**Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?**

Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur GraphSearch.

Concept# Théorème de fluctuation-dissipation

Résumé

En théorie de la réponse linéaire, il existe une relation entre la fonction de réponse \chi(\omega)
et la fonction de corrélation S(\omega). Celle-ci a été établie par Herbert Callen et Theodore Welton en 1951, et pour cette raison le théorème de fluctuation-dissipation est aussi appelé théorème de Callen-Welton. Selon ce théorème,
S(\omega)=\hbar \coth\left(\frac{\hbar\omega} {2 k_B T}\right) \mathrm{Im}\chi(\omega) .
Le nom de théorème de fluctuation-dissipation vient de ce que la partie imaginaire de la fonction de réponse mesure la dissipation, alors que la fonction de corrélation S(\omega) mesure l'intensité des fluctuations. On peut reformuler ce théorème en introduisant une force fluctuante f(\omega) par x(\omega)=\chi(\omega) f(\omega) où x(\omega) est la grandeur fluctuante. En introduisant cette expression dans la définition de S(\omega)=\langle x(\omega) x(-\omega)\rangle

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Publications associées

Chargement

Personnes associées

Chargement

Unités associées

Chargement

Concepts associés

Chargement

Cours associés

Chargement

Séances de cours associées

Chargement

Unités associées

Aucun résultat

Publications associées (16)

Chargement

Chargement

Chargement

Personnes associées (1)

Concepts associés (12)

Mouvement brownien

vignette|Simulation de mouvement brownien pour cinq particules (jaunes) qui entrent en collision avec un lot de 800 particules. Les cinq chemins bleus représentent leur trajet aléatoire dans le fluid

Bruit thermique

Le bruit thermique, également nommé bruit de résistance, bruit Johnson ou bruit de Johnson-Nyquist, est le bruit généré par l'agitation thermique des porteurs de charges, c'est-à-dire des électrons da

Théorie cinétique des gaz

La théorie cinétique des gaz a pour objet d'expliquer le comportement macroscopique d'un gaz à partir des caractéristiques des mouvements des particules qui le composent. Elle permet notamment de donn

Cours associés (13)

ME-426: Micro/Nanomechanical devices

In this course we will see an overview of the
exciting field of Micro and Nanomechanical systems. We will go over the dfferent scaling laws that dominate the critical parameters, how size affects material properties, how these devices are manufactured, designed and later used.

PHYS-405: Experimental methods in physics

The course's objectivs are: Learning several advenced methods in experimental physics, and critical reading of experimental papers.

PHYS-436: Statistical physics IV

This first part of the course covers non-equilibrium statistical processes and the treatment of fluctuation dissipation relations by Einstein, Boltzmann and Kubo. Moreover, the fundamentals of Markov processes, stochastic differential and Fokker Planck equations, mesoscopic master equation, noise s

Granular matter submitted to external perturbations exhibits various behaviors depending on the vibration intensity: when strongly vibrated, the granular system has a fluid aspect, whereas under low intensity perturbations, it is in a quasi-solid phase. In this work, we clarify these analogies: we discuss to what extent a "granular fluid" is close to a standard liquid and, on the other hand, investigate the similarities between weakly perturbed granular materials and supercooled liquids undergoing a glass transition. We first consider the case where a granular medium undergoes vibrations of high intensity (with accelerations of about 1 to 10 times that of gravity). This vibrated system is investigated using an immersed torsion oscillator which is sensitive to the granular agitation and which, as a result, exhibits irregular angular deflections that can be analyzed to give information on the grains' motion. This oscillator can also be used in forced mode: applying a torque allows measures of the mechanical susceptibility, which displays a resonance peak similar to that of a damped oscillator. It is thus possible to introduce a viscosity parameter for the system, which is found to be inversely proportional to the vibration acceleration imposed to the container. Moreover, by considering both the fluctuations (given by the diffusive noise, with the oscillator in free mode) and the susceptibility (forced mode), the validity of the fluctuation-dissipation theorem can be tested. Surprisingly, it turns out that very complicated system, even though far from equilibrium, satisfies this basic law of equilibrium statistical mechanics in first approximation, and therefore behaves very much like a plain liquid. The immersed oscillator can thus be compared to a pollen particle exhibiting Brownian motion due to the continuous molecular – here, the granular – agitation. We can thus also introduce an "effective temperature" parameter for vibration-fluidized granular matter and discuss its properties. In particular, we observe that the temperature defined is inhomogeneous and anisotropic, contrary to usual liquids. An interesting issue is also studied carefully: when the damping becomes large (for when the imposed vibrations are lower, or when the oscillator is deeply immersed), a stiffening phenomenon is observed, in which the apparent elastic constant increases linearly with the friction. An elementary rheological model suggests that this may be caused by the appearance of a force chains fretwork resisting the probe rotation. Various granular materials are analyzed with this met hod. The grain mass seems to be an important parameter, as well as the grain surface state. Experiments in which the grains are etched with acid in order to modify the surface roughness are also discussed. While we know what happens to a liquid when its temperature is decreased one can wonder what happens to a granular system when the perturbations become critically low. By using vibrations of weak intensity (with accelerations below that of gravity), we study how the system reaches a "frozen" static configuration when the perturbation intensity is decreased. The observed diffusive noise appears to approach the final jammed state according to the Vogel-Fulcher-Tammann law, that also describes the temperature dependence of viscosity or diffusivity in supercooled liquids, thus showing strong analogies with the vitrification process. Here, the parameter playing the role of temperature in the equations is found to depend only on the vibration amplitude. Finally, we briefly discuss how a small modification of the experimental setup may allow to create a "Brownian motor" generating useful work out of the random agitation of the grains.

