HYPOTHESIS AND THEORY article
Olbertian Partition Function in Scalar Field Theory
- 1International Space Science Institute (ISSI), Bern, Switzerland
- 2Department Geoscience and Environment, Munich University (LMU), Munich, Germany
- 3Space Research Institute (IWF), Austrian Academy of Sciences, Graz, Austria
The Olbertian partition function is reformulated in terms of continuous (Abelian) fields described by the Landau–Ginzburg action, respectively, Hamiltonian. In order to make some progress, the Gaussian approximation to the partition function is transformed into the Olbertian prior to adding the quartic Landau–Ginzburg term in the Hamiltonian. The final result is provided in the form of an expansion suitable for application of diagrammatic techniques once the nature of the field is given, that is, once the field equations are written down such that the interactions can be formulated.
Particle spectra in near-Earth space (e.g., see [1–3]) as well as in cosmic rays very frequently exhibit power law tails at high energies which since their introduction by  have been interpreted as Olbert distributions (κ-distributions).1 Cosmic ray spectra in particular extend as power laws over many decades reminding of several ultra-relativistic Olbert distributions adding up continuously . Olbert distributions have been inferred in plasma turbulence and many other occasions as for instance in front  and behind  collisionless shocks  as also, for example, in the heliosphere and its heliosheath [9, 10], which may serve as the paradigm of a stellar wind that is terminated by its interaction with the interstellar galactic medium. They were also derived in plasma wave–wave interaction theory [11–13]. Physically, they represent quasi-stationary states far from equilibrium [14–18]. To some degree, they are related to Tsallis' thermostatistics . We recently  investigated their connection to Olbert’s entropy.2 Here, we are interested in the role they might play in field theory which is the continuous version of the partition function [21–26]. We do not go into the definition what is meant by the term continuous. Fields are a basic concept of physics when many degrees of freedom come into play. In this case, one refers to action and Hamiltonian densities which are distributed in space–time, and for the statistical interacting fields and distributed sources, one refers to the field partition function from which the effects of the interaction between fields and particles can be deduced. In the following, we derive the Olbertian partition function for scalar fields, the so-called generating functional. One particularly interesting application which could be anticipated is in the cosmological theory of the early universe where phase transition is the rule . As it turns out, this is a highly nontrivial problem whose difficulties go substantially beyond those encountered with the Gibbsian partition function. Nevertheless, we show how the Olbertian field partition function can be constructed giving it an operational representation that is suitable for application. This is interesting in as far as the extra parameter κ which is fundamental to the Olbert theory provides an external degree of freedom which may become useful in applications like renormalization and phase transition where convergence is obliterated.
2. Olbertian Distribution: A Brief Review
For finite κ, this function is essentially a skewed Maxwellian (or Boltzmann) function with power law wings in momentum space (in energy space tails). It was widely discussed in the literature and applied to observe high energy tails on particle distribution functions as well as formally, interpreting it as a statistical not a physical distribution, in statistical data analysis to demonstrate the existence of correlations which give rise to higher-order moments like kurtosis. Sometimes, it is claimed that the above probability (neglecting the exponential cutoff at energy
which is obtained (in fact trivially) when expressing the first derivative of
Distribution functions similar to the Olbert kind have been obtained in theory in various different ways. The original theoretical attempt to find a plausible explanation for them almost immediately referred to a Fokker–Planck approach. The Fokker–Planck equation was naturally assigned to describe the formation of deformed nonstationary distributions by stochastic diffusion in phase space caused by electromagnetic fluctuations. It requires prescription of a phase space diffusion coefficient and, to some extent, corresponds to quasilinear theory where under collisionless conditions first-order wave-particle interactions scatter some of the resonant particles into higher energy states while confining the lower energy bulk. This process, for an energy-dependent diffusion coefficient, maintains the low-energy particles in the Maxwellian, while the unconfined higher energy particles run away into a skewed energetic tail that evolves with time. Fokker–Planck approaches are time dependent. One particular such diffusion coefficient is the Coulomb scattering (Spitzer conductivity) which is energy dependent and thus may be considered basic to scattering and power law tails. Coulomb scattering times are, however, very long such that it takes much time until an appreciable tail evolves. It is widely used for this property in cosmic ray physics where infinite time is available. In order to obtain a stationary state, Fokker–Planck approaches require continuous injection and some additional mechanism which cuts the distribution at some energy by removing energetic particles causing particle losses. Stationarity is achieved, for instance, when particles escape from the system or by imposing other particle sinks like, for instance, charge exchange with an atmosphere.
