Escaperatetheoryfornoisydynamicalsystems UniversitéLibredeBruxelles

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Université Libre de Bruxelles

Center for Nonlinear Phenomena and Complex Systems

& Department of Physics

Escape rate theory for noisy dynamical systems

Jonathan Demaeyer

Academic year 2012-2013

A thesis presented in partial fulfilment of the requirements for the degree of Doctor in Science.

Supervisor : prof. Pierre Gaspard

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In a world where no one is compelled to work more than four hours a day, every person possessed of scientiﬁc curiosity will be able to indulge it, and every painter will be able to paint without starving, however excellent his pictures may be.

Bertrand Russel,In Praise of Idleness

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à Florence, à mes parents, à mes grand-parents.

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Remerciements

Cette thèse est l’aboutissement d’un travail commencé il y a quelques années.

Commencé le jour où, étudiant en première licence et à la recherche d’un stage, j’ai poussé la porte du professeur Pierre Gaspard. J’appréciais particulièrement ses cours, où l’on se sentait libre de poser des questions. Le stage qu’il me proposa alors contenait déjà, en arrière-plan, la problématique du présent travail. Ainsi commença donc ce voyage dans la théorie du taux d’échappement, les questions apparaissant au cours de chaque étape (stage - mémoire - thèse) induisant na- turellement le sujet de l’étape suivante. Je suis reconnaissant à Pierre Gaspard de m’avoir initié à cette théorie et de m’avoir guidé à travers ses méandres. Je me souviens encore de la discussion initiale préalable au début du mémoire où il m’indiquait : “Il faudrait étudier cette structure et élucider son rôle dans le problème du taux d’échappement.”, parlant déjà de la structure hétérocline qui deviendra capitale à la ﬁn de la thèse.

Je suis reconnaissant au professeur Malek-Mansour de nous avoir communi- qué la référence [28] qui fut déterminante par la suite. Je suis aussi reconnaissant au professeur Peter Tinyakov d’avoir pris un peu de son temps pour m’expliquer certaines subtilités concernant les instantons et les zero modes.

Je remercie Bernard Knaepen, Jean Bricmont, Yannick De Decker, Musta- pha Tlidi et Thomas Gilbert d’avoir accepté de faire partie de mon jury.

Je pense que Pierre de Buyl ne s’oﬀusquera pas (on ne sait jamais) si je dis que cette thèse lui doit beaucoup. L’autre (ex-)assistant du service partage avec moi “un intérêt excessif pour les librairies graphiques et numériques” mais aussi le goût du café serré. Ces deux caractéristiques incontournables s’ajoutent à son amitié et à son soutien sans failles à la cause désespérée que j’ai pu être parfois.

Je remercie les collègues d’enseignement rencontrés au cours de ces années : Claire, Tiziana, Aurore, Frank, Michel, Pierre, Pasquale, Paul-Henri, Serge, Mustapha, Marc, Henri et Dominique. Je remercie aussi le staﬀ du labo Phys.

BA1 pour les mémorables discussions qui agrémentaient nos après-midis stu- dieuses : Malek, Laurent, Pascal, Yassin, Laura L.-H., Laura D., Stéven, Nicolas, Priscilla, Mathieu, Francis, Étienne ainsi que les techniciens sans qui l’organi- sation de ces labos serait purement et simplement impossible : Paul, Michaël et Anastase.

Au cours de ces années, l’expérimentarium de physique de l’ULB dirigé par Philippe Léonard - et avec le soutien de Manu - fut un lieu génial qui me sortait du train parfois monotone des cours et de la recherche. Je remercie Philippe et Manu pour leurs explications toujours limpides et pour leur patience.

Je remercie David Andrieux, mon voisin de bureau dans le service, pour les innombrables parties de dakao jouées ensemble et pour ses conseils toujours avisés. Je remercie également les autres membres du services : Thomas Gil- bert, Cem Servantie, Sebastien Viscardy, Jean-Sabin McEwen, Nathan “Nath Goldman” Goldman, Grégory “Greg” Bulnes Cuetara, Alexandre Dauphin, Eric Gerritsma, Jim Lutsko, Vasileios Basios, Patrick Grosﬁls, Massimiliano “Massi”

Esposito, Fabian Trillet, Delphine Vantighem, Nicole Aelst et Daniel Kosov.

Durant mes études à l’ULB, j’ai fait la connaissance de personnes vraiment formidables et tous les citer ici serait un peu long mais en particulier je pense à Julie, Ella, Vincent, Nassiba, Ariane, Michael, Thomas, Jérôme, Stéphane, Audrey, Renaud, Philippe, Jean-Rémy, Quentin, Cyril, Olivier, Nir, Yass, Nath,

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discussions sympas dans la salle café : Laurence, Yannick, Lorena, Lauren, Do- menico, Jessica, Geneviève et Anne. En outre, grâce à eux, il est impossible d’avoir faim un vendredi après-midi au cinquième étage du bâtiment NO.

Je remercie Jo et Marco qui sont là depuis le début et les boys, qui m’ont extrait la thèse de la tête (enﬁn, d’une de mes têtes) quand c’était nécessaire : Jérôme, Julien, Florian, Pierre, Max et Greg.

Je remercie aussi Tyl, François, Céline D., Céline V., Gilou S., Florence F., Laurence D., Morgan, Thanh, Jonathan, Philibert, Behrouz, Myr, Zyvou-colloc et Marilou.

Je remercie ma famille, qui ne m’a jamais laissé me sentir seul.

Je remercie mes parents, qui ont éclairé là où cela était possible et accepté de laisser le mystère là où cela était nécessaire.

Enﬁn, je pense que je n’aurais pas pu achever cette thèse sans le soutien de Florence, qui m’a épaulé pendant ces dernières années. Merci à elle, pour tout.

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Abstract

The escape of trajectories is a ubiquitous phenomenon in open dynamical systems and stochastic processes. If escape occurs repetitively for a statistical ensemble of trajectories, the population of remaining trajectories often undergoes an exponential decay characterised by the so-called escape rate. Its inverse deﬁnes the lifetime of the decaying state, which represents an intrinsic property of the system. This paradigm is fundamental to nucleation theory and reaction-rate theory in chemistry, physics, and biology.

In many circumstances, escape is activated by the presence of noise, which may be of internal or external origin. This is the case for thermally activated escape over a potential energy barrier and, more generally, for noise-induced escape in continuous-time or discrete-time dynamics.

