# 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. ?).

## Classical thermodynamics

In classical thermodynamics one has the entropy, $S$,

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

where $Q$ is the heat and $T$ is the temperature.

## Statistical mechanics

In statistical mechanics entropy is defined by

$\left. S \right. := -k_B \sum_{i=1}^W p_i \ln p_i$

where $k_B$ is the Boltzmann constant, $i$ is the index for the microstates, and $p_i$ is the probability that microstate i is occupied. In the microcanonical ensemble this gives:

$\left.S\right. = k_B \ln W$

where $W$ (sometimes written as $\Omega$) 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 [3] is defined as (Eq. 1)

$S_q:= k_B \frac{1-\sum_{i=1}^W p_i^q}{q-1}$

where $q$ is the Tsallis index [4]. As $q \rightarrow 1$ 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