Heat full statistics: heavy tails and fluctuations control

Transcription

1 Heat full statistics: heavy tails and fluctuations control Annalisa Panati, CPT, Université de Toulon joint work with T.Benoist, R. Raquépas

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3 Full statistics -confined systems Full (counting) Statistics [Lesovik,Levitov 93][Levitov, Lee,Lesovik 96]

4 Full statistics -confined systems Full (counting) Statistics [Lesovik,Levitov 93][Levitov, Lee,Lesovik 96] Confined systems: (H, H, ρ) dimh < Given an observable A: A = j a jp aj where a j σ(a) P ej associated spectral projections At time 0 we measure A with outcome a j with probability tr(ρp aj ). Then the reduced state is 1 tr(ρp aj ) P a j ρp aj. Let evolve for time t, and measure again. The outcome will be a k with probability 1 tr(ρp aj ) tr(e ith P aj ρp aj e ith P ak )

5 Full statistics -confined systems Joint probability of measuring a j at time 0 and a k at time t is: tr(e ith P aj ρp aj e ith P ek ) Full (counting) statistic is the atomic probability measure on R defined by P t (φ) = tr(e ith P aj ρp aj e ith P ak ) a k a j =φ (probability distribution of the change of A measured with the protocol above)

6 Full statistics and fluctuation relations Quantum extention of classical fluctuation relations Classical case:[evans-cohen-morris 93] [Evans-Searls 94] [Gallavotti-Cohen 94]

7 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 = 1l, 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 βh ).). Set P t (φ) := P t ( φ). Then for any φ in R, d P t dp t (φ) = e tφ.

8 1 Given a classical observable C and an intial state ρ, we call C-statistics the probability measure P C such that f (s)dp C (s) = f (C)dρ for all f B(R)

9 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 probability measure P C such that f (s)dp C (s) = f (C)dρ for all f B(R)

10 Classical system (M, H, ρ) H = H 0 + V Full statistics are equivalent to the law P At A t := A t A. 1 associated to 1 Given a classical observable C and an intial state ρ, we call C-statistics the probability measure P C such that f (s)dp C (s) = f (C)dρ for all f B(R)

11 Classical system (M, H, ρ) H = H 0 + V Full statistics are equivalent to the law P At 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 probability measure P C such that f (s)dp C (s) = f (C)dρ for all f B(R)

12 Classical system (M, H, ρ) H = H 0 + V Full statistics are equivalent to the law P At 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 H0,t = P Vt In particular if V is bounded by C: sup t H 0,t < 2C and supp(p H0,t ) bounded 1 Given a classical observable C and an intial state ρ, we call C-statistics the probability measure P C such that f (s)dp C (s) = f (C)dρ for all f B(R)

13 Quantum system (H, H, ρ) H = H 0 + V

14 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 H0,t P V,t

15 Mathematical setting and results: bounded perturbations General defintion (O, τ t, ω) C -dynamical system τ t = τ0 t + i[, V ] and ω is a τ 0 t invariant state π ω : O B(H ω ) a GNS representation ω(a) = (Ω ω, AΩ ω ) Hω Liouvillean: τ t 0 (A) = e itl π ω (A)e itl and LΩ ω = 0 Definition We define the energy full statistics (FS) measure for time t, denoted P t, to be the spectral measure for the operator with respect to the vector Ω ω L + π ω (V ) π ω (τ t (V )),

16 Mathematical setting and results: bounded perturbation Notions of regularity (γa) t τ t 0(V ) admits a bounded analytic extention to the strip {z C : Im z < 1 2 γ}. (nd) t τ t 0(V ) is n times norm-differentiable,

17 Theorem (Benoist, P., Raquépas 2017) Let (O, τ, ω) be a C -dynamical system as above and P t be the energy full statistics measure associated to a self-adjoint perturbation V O. Then (nd) sup E t [ E 2n+2 ] <. t R Corollary (γa) sup E t [e γ E ] 2e 2γv0. t R Under the conditions of the previous theorem, (nd) P t ( 1 t E R) C n(rt) 2n+2 (γa) P t ( 1 t E R) C γe Rt.

18 Regularity condition optimality: Fermi gas with impurity H = Γ a (h) with h = C L 2 (R +, de) H 0 = dγ(h 0 ) with h 0 = ε 0 ê H = dγ(h 0 + ψ 1 ψ f + ψ f ψ 1 ) = dγ(h 0 ) + a (ψ 1 )a(ψ f ) + a (ψ f )a(ψ 1 ) with ψ 1 = 1 0, ψ f = 0 f.

19 Regularity condition optimality: Fermi gas with impurity H = Γ a (h) with h = C L 2 (R +, de) H 0 = dγ(h 0 ) with h 0 = ε 0 ê H = dγ(h 0 + ψ 1 ψ f + ψ f ψ 1 ) = dγ(h 0 ) + a (ψ 1 )a(ψ f ) + a (ψ f )a(ψ 1 ) with ψ 1 = 1 0, ψ f = 0 f. Theorem (Benoist, P., Raquépas 2017) For the above model the following are equivalent: 1 sup t R E t [ E 2n+2 ] < ; 2 for a non-trivial time interval [t 1, t 2 ] t 2 t 1 E t [ E 2n+2 ]dt < ; 3 (nd)

