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C. -A. Pillet

Publications and source records attributed to C. -A. Pillet.

14 recordsLinked to original sources
arXiv2025

Entropic Fluctuations in Statistical Mechanics II. Quantum Dynamical Systems

The celebrated Evans-Searles, respectively Gallavotti-Cohen, fluctuation theorem concerns certain universal statistical features of the entropy production rate of a classical system in a transient, respectively steady, state. In this paper, we consider and compare several possible extensions of these fluctuation theorems to quantum systems. In addition to the direct two-time measurement approach whose discussion is based on (LMP 114:32 (2024)), we discuss a variant where measurements are performed indirectly on an auxiliary system called ancilla, and which allows to retrieve non-trivial statistical information using ancilla state tomography. We also show that modular theory provides a way to extend the classical notion of phase space contraction rate to the quantum domain, which leads to a third extension of the fluctuation theorems. We further discuss the quantum version of the principle of regular entropic fluctuations, introduced in the classical context in (Nonlinearity 24, 699 (2011)). Finally, we relate the statistical properties of these various notions of entropy production to spectral resonances of quantum transfer operators. The obtained results shed a new light on the nature of entropic fluctuations in quantum statistical mechanics.

math-ph↗
arXiv2025

What is the absolutely continuous spectrum?

We summarize (and comment on) the research program carried out in (CMP 319, 501 (2013)), (CMP 338, 347 (2015)), (CMP 344, 959 (2016), (LMP 106, 787 (2016)). This program is devoted to the characterization of the absolutely continuous spectrum of a self-adjoint operator H in terms of the transport properties of a suitable class of open quantum systems canonically associated to H.

math-ph↗
arXiv2024

Two-Time Measurement of Entropy Transfer in Markovian Quantum Dynamics

We consider a protocol for the two-time measurement of entropic observables in quantum open systems driven out of thermal equilibrium by coupling to several heat baths. We concentrate on the Markovian approximation of the time-evolution and relate the expected value of the so defined entropy variations with the well-known expression of entropy production due to Lebowitz and Spohn. We do so under the detailed balance condition and, as a byproduct, we show that the probabilities of outcomes of two-time measurements are given by a continuous time Markov process determined by the Lindblad generator of the Markovian quantum dynamics.

math-ph↗
arXiv2019

A Dyson equation for non-equilibrium Green's functions in the partition-free setting

We consider a small interacting sample coupled to several non-interacting leads. Initially, the system is at thermal equilibrium. At some instant $t_0$ the system is set into the so called partition-free transport scenario by turning on a bias on the leads. Using the theory of Volterra operators we rigorously formulate a Dyson equation for the retarded Green's function and we establish a closed formula for the associated proper interaction self-energy.

math-ph↗
arXiv2017

A mathematical account of the NEGF formalism

The main goal of this paper is to put on solid mathematical grounds the so-called Non-Equilibrium Green's Function (NEGF) transport formalism for open systems. In particular, we derive the Jauho-Meir-Wingreen formula for the time-dependent current through an interacting sample coupled to non-interacting leads. Our proof is non-perturbative and uses neither complex-time Keldysh contours, nor Langreth rules of 'analytic continuation'. We also discuss other technical identities (Langreth, Keldysh) involving various many body Green's functions. Finally, we study the Dyson equation for the advanced/retarded interacting Green's function and we rigorously construct its (irreducible) self-energy, using the theory of Volterra operators.

math-ph↗
arXiv2014

A Geometric Approach to the Landauer-Büttiker Formula

We consider an ideal Fermi gas confined to a geometric structure consisting of a central region -- the sample -- connected to several infinitely extended ends -- the reservoirs. Under physically reasonable assumptions on the propagation properties of the one-particle dynamics within these reservoirs, we show that the state of the Fermi gas relaxes to a steady state. We compute the expected value of various current observables in this steady state and express the result in terms of scattering data, thus obtaining a geometric version of the celebrated Landauer-Büttiker formula.

