Mesoscopic Systems: Statistics and Entropy Interpretation

The statistics of mesoscopic systems and the physical interpretation of extensive and non-extensive entropies Dragos-Victor Anghel, IFIN-HH, Magurele, Romania Alexandru S. Parvan, BLTP, JINR, and IFIN-HH

Motivation 5.03.2018 BLTP seminar 3

Comparison of Tsallis-factorized statistics with Tsallis-1 statistics pp - collision Tsallis-1 (Model B): Charged massless pions A.S.P., Eur. Phys. J. A 52 (2016) 355 Ex. NA61/SHINE, Eur. Phys. J. C 74, 2794 (2014) Ex. PHENIX, Phys. Rev. C 83, 064903 (2011) Ultrarelativistic distributions of the Tsallis-1 statistics: 3 N 3 N 1 q 1 q 3 N 3 N y y N 2 N 1 1 1 1 q q d N dp dy gV 1 N 2 N 1 1 1 q q 0 1 d N dp dy gV 0 = 2 T cosh p dy y = cosh p y ( ) 2 ( ) ! N T 3 2 2 ! p N 2 = 0 N T = y 0 N y T T 0 0 1 q 1 q 1 1 + 3 N + 3 N + + + + 1 cosh ( 1) q p y N 1 cosh ( 1) q p y N 1 q 1 q + + 1 T 1 T q T q T

Comparison of Tsallis-factorized statistics with Tsallis-1 statistics pp- collision Tsallis-1 (Model B): Charged massless pions A.S.P., Eur. Phys. J. A 52 (2016) 355 Ex. ALICE, Eur. Phys. J. C 71, 1655 (2011); Eur. Phys. J. C 75, 226 (2015) Ex. ALICE, Phys. Lett. B 736, 196 (2014) Ultrarelativistic distributions of the Tsallis-1 statistics: 3 3 N N 1 q 1 q 3 N 3 N y y y y N 2 1 N 1 1 1 q q N 1 2 1 N d N dp dy gV 1 1 1 q q d N dp dy gV 1 1 0 0 = cosh = p dy y 2 T cosh p dy y ( ) T 3 ( ) 2 ! p N 2 ! 2 N 2 = 0 N T T = y 0 N y T y y 0 0 0 0 1 q 1 q 1 1 + 3 N + + + 3 N 1 cosh ( 1) q p y N 1 q + + 1 cosh ( 1) q p y N 1 q + 1 T + 1 T q T q T

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 5.03.2018 BLTP seminar 6

Basic considerations and general postulates 5.03.2018 BLTP seminar 7

Existence of the function entropy The state of a homogeneous system, defined by the extensive parameters (?,?). Reversible, adiabatic processes The entropy variation ? is a measure of the irreversibility of the process ? This formalism is rigorously defined only for macroscopic systems. 5.03.2018 BLTP seminar 8

Tisza-Callen postulates of thermodynamics Postulate 1 (existence of equilibrium states): Any isolated system has equilibrium states that are characterized by a small number of extensive variables ?,? . Postulate 2 (existence of entropy): There exists a function (called entropy ?) of the extensive parameters, defined for all equilibrium states and having the following property. The values assumed by the extensive parameters in the absence of a constraint are those that maximize the entropy over the manifold of constrained equilibrium states. This postulate is an expression of the Second Law of Thermodynamics. Postulate 3 (additivity and differentiability of ?): The entropy of a composite system is additive over the constituent subsystems (whence the entropy of each constituent system is a homogeneous first order function of the extensive parameters). The entropy is continuous and differentiable. The observer appears the anthropomorphism of the entropy (Jaynes, Am. J. Phys. 33, 391, 1965). Macroscopic systems 5.03.2018 BLTP seminar 9

Properties of the entropy Is a function of the extensive parameters ? and ?: ? ?,? The entropy of a macroscopic system is a homogeneous function ? ??,?? = ?? ?,? Is a concave downwards function ? ?1,?1 + ? ?2,?2 ? ?1+ ?2,?1+ ?2 S S 1 ?=?? ?? U U ? > 0 ? = ? < 0 BLTP seminar 5.03.2018 10

Definitions and properties of the entropy 5.03.2018 BLTP seminar 11

Statistical mechanics description of the entropy Extensive parameters (?,?) -> microstates ? , of probabilities ?? The entropy should be a function of the set ??. The average value of any physical quantity ? is ? ???? ? Usually, the microstates are eigenstates of the Hamiltonian 5.03.2018 BLTP seminar 12

