# Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory

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## Abstract

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## 1. Introduction

## 2. Basic Results on Group Entropies

## 3. From Phase Space Volume to Group Entropies

(I) | Algebraic | $W\left(N\right)={N}^{a}$ | with | ${W}^{-1}\left(t\right)={t}^{\frac{1}{a}}$, |

(II) | Exponential | $W\left(N\right)={k}^{N}$ | with | ${W}^{-1}\left(t\right)=\frac{lnt}{lnk}$, |

(III) | Super-exponential | $W\left(N\right)={N}^{\gamma N}$ | with | ${W}^{-1}\left(t\right)=exp[L(\frac{lnt}{\gamma})]$. |

#### 3.1. From $W\left(N\right)$ to $G\left(t\right)$

- Trace-form case
- (I)
- Algebraic, $W\left(N\right)={N}^{a}$:$$\begin{array}{c}\hfill S[p]=a{\sum}_{i=1}^{W\left(N\right)}{p}_{i}\left(\right)open="["\; close="]">{(\frac{1}{{p}_{i}})}^{\frac{1}{a}}-1\end{array}$$$$\begin{array}{c}=\frac{1}{q-1}(1-{\sum}_{i=1}^{W\left(N\right)}{p}_{i}^{q}).\end{array}$$
- (II)
- Exponential, $W\left(N\right)={k}^{N}$, $k>0$:$$S\left[p\right]=\sum _{i=1}^{W\left(N\right)}{p}_{i}ln\frac{1}{{p}_{i}}.$$
- (III)
- Super-exponential, $W\left(N\right)={N}^{\gamma N}$, $\gamma >0$:$$S\left[p\right]=\sum _{i=1}^{W\left(N\right)}{p}_{i}\left(\right)open="\{"\; close="\}">exp\left(\right)open="["\; close="]">L(-\frac{ln{p}_{i}}{\gamma}).$$

- Non-trace-form case
- (I)
- Algebraic, $W\left(N\right)={N}^{a}$:$$S\left[p\right]=\left(\right)open="\{"\; close="\}">exp\left(\right)open="["\; close="]">\frac{ln({\sum}_{i=1}^{W\left(N\right)}{p}_{i}^{\alpha})}{a(1-\alpha )}.$$
- (II)
- Exponential, $W\left(N\right)={k}^{N}$:$$S\left[p\right]=\frac{ln({\sum}_{i=1}^{W\left(N\right)}{p}_{i}^{\alpha})}{1-\alpha}.$$
- (III)
- Super-exponential, $W\left(N\right)={N}^{\gamma N}$:$$S\left[p\right]=\left(\right)open="\{"\; close="\}">exp\left(\right)open="["\; close="]">L\left(\frac{ln{\sum}_{i=1}^{W\left(N\right)}{p}_{i}^{\alpha}}{\gamma (1-\alpha )}\right).$$

#### 3.2. The Composition Law $\varphi (x,y)$

- Non-trace-form case$$\varphi (x,y)=\frac{1}{1-\alpha}G[{G}^{-1}((1-\alpha )x)+{G}^{-1}((1-\alpha )y)].$$

- (I)
- Algebraic, $W\left(N\right)={N}^{a}$:$$\varphi (x,y)=x+y+\frac{1}{\lambda}xy=x+y+(1-q)xy.$$
- (II)
- Exponential, $W\left(N\right)={k}^{N}$:$$\varphi (x,y)=x+y.$$
- (III)
- Super-exponential, $W\left(N\right)={N}^{\gamma N}$:$$\varphi (x,y)=\lambda \left(\right)open="\{"\; close="\}">exp\left(\right)open="["\; close="]">L((1+\frac{x}{\lambda})ln(1+\frac{x}{\lambda})+(1+\frac{y}{\lambda})ln(1+\frac{y}{\lambda}))$$

## 4. Maximum Entropy Ensembles

#### 4.1. Trace-Form Entropies

- (I)
- Algebraic – $W\left(N\right)={N}^{a}$:$$\frac{\delta S}{\delta {p}_{i}}=a\left(\right)open="("\; close=")">1-\frac{1}{a}.$$
- (II)
- Exponential – $W\left(N\right)={k}^{N}$:$$\frac{\delta S}{\delta {p}_{i}}=\left(\right)open="("\; close=")">ln\frac{1}{{p}_{i}}-1$$
- (III)
- Super-exponential – $W\left(N\right)={N}^{\gamma N}$:$$\frac{\delta S}{\delta {p}_{i}}=exp\left(\right)open="["\; close="]">L({X}_{i})$$

- (I)
- Algebraic – $W\left(N\right)={N}^{a}$:$${p}_{i}=\frac{{[1+\beta (\Delta {E}_{i}-\overline{E})]}^{-a}}{Z},$$
- (II)
- Exponential – $W\left(N\right)={k}^{N}$:$${p}_{i}=\frac{exp[-\beta (\Delta {E}_{i}-\overline{E})]}{Z}$$

#### 4.2. Non-Trace-Form Entropies

## 5. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References and Notes

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Jeldtoft Jensen, H.; Tempesta, P.
Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory. *Entropy* **2018**, *20*, 804.
https://doi.org/10.3390/e20100804

**AMA Style**

Jeldtoft Jensen H, Tempesta P.
Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory. *Entropy*. 2018; 20(10):804.
https://doi.org/10.3390/e20100804

**Chicago/Turabian Style**

Jeldtoft Jensen, Henrik, and Piergiulio Tempesta.
2018. "Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory" *Entropy* 20, no. 10: 804.
https://doi.org/10.3390/e20100804