Heat fluctuations in the two-time CPT, Universit de Toulon - - PowerPoint PPT Presentation

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Heat fluctuations in the two-time measurement framework and ultraviolet regularity

joint work with T.Benoist, (Y. Pautrat) and R. Raquépas Annalisa Panati, CPT, Université de Toulon

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Context: quantum statistical mechanics Fluctuation relations Two-time measurment statistics Conservation laws Heat fluctuations: theorems and results Bounded perturbations Conclusions

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Context: quantum statistical mechanics

Fluctuation relations

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Context: quantum statistical mechanics

Fluctuation relations

Classical case: [Evans-Cohen-Morris ’93] numerical experiences [Evans-Searls ’94] [Gallavotti Cohen ’94] theoretical explanation

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Context: quantum statistical mechanics

Fluctuation relations

Classical case: [Evans-Cohen-Morris ’93] numerical experiences [Evans-Searls ’94] [Gallavotti Cohen ’94] theoretical explanation Statistical refinement of thermodynamics second law

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Context: quantum statistical mechanics

Fluctuation relations

Classical case: [Evans-Cohen-Morris ’93] numerical experiences [Evans-Searls ’94] [Gallavotti Cohen ’94] theoretical explanation Statistical refinement of thermodynamics second law work in driven system [Bochkov-Kuzovlev ’70s] [Jaryzinski ’97] [Crooks ’99] etc

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Context: quantum statistical mechanics

Fluctuation relations

Classical case: [Evans-Cohen-Morris ’93] numerical experiences [Evans-Searls ’94] [Gallavotti Cohen ’94] theoretical explanation Statistical refinement of thermodynamics second law work in driven system [Bochkov-Kuzovlev ’70s] [Jaryzinski ’97] [Crooks ’99] etc Quantum case: ??

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Quantization of fluctuation relations

Quantum case: Attempt 1: "Naive quantization" Underlying idea : define an observable Σt = 1

t (St − S) on H and

consider the spectral measure µΣt

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Quantization of fluctuation relations

Quantum case: Attempt 1: "Naive quantization" Underlying idea : define an observable Σt = 1

t (St − S) on H and

consider the spectral measure µΣt —attempted in work related litterature [Bochkov-Kuzovlev ’70s-’80s]) —attempted in the ’90, called "naive quantization"

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Quantization of fluctuation relations

Quantum case: Attempt 1: "Naive quantization" Underlying idea : define an observable Σt = 1

t (St − S) on H and

consider the spectral measure µΣt —attempted in work related litterature [Bochkov-Kuzovlev ’70s-’80s]) —attempted in the ’90, called "naive quantization" leads to NO-fluctuation relations!!!!

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Quantization of fluctuation relations

Quantum case: Attempt 1: "Naive quantization" Underlying idea : define an observable Σt = 1

t (St − S) on H and

consider the spectral measure µΣt —attempted in work related litterature [Bochkov-Kuzovlev ’70s-’80s]) —attempted in the ’90, called "naive quantization" leads to NO-fluctuation relations!!!! Attempt 2: Measurement has ben neglected. Associate to S the two-time measurment statistics PS

t defined as difference between

two measurement

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Quantization of fluctuation relations

Quantum case: Attempt 1: "Naive quantization" Underlying idea : define an observable Σt = 1

t (St − S) on H and

consider the spectral measure µΣt —attempted in work related litterature [Bochkov-Kuzovlev ’70s-’80s]) —attempted in the ’90, called "naive quantization" leads to NO-fluctuation relations!!!! Attempt 2: Measurement has ben neglected. Associate to S the two-time measurment statistics PS

t defined as difference between

two measurement

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Quantization of fluctuation relations

Quantum case: Attempt 1: "Naive quantization" Underlying idea : define an observable Σt = 1

t (St − S) on H and

consider the spectral measure µΣt —attempted in work related litterature [Bochkov-Kuzovlev ’70s-’80s]) —attempted in the ’90, called "naive quantization" leads to NO-fluctuation relations!!!! Attempt 2: Measurement has ben neglected. Associate to S the two-time measurment statistics PS

t defined as difference between

two measurement

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Quantization of fluctuation relations

Quantum case: Attempt 1: "Naive quantization" Underlying idea : define an observable Σt = 1

t (St − S) on H and

consider the spectral measure µΣt —attempted in work related litterature [Bochkov-Kuzovlev ’70s-’80s]) —attempted in the ’90, called "naive quantization" leads to NO-fluctuation relations!!!! Attempt 2: Measurement has ben neglected. Associate to S the two-time measurment statistics PS

t defined as difference between

two measurement —Key result by [Kurchan’00] leads to fluctuation relations

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Quantization of fluctuation relations

Quantum case: Attempt 1: "Naive quantization" Underlying idea : define an observable Σt = 1

t (St − S) on H and

consider the spectral measure µΣt —attempted in work related litterature [Bochkov-Kuzovlev ’70s-’80s]) —attempted in the ’90, called "naive quantization" leads to NO-fluctuation relations!!!! Attempt 2: Measurement has ben neglected. Associate to S the two-time measurment statistics PS

t defined as difference between

two measurement —Key result by [Kurchan’00] leads to fluctuation relations At the level of averages and variances, there is no difference!