The Olbert distribution, on the other hand, is a basic stationary quasi-thermodynamic state far from thermal equilibrium. It has been given a derivation from basic statistical mechanics where it can evolve under particular conditions, caused by the intrinsic structure of the distribution. It has been discussed in two forms from two different points of view, as thermostatistics , where an approximate form has been found, and in Lorentzian Gibbs–Boltzmann statistical mechanics [17, 28]. The latter leads directly to the Olbert distribution; the former constructs a similar distribution in suitable approximation. For a collection of other properties of the Olbert distribution, the reader is directed to the extended literature on κ distributions (cf., e.g., [15, 16]; and references therein).
It is clear that the microphysics of the formation of the Olbert distribution is contained in the free parameter
3. Gauss–Olbertian Theory
The Olbertian partition function based on the Olbert distribution when normalizing it according to the Gibbs prescription is given by
Olbertian Partition Function
We here for simplicity generalize the Olbert partition function to given scalar fields
The integration is over all realizations of the field
with the Landau–Ginzburg Hamiltonian density
originally introduced in superconductivity theory where
In the above expressions,
We also note that the Hamiltonian is an energy density. Since energy is additive, one can simply add any interacting external field as, for instance, the energy density of the electromagnetic field
As usual, the Landau–Ginzburg Hamiltonian, even though it is an expansion just up to second order in the modulus of the field
So, in order to proceed, we provisionally drop the quartic term in the absence of any external field and self-interaction, and consider the Gauss–Olbertian partition function
This Hamiltonian is assumed to hold for any, not particularly, specified field
where the wave number
where the integration is with respect to the decoupled real and imaginary fields. The unity in the bracket can in principle be absorbed into the Hamiltonian which is a sum. It is however rather inconvenient to work with this form because handling the finite sum in the denominator at this stage prevents any further progress unless it can be summed up, which however requires a functional form of the functions
For any external source
where the new field is
which maintains the symmetry of the Hamiltonian. The field integration in the partition function is then done with respect to
Gauss–Olbertian Partition Function
Let us return to the Gibbs–Boltzmann partition function
by writing it in terms of the field
with the definition
where V is the normalizing volume (for instance of a large box). These expressions transform directly into an explicit form of the Gauss–Olbertian partition function for the fields
From here, one observes that in the absence of an external source field, the function in the second line reduces to one, and only the zero-order partition function remains. The free energy then follows immediately from
a form that can be used to derive further quantities of thermodynamic interest like the energy and the specific heat. The form given is not yet suited for calculating correlation functions because it contains the Fourier-transformed sources. Returning in the partition function to the original sources J via the inverse transformation and using the particular form of the Dirac function
used in field theory at finite temperatures which results from the two-point correlation function, whose properties we give below, and the source integral can be rewritten in a form suitable for taking derivatives with respect to J
This is to be used in the partition function
This last expression is the definition of the correlation functions. Forming the one-point correlation function yields
showing that the Dirac function (18) is nothing else but the representation of the two-point correlation function (21) which is an identity and of practical use in the following. It is otherwise easy to show that (18) is indeed a representation of the Dirac function yielding the self-correlation of the field. Higher-order correlations do not exist in this approximation but may indeed occur in the more precise Olbert theory below.
We briefly note as a side remark that explicit calculation confirms that Eq. (18) is indeed a Dirac function. Since its denominator is symmetric in k, the integral can be analytically continued into the negative domain with the poles shifted along one of the axes by the amount
The above approximate Gauss–Olbertians are nevertheless in a still fairly inconvenient form. This can be made more explicit by writing them as exponentials. For instance, we have
expanding both the logarithm in the argument of the exponential and subsequently the exponential itself produced in Eq. (15)
This, with the above inverse Fourier representation, assumes the final form:
as an approximate representation of the Gauss–Olbertian partition function. Alhough this is just a simplified form which we below will make more precise, it will in most cases be sufficient in applications.