math-ph↗
arXiv2013

Entropic fluctuations in XY chains and reflectionless Jacobi matrices

We study the entropic fluctuations of a general XY spin chain where initially the left(x<0)/right(x>0) part of the chain is in thermal equilibrium at inverse temperature Tl/Tr. The temperature differential results in a non-trivial energy/entropy flux across the chain. The Evans-Searles (ES) entropic functional describes fluctuations of the flux observable with respect to the initial state while the Gallavotti-Cohen (GC) functional describes these fluctuations with respect to the steady state (NESS) the chain reaches in the large time limit. We also consider the full counting statistics (FCS) of the energy/entropy flux associated to a repeated measurement protocol, the variational entropic functional (VAR) that arises as the quantization of the variational characterization of the classical Evans-Searles functional and a natural class of entropic functionals that interpolate between FCS and VAR. We compute these functionals in closed form in terms of the scattering data of the Jacobi matrix h canonically associated to the XY chain. We show that all these functionals are identical if and only if h is reflectionless (we call this phenomenon entropic identity). If h is not reflectionless, then the ES and GC functionals remain equal but differ from the FCS, VAR and interpolating functionals. Furthermore, in the non-reflectionless case, the ES/GC functional does not vanish at 1 (i.e., the Kawasaki identity fails) and does not have the celebrated ES/GC symmetry. The FCS, VAR and interpolating functionals always have this symmetry. In the cases where h is a Schrödinger operator, the entropic identity leads to some unexpected open problems in the spectral theory of one-dimensional discrete Schrödinger operators.

math-ph↗
arXiv2012

Quantum Hypothesis Testing and Non-Equilibrium Statistical Mechanics

We extend the mathematical theory of quantum hypothesis testing to the general $W^*$-algebraic setting and explore its relation with recent developments in non-equilibrium quantum statistical mechanics. In particular, we relate the large deviation principle for the full counting statistics of entropy flow to quantum hypothesis testing of the arrow of time.

math-ph↗
arXiv2012

Temperature and Voltage Probes Far from Equilibrium

We consider an open system of non-interacting electrons consisting of a small sample connected to several reservoirs and temperature or voltage probes. We study the non-linear system of equations that determines the probe parameters. We show that it has a unique solution, which can be computed with a fast converging iterative algorithm. We illustrate our method with two well-known models: the three-terminal system and the open Aharovov-Bohm interferometer.

cond-mat.stat-mech↗
arXiv1996

Zeta functions with Dirichlet and Neumann boundary conditions for exterior domains

We generalize earlier studies on the Laplacian for a bounded open domain $Ω\in \real^2$ with connected complement and piecewise smooth boundary. We compare it with the quantum mechanical scattering operator for the exterior of this same domain. Using single layer and double layer potentials we can prove a number of new relations which hold when one chooses {\em independently} Dirichlet or Neumann boundary conditions for the interior and exterior problem. This relation is provided by a very simple set of $ζ$-functions, which involve the single and double layer potentials. We also provide Krein spectral formulas for all the cases considered and give a numerical algorithm to compute the $ζ$-function.

chao-dyn↗
arXiv1995

Ergodic Properties of the Spin - Boson System

We investigate the dynamics of a 2-level atom (or spin-1/2) coupled to a mass-less bosonic field at positive temperature. We prove that, at small coupling, the combined quantum system approaches thermal equilibrium. Moreover we establish that this approach is exponentially fast in time. We first reduce the question to a spectral problem for the Liouvillean, a self-adjoint operator naturally associated with the system. To compute this operator, we invoke Tomita-Takesaki theory. Once this is done we use complex deformation techniques to study its spectrum. The corresponding zero temperature model is also reviewed and compared.

adap-org↗
arXiv1995

Scattering Phases and Density of States for Exterior Domain

For a bounded open domain $Ω\in \real^2$ with connected complement and piecewise smooth boundary, we consider the Dirichlet Laplacian $-\DO$ on $Ω$ and the S-matrix on the complement $Ω^c$. Using the restriction $A_E$ of $(-Δ-E)^{-1}$ to the boundary of $Ω$, we establish that $A_{E_0}^{-1/2}A_EA_{E_0}^{-1/2}-1$ is trace class when $E_0$ is negative and give bounds on the energy dependence of this difference. This allows for precise bounds on the total scattering phase, the definition of a $ζ$-function, and a Krein spectral formula, which improve similar results found in the literature.

chao-dyn↗
arXiv1994

Spectral Duality for Planar Billiards

For a bounded open domain $Ω$ with connected complement in ${\bf R}^2$ and piecewise smooth boundary, we consider the Dirichlet Laplacian $-Δ_Ω$ on $Ω$ and the S-matrix on the complement $Ω^c$. We show that the on-shell S-matrices ${\bf S}_k$ have eigenvalues converging to 1 as $k\uparrow k_0$ exactly when $-Δ_Ω$ has an eigenvalue at energy $k_0^2$. This includes multiplicities, and proves a weak form of ``transparency'' at $k=k_0$. We also show that stronger forms of transparency, such as ${\bf S}_{k_0}$ having an eigenvalue 1 are not expected to hold in general.

chao-dyn↗