Physics at the mesoscale toroidal moments and nonextensive

Physics at the meso-scale: toroidal moments and nonextensive entropies Dragos-Victor Anghel, IFIN-HH, Magurele, Romania Alexandru S. Parvan, BLTP, JINR, and IFIN-HH

Summary • Motivation • Non-extensive entropies • The toroidal momentum operator

Motivation • Few-body systems • Thermodynamic systems • The “middle range” of mesoscopic systems

1. Non-extensive statistics

Why non-extensive statistics? 4. 12. 2018 BLTP seminar 6

Comparison of Tsallis-factorized statistics with Tsallis-1 statistics Tsallis-1 (Model B): Charged massless pions - collision Ex. NA 61/SHINE, Eur. Phys. J. C 74, 2794 (2014) A. S. P. , Eur. Phys. J. A 52 (2016) 355 Ex. PHENIX, Phys. Rev. C 83, 064903 (2011) Ultrarelativistic distributions of the Tsallis-1 statistics: 4. 12. 2018 BLTP seminar 7

Comparison of Tsallis-factorized statistics with Tsallis-1 statistics Tsallis-1 (Model B): Charged massless pions - collision Ex. ALICE, Eur. Phys. J. C 71, 1655 (2011); Eur. Phys. J. C 75, 226 (2015) A. S. P. , Eur. Phys. J. A 52 (2016) 355 Ex. ALICE, Phys. Lett. B 736, 196 (2014) Ultrarelativistic distributions of the Tsallis-1 statistics: 4. 12. 2018 BLTP seminar 8

Summary • Basic considerations and general postulates • Definitions and properties of the entropy • Three methods to find the equilibrium properties of compound systems • Applications of the methods and comparison of the results • Conclusions 4. 12. 2018 BLTP seminar 9

Basic considerations and general postulates 4. 12. 2018 BLTP seminar 10

Existence of the function entropy Reversible, adiabatic processes This formalism is rigorously defined only for macroscopic systems. 4. 12. 2018 BLTP seminar 11

Tisza-Callen postulates of thermodynamics • The observer appears – the anthropomorphism of the entropy (Jaynes, Am. J. Phys. 33, 391, 1965). Macroscopic systems 4. 12. 2018 BLTP seminar 12

Properties of the entropy • S S U 4. 12. 2018 U BLTP seminar 13

Definitions and properties of the entropy 4. 12. 2018 BLTP seminar 14

Statistical mechanics description of the entropy • 4. 12. 2018 BLTP seminar 15

Definitions of the entropy Extensive entropies Non-extensive entropies • • 4. 12. 2018 BLTP seminar 16

Boltzmann-Gibbs-Shanon-Rényi entropy • 4. 12. 2018 BLTP seminar 17

Boltzmann-Gibbs-Shanon-Rényi entropy – additivity property • The Boltzmann-Gibbs-Shanon-Rényi entropies have the right extensivity property, but what are we suppose to do with non-extensive systems? 4. 12. 2018 BLTP seminar 18

Entropies in isolated systems (microcanonical ensemble) Extensive entropies Non-extensive entropies • • 4. 12. 2018 BLTP seminar 19

Probabilities in compound systems • 4. 12. 2018 BLTP seminar 20

Systems composed of isolated subsystems • What happens when two systems, initially isolated, are put into contact? 4. 12. 2018 BLTP seminar 21

Additive, subadditive and superadditive entropies Extensive entropies Non-extensive entropies • • superadditive subadditive Additive entropies 4. 12. 2018 BLTP seminar 22

Three methods to find the equilibrium properties of compound systems 4. 12. 2018 BLTP seminar 23

Two systems in contact: Method 1 • 4. 12. 2018 BLTP seminar 24

Method 1 – continuation • 2. Configurations faraway from equilibrium are not taken into account. Fortunately, in macroscopic systems they don’t count, which is considered a justification—but it’s not. This is the influence of the observer on the definition of the entropy. 4. 12. 2018 BLTP seminar Temperature for Method 1 25

Equilibrium probability distribution Method 2 • Lagrange multiplier Temperature for Method 2 4. 12. 2018 BLTP seminar 26

Two systems in contact – generalization Method 3 • 4. 12. 2018 BLTP seminar 27

Reminder: additive, subadditive and superadditive entropies Extensive entropies Non-extensive entropies • • 4. 12. 2018 BLTP seminar 28

Method 3 – definition of the entropy of a composed system • 4. 12. 2018 BLTP seminar 29

Applications of the methods and comparison of the results 4. 12. 2018 BLTP seminar 30

Application of Method 1 – Boltzmann. Gibbs and Rényi entropies • 4. 12. 2018 BLTP seminar 31

Application of Method 1 – HCDT and LV entropies HCDT entropy LV entropy • • 4. 12. 2018 BLTP seminar 32

Application of Method 2 – Boltzmann. Gibbs and Rényi entropies • 4. 12. 2018 BLTP seminar 33

Application of Method 2 – HCDT and LV entropies HCDT entropy LV entropy • • 4. 12. 2018 BLTP seminar 34

Conclusions 4. 12. 2018 BLTP seminar 35

• We analyzed 4 definitions of entropies, two extensive (Boltzmann. Gibbs and Rényi) and two non-extensive (Havrada-Charvát-Daroczy. Tsallis and Landsberg-Vedral). • We discussed their general properties applicability to macroscopic and mesoscopic systems. • We proposed three different methods to calculate the equilibrium properties of systems. • We applied these methods to the entropies studied and we analyzed the results. • In conclusion, there are two possible scenarios for the interpretation of the entropy of a composed mesoscopic system, one being expressed in Method 1 and the other one expressed in Method 3 (equivalent to Method 2). 4. 12. 2018 BLTP seminar 36

2. The toroidal momentum operator 4. 12. 2018 BLTP seminar 38

Theoretical observation of “anapole” • 4. 12. 2018 BLTP seminar 39

Multipoles expansion • 4. 12. 2018 BLTP seminar 40

Parallel and longitudinal current densities OR 3 independent families of multipoles 4. 12. 2018 BLTP seminar 41

Charge and current multipoles 4. 12. 2018 BLTP seminar 42

The energy and force that act on the toroidal moment 4. 12. 2018 BLTP seminar 43

Toroidal moment in quantum mechanics • 4. 12. 2018 BLTP seminar 44

Incomplete beta functions 4. 12. 2018 BLTP seminar 45

Eigenvalues and eigenfunctions of the toroidal momentum operator 4. 12. 2018 BLTP seminar 46

Conclusions • 4. 12. 2018 BLTP seminar 47