Non-Extensive Quantum Statistics with Particle Hole Symmetry T.S. - PowerPoint PPT Presentation

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q - and κ -Statistics Summary Chinese-Hungarian bilateral research project Non-Extensive Quantum Statistics with Particle – Hole Symmetry T.S. Biró 1 K. M. Shen 2 B. W. Zhang 2 1 Heavy Ion Research Group MTA Research Centre for Physics, Budapest 2 Institute of Particle Physics Central China Normal University, Wuhan October 7, 2014 Talk given by T. S. Biró, ACHT 2014, Oct.08., Balatonfüred, Hungary TSB KMS BWZ NEXT Q STATISTICS 1 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle-Hole Symmetry within q - and κ -Statistics Summary Content Kubo–Martin–Schwinger Relation 1 Generalized Thermodynamics 2 Particle-Hole Symmetry within q - and κ -Statistics 3 TSB KMS BWZ NEXT Q STATISTICS 2 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle to Hole Ratio Particle-Hole Symmetry within q - and κ -Statistics Boltzmann - Gibbs ”knows it” Summary Content Kubo–Martin–Schwinger Relation 1 Particle to Hole Ratio Boltzmann - Gibbs ”knows it” Generalized Thermodynamics 2 Particle-Hole Symmetry within q - and κ -Statistics 3 TSB KMS BWZ NEXT Q STATISTICS 3 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle to Hole Ratio Particle-Hole Symmetry within q - and κ -Statistics Boltzmann - Gibbs ”knows it” Summary KMS relation and quantum statistics 1 Kubo – Martin – Schwinger: � A t B 0 � = � � e − β H e itH A e − itH B = Tr � � e i ( t + i β ) H A e − i ( t + i β ) H e − β H B = Tr � e − β H B e i ( t + i β ) H A e − i ( t + i β ) H � = Tr � � = B 0 A t + i β (1) TSB KMS BWZ NEXT Q STATISTICS 3 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle to Hole Ratio Particle-Hole Symmetry within q - and κ -Statistics Boltzmann - Gibbs ”knows it” Summary KMS relation and quantum statistics 2 Apply KMS for A t = e − i ω t a and B t = A † t = e i ω t a † . � a a † � � a † a � e βω = (2) meaning 1 + n ( ω ) = n ( ω ) e βω . (3) delivers Bose statistics 1 n ( ω ) = e βω − 1 . TSB KMS BWZ NEXT Q STATISTICS 4 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle to Hole Ratio Particle-Hole Symmetry within q - and κ -Statistics Boltzmann - Gibbs ”knows it” Summary CPT Missing negative energy particle = positive energy hole. − n ( − ω ) = ( 1 + n ( ω )) > 0 . (4) Connection to canonical thermodynamical weight − n ( − ω ) = e βω − → e q ( βω ) (5) n ( ω ) to be generalized... TSB KMS BWZ NEXT Q STATISTICS 5 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Particle to Hole Ratio Particle-Hole Symmetry within q - and κ -Statistics Boltzmann - Gibbs ”knows it” Summary KMS in general reaction to Cheng-Bin’s question The KMS relation - based on re-shuffling - holds very generally. ρ UAU − 1 B � � � A t B 0 � = = Tr ρ UAU − 1 ρ − 1 ρ B � � = Tr = � ρ B ( ρ U ) A ( ρ U ) − 1 � � � = Tr = B 0 A t ⊕ i β (6) For nonlinear H-dependence the ⊕ operation is energy dependent → curved spectra. TSB KMS BWZ NEXT Q STATISTICS 6 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Superstatistics: Fluctuating Temperature Particle-Hole Symmetry within q - and κ -Statistics Deformed Exponentials Summary Content Kubo–Martin–Schwinger Relation 1 Generalized Thermodynamics 2 Superstatistics: Fluctuating Temperature Generalized Entropy Formulas Deformed Exponentials Particle-Hole Symmetry within q - and κ -Statistics 3 TSB KMS BWZ NEXT Q STATISTICS 7 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Superstatistics: Fluctuating Temperature Particle-Hole Symmetry within q - and κ -Statistics Deformed Exponentials Summary Continous β scale Euler - integral: ∞ � k ! Θ k e − α Θ d Θ = (7) α k + 1 0 Gamma distribution: γ (Θ) = α k + 1 � Θ � = k + 1 k ! Θ k e − α Θ : . (8) α Char. fct.: � − ( k + 1 ) � 1 + � β Θ � ω � e − βω Θ � = (9) k + 1 this is a q = 1 + 1 / ( k + 1 ) Tsallis – Pareto distribution. TSB KMS BWZ NEXT Q STATISTICS 7 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Superstatistics: Fluctuating Temperature Particle-Hole Symmetry within q - and κ -Statistics Deformed Exponentials Summary KMS with Generalized Weights Let it be x = � β Θ � ω and K = k + 1. n 1 + x � − K 1 � 1 + n = =: e q ( x ) . (10) K From this the Tsallis – Bose would be given by 1 n = e q ( x ) − 1 . (11) TSB KMS BWZ NEXT Q