In the weak-noise limit, the escape rate is often observed to decrease expo- nentially with the inverse of the noise amplitude, a behaviour which is given by the van’t Hoﬀ-Arrhenius law of chemical kinetics. In particular, the two important quantities to determine in this case are the exponential dependence (the “activation energy”) and its prefactor.

The purpose of the present thesis is to develop an analytical method to determine these two quantities. We consider in particular one-dimensional continuous and discrete-time systems perturbed by Gaussian white noise and we focus on the escape from the basin of attraction of an attracting ﬁxed point.

In both classes of systems, using path-integral methods, a formula is de- duced for the noise-induced escape rate from the attracting ﬁxed point across an unstable ﬁxed point, which forms the boundary of the basin of attraction.

The calculation starts from the trace formula for the eigenvalues of the operator ruling the time evolution of the probability density in noisy maps. The escape rate is determined by the loop formed by two heteroclinic orbits connecting back and forth the two ﬁxed points in a two-dimensional auxiliary deterministic dynamical system. The escape rate is obtained, including the expression of the prefactor to van’t Hoﬀ-Arrhenius exponential factor.

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L’échappement des trajectoires est un phénomène omniprésent dans les sys- tèmes dynamiques ouverts et les processus stochastiques. Si l’échappement se produit de façon répétitive pour un ensemble statistique de trajectoires, la population des trajectoires restantes subit souvent une décroissance exponentielle caractérisée par le taux d’échappement. L’inverse du taux d’échappement déﬁnit alors la durée de vie de l’état transitoire associé, ce qui représente une propriété intrinsèque du système. Ce paradigme est fondamental pour la théorie de la nucléation et, de manière générale, pour la théorie des taux de transitions en chimie, en physique et en biologie.

Dans de nombreux cas, l’échappement est induit par la présence de bruit, qui peut être d’origine interne ou externe. Ceci concerne en particulier l’échap- pement activé thermiquement à travers une barrière d’énergie potentielle, et plus généralement, l’échappement dû au bruit dans les systèmes dynamiques à temps continu ou à temps discret.

Dans la limite de faible bruit, on observe souvent une décroissance exponentielle du taux d’échappement en fonction de l’inverse de l’amplitude du bruit, un comportement qui est régi par la loi de van’t Hoﬀ-Arrhenius de la cinétique chimique. En particulier, les deux quantités importantes de cette loi sont le co- eﬃcient de la dépendance exponentielle (c’est-à-dire “l’énergie d’activation”) et son préfacteur.

L’objectif de cette thèse est de développer une théorie analytique pour déter- miner ces deux quantités. La théorie que nous présentons concerne les systèmes unidimensionnels à temps continu ou discret perturbés par un bruit blanc gaus- sien et nous considérons le problème de l’échappement du bassin d’attraction d’un point ﬁxe attractif. Pour s’échapper, les trajectoires du système bruité ini- tialement contenues dans ce bassin d’attraction doivent alors traverser un point ﬁxe instable qui forme la limite du bassin.

Dans le présent travail, et pour les deux types de systèmes, une formule est dérivée pour le taux d’échappement du point fixe attractif en utilisant des méthodes d’intégrales de chemin. Le calcul utilise la formule de trace pour les valeurs propres de l’opérateur gouvernant l’évolution temporelle de la densité de probabilité dans le système bruité. Le taux d’échappement est déterminé en considérant la boucle formée par deux orbites hétéroclines liant dans les deux sens les deux points fixes dans un système dynamique auxiliaire symplec- tique et bidimensionnel. On obtient alors le taux d’échappement, comprenant l’expression du préfacteur de l’exponentielle de la loi de van’t Hoff-Arrhenius.

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Copyright notice

The present thesis is copyrighted to Jonathan Demaeyer.

Most of the results and ﬁgures in Chapter 10 have appeared in Demaeyer and Gaspard,Physical Review E80 031147 (2009). See Ref. [39].

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Introduction, Theory and

Physical motivations

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Chapter 1

Introduction

1.1 The van’t Hoff-Arrhenius law

At the end of the 19th century, van’t Hoﬀ [172] proposed an equation that gives the temperature dependence of the ratek at which a chemical reaction occurs:

k=A e⁻^E/RT (1.1)

whereRis thegas constant,Eis the so-calledactivation energyandT is the ab- solute temperature. In the simplest form, the pre-exponential factorAand the activation energyE are not depending on the temperature. At the time where Eq. (1.1) was first considered, several equations for the rate were promoted by other chemists [105]. However, all these equations being empirical, they were theoretically sterile. On the contrary, the law (1.1) contained theoretical in- sight. Indeed, Pfaundler [146] had previously showed, with the Maxwell law of distribution of molecular speed, that the fraction of molecules of the reactant having an energy greater than a critical value E is proportional to e⁻Ê/RT. He suggested that only the molecules with at least this critical energy could undergo the chemical transition. Arrhenius [6] resumed this idea and, while discussing rate data, he concluded that the molecules have to be “activated” in order to undergo a change of chemical state. These activated molecules, com- monly called “activated complex”, are distinct from the others. The rate at which the molecule becomes activated is thus given by the so-called van’t Hoff- Arrhenius law - Eq. (1.1). Following these developments, the discipline of rate theory was born [76]. Even nowadays, this discipline is mainly concerned with the estimation of the activation energyE and the prefactorA. As we shall see, it has been fertilised by the various scientific fields that have interacted with it.

At the time of Arrhenius, the mechanism by which a molecule becomes

“activated” was not clear. At the end of the century, it was realised that the fluctuations of energy were responsible for the activation. Indeed, in every physical system composed of a great number of particles, such as a gas or a solvent, the energy is fluctuating due to the individual behaviours deviating from the average. The statistical treatment of these fluctuations was later formalised, in connection with the work on Brownian motion.

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1.2 Brownian motion

The name of the individual who did the first observation of the Brownian motion is not known for sure, as many candidates are reported [1]. For instance, many people, at different times, have observed the rather chaotic motion of a small particle¹ in a fluid. However, the botanist Robert Brown was the first to give a clue towards a correct interpretation. It is interesting to note that it is by his quality of botanist that he became interested in the problem, trying to stabilise a grain of pollen under a microscope. After performing multiple experimental observations, he was able to rule out the possible organic origin of the motion - an explanation that had muddled the debate during the 18th century and the early 19th. He published his result in 1828 [25]. He was not alone at that time in reaching this conclusion, but history kept his name.

