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Derivation of the density operator with quantum analysis for the generalized Gibbs ensemble in quantum statistics

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  • Ishihara, Masamichi

Abstract

We derive the equation of the density operator for generalized entropy and generalized expectation value with quantum analysis when conserved quantities exist. The derived equation is simplified when the conventional expectation value is employed. The derived equation is also simplified when the commutation relations, [ρˆ,Hˆ] and [ρˆ,Qˆ[a]], are the functions of the density operator ρˆ, where Hˆ is the Hamiltonian, and Qˆ[a] is the conserved quantity. We derive the density operators for the von Neumann, Tsallis, and Rényi entropies in the case of the conventional expectation value. We also derived the density operators for the Tsallis and Rényi entropies in the case of the escort average (the normalized q-expectation value), when the density operator commutes with the Hamiltonian and the conserved quantities. We find that the argument of the density operator for the canonical ensemble is simply extended to the argument for the generalized Gibbs ensemble in the case of the conventional expectation value, even when conserved quantities do not commute. The simple extension of the argument is also shown in the case of the escort average, when the density operator ρˆ commutes with the Hamiltonian Hˆ and the conserved quantity Qˆ[a]: [ρˆ,Hˆ]=[ρˆ,Qˆ[a]]=0. These findings imply that the argument of the density operator for the canonical ensemble is simply extended to the argument for the generalized Gibbs ensemble in some systems.

Suggested Citation

  • Ishihara, Masamichi, 2021. "Derivation of the density operator with quantum analysis for the generalized Gibbs ensemble in quantum statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 583(C).
    • Handle: RePEc:eee:phsmap:v:583:y:2021:i:c:s037843712100594x
    DOI: 10.1016/j.physa.2021.126321
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    References listed on IDEAS

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    1. Abe, Sumiyoshi, 2001. "Heat and entropy in nonextensive thermodynamics: transmutation from Tsallis theory to Rényi-entropy-based theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 300(3), pages 417-423.
    2. Masuo Suzuki, 1999. "Computationics And Quantum Analysis," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 10(08), pages 1385-1390.
    3. Tsallis, Constantino & Mendes, RenioS. & Plastino, A.R., 1998. "The role of constraints within generalized nonextensive statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 261(3), pages 534-554.
    4. Ishihara, Masamichi, 2020. "Derivation of density operators for generalized entropies with quantum analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 543(C).
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