Effective Gibbs State for Averaged Observables
Abstract
:1. Introduction
2. Effective Hamiltonian
- is completely positive.
- is a self-adjoint (with respect to trace scalar product ) projector
- Let the spectral decomposition of have the form , where are (distinct) eigenvalues of and are orthogonal projectors , . Then,
3. Effective Hamiltonian as Analog of Mean Force Hamiltonian
4. Mean Force Hamiltonian for Effective Gibbs State
5. Examples
5.1. Two Interacting Two-Level Systems
5.2. Two Interacting Harmonic Oscillators
5.3. Two-Level System Interacting with Harmonic Oscillator
6. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
RWA | Rotating Wave Approximation |
Appendix A. Properties of Averaging Projector
Appendix B. Perturbative Expansion for Effective Hamiltonian
Appendix C. Eigenprojector Expansion
Appendix D. Average of Second Correction with Respect to Gibbs State for Free Hamiltonian
Appendix E. Perturbative Expansion of Mean Force Hamiltonian for Effective Gibbs State
Appendix F. Calculation of Mean Force Hamiltonian
Appendix G. Calculations for the Examples
Appendix G.1. Two Two-Level Systems
Appendix G.2. Two Oscillators
Appendix G.3. Two-Level System and Oscillator
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Teretenkov, A.E. Effective Gibbs State for Averaged Observables. Entropy 2022, 24, 1144. https://doi.org/10.3390/e24081144
Teretenkov AE. Effective Gibbs State for Averaged Observables. Entropy. 2022; 24(8):1144. https://doi.org/10.3390/e24081144
Chicago/Turabian StyleTeretenkov, Alexander Evgen’evich. 2022. "Effective Gibbs State for Averaged Observables" Entropy 24, no. 8: 1144. https://doi.org/10.3390/e24081144
APA StyleTeretenkov, A. E. (2022). Effective Gibbs State for Averaged Observables. Entropy, 24(8), 1144. https://doi.org/10.3390/e24081144