Entropy

Entropy was first described by Rudolf Julius Emanuel Clausius in 1865 (Ref. 1). 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 the entropy, S, is defined by

$\left.S\right.=-k_{B}\sum _{m}p_{m}\ln p_{m}$

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

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

where $\Omega$ (sometimes written as $W$) 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

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
• Shannon entropy
• Tsallis entropy
• H-theorem

Interesting 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)

References

1. R. Clausius "Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie", Annalen der Physik und Chemie 125 pp. 353-400 (1865)
2. Ya. G. Sinai, "On the Concept of Entropy of a Dynamical System," Doklady Akademii Nauk SSSR 124 pp. 768-771 (1959)
3. William G. Hoover "Entropy for Small Classical Crystals", Journal of Chemical Physics 49 pp. 1981-1982 (1968)

Classical thermodynamics