Definitions of the entropy Extensive entropies Non-extensive entropies Boltzmann-Gibbs ???= ?? Havrada-Charv t-Daroczy-Tsallis (HCDT) ??ln ?? ? 1 ??? 1 ? ????= ?? ?? ? R nyi Landsberg-Vedral (LV) ? ln ??? 1 ? ?= ?? ?? ?? 1 ??= ?? 1 ? 1 ? ??? ??? ? ???? ?? ?? = lim ?=1 ?? ?? = lim ?=1 ?? ?? = lim ?=1 ?? ?? 5.03.2018 BLTP seminar 13

Boltzmann-Gibbs-Shanon-Rnyi entropy ???is also called the amount of uncertainty concerning the outcome of an experiment. ???is the unique quantity (up to a multiplicative constant) that satisfies the postulates (A. R nyi, 4th Berkeley Symposium 1, 547, 1961; D. K. Fadeev, Arbeiten zur Informationstheorie I, 1957). a. ??? ?? is a symmetric function in its variables. b. ????,1 ? is a continuous function of ? for 0 ? 1. 1 2,1 c. ??? = 1 2 d. ?????1, 1 ? ?1,?2, ,?? = ????1,?2, ,?? + ?1????,1 ? 5.03.2018 BLTP seminar 14

Boltzmann-Gibbs-Shanon-Rnyi entropy additivity property (1) and ?? (2) We have 2 sets of probabilities ?? (1+2), where ??? (1+2)= ?? (1)?? (2) We define the direct product ??? 1+2 1 2 Then ??? = ??? + ??? ??? ?? ?? (additivity property) Postulate d: ?????1, 1 ? ?1,?2, ,?? = ????1,?2, ,?? + ?1????,1 ? Replaced by additivity The Boltzmann-Gibbs-Shanon-R nyi entropies have the right extensivity property, but what are we suppose to do with non-extensive systems? ? ln ??? 1 ? ?= ?? R nyi: ?? 5.03.2018 BLTP seminar 15

Entropies in isolated systems (microcanonical ensemble) For given ?,? , the number of accessible microstates of the system is ?,? All accessible microstates are equiprobable, i.e. ??= 1/ ?,? for any ? Extensive entropies Non-extensive entropies Boltzmann-Gibbs ????,? = ??ln ?,? R nyi ?? Havrada-Charv t-Daroczy-Tsallis (HCDT) 1 ??,? 1 1 ? ?????,? = ?? ?? ??,? = ??ln ?,? Landsberg-Vedral (LV) ?? ???,? = 1 ?1 ? 1?,? ?? 5.03.2018 BLTP seminar 16

Probabilities in compound systems 1 System 1, of microstates ? and probabilities ?? 2 System 2, of microstates ? and probabilities ?? 1+2 System 1 + 2 , of probabilities ??? 1+2= If systems 1 and 2 remain isolated of each other, ??? ?? otherwise we can t say much about the probabilities of the microstates of the system 1 + 2 . 1?? 2, 5.03.2018 BLTP seminar 17

Systems composed of isolated subsystems System 1, of parameters ?1,?1, is isolated 1 1= ?? , ? 1?1,?1 System 2, of parameters ?2,?2, is isolated 1 2= ?? , ? 2?2,?2 System 1 + 2 , of parameters ?1,?2,?1,?2, 1+2= 1?1,?1 2?2,?2 1 ??? , ?,? What happens when two systems, initially isolated, are put into contact? 5.03.2018 BLTP seminar 18

Additive, subadditive and superadditive entropies Extensive entropies Non-extensive entropies 1 ??,? 1 1 ? ????,? = ??ln ?,? ?1+2 = ?1 ?????,? = ?? ???? ?1,?2,?1,?2 ?????1,?1 + ?? 2 ?????1,?1?? 2 ?? ?? 1+2 = ?? 1 +1 ? ???1,?2,?1,?? ???1,?1 + ?2 ???2,?? ?????2,?2 ?????2,?2 ?? 1 ??,? = ??ln ?,? ?1+2 ?1,?2,?1,?? = ?1 ?? ?? ? ?? 1 ?1 ? 1?,? ?1,?2,?1,?2 ?1,?1 + ?? 2 ?? ?1,?1?? 2 ??2,?? ??1,?1 + ?2 ???,? = ?? ?? ?? 1+2 = ?? 1 1 ? ?? ?? ?? superadditive subadditive ?2,?2 ? < 1 ? > 1 ?? ?? 1 ?2,?2 Additive entropies 5.03.2018 BLTP seminar 19