Later in the 19th century, the center of interest shifted to the question of the origin of the ﬂuctuations inducing the chaotic motion. The possible candidates proposed were heat, light, electrical forces,... At the end of this century, it was also suggested that the disordered motion of the Brownian particle could be the result of its impact with the solvent molecules. Despite the accumulation of ev- idence in favour of this latter explanation, it kept its unresolved status until the theoretical works by Lord Rayleigh [119], Smoluchowski [173] and Einstein [44]

allowed a breakthrough. Einstein’s work in particular, has played a key role in linking the statistical theory he had developed with the relevant physical quantities of the problem. The experimental observations made by Jean Bap- tiste Perrin [145] conﬁrmed the predictions of Einstein and therefore deﬁnitively favoured the atomistic description of the matter.

The Einstein approach to the problem only requires two assumptions on the Brownian movement [57]:

i) The motion of every particle is independent from the motion of the other particles.

ii) The motion of a particle on a time interval is independent from the motion on another time interval, provided that all those intervals are not too small (like the 30 seconds interval in Fig. 1.1).

Deﬁning the intensive variable ν =f(x, t) as the number of particles lying at the time t in a volume element centered on the coordinate xof the Cartesian coordinates, he found that the distribution f is ruled by the following partial diﬀerential equation:

∂tf =D ∂_x²f (1.2)

Computing the solution f explicitly is a classic textbook exercise. With the initial condition f(x,0) = δ(x) whereδ(x) is the Dirac delta distribution, one obtains

f(x, t) = 1

√4πDt exp ï

− x² 4Dt

ò

. (1.3)

Therefore, the mean square displacement is a linear function of time:

x²(t)

= ˆ

dx x²f(x, t) = 2D t (1.4)

1For example a dust or a pollen grain.

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1.2 Brownian motion 21

Figure 1.1: Reproduction of the book of Jean Baptiste Perrin [145]. Three diﬀerent trajectories of particles in suspension, as seen under the microscope.

Successive positions every 30 seconds are joined by straight line segments.

From these results, Einstein obtained the relation between thediffusion coeffi- cientD and the physical observables.

Three years after Einstein’s paper, Langevin published a note [112] where he reached similar conclusions as Einstein, but by a diﬀerent path. Langevin does not start with the distribution of the particles but directly with its Newtonian equation of motion. He supposes that the particle feels, in the direction x, a viscosity force - given by the Stokes formula - and an unknowncomplementary force X accounting for the collision of the particle with the molecules of the ﬂuid. The equation of motion thus reads

md²x

dt² =−γdx

dt +X (1.5)

where γ is a proportionality coeﬃcient given by the Stokes formula and m is the mass of the particle considered. By multiplying the whole equation by x, taking the average over a huge number of particles and using the equipartition of the kinetic energy, he ﬁnally recovers Einstein’s results. The lack of knowledge about the forceX is bypassed by its allegedirregularity, giving a vanishing time average of the quantityXx:

X(t)x(t)= 0.

The equation (1.5) is now famously known as the Langevin equation. Al- though the methods of Einstein and Langevin reach the same conclusions, they are by essence very diﬀerent [116]. Einstein works with the Markovian hypothe- sis that we shall describe in details later. He also uses a probabilistic description of the position of the particle in terms of the partial diﬀerential equation (1.2).

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The Langevin method is apparently a simple derivation of the equation of motion of the particle, but involves a mysterious and irregular force. This irregularity requires the introduction of a new way of performing the integrals, thestochastic integral calculus, and the Langevin equation gives us an example of stochastic differential equation.

1.3 The Kramers theory and recent developments

Before and after the Einstein’s paper, several workers gave important contri- butions to the rate theory and the Brownian motion theory, including Fokker, Planck, Ornstein and Smoluchowski. However, the early progress related to the

“activated” rate theory were made in the theory of nucleation, more particularly about the study of homogeneous nucleation [53]. In 1927, Farkas [47] considered the dynamics of atoms arriving or leaving a droplet in a supersaturated gas. His contribution had many pioneering ideas [48, 76]², including one which is now underlying any rate calculation: the rate of escape from a metastable state is given by the ﬂux of particles that pass through the potential barrier separating the products from the reactants. Note that Farkas credits the basic idea for this treatment to Szilard [47, 76].

The next step was the development of the transition-state theory by Marcelin, Eyring, Evans, Polanyi and Wigner [105, 76]. In the framework of this theory, the rate of a chemical reaction was expressed in terms of quantities related to the underlying potential energy surface. This theory presented the advantage that the quantum eﬀects could be easily incorporated [170].

In 1940, Kramers published his paper “Brownian Motion in a Field of Force and the Diffusion Model of Chemical Reactions” [101]. He summarised the pre- vious research by describing a chemical reaction as a Brownian particle in a bistable potential, one of the metastable state representing the reactants and the other the products. By doing so, he definitively linked the rate theory to the Brownian motion theory, and the theory of noise-induced escape from metastable states was created. The Brownian - chaotic - motion of the particle accounted for the thermal forces. These thermal forces themselves depend on the temperatureT and on the frictionγ, through the fluctuation-dissipation theorem. Kramers obtained the Fokker-Planck equation governing the time evolution of the probability distribution and solved it separately for the cases of strong and weak friction. His treatment allowed him to obtain a general formula for the prefactorAin Eq. (1.1) in the case of moderate-to-strong friction. This formula is still used extensively nowadays, as we shall show in Chapter 2.

In the second part of the 20th century, people involved in the rate theory were mainly interested in the multidimensional transition-state theory [170, 148, 169, 149] and the quantum theory of reaction rate [126, 127]. In addition, several research were performed on new - more sophisticated - techniques to compute the escape rate. These methods included for example the concept of themean

2In this thesis, multiple references are organised in chronological order.

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1.4 Purpose of the thesis 23

first-exit time[167] and the study of the spectrum of the evolution operator acting on the probability distribution. In particular, this latter method presents the advantage that it provides more information on the dynamics than the sole escape rate. Indeed, the leading eigenvalue gives the escape rate while the others also give the transient behaviours in the system. The determination of the rate through the evaluation of the leading eigenvalue has been done with various techniques, from WKB methods [27] and path-integral methods [28] to super- symmetry techniques [17]. More occasionally, the spectrum has been obtained, for example for the Ornstein-Uhlenbeck process and the bistable potential [63], and also for oscillating systems [59] by a method called thetrace formula.

