamente dividida entre sistema e meio ambiente, permite que se contorne a necessidade da hipótese de acoplamento súbito sistema-reservatório para o tratamento exato da evolução do sistema. Em seguida, mostramos que essa evolução pode ser sempre descrita por uma equação mestra na forma usual de Lindblad, embora os coeficientes que a definem possam ser dependentes do tempo. Isso abre novas possibilidades para a dinâmica do sistema, e leva a efeitos que podem ser classificados de não-Markovianos, embora sejam descritos por uma equação mestra completamente local no tempo. Ressaltamos que, por ser baseado em uma solução exata, o método pode ser aplicado para qualquer intensidade de acoplamento, e é consideravelmente mais simples do que outros métodos disponíveis para esse fim, como os baseados em integrais de trajetória. Por fim, utilizamos simulações computacionais para explorar a validade das aproximações de ondas girantes e de Born-Markov, e os fenômenos que podem ser observados nos regimes em que elas deixam de ser válidas. / We present a comprehensive treatment of the interaction of a quantum system with an environment modeled as a set of harmonic oscillators. We start from a previous treatment of a network of quantum dissipative harmonic oscillators. Using the characteristic function, we transform the von Neumann equation in a differential equation, and exploring its linearity, this is transformed in a vector equation, whose solution is computationally efficient. Our method, whose origin lies on a network not necessarily divided into system and reservoir, allows us to circumvent the necessity of the sudden-coupling approximation for the exact treatment of the system evolution. After this, we show that this dynamics can always be described by a master equation in standard Lindblad form, although its coefficients may be functions of time. This opens new possibilities for the system dynamics, and lead to effects that may be called non-Markovian, even if they are described by a completely local-in-time master equation. It should be emphasized that, as it is based on an exact solution, the method may be applied for any strength of the system-reservoir interaction, and it is considerably simpler than other available methods, such as those based on path integrals. Finally, we employ computer simulations to investigate the validity of the rotating-wave and Born-Markov approximations, and the phenomena that migth be observed in regimes in which they fail to be valid.

O presente trabalho apresenta a teoria de sistemas dinâmicos dissipativos impulsivos. Apresentamos resultados suficientes e necessários para obtermos dissipatividade para sistemas impulsivos autônomos e não autônomos utilizando funções de Lyapunov. No que segue, desenvolvemos a teoria de estabilidade para a seção nula de um sistema dinâmico não autônomo com impulsos. Utilizando os resultados da teoria abstrata para sistemas não autônomos com impulsos, apresentamos o estudo da estabilidade de um modelo presa-predador com controle e impulsos. / The present work presents the theory of impulsive dissipative dynamical systems. We present necessary and sufficient conditions to obtain dissipativity for autonomous and non-autonomous impulsive dynamical systems via Lyapunov functions. In the sequel, we develop the theory of stability for the null section of non-autonomous dynamical systems with impulses. Using the results from the abstract theory we present the study of stability for a controlled prey-predator model under impulse conditions.

Dans ce travail, nous avons présenté notre contribution portant sur la modélisation et le contrôle a priori de congestion des réseaux de carrefours signalisés. De point de vue de la modélisation, nous avons introduit un nouveau regard sur les systèmes de transport en proposant un premier travail sur la manière dont les liens se tissent entre ces systèmes et la thermodynamique. Le point de vue dominant est l'assimilation des véhicules à l'énergie fournie ou/et échangée entre les intersections signalisées. L'avantages majeur de la modélisation thermodynamique est l'introduction de la notion d'entropie du transport mesurant le désordre du système. Elle peut être considérée non seulement comme un moyen pour la compréhension des phénomènes du trafic, mais aussi comme un outil d'évaluation surtout lorsqu'il s'agit de traiter des réseaux de grande taille. De point de vue du contrôle, nous nous sommes intéressés essentiellement à un travail en amont permettant d'éviter la congestion en forçant les files d'attente à ne pas dépasser le niveau du trafic correspondant à l'optimum opérationnel des lignes. Nous avons traité le problème de deux façons différentes. La première fait appel à l’approche de la commande dissipative. Nous avons exploité cet outil pour arriver à des résultats théoriques dont la vérification permet de conclure sur la possibilité de dissiper les véhicules au moyen d'une action de la commande adéquate. L'existence d'une commande dissipative est caractérisée par la faisabilité de certaines inégalités matricielles linéaires (LMI). La deuxième façon de traiter notre problème de commande fait appel à la commande H∞. Nous avons tiré profit de cet outil pour développer des résultats assurant l'invariance positive en boucle fermée d'un domaine ellipsoïdal contenu dans l'ensemble des contraintes. Le test d'existence et le calcul d'une loi de commande robuste par retour d'état peut alors se faire de façon simple par la résolution d'un problème de programmation linéaire convexe. Enfin, ses travaux ont été appliqués sur deux types de réseaux de carrefours, artériel et en grille afin de montrer l'intérêt des résultats. / In this work, we have presented a thermodynamic point of view for the transportation network. Analogies have been drawn between thermodynamic and transportation systems by considering traffic lanes as thermodynamic sub-systems and the vehicles as the abstract energy supplied to them. In addition, the concepts of thermal capacity and temperature are also introduced into transportation context to correspond to lane capacity and occupancy respectively. Then, it has been demonstrated that the first law of thermodynamics corresponds to the conservation of vehicles. It is also demonstrated that the transportation network can have a similar notion of entropy. Such transportation entropy is a measure of disorder of the system and hence may provide deep insight in the analysis of transportation control problems. In particular, this work has presented a dissipativity phenomenon of transportation entropy that reduces the system disorder and hence renders the system better organized. Though this phenomenon doesn’t exist naturally in transportation context, the ways to construct feedback control strategies have been proposed to achieve such objective by means of Linear Matrix Inequalities (LMIs). However, since transportation systems involve massive complex human activities, there exist substantial unpredictable uncertainties of the traffic demands. In this context, we have proposed a robust controller for disturbance attenuation of transportation network. The errors between input flows and the nominal ones are considered as disturbances and a constrained H∞ control has been formulated in terms of maximization of the tolerance under control constraints. The problem of disturbance attenuation is solved by means of a convex optimization with Linear Matrix Inequality. Finally, two types of networks (arterial and grid) are carried out to illustrate the performances of our strategies.

