# Entropy

"Energy has to do with possibilities. Entropy has to do with the probabilities of those possibilities happening. It takes energy and performs a further epistemological step." Constantino Tsallis [1]

Entropy was first described by Rudolf Julius Emanuel Clausius in 1865 [2]. The statistical mechanical desciption is due to Ludwig Eduard Boltzmann (Ref. ?). The word entropy originated from the Greek word meaning a turning or transformation "τροπή" [3].

## Classical thermodynamics

In classical thermodynamics one has the entropy, ${\displaystyle S}$,

${\displaystyle {\mathrm {d} }S={\frac {\delta Q_{\mathrm {reversible} }}{T}}}$

where ${\displaystyle Q}$ is the heat and ${\displaystyle T}$ is the temperature.

## Statistical mechanics

In statistical mechanics entropy is defined by

${\displaystyle \left.S\right.:=-k_{B}\sum _{i=1}^{W}p_{i}\ln p_{i}}$

where ${\displaystyle k_{B}}$ is the Boltzmann constant, ${\displaystyle i}$ is the index for the microstates, and ${\displaystyle p_{i}}$ is the probability that microstate i is occupied. In the microcanonical ensemble this gives:

${\displaystyle \left.S\right.=k_{B}\ln W}$

where ${\displaystyle W}$ (sometimes written as ${\displaystyle \Omega }$) is the number of microscopic configurations that result in the observed macroscopic description of the thermodynamic system. This equation provides a link between classical thermodynamics and statistical mechanics

## Tsallis entropy

Tsallis (or non-additive) entropy [4] is defined as (Eq. 1)

${\displaystyle S_{q}:=k_{B}{\frac {1-\sum _{i=1}^{W}p_{i}^{q}}{q-1}}}$

where ${\displaystyle q}$ is the Tsallis index [5]. As ${\displaystyle q\rightarrow 1}$ one recovers the standard expression for entropy. This expression for the entropy is the cornerstone of non-extensive thermodynamics.

## Arrow of time

Articles:

Books:

• Steven F. Savitt (Ed.) "Time's Arrows Today: Recent Physical and Philosophical Work on the Direction of Time", Cambridge University Press (1997) ISBN 0521599458
• Michael C. Mackey "Time's Arrow: The Origins of Thermodynamic Behavior" (1992) ISBN 0486432432
• Huw Price "Time's Arrow and Archimedes' Point New Directions for the Physics of Time" Oxford University Press (1997) ISBN 978-0-19-511798-1