On the putative essential discreteness of q-generalized entropies

Abstract

It has been argued in [EPL 90 (2010) 50004], entitled Essential discreteness in generalized thermostatistics with non-logarithmic entropy, that "continuous Hamiltonian systems with long-range interactions and the so-called q-Gaussian momentum distributions are seen to be outside the scope of non-extensive statistical mechanics". The arguments are clever and appealing. We show here that, however, some mathematical subtleties render them unconvincing

Toward a relative q-entropy

“…We presented a possible resolution in [15] among the other recent proposals. The present paper can be considered as a continuation along the line of these works [5,6,7,8,9,10,11,12,13,14,15].…”

“…We presented a possible resolution in [15] among the other recent proposals. The present paper can be considered as a continuation along the line of these works [5,6,7,8,9,10,11,12,13,14,15]. The present work relies very heavily on, and provides some physical context and interpretation of, the results of [16].…”

“…An easier to address, and somewhat related, question has to do with the validity, or even existence, the q-entropy for systems described by variables taking values in a continuous set, as opposed to discrete sets of variables. Some controversy has arisen related to this issue during the last decade [5,6,7,8,9,10,11,12,13,14]. We presented a possible resolution in [15] among the other recent proposals.…”

“…The controversy regarding the suitability, or possibility of extending the q-entropy for continuous systems [5,6,7,8,9,10,11,12,13,14], is related to the relative version of (11) which, following (4),( 8) is taken, by most authors working on this issue, to be…”

Characterising heavy-tailed networks using q-generalised entropy and q-adjacency kernels

“…( 19) as a Riemann integral. These replacements are not trivial but can, however, be justified for q-exponentials [48]. The resulting expression for the q-divergence is then The parameter η = (α − α )/α > 0 is the relative difference of the q-indices, characterising the distribution, while the parameter is assumed to be the same for both distributions.…”