Plan Full statistics and quantum fluctuation relations Heat fluctuations: classical VS quantum Mathematical settings and results Heat full statistics: heavy tails and fluctuations control Annalisa Panati, CPT, Université de Toulon joint work with T.Benoist, R. Raquépas Annalisa Panati, CPT, Université de Toulon [3mm] joint work with T.Benoist, R. Raquépas Heat full statistics: heavy tails and fluctuations control
Plan Full statistics and quantum fluctuation relations Heat fluctuations: classical VS quantum Mathematical settings and results Full statistics and quantum fluctuation relations 1 Heat fluctuations: classical VS quantum 2 Mathematical settings and results 3 Bounded perturbations Unbounded perturbations Annalisa Panati, CPT, Université de Toulon [3mm] joint work with T.Benoist, R. Raquépas Heat full statistics: heavy tails and fluctuations control
Plan Full statistics and quantum fluctuation relations Heat fluctuations: classical VS quantum Mathematical settings and results Full statistics -confined systems Full (counting) Statistics [Lesovik,Levitov ’93][Levitov, Lee,Lesovik ’96] Annalisa Panati, CPT, Université de Toulon [3mm] joint work with T.Benoist, R. Raquépas Heat full statistics: heavy tails and fluctuations control
Plan Full statistics and quantum fluctuation relations Heat fluctuations: classical VS quantum Mathematical settings and results Full statistics -confined systems Full (counting) Statistics [Lesovik,Levitov ’93][Levitov, Lee,Lesovik ’96] Confined systems: ( H , H , ρ ) dim H < ∞ Given an observable A : A = � j a j P a j where a j ∈ σ ( A ) P e j associated spectral projections At time 0 we measure A with outcome a j with probability tr ( ρ P a j ) . Then the reduced state is 1 tr ( ρ P a j ) P a j ρ P a j . Let evolve for time t , and measure again. The outcome will be a k with probability 1 tr ( ρ P a j ) tr ( e − i tH P a j ρ P a j e i tH P a k ) Annalisa Panati, CPT, Université de Toulon [3mm] joint work with T.Benoist, R. Raquépas Heat full statistics: heavy tails and fluctuations control
Plan Full statistics and quantum fluctuation relations Heat fluctuations: classical VS quantum Mathematical settings and results Full statistics -confined systems Joint probability of measuring a j at time 0 and a k at time t is: tr ( e − i tH P a j ρ P a j e i tH P e k ) Full (counting) statistic is the atomic probability measure on R defined by � tr ( e − i tH P a j ρ P a j e i tH P a k ) P t ( φ ) = a k − a j = φ (probability distribution of the change of A measured with the protocol above) Annalisa Panati, CPT, Université de Toulon [3mm] joint work with T.Benoist, R. Raquépas Heat full statistics: heavy tails and fluctuations control
Plan Full statistics and quantum fluctuation relations Heat fluctuations: classical VS quantum Mathematical settings and results Full statistics and fluctuation relations Quantum extention of classical fluctuation relations Classical case:[Evans-Cohen-Morris’93] [Evans-Searls ’94] [Gallavotti-Cohen’94] Annalisa Panati, CPT, Université de Toulon [3mm] joint work with T.Benoist, R. Raquépas Heat full statistics: heavy tails and fluctuations control
Plan Full statistics and quantum fluctuation relations Heat fluctuations: classical VS quantum Mathematical settings and results Full statistics and fluctuation relations Quantum extention of classical fluctuation relations Classical case:[Evans-Cohen-Morris’93] [Evans-Searls ’94] [Gallavotti-Cohen’94] Quantum case: Definition: ( H , H , ω ) is TRI iff there exists an anti-linear ∗ -automorphism, Θ 2 = 1 l , τ t ◦ Θ = Θ ◦ τ − t and ω (Θ( A )) = ω ( A ∗ ) . A = S entropy Proposition (Kurchan ’00, Tasaki-Matsui ’03) Assume ( H , H , ρ ) is TRI, (and ρ = e − β · H / tr ( e − β H ) . ) . Set ¯ P t ( φ ) := P t ( − φ ) . Then for any φ in R , d ¯ P t ( φ ) = e − t φ . d P t Annalisa Panati, CPT, Université de Toulon [3mm] joint work with T.Benoist, R. Raquépas Heat full statistics: heavy tails and fluctuations control
Plan Full statistics and quantum fluctuation relations Heat fluctuations: classical VS quantum Mathematical settings and results Heat fluctuations: classical VS quantum 1 Given a classical observable C and an intial state ρ , we call C -statistics the � f ( s ) d P C ( s ) = � f ( C ) d ρ for all f ∈ B ( R ) probability measure P C such that Annalisa Panati, CPT, Université de Toulon [3mm] joint work with T.Benoist, R. Raquépas Heat full statistics: heavy tails and fluctuations control
