Entropy production in classical and quantum systems

International audienceKoopmanism -- the spectral theory of dynamicalsystems -- reduces the study of dynamical properties of a classical or quantum system S to the spectral analysis of its Liouvillean L_S . By definition, the operator L_S implements the dynamics on a suitable representation of the observable algebra of S. Near thermal equilibrium,this representation can often be constructed explicitely. Recent developments have shown that, in this situation, spectral analysis becomes a powerful tool in the study of thermal relaxation processes. Far from thermal equilibrium, the explicit construction of stationary states and of the corresponding representations is usually not possible. Nevertheless, important physical properties of the system S can be obtained from a fairly simple mathematical analysis. In this work, I investigate entropy production in open systems driven away from equilibrium by thermodynamic forces

Stabilization of needle-crystals in the symmetric model of solidification

International audienceWe present the results of a careful numerical analysis of the stability problem for stationary needle-crystal solutions of the symmetric model of dentritic solidification. The major outcome is that such needle crystals are stable, at least on a time scale relevant to side-branching phenomena. Our study also indicates that the tip of the needle-crystal is very sensitive to external noises, thus supporting the selective amplification mecanism advocated by Langer, Barbieri and Barber on the basis of a WKB analysis

Entropic fluctuations in thermally driven harmonic networks

We consider a general network of harmonic oscillators driven out of thermal equilibrium by coupling to several heat reservoirs at different temperatures. The action of the reservoirs is implemented by Langevin forces. Assuming the existence and uniqueness of the steady state of the resulting process, we construct a canonical entropy production functional which satisfies the Gallavotti--Cohen fluctuation theorem, i.e., a global large deviation principle with a rate function I(s) obeying the Gallavotti--Cohen fluctuation relation I(-s)-I(s)=s for all s. We also consider perturbations of our functional by quadratic boundary terms and prove that they satisfy extended fluctuation relations, i.e., a global large deviation principle with a rate function that typically differs from I(s) outside a finite interval. This applies to various physically relevant functionals and, in particular, to the heat dissipation rate of the network. Our approach relies on the properties of the maximal solution of a one-parameter family of algebraic matrix Riccati equations. It turns out that the limiting cumulant generating functions of our functional and its perturbations can be computed in terms of spectral data of a Hamiltonian matrix depending on the harmonic potential of the network and the parameters of the Langevin reservoirs. This approach is well adapted to both analytical and numerical investigations

A Detailed Fluctuation Theorem for Heat Fluxes in Harmonic Networks out of Thermal Equilibrium

We continue the investigation, started in [J. Stat. Phys. 166, 926-1015 (2017)], of a network of harmonic oscillators driven out of thermal equilibrium by heat reservoirs. We study the statistics of the fluctuations of the heat fluxes flowing between the network and the reservoirs in the nonequilibrium steady state and in the large time limit. We prove a large deviation principle for these fluctuations and derive the fluctuation relation satisfied by the associated rate function

On the steady state correlation functions of open interacting systems

We address the existence of steady state Green-Keldysh correlation functions of interacting fermions in mesoscopic systems for both the partitioning and partition-free scenarios. Under some spectral assumptions on the non-interacting model and for sufficiently small interaction strength, we show that the system evolves to a NESS which does not depend on the profile of the time-dependent coupling strength/bias. For the partitioned setting we also show that the steady state is independent of the initial state of the inner sample. Closed formulae for the NESS two-point correlation functions (Green-Keldysh functions), in the form of a convergent expansion, are derived. In the partitioning approach, we show that the 0th order term in the interaction strength of the charge current leads to the Landauer-Buettiker formula, while the 1st order correction contains the mean-field (Hartree-Fock) results

Non-equilibrium steady-states for interacting open systems: exact results

Under certain conditions we prove the existence of a steady-state transport regime for interacting mesoscopic systems coupled to reservoirs (leads). The partitioning and partition-free scenarios are treated on an equal footing. Our time-dependent scattering approach is {\it exact} and proves, among other things the independence of the steady-state quantities from the initial state of the sample. Closed formulas for the steady-state current amenable for perturbative calculations w.r.t. the interaction strength are also derived. In the partitioning case we calculate the first order correction and recover the mean-field (Hartree-Fock) results.Comment: To appear in Phys. Rev.

A note on the entropy production formula

International audienceWe give an elementary derivation of the entropy production formula of [http://hal.archives-ouvertes.fr/hal-00005457] based on Araki Perturbation Theory of KMS states. Using this derivation we show that the entropy production of any normal, stationary state is zero

Conductance and absolutely continuous spectrum of 1D samples

We characterize the absolutely continuous spectrum of the one-dimensional Schr\"odinger operators $h=-\Delta+v$ acting on $\ell^2(\mathbb{Z}_+)$ in terms of the limiting behavior of the Landauer-B\"uttiker and Thouless conductances of the associated finite samples. The finite sample is defined by restricting $h$ to a finite interval $[1,L]\cap\mathbb{Z}_+$ and the conductance refers to the charge current across the sample in the open quantum system obtained by attaching independent electronic reservoirs to the sample ends. Our main result is that the conductances associated to an energy interval $I$ are non-vanishing in the limit $L\to\infty$ iff ${\rm sp}_{\rm ac}(h)\cap I=\emptyset$. We also discuss the relationship between this result and the Schr\"odinger Conjecture