Internal Energy, Fundamental Thermodynamic Relation, and Gibbs' Ensemble Theory as Laws of Statistical Counting
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866915097697320960 |
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| author | Qian, Hong |
| author_facet | Qian, Hong |
| contents | Counting ad infinitum is the holographic observable to a statistical dynamics with finite states under independent repeated sampling. Entropy provides the infinitesimal probability for an observed frequency $\hat{\boldsymbolν}$ w.r.t. a probability prior ${\bf p}$. Following Callen's postulate and through Legendre-Fenchel transform, without help from mechanics, we show an internal energy $\boldsymbolμ$ 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. $\boldsymbolμ$ is to $\hat{\boldsymbolν}$ what $ω$ is to $t$ in Fourier analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_14337 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Internal Energy, Fundamental Thermodynamic Relation, and Gibbs' Ensemble Theory as Laws of Statistical Counting Qian, Hong Statistical Mechanics Probability Counting ad infinitum is the holographic observable to a statistical dynamics with finite states under independent repeated sampling. Entropy provides the infinitesimal probability for an observed frequency $\hat{\boldsymbolν}$ w.r.t. a probability prior ${\bf p}$. Following Callen's postulate and through Legendre-Fenchel transform, without help from mechanics, we show an internal energy $\boldsymbolμ$ 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. $\boldsymbolμ$ is to $\hat{\boldsymbolν}$ what $ω$ is to $t$ in Fourier analysis. |
| title | Internal Energy, Fundamental Thermodynamic Relation, and Gibbs' Ensemble Theory as Laws of Statistical Counting |
| topic | Statistical Mechanics Probability |
| url | https://arxiv.org/abs/2209.14337 |