Temperature Fluctuations and Mixtures of Equilibrium States in the Canonical Ensemble
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Abstract
It has been suggested recently that "Q-exponential" distributions, which form the basis of Tsallis' nonextensive thermostatistical formalism, may be viewed as mixtures of exponential (Gibbs) distributions characterized by a fluctuating inverse temperature. In this chapter, we revisit this idea in connection with a detailed microscopic calculation of the energy and temperature fluctuations present in a finite vessel of perfect gas thermally coupled to a heat bath. We find that the probability density related to the inverse temperature of the gas has a form similar to a x<sup>2</sup> density, and that the "mixed" Gibbs distribution inferred from this density is non-Gibbsian. These findings are compared with those obtained by a number of researchers who worked on mixtures of Gibbsian distributions in the context of velocity difference measurements in turbulent fluids as well as secondary distributions in nuclear scattering experiments…. Most, if not all, textbooks on thermodynamics and statistical physics define temperature as being a quantity which, contrary to other thermodynamic observables like energy or pressure, does not admit fluctuations. Because of that, it is somewhat surprising to see papers with the expression "temperature fluctuations" in their titles appearing from time to time in serious scientific journals on subjects as various as particle physics and fluid dynamics (see, e.g., Ashkenazi and Steinberg [3], Ching [9], Chiu et al. [10], and Stodolsky [24]). Indeed, how can the temperature of a system, however small, fluctuate if one defines it "as equal to the temperature of a very large heat reservoir with which the system is in equilibrium and in thermal contact" [18]? Also, in the case of the reservoir, how can temperature be a fluctuating parameter if its definition requires one to assume the thermodynamic limit, in other words, to assume that the system acting as a reservoir is composed of an infinite number of particles or degrees of freedom? Presumably, the thermodynamic limit should rule out any fluctuations of thermodynamic quantities like the mean energy or the pressure, so that if temperature is related to these quantities, how can it fluctuate?
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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