On the entropy of plasmas described with regularized -distributions
In classical thermodynamics the entropy is an extensive quantity, i.e. the sum of the entropies of two subsystems in equilibrium with each other is equal to the entropy of the full system consisting of the two subsystems. The extensitivity of entropy has been questioned in the context of a theoretical foundation for the so-called -distributions, which describe plasma constituents with power-law velocity distributions. We demonstrate here, by employing the recently introduced regularized -distributions, that entropy can be defined as an extensive quantity even for such power-law-like distributions that truncate exponentially.
I Introduction and Motivation
The so-called -distributions have become popular (e.g., Pierrard and Lazar, 2010) to quantitatively describe the power law behaviour of velocity, momentum or energy distributions of various energetic particle populations, reaching from flare-accelerated electrons (e.g., Kašparová and Karlický, 2009; Oka et al., 2013; Effenberger et al., 2017) via suprathermal electrons and ions in the interplanetary medium (e.g., Fisk and Gloeckler, 2012; Maksimovic et al., 2005; Štverák et al., 2009; Vocks, 2012) as well as in the outer heliosphere Fahr et al. (2016, 2017) even to laboratory laser physics (e.g., Webb et al., 2012; Elkamash and Kourakis, 2016). These distributions have been employed in most cases as useful tools, i.e., in the pragmatic spirit with which they were introduced 50 years ago in the context of magnetospheric physics Olbert (1968); Vasyliunas (1968).
Attempts to physically justify these special power laws can be divided into two groups. On the one hand, it is possible to rigorously derive -distributions for specific systems where particles interact with external radiation Hasegawa et al. (1985), with plasma fluctuations (e.g, Ma and Summers, 1998; Shizgal, 2007; Yoon, 2014; Effenberger et al., 2017), or with a constant temperature heat bath Shizgal (2018). On the other hand, -distributions should be motivated on the basis of more fundamental considerations related to generalizations of the concept of entropy Tsallis (1988) or Gibbsian theory Treumann and Jaroschek (2008). Both approaches face limitations: While the former appears to be valid for systems with special constraints resulting in a special class of -distributions (termed ‘Kappa A’, see below) and only allows specific -values as discussed in Lazar et al. (2016), the latter is requiring a generalized, non-extensive entropy, apparently implying internal inconsistencies Nauenberg (2003) that have as yet not been resolved Tsallis (2004); Nauenberg (2004).
It has been pointed out recently by Scherer et al. Scherer et al. (2017) that, even if these difficulties could eventually be overcome, the resulting -distributions would still be hampered by an only finite number of non-diverging velocity moments, i.e. the condition that for the velocity moment of order to exist. This implies, in particular, that the definition of the -distribution itself, requiring the existence of the second-order moment, i.e., kinetic temperature, is valid only for . Moreover, the heat flux is given by the third-order moment and requires even larger values (see, e.g., Shaaban et al., 2018a, b), while the convergence of higher-order moments should ensure closure schemes for a macroscopic description. These motivated Scherer et al. Scherer et al. (2017) to introduce the regularized -distribution (RKD). The suggested regularization removes all divergences, allows to analytically calculate all (isotropic) velocity moments for all positive -values, and may adjust to power-law distributions observed in the solar wind with clear evidences of exponential cutoffs. As we demonstrate in the present paper, these improvements are not the only advantages: the RKD also possesses an additive entropy, which is, thus, an extensive quantity.
The paper is organized as follows. In the next two sections II and III we define the different forms of the -distributions discussed in the literature, consider the (Boltzmann-)Gibbs entropy, and calculate the entropy of a spatially homogeneous RKD plasma. In section IV we demonstrate explicitly, with an application to isolated plasmas, the additivity, i.e., extensitivity of the RKD’s entropy. Finally, in section V, we consider an inhomogeneous system that is better described with the entropy density rather than the entropy itself. All results are summarized and discussed in a concluding section VI.
Ii -distributions: definitions
where denotes the number density of the considered particle species, the gamma function, the particle speed, and . The reference speed , introduced as the most probable speed Vasyliunas (1968), is related to a kinetic temperature via the second-order moment
One distinguishes two choices. The first is to consider the temperature in Eq.(2) to be always equal to that of the associated Maxwellian (e.g., Hellberg et al., 2009; Yoon, 2014; Astfalk et al., 2015; Livadiotis, 2017),
which enables to extend the concept of temperature of a -distribution in the strict sense of thermodynamics. This choice naturally implies a distribution that is not only above the associated Maxwellian at high but also at low speeds, see ‘Kappa A’ in Figure 1.
The alternative is obtained in the limit , when the speed is independent of (and equals the thermal speed of Maxwellian limit) allowing for the modelling of suprathermal wings of a distribution, denoted ‘Kappa B’ in Figure 1, on the expense of its core population (e.g., Olbert, 1968; Hau and Fu, 2007; Fahr et al., 2014; Lazar et al., 2016), i.e. without any enhancement at low speeds relative to the associated Maxwellian. While systems properly described with Kappa A are usually specifically set up and consistent with special (isolated) -values, and exhibit -dependent speeds , those obeying Kappa B have less constraints Hau et al. (2009) and describe total populations with temperatures increasing for decreasing -value. For the ongoing debate about which choice is correct or, at least, represents an appropriate description of a given system, see Lazar et al. (2016) and Livadiotis (2017).