Recently, the focus has shifted to more complex systems, with metastable states such as periodic states [120] and chaotic attractors [162, 103, 163]. The latter ones have in general been considered through the study ofnoisy maps, i.e. discrete-time dynamical systems perturbed by noise. The study of such kind of systems began in the 1980s when the need for simple models with complex behaviours was stressed to study physical systems such as for instance the Josephson junction [93, 12] and the laser ring cavity [81]. New concepts, such as themost probable escape path, were introduced to compute the escape rate.

This path in the phase space of the considered system is the one for which the probability of occurrence is larger than all the other paths [42]. Another method uses the aforementioned leading eigenvalue of the operator which governs the time evolution of the probability distribution. Here, this operator is called the Perron-Frobenius operator. The eigenvalues of this operator can be computed thanks to atrace formula. In particular, this method has been applied to maps with a repellor [40, 36, 142]. For the particular case of one-dimensional bistable maps perturbed by noise with a spatial dependence, a discrete-time analog of the original Kramers method was developed by Reimann and Talkner to obtain the escape rate [155].

1.4 Purpose of the thesis

Discrete-time dynamics concerns dissipative dynamical systems that are peri- odically driven or subjected to some cybernetic feedback [69, 70, 164, 131, 132].

Discrete-time dynamics is also of application in oscillatory regimes dominated by a sufficiently well-defined period so that the continuous-time dynamics can be modeled with a Poincaré first-return map. In the presence of a source of noise, such systems can be described by noisy maps, which have been the topic of several studies [153, 152, 155, 154, 40, 36, 142, 16, 163, 50, 39, 157].

At the deterministic level of description, such dynamics may present an attractor surrounded by a basin of attraction. In the presence of noise, leakage would occur at the border of the basin, inducing the escape of trajectories to infinity. Typically, the rate of such activated escape processes vanishes with the noise amplitude in a non-analytic way, which is given by the van’t Hoff-Arrhenius law (1.1). This non-analytic dependence expresses the fact that escape does not preexist in the corresponding deterministic system and is a novel phenomenon entirely generated by the noise. This situation is thus very different from the situation where an escape is already present in the noiseless system, for instance

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the escape from a repellor. Indeed, in this latter case, the escape rate is given by the deterministic escape rate plus corrections in powers of the noise amplitude [40, 36, 142].

The purpose of the present thesis is to deduce a mathematical formula for the rate of activated escape in discrete-time dynamics. In particular, we consider the case of the escape from an attracting ﬁxed point. As aforementioned, the escape rate is identiﬁed as the leading eigenvalue of the Perron-Frobenius operator ruling the time evolution of probability densities in the process [40, 36, 142].

The spectrum of eigenvalues is given by a trace formula that is calculated with path-integral methods in the weak-noise limit. In this limit, the path integral selects the classical orbits of a symplectic map, which is defined in a phase space extended to include momenta canonically conjugated to the variables of the deterministic map [50, 39]. Taking the trace of the iterates of the Perron-Frobenius operator is known to select closed orbits [35, 40, 36, 142, 60]. Here, a challenge arises because the closed orbits are linked to the fixed points representing, on the one hand, the attractor and, on the other hand, the top of the barrier over which escape is activated by the noise. The closed orbits are thus forming a pair of heteroclinic orbits asymptotic to both fixed points. For continuous-time dynamical systems, such heteroclinic orbits are called instantons or kinks and their effect has been much studied in the literature [111, 28, 161, 31]. Here, our purpose is to consider discrete-time dynamics, for which we obtain the escape rate (1.1) including the expression of the prefactorAto the van’t Hoff-Arrhenius exponential factore⁻Ê/RT.

1.5 Outline of the thesis

The present work is divided into four parts. The ﬁrst part, which includes the current chapter, is devoted primarily to the introduction of the topic of this thesis, as well as some elements of theory:

Chapter 2 motivates the study of the noise-induced escape from attractors by introducing some of the areas were the occurrence of the phenomenon is crucial.

Chapter 3 is an introduction to the theory of dynamical systems.

Chapter 4 introduces several notions from the theory of stochastic processes and random dynamical systems. In addition, we detail the aforementioned trace-formula method.

The path-integral method involved in the derivation of this trace formula is rather delicate. Therefore, in Part II of the thesis, we ﬁrst consider its application to the continuous-time random dynamical systems with a metastable potential. By this way, we recover the well-known Kramers result. This part decomposes as follows:

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1.5 Outline of the thesis 25

Chapter 5 presents the original Kramers problem and details the Kramers method to obtain the escape rate. The trace-formula approach applied to this case is discussed.

Chapter 6 introduces the path-integral formulation for Fokker-Planck equation. A detailed analysis of this path integral is a prerequisite to the trace formula approach.

Chapter 7 presents the main results of this part. In this chapter, we obtain the trace of the propagator for systems with a metastable potential. From the trace, we ﬁnally recover the escape rate of the Kramers problem introduced in Chapter 5.

The third part consider the escape in discrete-time systems.

Chapter 8 discusses the escape problem in one-dimensional noisy maps with attracting ﬁxed points.

Chapter 9 presents the path-integral formulation for the Perron-Frobenius operator. This formulation will allow us to obtain the trace of the Perron- Frobenius operator in Chapter 11.

Chapter 10 studies the activation energy E in the van’t Hoﬀ-Arrhenius law (1.1). In particular, it focus on the neighbourhood of the bifurcations of one-dimensional maps.

Chapter 11 presents the main results of this part. In this chapter, we compute the trace of the Perron-Frobenius operator and obtain the escape rate.

Therefore, we also obtain a formula for the prefactor of van’t Hoﬀ-Arrhenius law (1.1).

We conclude this thesis in Part IV and give some further possible developments.

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Chapter 2

Physical motivations

The present thesis is mainly concerned with noise-activated escape processes and the calculation of the rates thereon. The knowledge of such rates is of crucial importance to a large number of fields. These fields are diversified, ranging from electronics to biology. Therefore, we describe here some examples where the forthcoming results could be applied.