STATISTICS 8 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Superstatistics: Fluctuating Temperature Particle-Hole Symmetry within q - and κ -Statistics Deformed Exponentials Summary Check negative energy states 1 n ( − ω ) = (12) e q ( − x ) − 1 therefore 1 − ( 1 + n ( ω )) = (13) 1 / e q ( x ) − 1 generalized KMS e q ( x ) · e q ( − x ) = 1 . It does not hold for Tsallis! TSB KMS BWZ NEXT Q STATISTICS 9 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Occupancy of Negative Energy States Particle-Hole Symmetry within q - and κ -Statistics Properties of the Arithmetic Mean Summary Content Kubo–Martin–Schwinger Relation 1 Generalized Thermodynamics 2 Particle-Hole Symmetry within q - and κ -Statistics 3 Bosons, Fermions and ”Boltzmannons” Occupancy of Negative Energy States Properties of the Arithmetic Mean TSB KMS BWZ NEXT Q STATISTICS 10 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Occupancy of Negative Energy States Particle-Hole Symmetry within q - and κ -Statistics Properties of the Arithmetic Mean Summary Kappa Distribution G. Kaniadakis � 1 /κ �� 1 + κ 2 x 2 + κ x e κ ( x ) := (14) It knows: e κ ( − x ) · e κ ( x ) = 1 . (15) It automatically satisfies KMS statistics. TSB KMS BWZ NEXT Q STATISTICS 10 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Occupancy of Negative Energy States Particle-Hole Symmetry within q - and κ -Statistics Properties of the Arithmetic Mean Summary Linear Combination with Holes Assume n KMS ( x ) = A ( n q ( x ) + n q ∗ ( x )) + B . For Tsallis distr. e q ( − x ) = 1 / e q ∗ ( x ) with q ∗ = 2 − q . Finally we get B = A − 1 / 2 and for having ”zero for zero” we get KMS (Ke-Ming Shen) ansatz n KMS ( x ) = 1 � � n q ( x ) + n q ∗ ( x ) . 2 TSB KMS BWZ NEXT Q STATISTICS 11 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Occupancy of Negative Energy States Particle-Hole Symmetry within q - and κ -Statistics Properties of the Arithmetic Mean Summary KMS with deformed exponentials Requirements: ρ ( H ) is the analytic continuation of U ( H ) , 1 U ( H ) is unitary 2 For U = g ( H ) from unitarity, U − 1 = U † , and from time reversal CPT, it follows 1 g ( ω ) = g ∗ ( ω ) = g ( − ω ) . (16) TSB KMS BWZ NEXT Q STATISTICS 12 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Occupancy of Negative Energy States Particle-Hole Symmetry within q - and κ -Statistics Properties of the Arithmetic Mean Summary General CPT solution to KMS We need to have a function ( not necessarily the exponential ), g ( ω ) satisfying g ( ω ) · g ( − ω ) = 1 . The general solution with the best small ω - behavior is g ( ω ) = e q ( ω/ 2 ) e q ( − ω/ 2 ) . (17) TSB KMS BWZ NEXT Q STATISTICS 13 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Occupancy of Negative Energy States Particle-Hole Symmetry within q - and κ -Statistics Properties of the Arithmetic Mean Summary Proper Quantum Statistics with Deformed Exponentials Using the notation x = βω/ 2 we have e q ( − x ) n B , F ( ω ) = e q ( x ) ∓ e q ( − x ) . (18) Now either with Tsallis or Kaniadakis deformed exp e q ( x ) we have power-law tail and perfect CPT. But Tsallis e q ( − x ) has a maximal x -value. TSB KMS BWZ NEXT Q STATISTICS 14 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Occupancy of Negative Energy States Particle-Hole Symmetry within q - and κ -Statistics Properties of the Arithmetic Mean Summary Graphs of Tsallis – exponentials Tsallis – Pareto deformed exponential distributions (1 / e q ( x ) ) with q = 0 . 95 , 1 . 00 , 1 . 05. Curvature in the log-plot: sign of q − 1 TSB KMS BWZ NEXT Q STATISTICS 15 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Occupancy of Negative Energy States Particle-Hole Symmetry within q - and κ -Statistics Properties of the Arithmetic Mean Summary Graphs of original Bose distributions Blue: − 1 − n ( ω ) , Green: n ( − ω ) , Red: n ( ω ) Blue and Green coincide. TSB KMS BWZ NEXT Q STATISTICS 16 / 43

Kubo–Martin–Schwinger Relation Generalized Thermodynamics Occupancy of Negative Energy States Particle-Hole Symmetry within q - and κ -Statistics Properties of the Arithmetic Mean Summary Graphs of Tsallis – based Bose distributions q=1.05 Blue: − 1 − n ( ω ) , Green: n ( − ω ) , Red: n ( ω ) Differences between holes and antiparticles? TSB KMS BWZ NEXT Q STATISTICS 17 / 43