The final last forms of the Gauss–Olbertian partition function provide a feasible way to go but are not completely satisfactory as the transformation from the Gibbs–Gaussian to the Gauss–Olbertian forms is done at a late stage. It can be taken as a lowest order approximation to a more precise formulation of the Olbertian field partition function whose derivation is attempted in the following. We therefore tentatively return to the formal definition of the Gauss–Olbertian partition function (11) through the Hamiltonian
The logarithm in the exponential can then be expanded if taking care that the Hamiltonian has been appropriately normalized. This yields
We therefore return to the Hamiltonian
where the two forms of
The two Hamiltonian functions are themselves sums with respect to k, which here being multiplied with each other term by term such that their product becomes
Each sum consists of two further sum terms. Only the second of these two factors depends on the source function J. Hence, if we expand it again into a binomial series leaving the first sum in its compact form, we find
Again transforming the binomial sum into a product, we arrive at
The explicit dependence on J is now contained only in the two factors in the last line. This expression still contains the sums over wave numbers which in a continuum representation can be replaced by integrals by reintroducing the inverse Fourier transform in these terms and using the Dirac function as given above. Without further assumptions about the field, we are, however, stuck at this point.
The latter difficulty can be circumvented when assuming that we are dealing with an isotropic field. In this case, the Hamiltonian, given in (12), separates the source dependence out, and the source-dependent Hamiltonian becomes
an expression that can directly be introduced in (30) since now the summand in this Hamiltonian is a single term such that the p-product disappears and the partition function assumes the simpler version:
where unimportant constant factors in front of the partition function have been suppressed. The field-dependent Hamiltonian
as the wanted form of the Gauss–Olbertian partition function. Once the integration with respect to
where the sum over wave numbers has been replaced by the k-integration. One may note that
In this form, we have obtained the final form of the Gauss–Olbertian partition function. It does not yet account for the quartic term in the Landau–Ginzburg Hamiltonian and therefore neither contains self-interactions of the field nor spontaneous symmetry breaking which was the big progress and success in the Landau–Ginzburg theory nor does it account for the effects of the external source on symmetry breaking. It does, however, contain Gaussian phase transitions. Nevertheless, in its Gaussian form, it is already subject to application of the diagrammatic technique using Feynman diagrams term by term. Still, the last mixed integral term provides complications even here. As one observes, the form given to it here suggests that there is a huge number of terms which contribute, even though they contribute ordered by increasing power.
In order to complete the theory to include spontaneous symmetry breaking on the level of the Olbertian partition function, one needs to refer to the quartic term in the Landau–Ginzburg Hamiltonian. This is the content of the next section.
4. Landau–Ginzburg–Olbertian Theory
So far, we derived the Gauss–Olbertian partition function which, from the point of view of Landau–Ginzburg theory, can be considered as the Gaussian approximation to the full Hamiltonian. In non-Olbertian field theory, the Gibbs partition function is the usual exponential, and the quartic term in the Hamiltonian enters through the exponential in the integrand. It can be easily factorized such that the partition function is simply multiplied by an additional function however complicated that function appears. Expanding this exponential then gives an infinite series of correlations which are subject to Feynman representations of the hierarchy of interactions. In Olbertian theory, this is not anymore possible in this simple way, unless one chooses to simplify the theory substantially. The full Hamiltonian entering into (4) is
such that we can write
(with dummy integration variable
with the functional derivatives of order
where the exponential has been replaced by an Olbertian function.
The last expression can, in the usual way, be raised to an exponential which now retains the Olbertian functional form in the logarithm:
Expanding the logarithm in the argument of the exponential, we arrive at another infinite product of exponentials
The exponential can finally be expanded to give
This version of the Olbertian partition function contains the main features of the Olbertian theory and is thus a valid approximation when applied to scalar fields in the Landau–Ginzburg theory.