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Two-time measurment statistics

Full (Counting) Statistics [Lesovik, Levitov ’93][Levitov, Lee,Lesovik ’96]

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Two-time measurment statistics

Full (Counting) Statistics [Lesovik, Levitov ’93][Levitov, Lee,Lesovik ’96] Confined systems: described by (H, H, ρ) dimH < ∞ Given an observable A: A =

j ajPaj where aj ∈ σ(A) Paj

associated spectral projections

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Two-time measurment statistics

Full (Counting) Statistics [Lesovik, Levitov ’93][Levitov, Lee,Lesovik ’96] Confined systems: described by (H, H, ρ) dimH < ∞ Given an observable A: A =

j ajPaj where aj ∈ σ(A) Paj

associated spectral projections Procedure:

• t = 0, we measure A (outcome aj)
• evolve for time t
• measure again at time t (outcome ak)

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Two-time measurment statistics

Full (Counting) Statistics [Lesovik, Levitov ’93][Levitov, Lee,Lesovik ’96] Confined systems: described by (H, H, ρ) dimH < ∞ Given an observable A: A =

j ajPaj where aj ∈ σ(A) Paj

associated spectral projections Procedure:

• t = 0, we measure A (outcome aj)
• evolve for time t
• measure again at time t (outcome ak)

Two-time measurement distribution of A: PA,t(φ)= probability of measuring a change in A equal to φ.

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Procedure:

• t = 0, we measure A (outcome aj)
• evolve for time t
• measure again at time t (outcome ak)

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Procedure:

• t = 0, we measure A (outcome aj)
• evolve for time t
• measure again at time t (outcome ak)

PA,t(φ)= probability of measuring a change in A equal to φ

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Procedure:

• t = 0, we measure A (outcome aj)
• evolve for time t
• measure again at time t (outcome ak)

PA,t(φ)= probability of measuring a change in A equal to φ In confined system: PA,t(φ) =

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Procedure:

• t = 0, we measure A (outcome aj)
• evolve for time t
• measure again at time t (outcome ak)

PA,t(φ)= probability of measuring a change in A equal to φ In confined system: PA,t(φ) = tr(ρPaj)

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Procedure:

• t = 0, we measure A (outcome aj)
• evolve for time t
• measure again at time t (outcome ak)

PA,t(φ)= probability of measuring a change in A equal to φ In confined system: PA,t(φ) = tr(ρPaj) with ρam = 1 tr(ρPaj)PajρPaj.

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Procedure:

• t = 0, we measure A (outcome aj)
• evolve for time t
• measure again at time t (outcome ak)

PA,t(φ)= probability of measuring a change in A equal to φ In confined system: PA,t(φ) = tr(ρPaj)tr(e−itHρameitHPak) with ρam = 1 tr(ρPaj)PajρPaj. Fact/Problem: the measurment perturbes the state, the intial state reduces to ρam

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Procedure:

• t = 0, we measure A (outcome aj)
• evolve for time t
• measure again at time t (outcome ak)

PA,t(φ)= probability of measuring a change in A equal to φ In confined system: PA,t(φ) =

• ak−aj=φ

tr(ρPaj)tr(e−itHρameitHPak) with ρam = 1 tr(ρPaj)PajρPaj. Fact/Problem: the measurment perturbes the state, the intial state reduces to ρam Remark: supp(PA,t) is included on the set of possible A-differences

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Procedure:

• t = 0, we measure A (outcome aj)
• evolve for time t
• measure again at time t (outcome ak)

PA,t(φ)= probability of measuring a change in A equal to φ In confined system: PA,t(φ) =

• ak−aj=φ

tr(ρPaj)tr(e−itHρameitHPak) =

• ak−aj=φ

tr(e−itHPajρPajeitHPak) with ρam = 1 tr(ρPaj)PajρPaj. Fact/Problem: the measurment perturbes the state, the intial state reduces to ρam Remark: supp(PA,t) is included on the set of possible A-differences