It must be noted that both choices exhibit unphysical features. First, in the usual classical (as opposed to a relativistic) treatment the power law (1) extends to infinite speeds, implying an infinite number of diverging velocity moments. While this feature has been tried to be explained some while ago in the context of a finite sample size effect and the concept of self-organized criticality (e.g. Schertzer and Lovejoy, 1993), regarding the -distributions there is a second unphysical feature, namely that even formally existing moments are diverging for values , see, for example, Eq.(2). In order to remove these unphysical features Scherer et al. Scherer et al. (2017) defined the regularized -distribution (RKD)
by introducing a physically motivated exponential cut-off controlled via the parameter . For sufficient low values of the latter both the low-order velocity moments and the kinetic properties are virtually the same as for the corresponding standard -distributions. While, again in view of a finite sample size effect, it might be difficult to determine the value of this cut-off parameter in all cases, examples for such determination can be found in Scherer et al. (2017). Most importantly, the RKD allows an analytical calculation of all velocity moments for all positive -values. is the required normalization constant.
Iii Gibbs Entropy
A general definition of entropy that is valid both for equilibrium and non-equilibrium systems was given originally by Boltzmann Boltzmann (1872) and Gibbs Gibbs (1902) and specifically for a plasma constituent more recently, e.g., by Balescu Balescu (1975, 1988), and Cercignani Cercignani (1988):
where is the phase space distribution function of particles of the considered species and is the Planck constant. This definition of the Gibbs entropy (sometimes called Boltzmann-Gibbs entropy) not only takes into account the quantum mechanical lower limit of the phase space volume occupied by a single particle and also contains the Gibbs factor in order to avoid the Gibbs paradoxon Gibbs (1902) that is related to the indistinguishability of states after interchanging identical particles, it is also valid for non-equilibrium systems Balescu (1988); Cercignani (1988). While this definition is useful for the case of homogeneous plasmas, it is more appropriate to define an entropy density for spatially inhomogeneous systems, an example of which we discuss in section V.
In the following, we first briefly review the calculation of the entropy for a Maxwellian plasma constituent and then apply the above definition for a plasma constituent obeying an RKD Scherer et al. (2017). For both cases we assume stationary, isolated plasmas with vanishing spatial gradients, i.e., we assume spatial homogeneity.
iii.1 Maxwellian plasma
With the usual definition of the zeroth- and second-order moments
the Maxwellian entropy can be written as
which, upon introducing the so-called thermal de Broglie wavelength (e.g., Goeke, 2010), reads
As long as the above assumption of constant number density holds, is proportional to the total number of particles and, thus, it is an extensive quantity.
iii.2 RKD plasma
The (non-drifting) RKD Scherer et al. (2017) is given in Eq.(4). As before, all quantities are assumed to be independent of location. Note that the phase space distribution is normalized to while the velocity distribution is normalized to unity. Using the RKD in Eq.(5) leads to
The normalisation constant is chosen such that
still holds, so that
The two remaning integrals are (i) independent of location, (ii) finite functions of the parameters and , and (iii) independent of particle number . This allows to express the entropy for the RKD as:
where we have defined a generalized thermal de Broglie wavelength , and the two functions
This is the main result: Since all quantities in the square bracket in Eq.(14) are independent of particle number the entropy is proportional to and, thus, an extensive quantity.
Obviously, the Maxwellian case is obtained in the limit with . Then one has
so that correcty reduces to .
Note that, interestingly, this finding may not apply to the standard -distribution, which is obtained from Eq.(4) with for . This is because in the case the number of non-diverging velocity moments is finite and, thus, the entropy definition (5) may not apply (Balescu, 1988). Consequently, this ‘incompleteness’ maybe the reason for the non-extensitivity of entropy for the standard -distribution.
Iv Isolated, homogeneous plasmas
To further elucidate the entropy formula for the RKD let us consider the case of two plasma volumes and filled with and particles of the same species, see the left box in Figure 2. The two plasmas are in equilibrium with each other, i.e., have the same temperature and pressure, and, thus, also the same number density . These plasmas are then mixing and, eventually, fill the total volume with particles, see the right box in Figure 2.
As before, we first briefly recapitulate the case of two Maxwellian plasmas and, afterwards, that of two RKD plasmas. In both cases we consider thermal equilibrium, i.e., plasmas of the same temperature.
iv.1 Two Maxwellian plasmas
For the entropies of the individual plasmas, it is in shown in standard textbooks, e.g., Goeke (2010) that one has with formula (7) and the relation
This is valid, because with the given constraints, one also has
Under these assumptions, in (local) equilibrium the sum of the entropy of two Maxwellian plasma systems with identical particles in separate volumes is equal to the entropy of the mixed plasma filling the total volume.
iv.2 Two RKD plasmas
First, it is important to note the fact that equal temperature and equal pressure for two plasmas described with RKDs implies that the two distributions have the same - as well as -values. Second, given that Scherer et al. (2017), they also have the same number density, implying that Eq.(21) also holds for two RKD plasmas under the given constraints.
