On the Origin of Polytropic Behavior in Space and Astrophysical Plasmas

It is shown that the polytropic behavior—a specific power-law relationship among the thermal plasma moments—restricts the functional form of the distribution of particle velocities and energies. Surprisingly, the polytropic behavior requires the statistical mechanics of the plasma particles to obey the framework of kappa distributions. An already known interesting property of these distributions is that they can lead to the polytropic relationship. New results show that the reverse derivation is also true, thus, the polytropic behavior has the role of a mechanism generating kappa distributions. Ultimately, an observation of a polytropic behavior in plasma particle populations constitutes a possible indirect observation of kappa velocity or energy distributions. Finally, it is discussed how the derived equivalence between the polytropic behavior and the kappa distribution function can be used in further modeling and data analyses in space and astrophysical plasmas.

It has been shown that there is a strong connection between the polytropic index and the governing parameter of the kappa distributions, the kappa index (e.g., Scudder 1992;Meyer-Vernet et al. 1995;Moncuquet et al. 2002;Tsallis et al. 2004;Livadiotis 2017, Chapters 5 and 11;Livadiotis 2018b;Livadiotis et al. 2018). According to this, higher kappa indices, which characterize stationary states near the classical Maxwell-Boltzmann distribution, correspond to polytropic indices closer to the isothermal value of γ=1; on the other hand, lower kappa indices, which characterize states farther from the Maxwell-Boltzmann distribution, correspond to polytropic indices closer to the isobaric γ=0 or isochoric values g  +¥; (the concept of thermodynamic distance of a stationary state, as well as the characterization near/far the classical thermal equilibrium, was presented in Livadiotis & McComas 2010). In general, the relationship among the polytropic and kappa indices has been shown theoretically and verified in several cases such as, for instance, in the case of solar wind near Earth (e.g., Livadiotis et al. 2018;Livadiotis 2018b).
The particle potential energy varies along the streamlines and plays the role of the connecting link between the profiles of thermal observables, which leads to the polytropic behavior.
Density, temperature, and thermal pressure depend on the potential energy and vary along the streamlines in a way that must be determined by the energy distribution function. In the classical case of Boltzmann-Gibbs (BG) statistical mechanics, the distribution at thermal equilibrium is given by the Boltzmannian exponential function, which can be separated as a product of two independent distributions, that is, the marginal distributions of kinetic and potential energy. As a result, the extracted polytropic behavior that originates from the exponential distribution is the trivial isothermal process. Moreover, in the case of generalized thermal equilibrium (Abe 2001;Livadiotis 2018c), where correlations among particles may be significant , the distribution is given by the kappa function (Livadiotis 2015a(Livadiotis , 2017. The formalism of kappa distributions does not allow the separation of potential and kinetic energy, while it addresses the density and temperature profiles in such a way that predicts exactly the whole spectrum of polytropic behavior. The formalism of kappa distributions can lead to the polytropic relationship, but this does not imply the reverse reasoning, i.e., that the "true" velocity distribution is necessarily given by a kappa function. This argument was first stated by Moncuquet et al. (2002), and it leads us to the following question. What is the most general distribution function of particle velocities and energies consistent with plasmas exhibiting polytropic behavior? Or, in short, what is the origin of the polytropic behavior in space and astrophysical plasmas?
The paper shows that the kappa distribution function constitutes the most general formalism for describing polytropes. The origin of any polytropic behavior is interwoven with the generalized thermal equilibrium, in which the particle velocities and energies are described by kappa distributions. Section 2 presents the formalism of kappa distributions in the presence of a potential energy, which leads to the polytropic relationship expressed in terms of the kappa index. Section 3 presents the Euler's momentum equation (starting from the corresponding Navier-Stokes momentum equation in the presence of a conservative external field), and shows how the Maxwell-Boltzmann and kappa distributions obey this equation. Section 4 shows the derivation of the most general density profile function that is consistent with polytropes, using Euler's equation. Section 5 shows the same thing, but using the Bernoulli integral. Section 6 shows the derivation in the presence of a magnetic field. Section 7 puts the last piece from the puzzle by showing the most general phase-space distribution function that is consistent with polytropes. Section 8 discusses the derived equivalence between the polytropic behavior and the kappa distribution functions and several interesting applications. Our conclusions in Section 9 summarize the results.

