It is demonstrated that the statistical mechanical partition function can be
used to construct various different forms of phase space distributions. This
indicates that its structure is not restricted to the Gibbs–Boltzmann factor
prescription which is based on counting statistics. With the widely used
replacement of the Boltzmann factor by a generalised Lorentzian (also known
as the

Since its introduction by

Vasyliunas (1968) acknowledges that in applying the kappa distribution as an apparently useful fit to the observed energy dependence of low-energy electron fluxes in the geomagnetic-tail plasma sheet he followed a suggestion of its functional form by Stanislaw Olbert.

, the so-called kappa-distribution functionFor a recent compilation and in-depth discussion of the various aspects and applications of the kappa distribution the reader is referred to the extended presentations contained in Livadiotis and McComas (2009, 2013) as well as to the almost complete list of papers referenced therein. This list gives a historical record of the work done on and application of the kappa distribution as well as its relation to the celebrated Tsallis nonextensive thermostatistics (Tsallis, 1988; Tsallis et al., 1998; Gell-Mann and Tsallis, 2004).

In principle, the kappa distribution is a probability distribution function
which, mathematically, is identical to the generalised Lorentzian

Formal relations between the Tsallis statistical mechanics of non-extensive
entropies and the kappa distribution do indeed exist. This is not surprising,
because the parameter

Its most recent exposition is found in Zaburdaev et al. (2015).

, though not referring to Tsallis' non-extensive statistics such that the coincidence was somehow accidental. It was independently elaborated byFor the complete lists of references see again Livadiotis and McComas (2009, 2013) and Livadiotis (2015).

A heuristic generalisation of statistical mechanics to general entropies has
been proposed more recently

It is interesting that the kappa distribution understood as a probability
distribution also reproduces distributions that are obtained when analysing
intermittency

As for a typical example of intermittence in solar wind magnetic turbulence see, for instance, Brown et al. (2015).

in the data of chaotic processes. In these cases it sometimes properly maps the tails of the probability distributions allowing for the determination of the power indexIn the following we start from the general Gibbs–Boltzmann partition function as the accepted physical basis of statistical mechanics. We then transform it into a kappa partition function and proceed to the derivation of the equation of state and the physical distribution of occupation of states. In doing so we follow the prescription of statistical mechanics in deriving the physical distribution function. The idea is thus very simple. However, this process is physically motivated and provides some additional physical insight.

The grand partition function

We now violate, on this advanced level, the stochastic assumption. We assume
that the structure of the partition function will remain intact if we replace
the exponential with another function that in some limit reproduces the
exponential. Such a function is, as for an example, the

The above version of the partition function can also be written in another
form,

Now we violate the assumption of pure stochasticity by introducing the
Lorentzian replacing

In a Fermi system we can only have two occupations

From the partition function one obtains the ideal gas equation of state as

For

For the Boson distribution we refer to the function

For high-temperature high-energy classical gases both distributions above
become a reasonable classical limit known as the kappa distribution. In such
classical cases the chemical potential becomes negative. Then the complete
classical distribution that is in accord with the partition function assumes
the form

We note in passing that the relativistic equivalent of the above kappa distribution should become

It is interesting that the chemical potential cannot be extracted from this
expression. This makes its use as a physical distribution more difficult and
requires use of approximation methods to eliminate

The straightforward calculations by

A negative chemical potential which is expected in the classical case should
cause trapping and thus retarding the electrons, and also accumulating
them around the trapping potential, i.e. the chemical potential. This is
obviously not the case – at least in the weakly turbulent regime
investigated by

We note that this effect had already been observed earlier in a model where
electrons were put into a heat bath of radiation photons

These observations as well as the results of the rigorous calculations of

This observation also explains why the particle spectra measured by

In contrast, charged particle interaction with solitons, cavitons, holes, shocks indeed traps the low energy part of the population while it accelerates the passing energetic population into a tail. The soliton potential thus acts as a partial chemical potential in this case, while the passing distribution ignores it by overcoming and picking up energy which goes into tail formation. This separation of the distribution is obviously entropically favourable. Inspection of the split distribution function should provide information about the nature of these processes, the equivalent chemical potential and its relation to the density of trapped particles.

There is no known counting statistics in cases where the system is not stochastic but respects some internal correlations. It is not clear how such cases should be treated even then when the correlations have been specified from the very beginning. The application of Bayesian statistics could possibly offer a route to such systems. Statistical mechanics, however, seems not to have had any needs so far in non-stochastic states on the microscopic level. These are usually treated by numerical simulations or kinetic theory where the evolution of the one-particle distribution function is followed in time. This is clearly the right physical approach to non-stationary systems in evolution. Statistical mechanics just deal with the stationary state of a system.

