保存先:
| 第一著者: | |
|---|---|
| フォーマット: | Recurso digital |
| 言語: | |
| 出版事項: |
Zenodo
2025
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| オンライン･アクセス: | https://doi.org/10.5281/zenodo.15801410 |
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目次:
- <p dir="ltr">Probability is a measure of the uncertainty of a result, but we argue that there is more involved in the case of the Maxwell-Boltzmann, Tsallis and Kaniadakis distributions. In fact, it is not correct to simply list a Kaniadakis distribution because there are both expk and Gaussian-k distributions, which is directly related to the topic of this note. In particular, we suggest that probability distribution, instead of simply being described as p(ei/T), is actually a function of the conserved quantity in the problem, although matters are more complicated than this in the Kaniadkis case. Thus, it is no accident that the Maxwell-Boltzmann distribution, or the Tsallis exp-q or Kaniadakis exp-k involve linear expressions in -ei/T to a power (in the latter two cases). </p> <p dir="ltr">The Kaniadkis exp-k= (sqrt(1+kkeiei/TT) -kei/T) power 1/k and so it is the -kei/T term which demonstrates the conserved quantity, but we will show how all three approaches identify with a conserved quantity.The a priori conserved quantity is E = Sum over i ei p(ei) or E= Sum over i p(ei) power q ei in the Tsallis case. If the conserved quantity <EE>, then ei*ei is used and leads to a different distribution.</p> <p dir="ltr"> This leads to the next question. Probability p(ei) represents the uncertainty associated with ei, and one often wishes to have maximum uncertainty. For example, a uniform distribution, by not distinguishing between any specific outcomes, maximizes uncertainty, but there is no concept of a conserved quantity. We have argued that one requires p(ei) to explicitly show the conserved quantity in its functional form and so probability unfortunately cannot be the sole measure of uncertainty. In fact, it carries information about the conserved quantity, be it ei or ei*ei or some other moment or function f(ei). Thus, it seems that one is forced to introduce a second uncertainty function which should be maximized (remove as much information), but should be subject to the two constraints Sum over i p(ei) =1 and Sum over i f(ei) p(ei), where f(ei) is the constraint function, be it ei or ei*ei etc. The key assumption which seems to arise is that instead of having F( all p(ei)’s), one has an uncertainty function which is additive. I.e. Sum over i g(p(ei)) so that one may link each g(p(ei)) with the f(ei) information. In other words, the idea of this maximization is to treat each dp(ei) as independent in a sense in that that the overall coefficient of dp(ei) is 0 for each i even though Sum dp(ei)=0 meaning that in general dp(ei)’s are not independent. This approach allows one to force p(ei) to be a function of the constraint variable ei or ei*ei etc or function f(ei), while at the same time maximizing a loss of information.</p> <p dir="ltr"> In the Maxwell-Boltzmann,Tsallis and Kaniadakis cases, this approach leads to writing g(p(ei) = p(ei) h(p(ei) (which is general because p(ei)>0). We argue that this step is key to forcing p(ei) to include the form of the conserved quantity f(ei), but that is not the whole picture. A second key idea of these two distributions is that they match d/dp(ei) g(p(ei)) dp(ei) directly to the constraints number=Sum over i f(ei) p(ei) (or p(ei) power q in the Tsallis case) and Sum over i p(ei) = 1. The Kaniadakis case also makes use of g(p(ei) = p(ei)h(p(ei)) and also links the first derivative term: dp h(p(ei) to the constraint Sum over i f(ei) p(ei) = number, but does not link the second derivative term, p dh/dp to these two constraints, showing that one may functions: Sum over i q(p(ei) dp(ei) =0 without this being a linear combination of: Sum over i (c1 ei + c2) dp(ei). In terms of the probability demonstrating the f(ei) constraint piece, this is enforced in all three cases by h(p(ei)) = f(ei)*Lagrange multiplier where Lagrange multiplier = -1/T.</p> <p> </p>