Formalism of Kappa Distributions in the Presence of a Potential Energy
First, we briefly present the formalism of kappa distributions in the presence of a potential energy, which leads to the derivation of the polytropic index expressed in terms of the kappa index and the degrees of freedom (see also Livadiotis 2015b, 2017, Chapters 3 and4).
The Hamiltonian function is 2 is the kinetic energy and F( ) r the potential energy (that depends only on the position vector). The kappa phase-space distribution of a Hamiltonian gives the probability distribution of a particle having its position and velocity in the infinitesimal intervals and , respectively, that is, The mean Hamiltonian defines the total degrees of freedom or dimensionality, d, summing the kinetic and potential degrees of freedom, where the potential degrees of freedom are defined similar to the kinetic ones, Note that by definition d Φ can be either positive or negative; alternatively, it could be defined to be non-negative by

The kappa index depends on the dimensionality as
remains invariant under changes of the dimensionality d.
The physical meaning of the thermodynamic parameter kappa is better carried by its invariant value κ 0 , because this is independent of the degrees of freedom , 2013bLivadiotis 2015a, 2015c, 2017. Throughout this analysis, we use the notion of the invariant kappa index κ 0 , but the typical three-dimensional index can be easily retrieved, k k = + 3 0 3 2 . Then, the phase-space distribution (2) is rewritten as The marginal distributions come from the integration over the velocity or the positional spaces, and , , 5 which constitute the positional and velocity distributions, respectively. If we integrate over the velocity space, instead of the whole phase-space, then we derive the positional kappa distribution The density profile is proportional to the positional probability distribution, , with normalization given by the density at some reference position r 0 ; below, the density n 0 , temperature T 0 , and pressure p 0 , are given at that position r 0 where the potential turns zero: The temperature profile is determined by the local mean kinetic energy (that is, without the integration over the positional space): In the classical case of BG statistical mechanics, it is rather trivial to obtain the positional distribution and the positional dependence of thermal observables. The integration of the exponential of Hamiltonian (Boltzmann distribution) over the velocity space gives the exponential distribution of the potential energy (Boltzmannian density profile) Therefore, in the Boltzmannian case we obtain µ ( ) ( ) r r p n , hence γ=1 (or n  ¥), corresponding to a single thermodynamic process, the isothermal one.
In the case of kappa distributions, we obtain leading to the relationship between kappa and polytropic indices: Note that we may express this relationship in terms of the it becomes n k = -+ 1 2 , that is, the formulation derived in the earlier studies (e.g., Meyer-Vernet et al. 1995;Moncuquet et al. 2002); care must be shown though, as the involved kappa index k k = + d 0 1 2 differs from the standard three-dimensional one, k k = + 3 0 3 2 , because of the nonzero potential degrees of freedom F d 1 2 .

Hydrodynamics: Euler's Momentum Equation
The Navier-Stokes momentum equation in the presence of a conservative external field of nonzero potential energy Φ, is given by This becomes Euler's equation when the viscosity tensor R is neglected. If the velocity vector field is also independent of the position vector, then the convective acceleration term vanishes,  = ( · ) u u 0. Finally, if the velocity vector field is stationary, then ¶ ¶ = u t 0. Therefore, Equation (13) becomes We observe that both Boltzmannian and kappa distributions of potentials obey the hydrostatic relationship of Equation (14) and the polytropic relationship in Equation (1). Indeed, in the classical case of Boltzmann distribution, we have the thermal observables shown in Equation 11(c). Then, we find In the case of kappa distributions, the pressure is given by Equation (10); then, using also Equations (8), (9), we find  (1). It is an important and interesting question whether these are the only two distribution functions that obey Euler's equation. What is the most general stationary distribution of particle velocities and energies, consistent with an inviscid and polytropic particle flow under a conservative field? First, we will find the most general density profile, and then we derive the corresponding phase-space distribution.

Most General Density Profile Consistent with Polytropes
The density and temperature are respectively written in terms of an arbitrary function f , The polytropic relationship still holds: Then, the thermal pressure becomes and applying Equations 16(a), (c) in Euler's Equation (14), we find The right-hand side is independent of the position, and given that f (0)=1, we find Then, we have . Then, We denote the constant in Equation (18)

Using the Bernoulli Integral
The Bernoulli's integral of energy E for an incompressible hydrodynamic fluid (i.e., = ( ) r n n 0 and = ( ) r B 0), expressed in the comoving reference frame along with the fluid, for a point in the plasma flow streamline with position vector r, is given by For a compressible hydrodynamic fluid, the ratio of the thermal term, p/n, is substituted by the integral ò / dp n, i.e., applying the polytropic relationship between the two different streamline points with position vectors r and r 0 (for which Φ=0), that is, p=p 0 ·(n/n 0 ) γ , we obtain   The functions exp q (x) and ln q (x) are called "q-deformed" exponential and logarithm, respectively (e.g., Silva et al. 1998;Yamano 2002;Livadiotis & McComas 2009;Livadiotis 2017, Chapter 1), x q x ln 1 1 , exp 1 1 , 28 q q q 1 q 1 1 and they are inverse to each other, i.e., ln q [exp q (x)]= exp q [ln q (x)]=x.

