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Dependence of the existence of thermal equilibrium on the number of particles at low temperatures
1.L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed. (Butterworth-Heinemann, Singapore, 1999), Chap. XII.
2.B. B. Mandelbrot, “Temperature fluctuations: A well-defined and unavoidable notion,” Phys. Today0031-9228, 42(1), 71–73 (1989) and references therein.
3.G. D. Phillies, “The polythermal ensemble: A rigorous interpretation of temperature fluctuations in statistical mechanics,” Am. J. Phys.0002-9505 52, 629–632 (1984). It is the first rigorous interpretation of temperature fluctuations from the perspective of the “polythermal” ensemble, which is an extension of the canonical ensemble consisting of a collection of canonical ensembles with different temperatures.http://dx.doi.org/10.1119/1.13583
4.H. B. Prosper, “Temperature fluctuations in a heat bath,” Am. J. Phys.0002-9505 61, 54–58 (1993). This paper contains a practical method to calculate the temperature fluctuations within statistical mechanics.http://dx.doi.org/10.1119/1.17410
5.J. X. Hou, X. Wang, S. Huang, J. J. Lin, C. L. Wan, and Q. H. Liu, “On the divergence problem of temperature fluctuations in simple systems,” Acta Phys. Sin.1000-3290 55, 1616–1621 (2006).
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7.B. H. Lavenda, Statistical Physics: A Probabilistic Approach (Wiley, New York, 1991), p. 193.
8.T. C. P. Chui, D. R. Swanson, M. J. Adriaans, J. A. Nissen, and J. A. Lipa, “Temperature fluctuations in the canonical ensemble,” Phys. Rev. Lett.0031-9007 69, 3005–3008 (1992). The temperature resolution is of order as given in Table I of this paper.http://dx.doi.org/10.1103/PhysRevLett.69.3005
9.M. Hartmann, G. Mahler, and O. Hess, “Existence of temperature on the nanoscale,” Phys. Rev. Lett.0031-9007 93, 080402–1 (2004);http://dx.doi.org/10.1103/PhysRevLett.93.080402
9.M. Hartmann, G. Mahler, and O. Hess,“Local versus global thermal states: Correlations and the existence of temperatures,” Phys. Rev. E1063-651X 70, 066148–1 (2004).http://dx.doi.org/10.1103/PhysRevE.70.066148
10.C. Kittel, “Temperature fluctuations: An oxymoron,” Phys. Today0031-9228 41(5), 93 (1988). Kittel argues that the temperature by its definition does not fluctuate at all. As an immediate response to Kittel’s opinion, Mandelbrot reaffirmed his 1960s’ assertion with the intention “to explain it to a wide audience.” (Ref. 2)
11.Reference 1, p. 69. The fact that as is a straightforward consequence of Nernst’s theorem, which is one of the standard statements of the third law of thermodynamics.
12.K. Huang, Statistical Mechanics (Wiley, New York, 1987).
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