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- Entropy,
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Generalised information and entropy measures in physics

500 C. Beckp(u) with R 11pðuÞ du ¼ 1, rather than a discrete set ofprobabilities p i with P i p i ¼ 1. In this case one def**in**esS ðTÞq ¼ 1 Z þ1q 1 11dus ðspðuÞÞq; ð25Þwhere aga**in** s is a scale parameter that has the samedimension as the variable u. It is **in**troduced for asimilar reason as before, namely to make the **in**tegral **in**Equation (25) dimensionless so that it can besubtracted from 1. For q ! 1 Equation (25) reducesto the Shannon **entropy**S ðTÞ1¼ S ¼Z 1dupðuÞ lnðspðuÞÞ:1ð26ÞA fundamental property of the Tsallis entropies isthe fact that they are not additive for **in**dependentsubsystems. In fact, they have no chance to do so, s**in**cethey are different from the Re´nyi entropies, the onlysolution to Equation (9).To **in**vestigate this **in** more detail, let us considertwo **in**dependent subsystems I **and** II with probabilitiesp I i **and** p IIj , respectively. The probabilities of jo**in**tevents i, j for the comb**in**ed system I,II are p ij ¼ p I i pII j .We may then consider the Tsallis **entropy** for the firstsystem, denoted as S I q , that of the second system,denoted as S IIq , **and** that of the jo**in**t system, denoted as. One hasS I;IIqS I;IIq¼ S I q þ SII q ðq 1ÞS I q SII q :ð27Þvanishes for q ¼ 1 only, i.e. for the case where the Tsallis**entropy** reduces to the Shannon **entropy**. Equation (27)is sometimes called the ‘pseudo-additivity’ property.Equation (27) has given rise to the name nonextensivestatistical mechanics. If we formulate ageneralised statistical mechanics based on maximis**in**gTsallis entropies, then the (Tsallis) **entropy** of **in**dependentsystems is not additive (Figure 3). However, itturns out that for special types of correlated subsystems,the Tsallis entropies do become additive if thesubsystems are put together [28]. This means, for thesetypes of correlated complex systems a description **in**terms of Tsallis entropies **in** fact can make th**in**gssimpler as compared to us**in**g the Shannon **entropy**,which is non-additive for correlated subsystems.2.3. L**and**sberg–Vedral **entropy**Let us cont**in**ue with a few other examples of generalised**in**formation**measures**. Consider!S ðLÞq ¼ 1 1Pq 1 W1 : ð32Þi¼1 pq iThis measure was studied by L**and**sberg **and** Vedral[29]. One immediately sees that the L**and**sberg–Vedral**entropy** is related to the Tsallis **entropy** S ðTÞq byS ðLÞq ¼ SðTÞ qP Wi¼1 pq i; ð33ÞProof of Equation (27): We may writeXðp I i Þq ¼ 1 ðq 1ÞS I q ; ð28ÞiXXp q ij ¼ X ðp X I i Þqi;j i jjðp IIj Þ q ¼ 1 ðq 1ÞS IIq ;ð29Þðp IIj Þ q ¼ 1 ðq 1ÞS I;IIq : ð30ÞFrom Equations (28) **and** (29) it also follows thatXðp X I i Þq ðp IIj Þ q ¼ 1 ðq 1ÞS I q ðq 1ÞS IIqjiþðq1Þ 2 S I q SII q :ð31ÞComb**in****in**g Equations (30) **and** (31) one ends up withEquation (27). ¤Apparently, if we put together two **in**dependentsubsystems then the Tsallis **entropy** is not additive butthere is a correction term proportional to q71, whichcolour **in**pr**in**t & onl**in**eFigure 3. If the nonadditive entropies S q are used tomeasure ** information**, then the

Contemporary Physics 501**and** hence S ðLÞq is sometimes also called normalisedTsallis **entropy**. S ðLÞq also conta**in**s the Shannon **entropy**as a special caselimq!1 SðLÞ q ¼ S ð34Þ**and** one readily verifies that it also satisfies apseudo-additivity condition for **in**dependent systems,namelyS ðLÞI;IIq¼ S ðLÞIqþ Sq ðLÞII þðq 1ÞSqðLÞIS ðLÞIIq : ð35ÞThis means that **in** the pseudo-additivity relation (27)the role of (q71) **and** 7(q71) is exchanged.2.4. Abe **entropy**Abe [30] **in**troduced a k**in**d of symmetric modificationof the Tsallis **entropy**, which is **in**variant under theexchange q ! q 71 . This is given byS Abeq ¼ X ip q i p q 1iq q 1 : ð36ÞThis symmetric choice **in** q **and** q 71 is **in**spired by thetheory of quantum groups which often exhibits**in**variance under the ‘duality transformation’ q ! q 71 .Like Tsallis **entropy**, the Abe **entropy** is also concave.In fact, it is related to the Tsallis **entropy** S T q byS Abeq ¼ ðq 1ÞST q ðq 1 1ÞS T q 1q q 1 : ð37ÞClearly the relevant range of q is now just the unit**in**terval (0,1], due to the symmetry q ! q 71 : Replac**in**gq by q 71 **in** Equation (36) does not change anyth**in**g.2.5. Kaniadakis **entropy**The Kaniadakis **entropy** (also called k-**entropy**) isdef**in**ed by the follow**in**g expression [4]S k ¼X ip 1þki p 1 ik2k: ð38ÞAga**in** this is a k**in**d of deformed Shannon **entropy**,which reduces to the orig**in**al Shannon **entropy** fork ¼ 0. We also note that for small k, **and** by writ**in**gq ¼ 1 þ k, q 71 17k, the Kaniadakis **entropy** approachesthe Abe **entropy**. Kaniadakis was motivatedto **in**troduce this entropic form by special relativity:the relativistic sum of two velocities of particles ofmass m **in** special relativity satisfies a similar relation asthe Kaniadakis **entropy** does, identify**in**g k ¼ 1/mc.Kaniadakis entropies are also concave **and** Leschestable (see Section 3.3).2.6. Sharma–Mittal entropiesThese are two-parameter families of entropic forms[31]. They can be written **in** the formS k;r ¼ p r p k i pik i: ð39Þ2kX iInterest**in**gly, they conta**in** many of the entropiesmentioned so far as special cases. The Tsallis **entropy**is obta**in**ed for r ¼ k **and** q ¼ 1–2k. The Kaniadakis**entropy** is obta**in**ed for r ¼ 0. The Abe **entropy** isobta**in**ed for k ¼ 1 2 ðq q 1 Þ **and** r ¼ 1 2 ðq þ q 1 Þ 1.The Sharma–Mittal entropies are concave **and** Leschestable.3. Select**in**g a suitable ** information** measure3.1. Axiomatic foundationsThe Kh

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