2.1 Chemical reactions

Historically, the theory of noise-induced escape from a metastable state was first introduced to describe chemical reactions. Indeed, a chemical reaction is conceived as the transformation of a stable chemical state into another stable one. This transformation can be seen as a two-steps process: the system first undergoes a transition from a stable state to an unstable state called the transition state (or activated complex) before performing finally the transition toward another stable state. The reaction of a given molecule is thus modeled as a path along a reaction coordinate, from a local minimum of the Gibbs free energy to another one. The transition state corresponds then to a local maximum of the Gibbs free energy: a saddle in the free energy landscape (see Fig. 2.1). This theory is known as thetransition state theory and was introduced first by Wigner and Eyring [76, 149] at the beginning of the 20th century. The process that drives molecules from one chemical kind to another is then identified with the energy fluctuations inside the media where the chemical reaction is happening.

The transition state theory has proved to be a reliable model for various kinds of reactions and compares well with other calculations and experiments [170, 169].

In a famous paper, Kramers proposed in 1940 a simple model for the chemical reactionA⇋C⇋BwhereCis the unstable transition state [101]. This model is a Brownian particle in a bistable potential with each state of the reaction represented by a corresponding state of the stochastic process (see Fig. 2.2).

The two statesAand B are associated with the two metastable states located at xA and xB. They are separated by an unstable state xC representing the transition state. The noise-driven hoping of the particle between the two wells accounts then for realisations of the chemical reaction.

Considering an ensemble of non-interacting Brownian particles in the po-

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Figure 2.1: Free energy landscape of an hypothetical reaction A+BC ⇋ (ABC)T ⇋AB+C. The dashed line is the reaction coordinate. Reproduction from Ref. [76]

Figure 2.2: The potential associated with the reactionA⇋C⇋B. Reproduc- tion from the original article of Kramers, Ref. [101].

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2.2 Nucleation 29

tential, Kramers obtained the rates of transition from one well to the other for various regimes of the friction experienced by the Brownian particle. These rates give estimates of the rates of the chemical reactions. His results show that these rates depend only on the friction¹ and on the curvature of the potential around the states². Consequently, since the work of Kramers, the determination of the escape rate from a metastable state via noise activation is usually called a Kramers problem. Further work in the 1980s connected the results of the transition state theory to those of Kramers [148, 149], thus validating the framework he proposed.

The elegant formulation of Kramers paved the way to numerous other results [76], in the ﬁeld of chemistry, but also, as we will see in this chapter, for various other disciplines. Moreover, we note that, since Kramers, stochastic processes have become an ubiquitous tool to describe chemical reactions [135, 65, 58].

2.2 Nucleation

The theory of nucleation deals with the dynamics of first-order phase transitions that occur between the gas, liquid and solid states. One of the first models introduced to study the kinetics of such transitions is the droplet model, applicable both to the homogeneous nucleation³ of liquid droplets in a supersaturated va- por and of gas bubbles in a superheated liquid. These droplets and bubbles arise in the region of the phase diagram where the two phases - gaseous and liquid - coexist. Indeed, if we consider for instance the isothermal compression of a gas into the coexistence region (see Fig. 2.3); there, the gas phase becomes metastable as the fluctuations of the density will induce locally the onset of small “nuclei” of the liquid phase [53, 48]. These nuclei will continue to appear until the system attains a new equilibrium state where the two phases coexist.

Compressing further the gas will induce a multiplication of the nuclei that will ultimately trigger the transition from the liquid to the gaseous phase. The same scheme applies to the isothermal relaxation of a liquid, which results in the for- mation of gas bubbles.

The droplet model deals with the kinetics of the apparition of the nuclei by considering that they are spherical. To detail this model, we use the example of liquid droplets appearing in a supersaturated gas but it applies also for overheated liquid. The model works as follows: Considering that the nuclei are small, the variation of energy induced by their onset is due to surface term [53].

The corresponding energy for one droplet of liquid of radius r is given by the surface of the droplet 4πr² multiplied by the surface tension σ. Therefore, if

“L” represents the liquid state and “G” the gaseous state, and considering the nucleation of one droplet, the thermodynamical potential of the whole system is given by:

Φ =NLφL+NGφG+ 4πr²σ (2.1)

1which accounts for the strength of the interaction between the reacting molecules and surrounding medium.

2See Chapter 5 for a derivation in the strong friction case.

3i.e. nucleation in a substance without impurities.

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Figure 2.3: Isothermal phase diagram with a liquid and gaseous phase. The region in between is bistable with the metastable branches bcand de and the coexistence state depicted by the dashed linebe. Reproduction from Ref. [107].

whereNL, φLandNG, φGare the numbers of particles and the chemical poten- tials of respectively the liquid and the gaseous phase. Moreover, we have that the total number of particles N =NL+NG is conserved. Now, starting from a conﬁguration of energy Φ0=N φG where all the particles are in the gaseous phase, the required amount of reversible work necessary to form a spherical liquid droplet of radiusr is [53]:

∆Φ = Φ−Φ0=−NL(φG−φL) + 4πr²σ . (2.2) All the particles belonging to the liquid phase are in the droplet, therefore one can writeNL= ⁴₃πr³/vLwherevLis the volume per particle in the liquid phase, and Eq. (2.2) can be rewritten:

∆Φ =−φG−φL

vL

4

3πr³+ 4πr²σ . (2.3)

Consequently, the potential Φ around the pure gaseous phase possesses a maximum (see Fig. 2.4) that corresponds to a droplet of radiusr^∗= 2σvL/(φG−φL).

A nucleus, created by the ﬂuctuations of the density, have to attain this critical size in order to nucleate. On the contrary, if it does not attain this size, it will ﬁnally disappear. We see thus that the “activation energy” necessary for the creation of the droplet is given by

∆Φmax=4πr^∗²

3 σ , (2.4)

a result obtained by Gibbs in 1878. The rate of nucleation of droplets is thus proportional to exp(−∆Φmax/kBT) wherekBis the Boltzmann’s constant. This result led to the introduction of a stochastic model to describe the dynamics of this nucleation [47, 13, 177]. Indeed, let n be the number of particles in a

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2.2 Nucleation 31

r r^∗

∆Φ

0

∆Φmax

Figure 2.4: The free energy (2.3) needed to create a spherical droplet of radius r. Forr < r^∗, the droplet will ﬁnally disappear; while forr > r^∗, the droplet has passed the barrier of energy needed to nucleate and has thus attained the critical size beyond which growth is irreversible.

droplet, and follow the variation ofnin time. Thennchanges by plus or minus one particle at a time. The number of particles inside the droplet can thus be viewed as a random walk in a potential. This random walk is described by a master equation which rules the concentration in the droplet and which can be approximated by a Fokker-Planck equation with a metastable potential such as the one treated by Kramers (see Fig. 2.2). The rate of nucleation is determined subsequently by using the Kramers theory [48].

