## Tsallis EntropyOctober 29, 2014 Thermodynamics has developed from contributions from scientists of many nationalities. As just a few examples, we have Sadi Carno (1796-1832) from France; Robert Boyle (1627-1691) and William Thomson, Lord Kelvin (1824-1907) from Ireland; James Prescott Joule (1818-1889) from England; Rudolf Clausius (1822-1888) from Germany; James Clerk Maxwell (1831-1879) from Scotland; Josiah Willard Gibbs (1839-1903) from the United States; and Ludwig Boltzmann (1844-1906) from Austria).
S is the entropy, K is the Boltzmann constant (1.38062 x 10_{B}^{-23} joule/kelvin), and Ω is the number of states accessible to the system. This equation is actually a simplification of the Boltzmann-Gibbs entropy when every state has the same probability p; viz,
_{i}S, given by the following equation:
_{q}q, is called the entropic-index. This equation becomes the Boltzmann-Gibbs entropy equation when we take the limit as q approaches one. It's been a while since I did limits in Calculus, so I'll just take this on faith.
There are presently 69 papers posted on arXiv having "Tsallis entropy" in their titles. What's so important about Tsallis entropy? Boltzmann entropy applies only to systems in equilibrium, and the Boltzmann entropy is an extensive function; that is, it depends on how much matter we have in our system. In non-equilibrium systems we need another way to look at entropy, and now we have Tsallis entropy, which is non-extensive. As can be imagined, a lot of people have a problem with a non-extensive entropy.
q acts merely as a fitting parameter. There's also an argument that Tsallis entropy violates the zeroth law of thermodynamics. That's the law that states that systems in thermal equilibrium with another system are in equilibrium with each other.[3]
In the end, it appears that physicists just need to know whether to apply one entropy or another to a given system. It's not unlike knowing that you should use classical mechanics for bowling balls and quantum mechanics for atoms.[3]
## References:- Constantino Tsallis, "Possible generalization of Boltzmann-Gibbs statistics," Journal of Statistical Physics, vol. 52, nos. 1-2 (July, 1988), pp. 479-487.
- H. V. Ribeiro, E. K. Lenzi, R. S. Mendes, G. A. Mendes, and L. R. da Silva, "Symbolic Sequences and Tsallis Entropy," arXiv Preprint Server, January 16, 2014.
- Jon Cartwright, "Roll over, Boltzmann," Physics World, May, 2014, pp. 31-35.
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