Questioning the validity of non-extensive thermodynamics for classical Hamiltonian systems

Abstract

We examine the non-extensive approach to the statistical mechanics of Hamiltonian systems with H = T + V where T is the classical kinetic energy. Our analysis starts from the basics of the formalism by applying the standard variational method for maximizing the entropy subject to the average energy and normalization constraints. The analytical results show (i) that the nonextensive thermodynamics formalism should be called into question to explain experimental results described by extended exponential distributions exhibiting long tails, i.e. q-exponentials with q > 1, and (ii) that in the thermodynamic limit the theory is only consistent in the range 0 ≤ q ≤ 1 where the distribution has finite support, thus implying that configurations with e.g. energy above some limit have zero probability, which is at variance with the physics of systems in contact with a heat reservoir. We also discuss the (q-dependent) thermodynamic temperature and the generalized specific heat.(a)

Tsallis power laws and finite baths with negative heat capacity

“…It is worth noting that a particular case of the above general result was also obtained by Lutsko and Boon [23] solely by considering the integration over momenta degrees of freedom so that E S /T B = DN S /2 [compare Eq. (17) above to the one below Eq.…”

“…(10) in Ref. [23]], where D and N S denote the dimensionality of the phase space and the number of particles in the system, respectively. It is indeed remarkable that Lutsko and Boon obtained this particular result just by checking the integrability conditions of the concomitant Tsallis distributions [23].…”

“…[23]], where D and N S denote the dimensionality of the phase space and the number of particles in the system, respectively. It is indeed remarkable that Lutsko and Boon obtained this particular result just by checking the integrability conditions of the concomitant Tsallis distributions [23]. However, the results of Lutsko and Boon include neither the finiteness of the bath nor the necessity of its negative heat capacity for the escort q distribution with fat tails to emerge.…”

Spectral and Pseudospectral Methods of Solution of the Fokker-Planck and Schrödinger Equations

“…In the limit, κ → ∞, the Kappa distribution tends to a Maxwellian. The Tsallis nonextensive entropy formalism is a controversial topic (Nauenberg 2003;Tsallis 2004;Lutsko and Boon 2011).…”

The validity of the third law of thermodynamics for the Rényi and homogeneous entropies

“…Concerning the question whether any information-theoretic entropy can be useful for constructing a generalized statistical mechanics, many candidates such as the Tsallis [1], Rényi [2], Kaniadakis [3] and homogeneous entropies [4][5][6] have recently been proposed. These generalized entropies have been applied to various fields of research such as quantum information [7][8][9][10], econophysics [11,12], high energy phenomenology [13] and black hole thermodynamics [14,15].…”