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws

1Given a classical observable C and an intial state ρ, we call C-statistics the

probability measure PC such that

• f (s)dPC (s) =
• f (C)dρ for all f ∈ B(R)

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws

Classical system:(M, H, ρ)

1Given a classical observable C and an intial state ρ, we call C-statistics the

probability measure PC such that

• f (s)dPC (s) =
• f (C)dρ for all f ∈ B(R)

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws

Classical system:(M, H, ρ) Two-time measurment statistics is equivalent to the law P△At associated to △At := At − A. 1

1Given a classical observable C and an intial state ρ, we call C-statistics the

probability measure PC such that

• f (s)dPC (s) =
• f (C)dρ for all f ∈ B(R)

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws

Classical system:(M, H, ρ) Two-time measurment statistics is equivalent to the law P△At associated to △At := At − A. 1 Assume A + B is conserved. The identity as classical observables △At = At − A = −(Bt − B) = −△Bt yiels the identity P△At = P−△Bt

1Given a classical observable C and an intial state ρ, we call C-statistics the

probability measure PC such that

• f (s)dPC (s) =
• f (C)dρ for all f ∈ B(R)

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws

Classical system:(M, H, ρ) Two-time measurment statistics is equivalent to the law P△At associated to △At := At − A. 1 Assume A + B is conserved. The identity as classical observables △At = At − A = −(Bt − B) = −△Bt yiels the identity P△At = P−△Bt Quantum system At − A = −(Bt − B) as operators yields identity between all spectral measures µ△At = µ−△Bt but PA,t = P−B,t

1Given a classical observable C and an intial state ρ, we call C-statistics the

probability measure PC such that

• f (s)dPC (s) =
• f (C)dρ for all f ∈ B(R)

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws-energy balance

Classical system (M, H, ρ) H = H0 + V ρ is invariant for the dynamics associated to H0 (H0 interpreted as heat)

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws-energy balance

Classical system (M, H, ρ) H = H0 + V ρ is invariant for the dynamics associated to H0 (H0 interpreted as heat) Energy conservation: △H0,t = H0,t − H0 = −(Vt − V ) = (−V )t − (−V ) as function on M

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws-energy balance

Classical system (M, H, ρ) H = H0 + V ρ is invariant for the dynamics associated to H0 (H0 interpreted as heat) Energy conservation: △H0,t = H0,t − H0 = −(Vt − V ) = (−V )t − (−V ) as function on M In particular if V is bounded by C then supt |△H0,t| < 2C and supp(P△H0,t) bounded

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws-energy balance

Classical system (M, H, ρ) H = H0 + V ρ is invariant for the dynamics associated to H0 (H0 interpreted as heat) Energy conservation: △H0,t = H0,t − H0 = −(Vt − V ) = (−V )t − (−V ) as function on M In particular if V is bounded by C then supt |△H0,t| < 2C and supp(P△H0,t) bounded Quantum system (H, H, ρ) H = H0 + V

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws-energy balance

Classical system (M, H, ρ) H = H0 + V ρ is invariant for the dynamics associated to H0 (H0 interpreted as heat) Energy conservation: △H0,t = H0,t − H0 = −(Vt − V ) = (−V )t − (−V ) as function on M In particular if V is bounded by C then supt |△H0,t| < 2C and supp(P△H0,t) bounded Quantum system (H, H, ρ) H = H0 + V Energy conservation: H0,t − H0 = −Vt + V = (−V )t − (−V ) as

• perator on H yields an identity between all spectral measures but

PH0,t = P−V ,t

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws-energy balance

Classical system (M, H, ρ) H = H0 + V ρ is invariant for the dynamics associated to H0 (H0 interpreted as heat) Energy conservation: △H0,t = H0,t − H0 = −(Vt − V ) = (−V )t − (−V ) as function on M In particular if V is bounded by C then supt |△H0,t| < 2C and supp(P△H0,t) bounded Quantum system (H, H, ρ) H = H0 + V Energy conservation: H0,t − H0 = −Vt + V = (−V )t − (−V ) as

• perator on H yields an identity between all spectral measures but

PH0,t = P−V ,t A priori one can have: supp(PV ,t) bounded but Et(φ2n) =

• φ2ndPH0,t(φ) = +∞

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws-energy balance

Classical system (M, H, ρ) H = H0 + V ρ is invariant for the dynamics associated to H0 (H0 interpreted as heat) Energy conservation: △H0,t = H0,t − H0 = −(Vt − V ) = (−V )t − (−V ) as function on M In particular if V is bounded by C then supt |△H0,t| < 2C and supp(P△H0,t) bounded Quantum system (H, H, ρ) H = H0 + V Energy conservation: H0,t − H0 = −Vt + V = (−V )t − (−V ) as

