Central Limit Theorem for Locally Interacting Fermi Gas

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

doi:10.1007/s00220-008-0610-6

Cited by 24 publications

(32 citation statements)

References 47 publications

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“…The last formula further yields the Onsager reciprocity relation L jk = L kj (see [JOP3,JPP] for details. Similar results hold for the energy currents).…”

Section: Existence Of the Nessmentioning

confidence: 98%

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On the Steady State Correlation Functions of Open Interacting Systems

Cornean

Moldoveanu

Pillet

2014

Commun. Math. Phys.

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Abstract. 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 0 th order term in the interaction strength of the charge current leads to the Landauer-Büttiker formula, while the 1 st order correction contains the mean-field (Hartree-Fock) results.

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“…The last formula further yields the Onsager reciprocity relation L jk = L kj (see [JOP3,JPP] for details. Similar results hold for the energy currents).…”

Section: Existence Of the Nessmentioning

confidence: 98%

“…Remark 5. The linear response theory of the partitioned NESS · β,µ,0 + was established in [JOP3,JPP]. In particular the Green-Kubo formula…”

Section: Existence Of the Nessmentioning

confidence: 99%

On the Steady State Correlation Functions of Open Interacting Systems

Cornean

Moldoveanu

Pillet

2014

Commun. Math. Phys.

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“…Let us take electrons in a one-dimensional lattice [96][97][98][99][100][101]. The Hamiltonian operator ruling this system reads…”

Section: E Quantum Systemsmentioning

confidence: 99%

Random paths and current fluctuations in nonequilibrium statistical mechanics

Gaspard

2014

Journal of Mathematical Physics

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An overview is given of recent advances in nonequilibrium statistical mechanics about the statistics of random paths and current fluctuations. Although statistics is carried out in space for equilibrium statistical mechanics, statistics is considered in time or spacetime for nonequilibrium systems. In this approach, relationships have been established between nonequilibrium properties such as the transport coefficients, the thermodynamic entropy production, or the affinities, and quantities characterizing the microscopic Hamiltonian dynamics and the chaos or fluctuations it may generate. This overview presents results for classical systems in the escape-rate formalism, stochastic processes, and open quantum systems.

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“…Relaxation to a NESS of a locally interacting Fermi gas in the partitioned scenario was first proved by Fröhlich et al [DFG, FMU]. Linear response theory (including a central limit theorem) for such NESS was developed in [JOP1,JOP2,JPP]. Using similar techniques, a mathematical theory of basic thermodynamic processes in ideal and locally interacting Fermi gases has been developed in [FMSU].…”

Section: Introductionmentioning

confidence: 99%

A geometric approach to the Landauer-Büttiker formula

Sâad

Pillet

2014

Journal of Mathematical Physics

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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.An operator on H is a linear map A : D → H , where D is a subspace of H . We say that D is the domain of A which we denote by Dom(A). A is densely defined if its domain is dense in H . The range and the kernel of A are the subspaces RanThe graph of an operator A on H is the the subspaceAn operator A is completely characterized by its graph. Moreover, a subspace G ⊂ H × H is the graph of an operator iff 〈0, v〉 ∈ G implies v = 0. If A and B are two operators such that Gr (A) ⊂ Gr (B ) we say that B is an extension of A and we write A ⊂ B . An operator A is closed if its graph is closed in H ×H , and this is the case iff Dom(A), equipped with the graph norm of A, is a Banach space. If A is both closed and bijective, then Gr (A −1 ) = KGr (A) and thus A −1 is also closed. If the closure Gr (A) cl of the graph of A in H × H is a graph we say that A is closable and we define its closure as the operator A cl such that Gr (A cl ) = Gr (A) cl . It is clear that A cl is the smallest closed extension of A, that is to say that if B is closed and A ⊂ B , then A cl ⊂ B . An operator A is densely defined iff J(Gr (A) ⊥ ) is a graph. In this case, the adjoint of A is the operator A * defined by Gr (A * ) = J(Gr (A) ⊥ ). A * is closed and its domain is given by(A * u, v) = (u, Av) holds for all 〈u, v〉 ∈ Dom(A * ) × Dom(A), in particular Ker (A * ) = Ran (A) ⊥ .A is closable iff A * is densely defined. In this case A cl = A * * and A cl * = A * . An operator A is bounded if there exists a constant C such that Gr (A) ⊂ {〈u, v〉 | v ≤ C u }. One easily verifies that A is continuous as a map from Dom(A) to H iff it is bounded. A bounded operator is obviously closable and its closure coincide with its unique continuous extension to the closure of Dom(A). In particular a bounded densely defined operator A has a unique continuous extension A cl with domain Dom(A cl ) = H . A cl is closed and bounded. The collection of all bounded operators with domain H is denoted by B(H ). It is a Banach algebra (actually a C * -algebra, see Section 2.2) with norm A ≡ sup u∈H , u =1Au .By the closed graph theorem, an operator A with domain H is bounded iff it is closed. If A is bounded and densely defined, then Dom(A * ) = H and A * is bounded. Furthermore, A * = A and A * A = A 2 .

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Central Limit Theorem for Locally Interacting Fermi Gas

Cited by 24 publications

References 47 publications

On the Steady State Correlation Functions of Open Interacting Systems

On the Steady State Correlation Functions of Open Interacting Systems

Random paths and current fluctuations in nonequilibrium statistical mechanics

A geometric approach to the Landauer-Büttiker formula

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