*Olivier Darrigol*

- Published in print:
- 2018
- Published Online:
- March 2018
- ISBN:
- 9780198816171
- eISBN:
- 9780191853661
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198816171.003.0003
- Subject:
- Physics, Atomic, Laser, and Optical Physics, History of Physics

This chapter is the first subset of a set of critical summaries Boltzmann’s writings on kinetic-molecular theory. It covers a first period in which he tried to construct the laws of thermal ...
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This chapter is the first subset of a set of critical summaries Boltzmann’s writings on kinetic-molecular theory. It covers a first period in which he tried to construct the laws of thermal equilibrium, including the existence of the entropy function and the Maxwell–Boltzmann law, by various means including the principle of least action, Maxwell’s collision formula, the ergodic hypothesis, and a procedure of adiabatic variation. This is an immensely fertile period in which Boltzmann introduced several of the basic concepts, problems, and difficulties of modern statistical mechanics.Less

This chapter is the first subset of a set of critical summaries Boltzmann’s writings on kinetic-molecular theory. It covers a first period in which he tried to construct the laws of thermal equilibrium, including the existence of the entropy function and the Maxwell–Boltzmann law, by various means including the principle of least action, Maxwell’s collision formula, the ergodic hypothesis, and a procedure of adiabatic variation. This is an immensely fertile period in which Boltzmann introduced several of the basic concepts, problems, and difficulties of modern statistical mechanics.

*J. S. Andrade Jr, M. P. Almeida, A. A. Moreira, A. B. Adib, and G. A. Farias*

- Published in print:
- 2004
- Published Online:
- November 2020
- ISBN:
- 9780195159769
- eISBN:
- 9780197562024
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195159769.003.0012
- Subject:
- Earth Sciences and Geography, Atmospheric Sciences

Since the pioneering work of Tsallis in 1988 [15] in which a nonextensive generalization of the Boltzmann-Gibbs (BG) formalism for statistical mechanics was proposed, intensive research has been ...
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Since the pioneering work of Tsallis in 1988 [15] in which a nonextensive generalization of the Boltzmann-Gibbs (BG) formalism for statistical mechanics was proposed, intensive research has been dedicated to the development of the conceptual framework behind this new thermodynamical approach and to its application to realistic physical systems. In order to justify the Tsallis generalization, it has been frequently argued that the BG statistical mechanics has a domain of applicability restricted to systems with short-range interactions and non-(multi)fractal boundary conditions [14]. Moreover, it has been recalled that anomalies displayed by mesoscopic dissipative systems and strongly non-Markovian processes represent clear evidence of the departure from BG thermostatistics. These types of arguments have been duly reinforced by recent convincing examples of physical systems that are far better described in terms of the generalized formalism than in the usual context of the BG thermodynamics (see Tsallis [14] and references therein). It thus became evident that the intrinsic nonlinear features present in the Tsallis formalism that lead naturally to power laws represent powerful ingredients for the description of complex systems. In the majority of studies dealing with the Tsallis thermostatistics, the starting point is the expression for the generalized entropy S<sub>q<sub>, where A; is a positive constant, q a parameter, and / is the probability distribution. Under a different framework, some interesting studies [8] have shown that the parameter q can be somehow linked to the system sensibility on initial conditions. Few works have been committed to substantiate the form of entropy (1) in physical systems based entirely on first principles [1, 13]. For example, it has been demonstrated that it is possible to develop dynamical thermostat schemes that are compatible with the generalized canonical ensemble [12].
Less

Since the pioneering work of Tsallis in 1988 [15] in which a nonextensive generalization of the Boltzmann-Gibbs (BG) formalism for statistical mechanics was proposed, intensive research has been dedicated to the development of the conceptual framework behind this new thermodynamical approach and to its application to realistic physical systems. In order to justify the Tsallis generalization, it has been frequently argued that the BG statistical mechanics has a domain of applicability restricted to systems with short-range interactions and non-(multi)fractal boundary conditions [14]. Moreover, it has been recalled that anomalies displayed by mesoscopic dissipative systems and strongly non-Markovian processes represent clear evidence of the departure from BG thermostatistics. These types of arguments have been duly reinforced by recent convincing examples of physical systems that are far better described in terms of the generalized formalism than in the usual context of the BG thermodynamics (see Tsallis [14] and references therein). It thus became evident that the intrinsic nonlinear features present in the Tsallis formalism that lead naturally to power laws represent powerful ingredients for the description of complex systems. In the majority of studies dealing with the Tsallis thermostatistics, the starting point is the expression for the generalized entropy S<sub>q<sub>, where A; is a positive constant, q a parameter, and / is the probability distribution. Under a different framework, some interesting studies [8] have shown that the parameter q can be somehow linked to the system sensibility on initial conditions. Few works have been committed to substantiate the form of entropy (1) in physical systems based entirely on first principles [1, 13]. For example, it has been demonstrated that it is possible to develop dynamical thermostat schemes that are compatible with the generalized canonical ensemble [12].