Entropy units9/2/2023 ![]() ![]() Wojtkowski, "Measure theoretic entropy of the system of hard spheres" Ergod. Walters, "An introduction to ergodic theory", Springer (1982) For several useful recent references concerning the computation of entropy, see. 288–300Įntropy the term sequence entropy is used in the English literature. Ruelle, "Mean entropy of states in classical statistical mechanics" Comm. Mañé, "A proof of Pesin's formula" Ergod. ![]() Pesin, "Characteristic Lyapunov exponents, and smooth ergodic theory" Russian Math. Millionshchikov, "A formula for the entropy of smooth dynamical systems" Differential Eq. ![]() Sucheston, "On convergence of information in spaces with infinite invariant measure" Z. Ionesco-Tulcea, "Contributions to information theory for abstract alphabets" Arkiv for Mat. Pitskeĺ, "Nonuniform distribution of entropy for processes with a countable set of states" Probl. Chung, "A note on the ergodic theorem of information theory" Ann. Breiman, "Correction to "The individual ergodic theorem of information theory" " Ann. Breiman, "The individual ergodic theorem of information theory" Ann. Kushnirenko, "Metric invariants of entropy type" Russian Math. Krengel, "Entropy of conservative transformations" Z. Kieffer, "A generalized Shannon–McMillan theorem for the action of an amenable group on a probability space" Ann. Safonov, "Information parts in groups" Math. Billingsley, "Ergodic theory and information", Wiley (1965)Ī.V. Rokhlin, "Lectures on the entropy theory of transformations with invariant measure" Russian Math. Sinai, "On the notion of entropy of dynamical systems" Dokl. Then we will derive the entropy formula for ideal gas, ' S(N V E) NkB ln 4 mE 3Nh2 32 + 5 from the microcanonical (NV E) ensemble. Kolmogorov, "On entropy per unit time as a metric invariant of automorphisms" Dokl. 2 3 6 6 10 In this lecture, we will rst discuss the relation between entropy and irreversibility. Kolmogorov, "A new metric invariant of transitive dynamical systems, and Lebesgue space automorphisms" Dokl. Topological dynamical system) new concepts such as "Gibbsian measures", the "topological pressure" (an analogue to the free energy) and the "variational principle" for the latter (see the references to $ Y $-Ī.N. The analogy with statistical physics was one of the stimuli for introducing in ergodic theory (even in a not-purely metric context and for topological dynamical systems, cf. The name "entropy" is explained by the analogy between the entropy of dynamical systems and that in information theory and statistical physics, right up to the fact that in certain examples these entropies are the same (see, for example,, ). where p is the pressure and V is the volume of the gas. Substituting for the definition of work for a gas. where E is the internal energy and W is the work done by the system. We begin by using the first law of thermodynamics: dE dQ - dW. )įor smooth dynamical systems with a smooth invariant measure a connection has been established between the entropy and the Lyapunov characteristic exponent of the equations in variations (see – ). For gases, there are two possible ways to evaluate the change in entropy. Has been proved for a certain general class of transformation groups. See Metric isomorphism) of a Lebesgue space $ ( X, \mu ) $.įor any finite measurable decomposition (measurable partition) $ \xi $ Basic is the concept of the entropy $ h ( S) $ Send us feedback about these examples.One of the most important invariants in ergodic theory. These examples are programmatically compiled from various online sources to illustrate current usage of the word 'entropy.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Sebastian Smee, Washington Post, After seven years in the Hammerskins, exhaustion and entropy were setting in. 2022 Succumbing to something closer to entropy than evolution, the watercolor starts to puddle and bruise. Ahmed Almheiri, Scientific American, 17 Aug. 2021 This is the island formula for the entanglement entropy of the Hawking radiation. ![]() Conor Feehly, Discover Magazine, 3 Nov. James Riordon, Scientific American, In short, the tendency for systems to move from low entropy to high entropy, the particular spacetime conditions of our solar system and the indeterminacy of the future combine to create our particular conception of time. James Riordon, Scientific American, The expansion allows the universe to smooth out, dissipating the entropy before collapsing again. Quanta Magazine, That flaw is entropy, which builds up as a universe bounces. 2022 If the entropy of the system decreases, the entropy of the environment must increase such that the sum of the two entropies can only increase or stay the same, but never decrease. Jennifer Ouellette, Ars Technica, 2 Dec. Recent Examples on the Web Jacob Bekenstein realized in 1974 that black holes also have entropy. ![]()
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