Entropic fluctuations in quantum statistical mechanics—an introduction

Jakšić, Vojkan; Ogata, Yoshiko; Pautrat, Yan; Pillet, Claude-Alain

doi:10.1093/acprof:oso/9780199652495.003.0004

Cited by 59 publications

(66 citation statements)

References 62 publications

(22 reference statements)

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“…It holds under an assumption of timereversal invariance. This symmetry and its connection with refinements of the second law of thermodynamics are well established ( [ES94], see also [RM07], [JOPP12] and references therein). This paper focuses on the other symmetry, which is a translation invariance.…”

Section: Introductionmentioning

confidence: 95%

“…surely, in quantum systems its level of validity depends on the definition of thermodynamic quantities. In the present article we consider the Two-Time Measurement (TTM) definition of thermodynamic quantities (see [JOPP12,EHM09,CHT11] for reviews and [ Kur00,Tas00] for the initial introduction of this definition). In this framework, while the derivation of the fluctuation relation becomes trivial, the equality of heat variation and work holds only for their first two moments (see Section 3.5 of [JOPP12]).…”

Section: Introductionmentioning

confidence: 99%

“…In the present article we consider the Two-Time Measurement (TTM) definition of thermodynamic quantities (see [JOPP12,EHM09,CHT11] for reviews and [ Kur00,Tas00] for the initial introduction of this definition). In this framework, while the derivation of the fluctuation relation becomes trivial, the equality of heat variation and work holds only for their first two moments (see Section 3.5 of [JOPP12]). In a related paper [BPR18], examples are provided where total heat variation distribution has infinite higher moments even if the interaction is bounded and its norm arbitrary small.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, in the TTM picture, thermodynamic quantities have a clear physical interpretation. In addition it is not difficult to show that with the appropriate choice of X, fluctuation relations hold (see [Kur00,Tas00], and also [EHM09,CHT11,JOPP12]).…”

Section: Introductionmentioning

confidence: 99%

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Heat Conservation and Fluctuations Between Quantum Reservoirs in the Two-Time Measurement Picture

2020

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This work concerns the statistics of the Two-Time Measurement definition of heat variation in each thermal bath of a thermodynamic quantum system. We study the cumulant generating function of the heat flows in the thermodynamic and large-time limits. It is well-known that, if the system is time-reversal invariant, this cumulant generating function satisfies the celebrated Evans-Searles symmetry. We show in addition that, under appropriate ultraviolet regularity assumptions on the local interaction between the baths, it satisfies a translationinvariance property, as proposed in [Andrieux et al. New J. Phys. 2009]. We particularly fix some proofs of the latter article where the ultraviolet condition was not mentioned. We detail how these two symmetries lead respectively to fluctuation relations and a statistical refinement of heat conservation for isolated thermodynamic quantum systems. As in [Andrieux et al. New J. Phys. 2009], we recover the Fluctuation-Dissipation Theorem in the linear response theory, short of Green-Kubo relations. We illustrate the general theory on a number of canonical models.

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Heat Conservation and Fluctuations Between Quantum Reservoirs in the Two-Time Measurement Picture

2020

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“…(2) According to Bryc's lemma (see [Br] or [JOPP,Section 4.8.4]) the Central Limit Theorem for the family (η t ) t>0 holds, provided that the generating function g t has an analytic continuation to the disc D = {α ∈ C | |α| < } for some > 0 and satisfies the estimate…”

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confidence: 99%

Entropic Fluctuations in Thermally Driven Harmonic Networks

2016

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Abstract. 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 S t which satisfies the Gallavotti-Cohen fluctuation theorem. More precisely, we prove that there exists κ c > 1 2 such that the cumulant generating function of S t has a large-time limit e(α) which is finite on a closed interval [ 1 2 − κ c , 1 2 + κ c ], infinite on its complement and satisfies the Gallavotti-Cohen symmetry e(1 − α) = e(α) for all α ∈ R. Moreover, we show that e(α) is essentially smooth, i.e., that e (α) → ∓∞ as α → 1 2 ∓ κ c . It follows from the Gärtner-Ellis theorem that S t satisfies 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 ∈ R. We also consider perturbations of S t 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 S t 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.

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Entropic Fluctuations in XY Chains and Reflectionless Jacobi Matrices

Jakšić

Landon

Pillet

2013

Ann. Henri Poincaré

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We study entropic functionals/fluctuations of the XY chain with Hamiltonianwhere initially the left (x ≤ 0)/right (x > 0) part of the chain is in thermal equilibrium at inverse temperature β l /βr. 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 hux = Jxux+1 + λxux + Jx−1ux−1 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 α = 1 (i.e., the Kawasaki identity fails) and does not have the celebrated α ↔ 1 − α symmetry. The FCS, VAR and interpolating functionals always have this symmetry.In the Schrödinger case, where Jx = J for all x, the entropic identity leads to some unexpected open problems in the spectral theory of one-dimensional discrete Schrödinger operators.,

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Entropic fluctuations in quantum statistical mechanics—an introduction

Cited by 59 publications

References 62 publications

Heat Conservation and Fluctuations Between Quantum Reservoirs in the Two-Time Measurement Picture

Heat Conservation and Fluctuations Between Quantum Reservoirs in the Two-Time Measurement Picture

Entropic Fluctuations in Thermally Driven Harmonic Networks

Entropic Fluctuations in XY Chains and Reflectionless Jacobi Matrices

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