Internal Energy, Fundamental Thermodynamic Relation, and Gibbs’ Ensemble Theory as Emergent Laws of Statistical Counting
Statistical counting <i>ad infinitum</i> is the holographic observable to a statistical dynamics with finite states under independent and identically distributed <i>N</i> sampling. Entropy provides the infinitesimal probability for an observed empirical frequency <inline-f...
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
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| Series: | Entropy |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1099-4300/26/12/1091 |
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| Summary: | Statistical counting <i>ad infinitum</i> is the holographic observable to a statistical dynamics with finite states under independent and identically distributed <i>N</i> sampling. Entropy provides the infinitesimal probability for an observed empirical frequency <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula> with respect to a probability prior <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">p</mi></semantics></math></inline-formula>, when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">^</mo></mover><mo>≠</mo><mi mathvariant="bold">p</mi></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. Following Callen’s postulate and through Legendre–Fenchel transform, without help from mechanics, we show that an internal energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">u</mi></semantics></math></inline-formula> emerges; it provides a linear representation of real-valued observables with full or partial information. Gibbs’ fundamental thermodynamic relation and theory of ensembles follow mathematically. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">u</mi></semantics></math></inline-formula> is to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula> what chemical potential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> is to particle number <i>N</i> in Gibbs’ chemical thermodynamics, what <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>=</mo><msup><mi>T</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></semantics></math></inline-formula> is to internal energy <i>U</i> in classical thermodynamics, and what <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> is to <i>t</i> in Fourier analysis. |
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| ISSN: | 1099-4300 |