Polymer networks are an important class of materials that are ubiquitously found in natural, biological, and man-made systems. The complex mesoscale structure of these soft materials has made it difficult for researchers to fully explore their properties. In this dissertation, we introduce a coarse-grained computational model for permanently cross-linked polymer networks than can properly capture common properties of these materials. We use this model to study several practical problems involving dry and solvated networks. Specifically, we analyze the permeability and diffusivity of polymer networks under mechanical deformations, we examine the release of encapsulated solutes from microgel capsules during volume transitions, and we explore the complex tribological behavior of elastomers. Our simulations reveal that the network transport properties are defined by the network porosity and by the degree of network anisotropy due to mechanical deformations. In particular, the permeability of mechanically deformed networks can be predicted based on the alignment of network filaments that is characterized by a second order orientation tensor. Moreover, our numerical calculations demonstrate that responsive microcapsules can be effectively utilized for steady and pulsatile release of encapsulated solutes. We show that swollen gel capsules allow steady, diffusive release of nanoparticles and polymer chains, whereas gel deswelling causes burst-like discharge of solutes driven by an outward flow of the solvent initially enclosed within a shrinking capsule. We further demonstrate that this hydrodynamic release can be regulated by introducing rigid microscopic rods in the capsule interior. We also probe the effects of velocity, temperature, and normal load on the sliding of elastomers on smooth and corrugated substrates. Our friction simulations predict a bell-shaped curve for the dependence of the friction coefficient on the sliding velocity. Our simulations also illustrate that at low sliding velocities, the friction decreases with an increase in the temperature. Overall, our findings improve the current understanding of the behavior of polymer networks in equilibrium and non-equilibrium conditions, which has important implications for synthesizing new drug delivery agents, designing tissue engineering systems, and developing novel methods for controlling the friction of elastomers.

The present thesis is devoted to the study of path probability of random motion on the basis of an extension of Hamiltonian/Lagrangian mechanics to stochastic dynamics. The path probability is first investigated by numerical simulation for Gaussian stochastic motion of non dissipative systems. This ideal dynamical model implies that, apart from the Gaussian random forces, the system is only subject to conservative forces. This model can be applied to underdamped real random motion in the presence of friction force when the dissipated energy is negligible with respect to the variation of the potential energy. We find that the path probability decreases exponentially with increasing action, i.e., P(A) ~ eˉγA, where γ is a constant characterizing the sensitivity of the action dependence of the path probability, the action is given by A = ∫T0 Ldt, a time integral of the Lagrangian L = K-V over a fixed time period T, K is the kinetic energy and V is the potential energy. This result is a confirmation of the existence of a classical analogue of the Feynman factor eiA/ħ for the path integral formalism of quantum mechanics of Hamiltonian systems. The above result is then extended to real random motion with dissipation. For this purpose, the least action principle has to be generalized to damped motion of mechanical systems with a unique well defined Lagrangian function which must have the usual simple connection to Hamiltonian. This has been done with the help of the following Lagrangian L = K - V - Ed, where Ed is the dissipated energy. By variational calculus and numerical simulation, we proved that the action A = ∫T0 Ldt is stationary for the optimal paths determined by Newtonian equation. More precisely, the stationarity is a minimum for underdamped motion, a maximum for overdamped motion and an inflexion for the intermediate case. On this basis, we studied the path probability of Gaussian stochastic motion of dissipative systems. It is found that the path probability still depends exponentially on Lagrangian action for the underdamped motion, but depends exponentially on kinetic action A = ∫T0 Kdt for the overdamped motion.