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Entropy

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Thermodynamics and statistical mechanics are among the most important formalisms in contemporary physics. They have overwhelming and intertwinedapplications in science and technology. They essentially rely on two basic concepts, namely energy and entropy . The mathematical expression that is used for the first one is well known to be nonuniversal; indeed, it depends on whether we are say in classical, quantum, or relativistic regimes. The second concept, andvery specifically its connection with the microscopic world, has been considered during well over one century as essentially unique and universalas a physical concept. Although some mathematical generalizations of the entropy have been proposed during thelast forty years, they have frequently been considered as mere practical expressions for disciplines such as cybernetics and control theory, with noparticular physical interpretation. What we have witnessed during the last two decades is the growth, among...

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Abbreviations

Absolute temperature:

Denoted T.

Clausius entropy:

Also called thermodynamic entropy. Denoted S.

Boltzmann–Gibbs entropy:

Basis of Boltzmann–Gibbs statistical mechanics. This entropy, denoted \( { S_\mathrm{BG} } \), is additive. Indeed, for two probabilistically independent subsystems A and B, it satisfies \( S_\mathrm{BG}(A+B)= S_\mathrm{BG}(A) + S_\mathrm{BG}(B) \).

Nonadditive entropy:

It usually refers to the basis of nonextensive statistical mechanics. This entropy, denoted S q , is nonadditive for \( { q \ne 1 } \). Indeed, for two probabilistically independent subsystems A and B, it satisfies \( { S_{q}(A+B)\ne S_{q}(A)+S_{q}(B) \;(q \ne 1) } \). For historical reasons, it is frequently (but inadequately) referred to as nonextensive entropy.

q‑logarithmic and q‑exponential functions:

Denoted \( { \ln_q x\, (\ln_1 x = \ln x) } \), and \( { \text{e}_q^x\, (\text{e}_1^x = \text{e}^x) } \), respectively.

Extensive system:

So called for historical reasons. A more appropriate name would be additive system. It is a system which, in one way or another, relies on or is connected to the (additive) Boltzmann–Gibbs entropy. Its basic dynamical and/or structural quantities are expected to be of the exponential form. In the sense of complexity, it may be considered a simple system.

Nonextensive system:

So called for historical reasons. A more appropriate name would be nonadditive system. It is a system which, in one way or another, relies on or is connected to a (nonadditive) entropy such as \( { S_q (q \ne 1) } \). Its basic dynamical and/or structural quantities are expected to asymptotically be of the power-law form. In the sense of complexity, it may be considered a complex system.

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Acknowledgments

Among the very many colleagues towards which I am deeply grateful for profound and long‐lasting comments along many years, it is a must to explicitly thank S. Abe, E.P. Borges, E.G.D. Cohen, E.M.F. Curado, M. Gell-Mann, R.S. Mendes, A. Plastino, A.R. Plastino, A.K. Rajagopal, A. Rapisarda and A. Robledo.

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  1. Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, Brazil

    Constantino Tsallis

  2. Santa Fe Institute, Santa Fe, USA

    Constantino Tsallis

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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Tsallis, C. (2009). Entropy. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_172

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