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Solution of the fractional Liouville equation by using Riemann–Liouville and Caputo derivatives in statistical mechanics

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Abstract

We solve the fractional Liouville equation by using Riemann–Liouville and Caputo derivatives for systems exhibiting noninteger power laws in their Hamiltonians. Based on the fractional Liouville equation, we calculate the density function (DF) of a classical ideal gas. If the Riemann–Liouville derivative is used, the DF is a function depending on both the momentum \(p\) and the coordinate \(q\), but if the derivative in the Caputo sense is used, the DF is a constant independent of \(p\) and \(q\). We also study a gas consisting of \(N\) fractional oscillators in one-dimensional space and obtain that the DF of the system depends on the type of the derivative.

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Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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Authors and Affiliations

  1. Department of Exact Sciences, École Normale Supérieure de Ouargla, Ouargla, Algeria

    Z. Korichi & A. Souigat

  2. Department of Physics, Kasdi Merbah University, Ouargla, Algeria

    R. Bekhouche & M. T. Meftah

Authors
  1. Z. Korichi
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  2. A. Souigat
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  3. R. Bekhouche
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  4. M. T. Meftah
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Correspondence to Z. Korichi.

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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2024, Vol. 218, pp. 389–399 https://doi.org/10.4213/tmf10545.

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Korichi, Z., Souigat, A., Bekhouche, R. et al. Solution of the fractional Liouville equation by using Riemann–Liouville and Caputo derivatives in statistical mechanics. Theor Math Phys 218, 336–345 (2024). https://doi.org/10.1134/S0040577924020107

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  • DOI: https://doi.org/10.1134/S0040577924020107

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