Three methods to find the equilibrium properties of compound systems 5.03.2018 BLTP seminar 20

Two systems in contact: Method 1 1 System 1: ?1,?1, 1?1,?1,?? System 2: ?2,?2, 2?2,?2,?? System 1 + 2 : ??= ?1+ ?2,?1,?2, 1+2,M1??,?1,?2, ?1,?1 = 1/ 1?1,?1 2 ?2,?2 = 1/ 2?2,?2 If the system 1 + 2 is completely isolated, we don t know anything about the evolution towards equilibrium. This is the end of the story. 1+2 M1??,?1,?2 = 1/ 1+2,M1??,?1,?2 ??? 1+2,M1??,?1,?2 = 1?1,?1 2?? ?1,?2 ?1 1+2 M1??,?1,?2 ?1+2,M1??,?1,?2 ? ??? 5.03.2018 BLTP seminar 21

1. The description of systems 1 and 2 remains in terms of the whole number of states, 1?1,?1 and 2?2,?2--we don t observe their internal equilibrium Method 1 continuation Postulate 2 (reminder): The values assumed by the extensive parameters in the absence of a constraint are those that maximize the entropy over the manifold of constrained equilibrium states. Here the observer reappears. He observes the evolution towards equilibrium and the averages: ?1 ?1,?2 ?2 We define ? ?1 1?1,?1 2?? ?1,?2/ 1+2,M1??,?1,?2 Maximum probability distribution ?? ?1 ??1 1+2,M1??,?1,?2 ?ln 1?1,?1 ??1 ?1=?1 ??2 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. 1 ? 1?1,?1 2?? ?1,?2 ??1 ?ln 2?2,?2 = 0 1 = ?M1 This is the influence of the observer on the definition of the entropy. ???M1 ?2=?2=?? ?1 Temperature for Method 1 5.03.2018 BLTP seminar 22

Equilibrium probability distribution Method 2 Maximum entropy principle: the entropy is maximized for fixed average values of (some of) the extensive parameters (Jaynes, Phys. Rev. 106, 620, 1957; Tsallis, J. Stat. Phys. 52, 479, 1988; Psaltino & Psaltino, Phys. Lett. A 281, 126, 1996, Parvan, arXiv:1711.09354) ? max ? ?,? ?M2 ? ?? ?,? ?? 1 = Lagrange multiplier ?M2 Temperature for Method 2 5.03.2018 BLTP seminar 23

Two systems in contact generalization Method 3 1= 1 1?1,?1, System 1, with ?1,?1, 1?1,?1,?? ?1 ?? ?1?1,?1 1 2= 2 1?2,?2, System 2, with ?2,?2, 2?2,?2,?? ?2 ?? ?2?2,?2 If the systems remain isolated: ??? ?1+2 The subsystems are isolated from each other 2 1+2= 1 1?1,?1 2 1+2 ?1+2?1,?2,?1,?2 1?2,?2, ??? 5.03.2018 BLTP seminar 24

Reminder: additive, subadditive and superadditive entropies Extensive entropies Non-extensive entropies 1 ??,? 1 1 ? ????,? = ??ln ?,? ?1+2 = ?1 ?????,? = ?? ???? ?1,?2,?1,?2 ?????1,?1 + ?? 2 ?????1,?1?? 2 ?? ?? 1+2 = ?? 1 +1 ? ???1,?2,?1,?? ???1,?1 + ?2 ???2,?? ?????2,?2 ?????2,?2 ?? 1 ??,? = ??ln ?,? ?1+2 ?1,?2,?1,?? = ?1 ?? ?? ? ?? 1 ?1 ? 1?,? ?1,?2,?1,?2 ?1,?1 + ?? 2 ?? ?1,?1?? 2 ??2,?? ??1,?1 + ?2 ???,? = ?? ?? ?? 1+2 = ?? 1 1 ? ?? ?? ?? ?2,?2 ?? ?? 1 ?2,?2 5.03.2018 BLTP seminar 25

Method 3 definition of the entropy of a composed system Systems 1 ?1,?1 and 2 ?2,?2 are put in (thermal) contact, so the system 1 + 2 has the parameters ??= ?1+ ?2,?1 We define the entropy of the compound system as ?1+2,M3??,?1,?2 = max ?1 ??1?1,?1 ??1 ?1,M3 ??2 If system 2 is a reservoir, ?2,M3 ?M3 is a constant and we recover Method 2 By construction, Method 1 and Method 3 are different (in mesoscopic systems). ?1?1,?1 + ?2?? ?1,?2 1 =??2?2,?2 1 ?2,M3 5.03.2018 BLTP seminar 26