The way of how to arrive at this result is now quite clear. In an even more precise theory than that given here, one returns to the original Landau–Ginzburg Hamiltonian and repeats the sequence of steps. We briefly sketch how this is done. To this purpose, one returns to (40), an expression which in the argument of the exponential contains the Gauss–Hamiltonian plus the Landau–Ginzburg term. The logarithm can advantageously be split into a sum of logarithms
When inserted into the partition function and evaluated, the first term on the right leads to the above expression
shows the effect of the Gaussian Hamiltonian just as a higher order correction factor. Thus, one observes that the final result given above is the best available version of the Landau–Ginzburg–Olbertian partition function in a form which is suitable for application. Neglect of the second term on the right is sufficient justification for our approximation made in the former subsection to arrive at
Field partition functions play a substantial role in field theory when interacting fields are under scrutiny. In particular, the partition function is the key to the identification of phase transitions on the one hand, and on the other in managing renormalization [23, 35] and elimination of divergences. Since Olbertian distribution functions have turned out to apparently building up frequently in nature where in the particle realm they can be treated by application of the partition function, it seems to make sense to investigate whether they can be reformulated as well in field theory. This has been done in the present note where it has been shown that with some substantial modifications, Olbertian field partition functions can be formulated and brought into a form which is suitable for application of diagrammatic techniques like Feynman diagrams if only the interactions can be identified. One can imagine that in various applications, the Olbertian partition function may become useful. This may happen in systems which turn out to exhibit non-Gaussian behavior. Classically, such cases are familiar from turbulence theory where power laws are the rule thinking of, for instance, Kolmogorov spectra. Other candidates for application are the very early universe and phase transitions therein. Here, we just provided the framework for application given in the formulation of the elaborated although convenient mathematical structure of the Olbertian partition function .
Data Availability Statement
The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding author.
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
This work was part of a brief Visiting Scientist Programme at the International Space Science Institute Bern. RT acknowledges the interest of the ISSI directorate as well as the generous hospitality of the ISSI staff, in particular the assistance of the librarians Andrea Fischer and Irmela Schweitzer, and the systems administrator Saliba F. Saliba. We acknowledge valuable discussions with M. Leubner, R. Nakamura, and Z. Vörös.
1They were invented by Stan Olbert in 1966 and applied first in the unpublished Ph.D. thesis of Binsack  who acknowledged its suggestion by Olbert. (We thank C. Tsallis for kindly bringing this reference to our attention.) Olbert also suggested it to Vasyliunas whose article  contained the first refereed, published, and thus multiply cited version of the Olbert (κ) distribution.
3In physics (and science in general), it is customary to assign the names of their inventors to theories or equations in order to be specific and make them indistinguishable. One speaks of the Boltzmann equation, Gibbs statistical mechanics, Tsallis statistics instead q-statistics, Heisenberg’s relations, Einstein’s theory, Feynman integrals, and so on, unless the theory has its own indistinguishable names like QED, QCD, and GRT. The name κ-distribution is inappropriate. The letter κ is nothing specific. The only appropriate name for the Olbert distribution and everything which follows from it would be Olbert’s.
4An idea first introduced by Einstein  in his approach to the stochastic motion of molecules.
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Keywords: Olbert distribution, partition function, Landau–Ginzburg theory, field theory, phase transitions, cosmology
Citation: Treumann RA and Baumjohann W (2020) Olbertian Partition Function in Scalar Field Theory. Front. Phys. 8:610625. doi: 10.3389/fphy.2020.610625
Received: 26 September 2020; Accepted: 02 November 2020;
Published: 08 December 2020.
Edited by:Marian Lazar, Ruhr-Universität Bochum, Germany
Reviewed by:Rudi Gaelzer, Federal University of Rio Grande do Sul, Brazil
Klaus Scherer, Ruhr University Bochum, Germany
Copyright © 2020 Treumann and Baumjohann. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Wolfgang Baumjohann, Wolfgang.Baumjohann@oeaw.ac.at