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

Entropy was first described by Rudolf Julius Emanuel Clausius in 1865 . The statistical mechanical desciption is due to Ludwig Eduard Boltzmann (Ref. ?).

Classical thermodynamics

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathrm d} S = \frac{\delta Q_{\mathrm {reversible}}} {T} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} is the heat and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is the temperature.

Statistical mechanics

In statistical mechanics entropy is defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. S \right. := -k_B \sum_{i=1}^W p_i \ln p_i}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B} is the Boltzmann constant, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} is the index for the microstates, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_i} is the probability that microstate i is occupied. In the microcanonical ensemble this gives:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.S\right. = k_B \ln W}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} (sometimes written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 is defined as (Eq. 1)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_q:= k_B \frac{1-\sum_{i=1}^W p_i^q}{q-1}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} is the Tsallis index . As Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

• T. Gold "The Arrow of Time", American Journal of Physics 30 pp. 403-410 (1962)
• Joel L. Lebowitz "Boltzmann's Entropy and Time's Arrow", Physics Today 46 pp. 32-38 (1993)
• Milan M. Ćirković "The Thermodynamical Arrow of Time: Reinterpreting the Boltzmann–Schuetz Argument", Foundations of Physics 33 pp. 467-490 (2003)

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

See also:

• Entropy of a glass
• H-theorem
• Non-extensive thermodynamics
• Shannon entropy

References

Related reading

• Karl K. Darrow "The Concept of Entropy", American Journal of Physics 12 pp. 183-196 (1944)
• E. T. Jaynes "Gibbs vs Boltzmann Entropies", American Journal of Physics 33 pp. 391-398 (1965)
• Daniel F. Styer "Insight into entropy", American Journal of Physics 86 pp. 1090-1096 (2000)
• S. F. Gull "Some Misconceptions about Entropy" in Brian Buck and Vincent A. MacAulay (Eds.) "Maximum Entropy in Action", Oxford Science Publications (1991)
• Efstathios E. Michaelides "Entropy, Order and Disorder", The Open Thermodynamics Journal 2 pp. (2008)
• Ya. G. Sinai, "On the Concept of Entropy of a Dynamical System," Doklady Akademii Nauk SSSR 124 pp. 768-771 (1959)
• William G. Hoover "Entropy for Small Classical Crystals", Journal of Chemical Physics 49 pp. 1981-1982 (1968)
• Arieh Ben-Naim "Entropy Demystified: The Second Law Reduced to Plain Common Sense", World Scientific (2008) ISBN 978-9812832252
• Arieh Ben-Naim "Farewell to Entropy: Statistical Thermodynamics Based on Information", World Scientific (2008) ISBN 978-981-270-707-9
• Arieh Ben-Naim "Discover Entropy and the Second Law of Thermodynamics: A Playful Way of Discovering a Law of Nature" World Scientific Publishing (2010) ISBN: 978-981-4299-75-6
• Arieh Ben-Naim "Entropy and the Second Law Interpretation and Misss-Interpretations", World Scientific Publishing (2012) ISBN 978-981-4407-55-7
• Arieh Ben-Naim "Information, Entropy, Life and the Universe: What We Know and What We Do Not Know" World Scientific Publishing (2015) ISBN 978-981-4651-66-0
• Jose M. G. Vilar and J. Miguel Rubi "System-size scaling of Boltzmann and alternate Gibbs entropies", Journal of Chemical Physics 140 201101 (2014)

External links

• entropy an international and interdisciplinary Open Access journal of entropy and information studies.
• Joel L. Lebowitz "Time's arrow and Boltzmann's entropy", Scholarpedia, 3(4):3448 (2008)