We study the large deviations of the power injected by the active force for an active Ornstein-Uhlenbeck particle (AOUP), free or in a confining potential. For the free-particle case, we compute the rate function analytically in d-dimensions from a saddle-point expansion, and numerically in two dimensions by (a) direct sampling of the active work in numerical solutions of the AOUP equations and (b) Legendre-Fenchel transform of the scaled cumulant generating function obtained via a cloning algorithm. The rate function presents asymptotically linear branches on both sides and it is independent of the system's dimensionality, apart from a multiplicative factor. For the confining potential case, we focus on two-dimensional systems and obtain the rate function numerically using both methods (a) and (b). We find a different scenario for harmonic and anharmonic potentials: in the former case, the phenomenology of fluctuations is analogous to that of a free particle, but the rate function might be non-analytic; in the latter case the rate functions are analytic, but fluctuations are realised by entirely different means, which rely strongly on the particle-potential interaction. Finally, we check the validity of a fluctuation relation for the active work distribution. In the free-particle case, the relation is satisfied with a slope proportional to the bath temperature. The same slope is found for the harmonic potential, regardless of activity, and for an anharmonic potential with low activity. In the anharmonic case with high activity, instead, we find a different slope which is equal to an effective temperature obtained from the fluctuation-dissipation theorem.

Large-eddy simulation (LES) is a very promising technique for the numerical computation of unsteady turbulent flows, and seems to be the perfect tool to simulate the compressible air flow around a high-speed train in a tunnel, providing unsteady results for aerodynamic and aeroacoustic analysis. To look into this possible future application of LES, two major lines of investigation are pursued in this thesis: first, the study of the effective ability of shock-capturing schemes to predict fundamental turbulent phenomena; second, the analysis of the aerodynamic phenomena induced by a high-speed train in a tunnel. The numerical simulation of compressible flows requires the use of shock-capturing schemes. These schemes can be relatively dissipative and mask the subgrid-scale contribution introduced in a large-eddy simulation to account for the unresolved turbulence scales. To estimate their effective dissipation and their ability to resolve turbulence phenomena, shock-capturing schemes widely used for aeronautical applications, from second- to fifth-order space accuracy, are employed here for simulating well-known fundamental flows in subsonic and supersonic regimes. Direct and large-eddy numerical simulations are undertaken for the inviscid and viscous Taylor-Green vortex decay problem, the freely decaying homogeneous and isotropic turbulence, and the fully developed channel flow. For all the turbulent flows investigated, several turbulence statistics are computed and the numerical dissipation of the schemes tested is interpreted in terms of subgrid-scale dissipation in a LES context, yielding an equivalent Smagorinsky or dynamic coefficient. This coefficient is for all schemes of the same order of magnitude as the commonly accepted values in LES for the subgrid-scale term. On the grounds of this analysis and of the comparisons of the turbulence statistics with accurate data obtained in the literature, the addition of explicit subgrid-scale models to the shock-capturing schemes tested is not recommended. It is thus concluded that the use of the LES technique for compressible turbulent flows is not yet suitable for industrial applications. The aerodynamic phenomena generated by a high-speed train travelling in a tunnel are also discussed, their importance on the design of high-speed lies is pointed out, and the analysis tools commonly employed for their study are reviewed. To study numerically the three-dimensional, compressible and turbulent air flow around a high-speed train accelerating in a tunnel, by accounting for the unsteady effects at inlet and outlet boundaries due to the propagation of pressure waves generated at the train departure, new coupling conditions between one-dimensional and three-dimensional domains are developed. These conditions are applied successfully to the numerical simulation of the unsteady wake developing behind two- and three-dimensional vehicles, where the averaged Navier-Stokes equations are solved with the turbulence modelling approach. The influence on the wake of the length of the vehicle tail is also discussed and results of multi-dimensional simulations are compared with one-dimensional data.

Séances de cours associées (16)