• perator on H yields an identity between all spectral measures but

PH0,t = P−V ,t A priori one can have: supp(PV ,t) bounded but Et(φ2n) =

• φ2ndPH0,t(φ) = +∞

We want to study the behaviour of PH0,t tails

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws-energy balance

Quantum system (H, H, ρ) H = H0 + V

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws-energy balance

Quantum system (H, H, ρ) H = H0 + V Energy conservation: H0,t − H0 = −Vt + V = (−V )t − (−V ) as

• perator on H

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws-energy balance

Quantum system (H, H, ρ) H = H0 + V Energy conservation: H0,t − H0 = −Vt + V = (−V )t − (−V ) as

• perator on H

Work definition (?): lack of notion of trajectory Work defined as energy/heat difference

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws-energy balance

Quantum system (H, H, ρ) H = H0 + V Energy conservation: H0,t − H0 = −Vt + V = (−V )t − (−V ) as

• perator on H

Work definition (?): lack of notion of trajectory Work defined as energy/heat difference ◮ Work as observable Defining W := H0,t − H0 = (−Vt) − (−V ) ([Bochkov-Kuzovlev ’70s-’80s]) leads to

• 1. no-fluctuation relations!!!!
• 2. µ∆H0 = µW

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Two-time measurment statistics

Conservation laws-energy balance

Quantum system (H, H, ρ) H = H0 + V Energy conservation: H0,t − H0 = −Vt + V = (−V )t − (−V ) as

• perator on H

Work definition (?): lack of notion of trajectory Work defined as energy/heat difference ◮ Work as observable Defining W := H0,t − H0 = (−Vt) − (−V ) ([Bochkov-Kuzovlev ’70s-’80s]) leads to

• 1. no-fluctuation relations!!!!
• 2. µ∆H0 = µW

◮ Work is not an observable Work defined as energy/heat difference between two measurments

• 1. fluctuation relations
• 2. PH0,t = P−V ,t

Underlying jump picture

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Heat fluctuations

Perturbation V bounded

General setting (O, τ t, ω) C ∗-dynamical system τ t = τ t

0 + i[−, V ] and ω is a τ t 0 invariant state

πω : O → B(Hω) a GNS representation ω(A) = (Ωω, AΩω)Hω Liouvillean: τ t

0(A) = e+itLπω(A)e−itL and LΩω = 0

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Heat fluctuations

Perturbation V bounded

General setting (O, τ t, ω) C ∗-dynamical system τ t = τ t

0 + i[−, V ] and ω is a τ t 0 invariant state

πω : O → B(Hω) a GNS representation ω(A) = (Ωω, AΩω)Hω Liouvillean: τ t

0(A) = e+itLπω(A)e−itL and LΩω = 0

Definition

We define the heat two-time measurement statistics for time t, denoted Pt, to be the spectral measure for the operator L + πω(V ) − πω(τ t(V )), with respect to the vector Ωω

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Heat fluctuations

Perturbation V bounded

General setting (O, τ t, ω) C ∗-dynamical system τ t = τ t

0 + i[−, V ] and ω is a τ t 0 invariant state

πω : O → B(Hω) a GNS representation ω(A) = (Ωω, AΩω)Hω Liouvillean: τ t

0(A) = e+itLπω(A)e−itL and LΩω = 0

Definition

We define the heat two-time measurement statistics for time t, denoted Pt, to be the spectral measure for the operator L + πω(V ) − πω(τ t(V )), with respect to the vector Ωω

Remark

Et(φ2n) = (L + πω(V ) − πω(τ t(V )))nΩω

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Heat fluctuations

Perturbation V bounded

Theorem (Benoist, P., Raquépas 2017)

Let (O, τ, ω) be a C ∗-dynamical system as above and Pt be the heat full statistics measure associated to a self-adjoint perturbation V ∈ O. Then (nD) ⇒ sup

t∈R

Et[φ2n+2] < +∞. (γA) ⇒ sup

t∈R

Et[eγ|φ|] ≤ +∞.

Corollary

Under the conditions of the previous theorem, (nD) ⇒ Pt(|φ| ≥ tR) ≤ Cn(Rt)−2n+2 (γA) ⇒ Pt(|φ| ≥ tR) ≤ Cγe−Rt.