Plan Full statistics and quantum fluctuation relations Heat fluctuations: classical VS quantum Mathematical settings and results Heat fluctuations: classical VS quantum Classical system ( M , H , ρ ) H = H 0 + V 1 Given a classical observable C and an intial state ρ , we call C -statistics the � f ( s ) d P C ( s ) = � f ( C ) d ρ for all f ∈ B ( R ) probability measure P C such that Annalisa Panati, CPT, Université de Toulon [3mm] joint work with T.Benoist, R. Raquépas Heat full statistics: heavy tails and fluctuations control
Plan Full statistics and quantum fluctuation relations Heat fluctuations: classical VS quantum Mathematical settings and results Heat fluctuations: classical VS quantum Classical system ( M , H , ρ ) H = H 0 + V Full statistics are equivalent to the law P △ A t associated to △ A t := A t − A . 1 1 Given a classical observable C and an intial state ρ , we call C -statistics the � f ( s ) d P C ( s ) = � f ( C ) d ρ for all f ∈ B ( R ) probability measure P C such that Annalisa Panati, CPT, Université de Toulon [3mm] joint work with T.Benoist, R. Raquépas Heat full statistics: heavy tails and fluctuations control
Plan Full statistics and quantum fluctuation relations Heat fluctuations: classical VS quantum Mathematical settings and results Heat fluctuations: classical VS quantum Classical system ( M , H , ρ ) H = H 0 + V Full statistics are equivalent to the law P △ A t associated to △ A t := A t − A . 1 Energy conservation: △ H 0 , t = H 0 , t − H 0 = V t − V as function on M 1 Given a classical observable C and an intial state ρ , we call C -statistics the � f ( s ) d P C ( s ) = � f ( C ) d ρ for all f ∈ B ( R ) probability measure P C such that Annalisa Panati, CPT, Université de Toulon [3mm] joint work with T.Benoist, R. Raquépas Heat full statistics: heavy tails and fluctuations control
Plan Full statistics and quantum fluctuation relations Heat fluctuations: classical VS quantum Mathematical settings and results Heat fluctuations: classical VS quantum Classical system ( M , H , ρ ) H = H 0 + V Full statistics are equivalent to the law P △ A t associated to △ A t := A t − A . 1 Energy conservation: △ H 0 , t = H 0 , t − H 0 = V t − V as function on M which yields P △ H 0 , t = P △ V t In particular if V is bounded by C : sup t |△ H 0 , t | < 2 C and supp ( P △ H 0 , t ) bounded 1 Given a classical observable C and an intial state ρ , we call C -statistics the � f ( s ) d P C ( s ) = � f ( C ) d ρ for all f ∈ B ( R ) probability measure P C such that Annalisa Panati, CPT, Université de Toulon [3mm] joint work with T.Benoist, R. Raquépas Heat full statistics: heavy tails and fluctuations control
Plan Full statistics and quantum fluctuation relations Heat fluctuations: classical VS quantum Mathematical settings and results Heat fluctuations: classical VS quantum Quantum system ( H , H , ρ ) H = H 0 + V Annalisa Panati, CPT, Université de Toulon [3mm] joint work with T.Benoist, R. Raquépas Heat full statistics: heavy tails and fluctuations control
Plan Full statistics and quantum fluctuation relations Heat fluctuations: classical VS quantum Mathematical settings and results Heat fluctuations: classical VS quantum Quantum system ( H , H , ρ ) H = H 0 + V Energy conservation: H 0 , t − H 0 = V t − V as operators on H implies equality of spectral measures but in general P H 0 , t � = P V , t Annalisa Panati, CPT, Université de Toulon [3mm] joint work with T.Benoist, R. Raquépas Heat full statistics: heavy tails and fluctuations control
Plan Full statistics and quantum fluctuation relations Bounded perturbations Heat fluctuations: classical VS quantum Unbounded perturbations Mathematical settings and results Mathematical setting and results: bounded perturbations General defintion ( O , τ t , ω ) C ∗ -dynamical system τ t = τ t 0 + i [ − , V ] and ω is a τ t 0 invariant state π ω : O → B ( H ω ) a GNS representation ω ( A ) = (Ω ω , A Ω ω ) H ω 0 ( A ) = e − i tL π ω ( A ) e − i tL and L Ω ω = 0 Liouvillean: τ t Definition We define the energy full statistics (FS) measure for time t , denoted P t , to be the spectral measure for the operator L + π ω ( V ) − π ω ( τ t ( V )) , with respect to the vector Ω ω Annalisa Panati, CPT, Université de Toulon [3mm] joint work with T.Benoist, R. Raquépas Heat full statistics: heavy tails and fluctuations control
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