20 Regularity condition optimality: Fermi gas with impurity H = Γ a (h) with h = C L 2 (R +, de) H 0 = dγ(h 0 ) with h 0 = ε 0 ê H = dγ(h 0 + ψ 1 ψ f + ψ f ψ 1 ) = dγ(h 0 ) + a (ψ 1 )a(ψ f ) + a (ψ f )a(ψ 1 ) with ψ 1 = 1 0, ψ f = 0 f. Theorem (Benoist, P., Raquépas 2017) For the above model the following are equivalent: 1 sup t R E t [ E 2n+2 ] < ; 2 for a non-trivial time interval [t 1, t 2 ] t 2 t 1 E t [ E 2n+2 ]dt < ; 3 (nd) For this model (nd) is equivalent to R s e isê f L 2 (R +, de) is n times norm- differentiable i.e f Dom(ê n )

21 Regularity condition optimality: Fermi gas with impurity H = Γ a (h) with h = C L 2 (R +, de) H 0 = dγ(h 0 ) with h 0 = ε 0 ê H = dγ(h 0 + ψ 1 ψ f + ψ f ψ 1 ) = dγ(h 0 ) + a (ψ 1 )a(ψ f ) + a (ψ f )a(ψ 1 ) with ψ 1 = 1 0, ψ f = 0 f. Theorem (Benoist, P., Raquépas 2017) For the above model the following are equivalent: 1 sup t R E t [ E 2n+2 ] < ; 2 for a non-trivial time interval [t 1, t 2 ] t 2 t 1 E t [ E 2n+2 ]dt < ; 3 (nd) For this model (nd) is equivalent to R s e isê f L 2 (R +, de) is n times norm- differentiable i.e f Dom(ê n ) Remark: decay of f controls how high energy frequencies contribute to the interaction (ultraviolet regularization)

22 Regularity condition optimality: bosonic systems Bose gas with impurity H = Γ s (h) with h = C L 2 (R +, de) H 0 = dγ(h 0 ) with h 0 = ε 0 ê H = dγ(h 0 + ψ 1 ψ f + ψ f ψ 1 ) = dγ(h 0 ) + a (ψ 1 )a(ψ f ) + a (ψ f )a(ψ 1 ) with ψ 1 = 1 0, ψ f = 0 f. Conditions: f Dom(ê) Dom(ê 1/2 ) and h 1/2 0 ψ f ε 1/2 0

23 Regularity condition optimality: bosonic systems Bose gas with impurity H = Γ s (h) with h = C L 2 (R +, de) H 0 = dγ(h 0 ) with h 0 = ε 0 ê H = dγ(h 0 + ψ 1 ψ f + ψ f ψ 1 ) = dγ(h 0 ) + a (ψ 1 )a(ψ f ) + a (ψ f )a(ψ 1 ) with ψ 1 = 1 0, ψ f = 0 f. Conditions: f Dom(ê) Dom(ê 1/2 ) and h 1/2 0 ψ f ε 1/2 0 Theorem (Benoist, P., Raquépas 2017) For the Bose gas with impurity model, assume (γa ) : t τ0 t (V )Ω admits a bounded analytic extention to the strip {z C : Im z < 1 2 γ}. Then 1 sup t R E t [e γ E ] < ; For this model (γa ) is equivalent to R s e isê f L 2 (R +, de) extends to an analitic function on the strip i.e f Dom(e 1 2 γê )

24 Theorem (Benoist, P., Raquépas 2017) For the Bose gas with impurity model the following are equivalent: 1 sup t R E t [ E 2n+2 ] < ; 2 for a non-trivial time interval [t 1, t 2 ] t 2 t 1 E t [ E 2n+2 ]dt < ; 3 (nd ) t τ0 t (V )Ω is n-times norm differentable (here Ω is the vacuum) For this model, (nd ) is equivalent to R s e isê f L 2 (R +, de) is n times differentiable in the norm sense i.e f Dom( eˆ n )

25 van Hove Hamiltonians H = Γ s (h) with h = L 2 (R +, de) H 0 = dγ(e), H = dγ(e) + a (f ) + a(f ) Conditions: f Dom(ê) Dom(ê 1/2 ). Theorem (Benoist, P., Raquépas 2017) For the van Hove bosonic models the following are equivalent: 1 sup t R E t [ E 2n+2 ] < ; 2 for a non-trivial time interval [t 1, t 2 ] t 2 t 1 E t [ E 2n+2 ]dt < ; 3 (nd ) s τ0 t (V )Ω is n-times norm differentable (here Ω is the vacuum) For this models, (nd ) is equivalent to R s e isê f L 2 (R +, de) is n times differentiable in the norm sense i.e f Dom( eˆ n )

26 Theorem (Benoist, P., Raquépas 2017) For the van Hove model the following are equivalent: 1 sup t R E t [e γ E ] < ; 2 for a non-trivial time interval [t 1 (γ), t 2 (γ)] t 2(γ) t 1(γ) E t[e γ E ]dt < ; 3 (γa ) For this model (γa ) is equivalent to R s e isê f L 2 (R +, de) extends to an analitic function on the strip i.e f Dom(e 1 2 γê )

27 Summary Classical statistical mechanics models: heat fluctuations are controlled by the interaction intensity. Particularly, no fluctuations exist when the interaction is bounded. Quantum statistical mechanics models: in the two time measurement picture, heat fluctuations are controlled by a notion of regularity. Particularly, large fluctuations may exists even if the interaction is bounded. In concrete models, regularity notion translate in contribtuon of high energy frequencies to the interaction (UV regularization).

28 Thank you for your attention!