The theory presented above is usually known as the classical nucleation theory, where one obtains the time evolution ruling the nucleus concentration on the form of a Fokker-Planck equation [90]. This equation can then be solved using the method of Kramers. Consequently, noise-activated escape from metastable state is the main framework available to model nucleation phenomena.

Despite its elegance, the classical nucleation theory presents some ﬂaws.

One of the principal problems is that it requires a very good estimate of the energy barrier that the system has to cross, otherwise the rate of nucleation is known to be many orders too large or too small [123] (a problem also known in the transition state theory for chemical reactions [169]). Other techniques to evaluate the barriers have then to be used, for example the density functional theory [141, 166, 122] or the steepest-descent method on the free-energy surface [121, 122]. More recently, the need for a nonequilibrium description of the nucleation process have been stressed, leading to a description in terms of fluctuating hydrodynamics [123, 124]. In this framework, the quantities of interest are the local ensemble-averaged densities, similar to the concentration variable in the classical nucleation theory. These densities are fields defined on each position of the system and described by stochastic differential equation. These equations can then be discretised and, in the particular high damping and weak noise limits, amost likely path can be defined. In this case, the Kramers theory

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Figure 2.5: (a) A circuit involving a resistance R and a tunnel diode (with a capacitanceC). (b) The characteristic of the diode with its load line. The inter- sectionsA, B and C of the characteristic with the load line are the stationary states of the system. Both panels are reproduced from Ref. [110].

has also to be used to obtain the rate of nucleation.

2.3 Electronic circuits

In the late 1920s, Johnson observed and measured the ﬂuctuations of the current in a resistor and correctly interpreted the thermal agitation as its main cause [84, 138]. These ﬂuctuations, called the Johnson noise, have the same properties as the Gaussian white noise perturbing the Brownian particle in the Kramers model [66]. Therefore, when the thermal agitation becomes a source of instability in electronic circuit, for instance when the Johnson noise is able to induce the circuit to escape a stable state, one can estimate the rate at which such events happen thanks to Kramers theory. In this case, the rate gives an estimate of the stability of the device with respect to thermal perturbations.

The theory of noise activation in electronic circuits has been used in the early 1960s when it was realised that dissipation is a key point in the design of computers, as a result of the logical irreversibility of the computational processes [109]. Consequently, bistable dissipative data storage units were required to develop the logical framework of computers⁴ [165]. Simple models for such storage units could be obtained by considering tunnel diode circuits [110] like the one depicted in Fig. 2.5 (a). The characteristics of the diode displays three stationary states given by the intersection with the load line [see Fig. 2.5 (b)].

Moreover, the characteristic curve is tantamount to minus the dissipative force acting in these systems [109], therefore A and B are stable states while C is

4The bistability was needed to store the binary values.

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2.4 Josephson tunnel junctions 33

unstable. Depending on the parameters of the circuits and on the temperature, the thermal ﬂuctuations can induce noise-activated hoping events. These systems can thus be modeled by a Brownian particle in a bistable potential like the one depicted in Fig. 2.2. These simple models enabled to answer questions on the limitations that the physical laws induce on the optimisation of computing devices [165]. They also allowed to compute the minimal dissipation needed to process digital data [109, 4].

2.4 Josephson tunnel junctions

The Josephson effect is the phenomenon characterised by the presence of a current flowing indefinitely long, but without any voltage applied. It was discovered by Josephson in the early 1960s [85, 86]. It occurs in devices called Josephson junctions which are composed of two superconductors coupled (separated) by a weak link. The basic equations governing the dynamics of the junction are then:

V(t) = φ0

2π dφ

dt (2.5)

IJ(t) = Ic sinφ(t) (2.6)

where V(t) and IJ(t) are respectively the voltage and the current across the junction. φ(t) is the phase difference between the two superconductors⁵. In addition, Ic is the Josephson critical current and φ0 =h/2e is the flux quantum, withhthe Planck’s constant and ethe elementary charge. Applications of the Josephson effect include superconducting quantum interference devices (which are very sensitive magnetometers) [30], single-electron transistors [55]

and quantum computation [22].

The kind of weak link used to separate the superconductors characterises the type of model describing the junction. This separation can for instance be a point-contact coupling [82], a superconductor-metal-superconductor sandwiches or a superconductor-insulator-superconductor sandwiches [54]. In the latter case, called a Josephson tunnel junction, the effect of the self-magnetic field is negligible, a fact which simplifies the model. A current-biased⁶Josephson tunnel junction can then be represented by an equivalent circuit [depicted on Fig. 2.6 (a)] in terms of the currentIJ =Ic sinφ, the bias DC currentI, a resistanceR and a capacityC [54]. Accordingly, the current-continuity equation is given by

x= sinφ(t) + 1 ω_J²γ

dφ dt + 1

ω_J² d²φ

dt² . (2.7)

where x = I/Ic, γ = RC and ω_J² = 2πIc/φ0C. Eq. (2.7) is the equation of motion of a particle in a potential of the formU(φ) =−(xφ+ cosφ) where the position of the particle represents the phaseφ. Accordingly, the behavior of this particle completely describes the behavior of the junction through Eqs. (2.5) and (2.6). The shape of the potential U depends on the value x. Indeed, as

5i.e. the argument of the complex order parameter in the Ginzburg-Landau theory of superconductivity [108].

6i.e. a Josephson junction subject to a given external current.

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Figure 2.6: (a) Equivalent-circuit model of the current-biased Josephson tunnel junction. (b) The potentialU(φ) =−(xφ+cosφ) for diﬀerent values ofx=I/Ic. Figure reproduced from Ref. [54].

shown in Fig. 2.6 (b), this parameter controls the tilting of this inﬁnite periodic potential. On the other hand, the quantity ωJγ controls the damping of the particle. The damping parameter depends, among other things, on how the Josephson tunnel junction was built.

