The paper dealt with quantum canonical ensembles by random walks, where state transitions are triggered by the connections between labels, not by elements, which are transferred. The balance conditions of such walks lead to emission rates of the labels. The labels with emission rates definitely lower than 1 are like modes. For labels with emission rates very close to 1, the quantum numbers are concentrated around a mean value. As an application I consider the role of the zero label in a quantum gas in equilibrium.
References
[1]
Huang, K. (1987) Statistical Mechanics. 2nd Edition, John Wiley & Sons, New York.
[2]
Kittel, C. and Kroemer, H. (1989) Physik der Wärme. 3rd Edition, R. Oldenbourg, Munic, Oldenburg.
[3]
Baez, J.C. and Biamonte, J.D. (2012) A Course on Quantum Techniques for Stochastic Mechanics. arXiv:1209.3632v1 [quant-ph].
[4]
Tuckerman, M.E. (2010) Statistical Mechanics: Theory and Molecular Simulation. Oxford University Press, Oxford.
[5]
Wolschin, G. (2018) Physica A, 499, 1-10. https://doi.org/10.1016/j.physa.2018.01.035
[6]
Kempe, J. (2003) Quantum Random Walks—An Introductory Overview. arXiv:quant-ph/0303081v1.
[7]
Miranda, E.N. (2015) Journal of Modern Physics, 6, 1051-1057. https://doi.org/10.4236/jmp.2015.68109
[8]
Einstein, A. (1917) On the Quantum Theory of Radiation. Physikalische Zeitschrift Bd. 18.
[9]
Plischke, M. and Bergerson, B. (2005) Equilibrium Statistical Physics. 3rd Edition, World Scientific Publishing, Hackensack.
[10]
Davies, E.B. (1972) Communications in Mathematical Physics, 28, 69-86. https://doi.org/10.1007/BF02099372
[11]
Bianconi, G., Ferretti, L. and Franz, S. (2009) Non-Neutral Theory of Biodiversity. Europhysics Letters, 87, P07028. https://doi.org/10.1209/0295-5075/87/28001
[12]
Damm, T., Schmitt, J., Qi Liang, D. Dung, F. Vewinger, M. Weitz and J. Klaers, (2016) Nature Communications, 7, Article Number: 11340. https://doi.org/10.1038/ncomms11340
[13]
Klaers, J., Schmitt, J., Vewinger, F. and Weitz, M. (2010) Nature, 468, 545-548.