That the introduction of correlations via the replacement of the exponential
by the Lorentzian on the level of the partition function nevertheless
reproduces the correct kappa statistics as derived intuitively from
assumptions that have nothing in common with stochasticity, suggests that the
structure of the partition function is more general than purely stochastic.
It is just the sum over all occupations in the probabilities of states –
quite a general notion. One may thus ask whether or not other functions exist
with the physical meaning that they include correlations when used in the
partition function. The requirement on them implies that they should behave
correctly at large energies, i.e. converge for

A particular function which seems to offer itself is the modified Bessel
function of the first kind

The most interesting case is the behaviour at zero temperature

Let us now try the function

As we have shown, the use of Bessel functions in order to obtain Bessel-Fermi distributions is not successful. What about the classical case? Does a formal Bessel–Boltzmann distribution exist? The case of the kappa distribution suggests that this would not be categorically excluded independent on whether the distribution found has any real application to physical problems. So, in the following, we check whether a classical limit exists for Bessel distributions.

The classical case requires that the chemical potential is negative and thus

Let us do this for the modified Bessel function of the first kind. We expect
that for large argument the Bessel-Fermi distribution should become the
classical equivalent of the Boltzmann distribution similar to the transition
from ordinary Fermi to the ordinary Boltzmann distribution. In the limit of
very large

One realises that the factor in front of the brackets is a simple Lorentzian distribution. One thus may note that the last expression can be interpreted as kind of a Bessel-modified Lorentzian distribution of states.

This suggests various further generalisations. The relativistic version is
obtained by mapping

One could even go further, interpreting the bracket as the expansion of an
exponential. This then yields

The last form suggests that in kappa distributions the energy term could be
replaced by any real continuous function

The functional form of the partition function as the sum over probabilities
of states obtained from simple counting of states thus allows for completely
different classical distributions which we have guessed, while it suppresses
the quantum distributions. This suppression is reasonable because quantum
physics relies solely on stochasticity. Quantum chaos is an unresolved
concept and is presumably absent at

Whether the classical Bessel–Boltzmann distribution and its further generalisations obtained have any physical meaning or not, is a completely different question. We just played with the possibility of a different kind of statistical mechanical distributions of occupation of states arbitrarily choosing Bessel functions for our experiment. Inferring whether a distribution like this one has physical meaning requires the derivation of the corresponding entropy and testing the thermodynamic relations.

We have used the Gibbs–Boltzmann prescription of the partition function in
application to different basis factors which replace the so-called
Gibbs–Boltzmann factor, the exponential function in the definition of
probability. The latter results from the assumption of complete stochasticity
in the processes underlying the interaction of the particles respectively
systems involved. Their foundation is Gauss' error distribution transformed
into energy and momentum space. Any replacement of the
Gibbs–Boltzmann factor by another more complicated function thus implies that
one uses non-stochastic probabilities which may involve correlations. This
has been discussed at other places. We have shown that such a replacement
works nicely for the generalised Lorentzian factor used in giving the
so-called kappa distribution a physical fundament. The kappa distribution
actually becomes a generalised Lorentzian distribution. Its derivation from
the generalised partition function results in a slightly different version
than used in its otherwise widely distributed applications. Bringing it into
complete accord with thermodynamics fixes the free parameter

Generalisation of the partition function to the generalised Lorentzian
implies that for large

At finite temperatures both distributions might exist. For one of them we have shown that, in the classical domain, it transforms into a reasonable though complicated Boltzmann–Bessel distribution. Whether it has any physical meaning or not is, however, unknown,. We do not attempt to check it here as the demonstration intends nothing more than to provide an example.

The new classical distribution turns out to belong to the family of

It would be very interesting in this respect of stepping down into the Gaussian error analysis trying to infer the effect of correlations. One possibility of doing this would be by reference to Bayesian statistics. Bayesian statistics imposes extra conditions – hypotheses – which could be physically motivated. Construction of a different Bayesian–Gauss–Gibbs–Boltzmann factor should then provide a physically motivated version of the partition function to be used by standard methods to infer about the resulting average occupation numbers of physical states.

The present note was part of work on superdiffusion and information theory performed during two short visits at the International Space Science Institute Bern in 2006/2007. Rudolf A. Treumann acknowledges the hospitality of that institution. Moreover, he thanks the two Referees for their thoughtful comments. He is particularly indebted to George Livadiotis for his valuable remarks on the manuscript of the present paper, addressing him to the relevant references concerning the kappa distribution, its history, mathematical and physical contents, and to its various applications in mathematics and space physics. Edited by: G. Balasis