The Tsallis Entropic Index q as a Measure of Distance from Thermal Equilibrium in Paramagnetic Spin Lattices

For super–systems <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">C</italic> + <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://ww...

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Detalles Bibliográficos
Autores principales:	Masthay, Mark, Benton, Charles, Guerrieri, Luke, Keil, Robert
Formato:	Capítulo de libro
Lenguaje:	inglés
Publicado:	IntechOpen 2024
Materias:	Physical Sciences, Engineering and Technology
Acceso en línea:	https://www.intechopen.com/chapters/89411
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

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Sumario:	<p id="p1">For super–systems <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">C</italic></em> + <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">D</italic></em> comprised of combinations of sub–systems <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">C</italic></em> and <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">D</italic></em> which obey Boltzmann thermo–statistics, the entropy <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">S</italic></em> is additive (<em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">S<sub>C</sub></italic></em> + <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">S<sub>D</sub></italic></em> = <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">S</italic></em><sub><em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">C</italic></em> + <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">D</italic></em></sub>) and extensive and the temperature <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">T</italic></em> is intensive. However, because of finite–size effects, the entropy becomes non–additive and non–extensive, and the temperature non–intensive, for very small systems. In such cases, the Tsallis entropic index <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">q</italic></em> quantifies the extent to which the entropy is non–additive and the temperature is non–intensive. In this paper, we use paramagnetic spin lattices (PSLs) as model systems to demonstrate that <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">q</italic></em> is not only a measure of entropic non–extensivity and temperature non–intensivity, but also the extent to which sub–PSL/super–PSL combinations of various sizes deviate from the global thermal equilibrium condition <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">T<sub>C</sub></italic></em> = <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">T<sub>D</sub></italic></em> = <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">T</italic></em><sub><em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">C</italic></em> + <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">D</italic></em></sub>. Our results demonstrate that <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">q</italic></em> → 1 as global thermal equilibrium is approached regardless of system size, thus indicating that <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">q</italic></em> is an effective measure of distance from equilibrium.</p>
ISBN:	978-0-85014-099-6
DOI:	10.5772/intechopen.114829

The Tsallis Entropic Index <em>q</em> as a Measure of Distance from Thermal Equilibrium in Paramagnetic Spin Lattices

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