Applications of the methods and comparison of the results 5.03.2018 BLTP seminar 27

Application of Method 1 Boltzmann- Gibbs and R nyi entropies We assume that system 2 is the reservoir ? The probability to have in system 1 the microstate ? of energy ?1 is 1 M1= ??? ? M1??,?1,?2 ??? ?1,?? = ??? ?1,?? ?????,?? ??? 1 ??1 ?? ?? = e ?? 1 BG,M1=e ??? ?? ???,?1,?? ?1= ????,?? ?1 ????,?? = ????,?? ? ??=?? 1 ?M1 ?1 ???M1 ?? ?=??? ?? ??= ??ln ???,?? ?? Identical results for R nyi entropy, ??. ????,?? ????,?? ???,?? = e ?1 e ?1? ???M1 ???M1 ?? , where ??? ?1 ? 5.03.2018 BLTP seminar 28

Application of Method 1 HCDT and LV entropies HCDT entropy LV entropy 1 1 1 ? 1 ? , ?= 1 +1 ? ?? 1 ?1 ? ? 1, ?= 1 1 ? 1 ? ? 1 ????= ?? ???? ??= ?? ?? ?? ???? ???? 1 ? 1 ??1=? ln ? ??1=? ln ? 1 Hypothesis 1: physical temperature ) Hypothesis 1: physical temperature ) ??? = constant (the ?1 ????1 ??? = constant (the ?1 ????1 1 ????,?1=? 1 ??,?1=? ?? ?? ?? ?? ??1 ??1 ???? ???? ?????=??? 1 ??,????=??? 1 Hypothesis 2: Hypothesis 2: =constant =constant ??? ??? 1 1 1 ? ? 1 1 ? ?1 ? 1 ?1 ????? ? ??? = ??? 1 ??? = ??? 1 1 ??? ? 1?? ??????? ? 1 1 1 ? ? 1 1 1 ? ?1 1 ?11 ? 1 ?1 ????? ? 1 ????,?1= 1 ??,?1= ? ? 1 ? ? ?1 1 ??? ? 1?? ????? ??? ??????? ? 1 ????,?1 and ? ? 1 ??,?1 converge to the Botlzmann-Gibbs distribution In the limit ? 1, both ? ? 5.03.2018 BLTP seminar 29

Application of Method 2 Boltzmann- Gibbs and R nyi entropies For the Boltzmann-Gibbs entropy we obtain the same result as in Method 1: 1 BG,M2=e ?1 ??? For the R nyi entropy, e ?1? ??? ??? ?2, ?2 ?? where ??? ? 1 1 1 ?21 +? 1 1 +? 1 ? 1 ? 1 1 R,M2= ?2= ?? ? ?1 ,?? ? ?1 ??? ??? ?? ? And the equation for the self-consistency 1 +? 1 ??? We observe that M1 and M2 give different results even for the extensive R nyi entropy. 1 ? 1 +? 1 ? 1 ? 1 ? ?1 = ? ? ?1 ??? ? ? 5.03.2018 BLTP seminar 30

Application of Method 2 HCDT and LV entropies HCDT entropy LV entropy 1 1 1 ????,?2= 1 +? 1 ???? ?1? ??? 1 ?21 + ? 1 ?? ?1? ??? ? 1 ? 1 1 ??,?2= ?? ?? , ? ??? 1 With the equation for self-consistency ?? ?1? ??? ? 1 1 ?2= ??? 1 + ? 1 1 +? 1 ???? ?1? ??? ? 1 = 1 ? ? With the equation for self-consistency ? 1 ?? ?1? ??? ? 1 1 + ? 1 ? 2 ? ?+1 1 ?? ?1? ??? ? 1 = ? 1 + ? 1 ? 1 ????,?2 and ?? 1 ??,?2 converge to the Botlzmann-Gibbs distribution In the limit ? 1, both ?? 5.03.2018 BLTP seminar 31

Conclusions 5.03.2018 BLTP seminar 32

We analyzed 4 definitions of entropies, two extensive (Boltzmann- Gibbs and R nyi) and two non-extensive (Havrada-Charv t-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). 5.03.2018 BLTP seminar 33