Two diﬀerent behaviors are possible for the particle: the particle is either located in a given well of the potential or rolling steadily downside. These two cases account for the two possible behaviors of the junction [54]: either the phase φis constant and the potential across the junction vanishes (V = 0), or the junction is “free-running” with a phase increasing according to dφ/dt= 2πIR/φ0, i.e. V =RI. The junction is then said to be respectively in alocked (V = 0) or arunning(V 6= 0) state. Depending on the average slopexof the potential and on the value ωJγ of the damping, the locked state and the running state are possible states of the junction. Domains of the parameters where both states of the junction are possible exist and form therefore regions of bistability [26].

Within the framework of this model, the thermal noise presented in the pre- vious paragraph can be accounted for by an additive Gaussian white noise term in Eq. (2.7). In the appropriate conditions, this noise induces transitions between the stationary states. Kramers theory has been used to obtain the rate of these transitions. In the overdamped case, the only kind of available states are the locked states and therefore, due to the ﬂuctuations, the junction can make transitions between two neighboring locked states [104, 82]. In the regions of bistability, where the locked states and the running states are possible, the noise can induce transitions from a locked state to a running state and vice versa. In both cases, the Kramers theory gives the rates of transitions and the probability

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2.5 Transport in biological systems 35

of occupation of each kind of states [14, 26].

If an AC component is added to the bias current (or to the noise), the system (2.7) typically exhibits new behaviors such as periodic and chaotic-running solutions as well as diﬀusive motions [80, 94, 91]. This kind of system can also be modeled, through the introduction of a Poincaré section, by a simple discrete-time map of the kind [159]:

φi+1=φi+λsin 2πφi , λ∈R. (2.8) One of the interesting problems arising then for such kinds of physical systems is the noise-induced escape from chaotic attractors [92, 120, 162].

2.5 Transport in biological systems

The various transport phenomena in biological systems are crucial properties to maintain and secure the functions that permit life. These transport phenomena can be described by stochastic models [24]. The classical example is the passive diﬀusion towards other cells of the synthesised products in the cell media. Due to the highly viscous nature of the media, this transport can be modeled by an overdamped Brownian particle and the escape from the cell is then tantamount to a Kramers problem. Another kind of phenomena is the active transport [24]

where small motorised carriers remove the products from the cells by travel- ing along ﬁlaments such as microtubules. The motion of these carriers is then studied through the dynamics of their molecular motors. Note that contrary to passive diﬀusion, such transports require a power source which is given here by the hydrolysis of adenosine triphosphate (ATP).

The molecular motors behave by cycling through sequences of conforma- tional states [24]. For instance, for the F1-ATPase motor [137, 176], several distinct states have been identiﬁed [62]. The behavior of the motors can be modeled by Brownian ratchets [151]: Let us consider the example of such a motor with two states “1” and “2”. The behavior of the motor in each statei can be described by a Brownian particle in a periodic potentialVi(see Fig. 2.7).

The probabilitypi to find the particle at a positionxat the timet in such potential is ruled by a Fokker-Planck equation. In addition, the system can make transitions from one state to the other and vice versa with probability per unit time given by the ratesω1 and ω2. This can be taken into account by adding source and sink terms in the Fokker-Planck equations [62, 24]. The probabilities pi are thus ruled by coupled Fokker-Planck equations and describe the complete state of the system. The shape differences of the potentialsViinduce then a net transport of probability in the system which results into the directed motion of the motor. The Kramers theory gives the transition ratesωi as well as the transport coefficient in the resulting systems [62].

2.6 Lasers

As a last example, we consider the ﬁeld of non-linear optics. A Fabry-Perot cavity (i.e. a laser cavity) containing a non-linear absorbing media composed of

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Figure 2.7: PotentialsV1 andV2 of the Brownian ratchet model of a molecular motor with two internal states. Figure reproduced from Ref. [24].

two-level systems is known to display bistability. This bistability phenomenon can be studied through the simpler model of a ring cavity [21], depicted in Fig. 2.8 (a). In this model, the amplitude of the electrical ﬁeld exiting the cavity is described in term of a delay-diﬀerential equation. In the late 1970s, discretising this latter equation, Ikeda introduced a paradigmatic recurrence equation involving a complex quantityznthat describes both the intensity and the phase of thenth pulse exiting the ring cavity [81, 103]:

zn+1=a+bzn exp ï

i Å

κ− η

1 +|zn|² ãò

(2.9) whereais the laser input amplitude. b,κandηare other parameters associated with the system (see Ref. [103]). Depending on the value of these parameters, the system can display behaviors such as bistability and chaotic dynamics. For given values of the parameter b, κand η, it is shown in Fig. 2.8 (b) that de- pending on the value ofa, various regions with diﬀerent stability properties may exist: regions with one stable state, regions with two stable states and even a region with a “strange” (chaotic) attractor.

In this framework, where multiple situations arise, a noise source is usually added to the model to take into account the spontaneous emissions that occur in the absorbing media [56]. In this case, the noise-activated escape from stationary states (ﬁxed points, periodic orbits or chaotic attractors) is an interesting and crucial feature, as it is observed in real laser [139, 130, 15]. For instance, recently, the noise-induced mechanism of the escape from a non-hyperbolic attractors have been investigated [103] by considering, among others, the model (2.9).

However, the search for a Kramers-like theory for discrete-time systems remains, even in the case of one-dimensional systems, a vast open problem that is partially addressed in this thesis.

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2.6 Lasers 37

Figure 2.8: (a) A ring cavity with mirrors 1 and 2 of reflectivityRand mirrors 3 and 4 of 100% reflectivity. The sample cell is filled with two-level absorbing media. (b) Fixed points of the map (2.9) for various values of the input light intensity|ǫI|(associated with the parameter aof the map). The (un)stability is depicted by a solid (dashed) line. For values such thatǫ1<|ǫI|< ǫ2, there is no stable fixed points. In this region of the parameter, chaotic behavior occurs and a chaotic attractor is thus present. Panels reproduced both from Ref. [81].

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Chapter 3

Introduction to the theory of dynamical systems

In this short chapter, we introduce some elementary notions that we will use subsequently in this thesis. In particular, we give the deﬁnition of adynamical systemas well as few related concepts.