In the Presence of a Magnetic Field
For a compressible magnetohydrodynamic fluid, the thermal pressure p also includes the magnetic pressure and magnetic energy density, both equal to m ( ) ( ) r B 2 2 ; namely, the Bernoulli integral becomes where the effective potential energy is defined by the summation of the external potential energy and the magnetic energy, Hereafter, the steps are those followed in the previous section, but now they are expressed in terms of the effective potential energy; then, we find

Most General Phase-space Distribution Consistent with Polytropes
Having shown that the polytropic behavior requires a density profile described by kappa distributions, it is straightforward now to search for the last piece of the puzzle, that is, to show that the kappa distribution density profile, shown in Equation (24) or (34), is coming from the standard phasespace kappa distribution. In other words, we need to show that the phase-space kappa distribution is the unique distribution function that has a positional marginal distribution described by the mentioned kappa distribution profile.
The positional kappa distribution is given after the integration in the velocity space, This integration can be expressed in terms of the kinetic and potential energies treated as variables: The latter can be written as Therefore, the question we wish to answer is, what is the most general functional form f, that is involved in the following integral equation: We can rewrite the above as follows: This is a special case of the Fredholm integral equation of the first kind (Arfken 1985), written as (g 0 includes the normalization constants). This type of integral equation is a convolution of f and h and it can be solved in order to find f as follows. We apply the Fourier transformation F in both sides of Equation (38), then, the function f is expressed as We have the identities where we defined the convolution function Hence, substituting Equation (46) into (43), we find that Namely, it is finally shown that the function f of Equation (36) is exactly the one shown in Equation 35(c), that is, the standard kappa distribution function. Therefore, the only function f involved in the density profile, Notes. Similar derivation steps can lead to a phase-space distribution with anisotropy in the velocities. Also, other nondynamical effects may also cause the polytropic behavior, e.g., the spherical expansion of the solar wind plasma, where the total density profile would be given by the convolution of the two terms.

Discussion
The derived equivalence between the polytropic behavior and the kappa distribution functions can be valuable for further modeling and data analyses. The following are some example applications: (1) Polytropic plasmas are described by kappa distributions (or combinations thereof). An observation of the polytropic behavior of plasma populations in the presence of an external field, constitutes an indirect observation of their distribution function.
(2) The existing Rankine-Hugoniot jump conditions at shock discontinuities do not include the kappa index, because the conservation of mass, momentum, and energy equations are independent of the kappa index (Livadiotis 2015d). The strong connection between polytropic and kappa indices introduces the new type of jump condition. Assuming the same potential upstream (1) and downstream (2) the shock, we find the new jump condition: k n k n + = + ( ) .
4 9 1 1 2 2 the plasma streamlines. 2 . The application requires the knowledge of the potential energy (Equation (33)); examples are the power-law attractive potentials, 1 2 r with d r indicating the dimensionality of the position vector (e.g., d r =3 for a 3D system); hence, (Livadiotis 2017, Chapters 3-5). Finding the polytropic indices requires only the moments of density and temperature. On the other hand, finding the kappa indices is a harder problem that involves analysis of the velocity/energy distributions. In addition, the data sets of distributions are usually not publicly accessible, making this methodology of finding kappa quite valuable.
(5) The estimation of the polytropic and kappa indices may lead to the determination of the potential degrees of freedom, which can help with figuring the local potential energy that applies in the examined plasma particles (e.g., Livadiotis 2018b). In this case, the analysis should be restricted to data sets with slope −1 between the estimated kappa and polytropic indices, according to k n = --- gives the value of b, from which we can determine the type of the potential.

Conclusions
Kappa distributions have been ambitiously used in studies of space and astrophysical plasmas. While it was known that these distributions can lead to polytropes-particle systems with a polytropic behavior-it was unexpected that polytropes could restrict the functional form of the distribution of particle velocities and energies. However, this paper has shown that the reverse is also true: polytropes are consistent only with kappa distributions! In other words, not only the kappa distributions lead to polytropic behavior, but also polytropic behavior leads to kappa distributions (Figure 2). In this way, polytropic behavior has the role of a mechanism generating kappa distributions (Livadiotis et al. 2018).
The work was supported in part by the project NNX17AB74G of NASA's HGI Program. 1 , respectively, and for various polytropic indices γ, where m k º ( ) r z B n kT 0 0 B , º ( ) r x n n 0 , and º ( ) r y T T 0 . The parameter "s" stands for the sign of the kappa index: standard kappa distributions are referred to as positive sign (s=+1), but the distributions of negative sign (s=−1) may also been used (e.g., see Livadiotis 2015b). (The locus of maxima or minima is depicted with dash black line.) Figure 2. Kappa distributions can lead to polytropic relationship through the kappa-distributed density n and temperature T profiles (positional dependence). Polytropic behavior can lead to kappa distributions through Euler's momentum or Bernoulli's energy conservation equations. Both directions of derivations consider the existence of a potential energy Φ, varying along plasma streamlines, at which kappa and polytropic indices are invariant constants.