Shannon's Entropy from Dynamic Considerations?
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| Format: | Recurso digital |
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Zenodo
2025
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| _version_ | 1866901768685748224 |
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| author | Ruggeri, Francesco R. |
| author_facet | Ruggeri, Francesco R. |
| contents | <p dir="ltr"> In a number of previous notes (1),(2), we have argued that the P(x,y,z) = 1/V volume probability for an ideal gas may be obtained through deterministic Newtonian scenarios for which the time measurement has been removed. In other words, one does not need the idea of states or maximization of entropy to find this probability. We have also argued that the Maxwell-Boltzmann Cexp(-ei/T) may be obtained through a consideration of collisions with energy conservation. These approaches only find the probability, not Shannon’s entropy.</p> <p dir="ltr"> Here we try to find the form of Shannon’s entropy through dynamic considerations. In particular, we use the notion of a function d F (1/V) matching - P dV. Pressure may be obtained from dynamic considerations of an ideal gas and involves the factor 1/V. Thus, -PdV yields dV/V and F should be ln. Now, ln(1/L) = Sum over dx/L ln(1/L) = ln(1/L) and so is equivalent to an average of ln(1/L), the probability. A similar result should hold for p(ei) such that “d” of this function yields dE. We show this is the case, i.e. that one uses: - Sum over i p(ei) ln(p(ei)). Thus, it seems that one does not have to consider factorial expressions and various state arrangements, but may obtain Shannon’s entropy directly from a dynamic expression, albeit one associated with an equilibrium gas.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15376328 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Shannon's Entropy from Dynamic Considerations? Ruggeri, Francesco R. <p dir="ltr"> In a number of previous notes (1),(2), we have argued that the P(x,y,z) = 1/V volume probability for an ideal gas may be obtained through deterministic Newtonian scenarios for which the time measurement has been removed. In other words, one does not need the idea of states or maximization of entropy to find this probability. We have also argued that the Maxwell-Boltzmann Cexp(-ei/T) may be obtained through a consideration of collisions with energy conservation. These approaches only find the probability, not Shannon’s entropy.</p> <p dir="ltr"> Here we try to find the form of Shannon’s entropy through dynamic considerations. In particular, we use the notion of a function d F (1/V) matching - P dV. Pressure may be obtained from dynamic considerations of an ideal gas and involves the factor 1/V. Thus, -PdV yields dV/V and F should be ln. Now, ln(1/L) = Sum over dx/L ln(1/L) = ln(1/L) and so is equivalent to an average of ln(1/L), the probability. A similar result should hold for p(ei) such that “d” of this function yields dE. We show this is the case, i.e. that one uses: - Sum over i p(ei) ln(p(ei)). Thus, it seems that one does not have to consider factorial expressions and various state arrangements, but may obtain Shannon’s entropy directly from a dynamic expression, albeit one associated with an equilibrium gas.</p> |
| title | Shannon's Entropy from Dynamic Considerations? |
| url | https://doi.org/10.5281/zenodo.15376328 |