3.1 Definition

A dynamical system is a semigroupGacting on a spaceM [168], i.e. it exists a mapT:

T : (t,x)7→T^t(x) :G×M 7→M (3.1) such that

T^t◦T^τ=T^t^◦^τ. (3.2)

IfGis a group, then the dynamical system is said to beinvertible. The spaceM is called thephase space. It is a space in which all possible states of the system are represented. The variablex∈M is then an initial state of the dynamical system that is mapped under the time evolution on the stateT^t(x).

Depending on the cardinality of G, the time in the dynamical system is said to be discrete or continuous and we shall now separate the presentation according to these two cases.

3.2 Continuous-time dynamical systems

Here, we consider here the semigroupGas uncountable and from now the reader can have in mind the usual exampleG=RorG=R⁺.

3.2.1 Time evolution

In general, the time evolution of a time-continuous dynamical system is described by a set of ﬁrst-order diﬀerential equations (ODE) [60]:

˙

x=f(x, t) where f :M×G7→TxM . (3.3)

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The space TxM is the tangent space ofM at the point x. The disjoint union of the tangent space ofM:

T M = [

x∈M

TxM (3.4)

is called the tangent bundle of M. The elements (x,x) of this space give a˙ complete description of the states that the system can achieve.

Many physical systems are ruled by ODEs that are autonomous, i.e., such that there is no time-dependent forcing: ∂tf(x, t) = 0. Hence, in the following, we shall mainly be interested in dynamical systems with equations that are independent of time:

˙

x=f(x). (3.5)

In this case, the general solution of Eq. (3.5) is given by a one-parameter group called the ﬂow:

x=Φ^t(x0) (3.6)

which is in general a nonlinear function of the timetand of the initial conditions x0 [60]. The flow defines a function like (3.1) and it is consistent with the fact that Eq. (3.5) rules the time evolution of a dynamical system. The flow defines also a group because

Φ^t+τ =Φ^t◦Φ^τ. (3.7)

For a given pointxof the phase spaceM, the set γx∈M of points related by the ﬂow acting onxis called theorbit ofx:

γx= Φ^t(x)

t∈G . (3.8)

The orbits of a dynamical system are geometrical objects embedded in the phase space while the functionx(t) = Φ^t(x0) generated by applying the ﬂow on an initial conditionx0is called atrajectory of the dynamical system.

3.2.2 Statistical description

In general, the study of complex dynamical systems on long time scales can be intractable if one observes independently the trajectories generated by the flow, particularly, if these trajectories present a sensitivity to initial conditions. One must then resort to a statistical description and consider an infinite number of copies of the system, with different initial conditions [134]. In this case, phase- space probability distributionρ(x, t) can be naturally defined. Indeed, taking a phase-space cell of volumeV centered onx, the probability distribution is given by the following limit of small volume:

ρ(x, t) = 1 Ntot

Vlim→0

N(t)

V = 1

Ntot

dN

dV (t) (3.9)

where Ntot is the total number of copies in the phase space and N(t) is the number of copies inV at the timet.

Because of the uniqueness of the solutions given by the ﬂow (3.6), the number of phase-space trajectories emanating from a given set of initial conditions is

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3.2 Continuous-time dynamical systems 41

conserved. Therefore, these trajectories behave like the trajectories of particles in a ﬂuid and the distributionρ is similar to a mass density. In particular, ρ satisﬁes the continuity equation:

∂tρ+∇· ρf= 0. (3.10)

One can check that the solution of this equation, also called the Liouville equation, reads [63]:

ρ(x, t) = ˆ

dx0δ x−Φ^t(x0)ρ(x0,0) (3.11) where we integrate over the whole phase space¹ and whereΦ^t is the ﬂow associated with f. The time evolution of the density is thus ruled by an integral operator called thePerron-Frobenius operator [32].

The Liouville equation describes the local² temporal evolution of the distribution ρ. On the other hand, if we want to describe the time evolution of the distribution along a given trajectory, we need to introduce the hydrodynamic derivative [106, 134] ofρ:

dρ

dt =∂tρ+ ˙x·∇ρ . (3.12)

The property∇· ρf=f·∇ρ+ρ∇·f allows us with Eqs. (3.5) and (3.10) to express the hydrodynamic derivative as

dρ

dt =−ρ∇·f (3.13)

which gives the time evolution ofρ. Integrating this equation from 0 tot, one obtains

ρt=ρ0exp ñ

− ˆ t

0

dτ∇·f ô

(3.14) where ρt is the distribution along a path x(t) of the system (3.5). Using Eq.

(3.9), we can rewrite this latter equation as dN

dV (t) = dN

dV(0) exp ñ

− ˆ t

0

dτ∇·f ô

. (3.15)

Now if we consider the evolution of the set of trajectories with initial conditions inside a volume V0 of the phase space, then the volume Vt occupied by these trajectories after a timetis obtained thanks to Eq. (3.15) and to the fact that the number of trajectories is constantN(t) =N(0) as

Vt= ˆ

x∈Vt

dx= ˆ

x0∈V0

dx0 exp ñˆ t

0

dτ∇·f ô

(3.16)

1In the present thesis, when none is specified, the domain of integration is the whole state space.

2i.e. in a fixed infinitesimal cell dV.

Escaperatetheoryfornoisydynamicalsystems UniversitéLibredeBruxelles

Université Libre de Bruxelles

Center for Nonlinear Phenomena and Complex Systems

& Department of Physics

Escape rate theory for noisy dynamical systems

Jonathan Demaeyer

Academic year 2012-2013

A thesis presented in partial fulfilment of the requirements for the degree of Doctor in Science.

Supervisor : prof. Pierre Gaspard

Remerciements

Abstract

Copyright notice

Table of contents

I Introduction, Theory and Physical motivations 17

II Continuous-time systems 87

III Discrete-time systems 177

IV Summary and perspectives 271

Part I

Introduction, Theory and

Physical motivations

Chapter 1

Introduction

1.1 The van’t Hoff-Arrhenius law

1.2 Brownian motion

1.3 The Kramers theory and recent developments

1.4 Purpose of the thesis

1.5 Outline of the thesis

Chapter 2

Physical motivations

2.1 Chemical reactions

2.2 Nucleation

2.3 Electronic circuits

2.4 Josephson tunnel junctions

2.5 Transport in biological systems

2.6 Lasers

Chapter 3

Introduction to the theory of dynamical systems

3.1 Definition

3.2 Continuous-time dynamical systems

3.2.1 Time evolution

3.2.2 Statistical description