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Learning with Boltzmann–Gibbs Statistical Mechanics

  • Constantino Tsallis

where \(x/\sigma \in{\mathbb R}^D\), \(D \ge 1\) being the dimension of the full space of microscopic states (called Gibbs \(\Gamma\) phase-space} for classical Hamiltonian systems). Typically x carries physical units. The constant \(\sigma\) carries the same physical units as x, so that \(x/\sigma\) is a dimensionless quantity (we adopt from now on the notation \([x]=[\sigma]\), hence \([x/\sigma]=1\)). For example, if we are dealing with an isolated classical N-body Hamiltonian system of point masses interacting among them in d dimensions, we may use \(\sigma=\hbar^{Nd}\). This standard choice comes of course from the fact that, at a sufficiently small scale, Newtonian mechanics becomes incorrect and we must rely on quantum mechanics. In this case, \(D=2dN\), where each of the d pairs of components of momentum and position of each of the N particles has been taken into account (we recall that \([momentum][position]=[\hbar]\)).

Keywords

Hamiltonian System Lyapunov Exponent Entropy Production Microcanonical Ensemble Lagrange Parameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Constantino Tsallis
    • 1
    • 2
  1. 1.Centro Brasileiro de Pesquisas FísicasRua Xavier Sigaud 150Brazil
  2. 2.Santa Fe InstituteSanta FeUSA

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