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Heat fluctuations

Regularity condition optimality

Quantum impurity in a free Fermi gas H = Γa(C) ⊗ Γa(L2(R+, de)) = C2 ⊗ Γa(L2(R+, de)) H0 = dΓ(ε0) ⊗ 1 l + 1 l ⊗ dΓ(ˆ e) H = H0 + (a∗(1) ⊗ 1 l)(1 l ⊗ a(f )) + (a(1) ⊗ 1 l)(1 l ⊗ a∗(f )) f ∈ L2(R+, de) ω is a (τ0, β) KMS state

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Heat fluctuations

Regularity condition optimality

Quantum impurity in a free Fermi gas H = Γa(C) ⊗ Γa(L2(R+, de)) = C2 ⊗ Γa(L2(R+, de)) H0 = dΓ(ε0) ⊗ 1 l + 1 l ⊗ dΓ(ˆ e) H = H0 + (a∗(1) ⊗ 1 l)(1 l ⊗ a(f )) + (a(1) ⊗ 1 l)(1 l ⊗ a∗(f )) f ∈ L2(R+, de) ω is a (τ0, β) KMS state

Theorem (Benoist, P., Raquépas 2017)

For the above model the following are equivalent:

• 1. supt∈R Et[φ2n+2] < ∞;
• 2. for a non-trivial time interval [t1, t2]

t2

t1 Et[φ2n+2]dt < ∞;

• 3. (nD)

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Heat fluctuations

Regularity condition optimality

Quantum impurity in a free Fermi gas H = Γa(C) ⊗ Γa(L2(R+, de)) = C2 ⊗ Γa(L2(R+, de)) H0 = dΓ(ε0) ⊗ 1 l + 1 l ⊗ dΓ(ˆ e) H = H0 + (a∗(1) ⊗ 1 l)(1 l ⊗ a(f )) + (a(1) ⊗ 1 l)(1 l ⊗ a∗(f )) f ∈ L2(R+, de) ω is a (τ0, β) KMS state

Theorem (Benoist, P., Raquépas 2017)

For the above model the following are equivalent:

• 1. supt∈R Et[φ2n+2] < ∞;
• 2. for a non-trivial time interval [t1, t2]

t2

t1 Et[φ2n+2]dt < ∞;

• 3. (nD)

For this model (nD) is equivalent to R ∋ s → eis ˆ

ef ∈ L2(R+, de)

is n times norm- differentiable i.e f ∈ Dom(ˆ en)

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Heat fluctuations

Regularity condition optimality

Quantum impurity in a free Fermi gas H = Γa(C) ⊗ Γa(L2(R+, de)) = C2 ⊗ Γa(L2(R+, de)) H0 = dΓ(ε0) ⊗ 1 l + 1 l ⊗ dΓ(ˆ e) H = H0 + (a∗(1) ⊗ 1 l)(1 l ⊗ a(f )) + (a(1) ⊗ 1 l)(1 l ⊗ a∗(f )) f ∈ L2(R+, de) ω is a (τ0, β) KMS state

Theorem (Benoist, P., Raquépas 2017)

For the above model the following are equivalent:

• 1. supt∈R Et[φ2n+2] < ∞;
• 2. for a non-trivial time interval [t1, t2]

t2

t1 Et[φ2n+2]dt < ∞;

• 3. (nD)

For this model (nD) is equivalent to R ∋ s → eis ˆ

ef ∈ L2(R+, de)

is n times norm- differentiable i.e f ∈ Dom(ˆ en) Remark: decay of f controls how high energy frequencies contribute to the interaction (ultraviolet regularity)

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Heat exchange between two reservoirs

H0 = H1 + H2, V small Classical case: P 1

t H0,t ≈ 0 and P 1 t H1,t ≈ P− 1 t H2,t

statements about control of fluctuation (large deviation principle) are trivially satisfied Quantum setting: P 1

t H0,t ≈ 0 NOT TRUE ANYMORE

P 1

t H1,t ≈ P 1 t H2,t has to be justified

Theorem ( Benoist, Pautrat, P. "19)

Under stronger analiticity properties (UV conditions), usual statement of control of fluctuation are still true (law of large number, central limit theorem, large deviation principle).

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Conclusions:

◮ Classical statistical mechanics models: energy fluctuations are controlled by the interaction intensity. Particularly, no large fluctuations exist when the interaction is bounded. ◮ Quantum statistical mechanics models: in the two time measurement picture, energy 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

• f high energy frequencies to the interaction (UV

regularization). In other words, in the quantum case the primary contribution to the fluctuations is given by energy transitions induced by the interaction rather then the interaction itself.

Heat fluctuations in the two-time measurement framework and ultraviolet regularity Annalisa Panati, CPT, Université de Toulon Plan Context: quantum statistical mechanics

Fluctuation relations

Two-time measurment statistics

Conservation laws

Heat fluctuations: theorems and results

Bounded perturbations

Conclusions

Thank you for your attention!