Then, upon introducing the abbreviation in Eq.(14), one has
The same formula yields for the situation when the plasmas have merged:
Exploiting Eq.(21) again results in the finding
Consequently, entirely analogous to the case of two Maxwellian plasmas, one finds that the RKD entropy is an extensive quantity. Again, as noted in section III, this is not necessarily including the case , i.e., the standard -distribution, for which non-extensitivity of entropy has been shown (e.g. Tsallis, 1988; Leubner and Vörös, 2005).
V Spatially inhomogeneous plasmas
We consider an example from space plasma physics, which is not only the origin of -distributions, but an area of their frequent application. The subsonic solar wind in the so-called inner heliosheath, i.e., the region between the shock transition that terminates the supersonic expansion of the solar wind and the heliopause that separates the solar from the interstellar plasma, is a spatially inhomogeneous plasma. While, due to the subsonic flow, incompressibilty is nearly fulfilled Fahr and Siewert (2015) and, thus, the density is constant, both the hydrodynamic bulk velocity and the kinetic velocity distributions of suprathermal particles are functions of the spatial coordinates. The corresponding proton and electron constituents have recently been treated on the basis of standard -distributions Fahr et al. (2014, 2016, 2017). The evolution of the proton distribution function was described by deriving a hydrodynamical differential equation for the parameter as a function of position.
Depending on the absence or presence of sources or sinks of energy, one can distinguish an isentropic and a non-isentropic case, respectively. Starting with the former, we consider the first law of thermodynamics
where the new quantities and denote the internal energy and pressure of the plasma component, and is the volume of a plasma parcel. If the flow isentropic and isothermal along a flowline with coordinate , i.e. and , the work done by the pressure at the expansion of a moving plasma volume is the only reason to change its internal energy . Consequently, one has
where we have introduced the energy density , and the second equality follows from its moment definition. This reduces to
Using , Eq.(2) and incompressibility, i.e. one obtains
Particle conservation implies along a given streamline (where is the solar wind plasma convection speed), which for constant reduces to . so that
For this reproduces the results obtained by Fahr et al. (2016) for the case of no sources or sinks and vanishing velocity diffusion: a constant convection speed along a streamline implies constant and an increasing (decreasing) speed results in an increasing (decreasing) . While constant yields, via Eq.(2), both constant temperature and constant reference speed , increasing (decreasing) would translate into decreasing (increasing) temperature Scherer et al. (2018). This, however, is excluded here by the above assumption of , so that cannot be considered constant. The latter combination is equivalent to Kappa A as discussed in section II. Given that the heliosheath is not isothermal (Richardson and Wang, 2012), however, it is more likely that Kappa B is the appropriate choice.
In case the flow along the streamlines developes non-isentropically due to presence of energy sources and sinks, i.e. if the entropy of the fluid changes with the flow line element , one has to consider the following relation for the entropy per volume
which follows from with and the incompressibility condition . The newly introduced quantitiy describes changes of the internal energy of a comoving volume element . As discussed in Fahr et al. (2016), these changes are due to (i) velocity diffusion with a diffusion coefficient proportional to and (ii) the so-called magnetic cooling. The related changes are, as calculated in Fahr et al. (2016) for standard -distributions, proportional to the thermal pressure:
with denoting a diffusion constant and the strength of the magnetic field. Since the same holds for the temperature via , the change of entropy density along a streamline
is independent of . Consequently, one obtains
which describes the change of entropy density along a given streamline. Note, first, that this expression is via still depending on and, second, that for other velocity diffusion models and other distribution functions (e.g., the RKD) also the entropy density change will depend on .
Vi Summary and conclusions
Starting from the general (Boltzmann-)Gibbs definition, we derived, first, a formula for the entropy of a spatially homogeneous plasma whose constituents can be modelled on the basis of the regularized -distribution. Second, we have demonstrated that for these distribution functions entropy is, analogous to a Maxwellian plasma, an extensive quantity. And, third, we have discussed the change of entropy (density) along streamlines in an incompressible, but otherwise inhomogeneous flow.
In conclusion, we state that within the framework of regularized -distributions entropy can be defined in such a way that it maintains – in difference to the case of the standard -distributions – its additivity, which appears mandatory in view of the fundamental laws of thermodynamics.
Acknowledgements.We are grateful for support from the Deutsche Forschungsgemeinschaft (DFG) via the grants SCHE 334/9-2, SCHL 201/35-1, and from FWO-Vlaanderen (Grant GOA2316N). We also appreciate the support from the International Space Science Institute (ISSI) for hosting the international ISSI team on Kappa Distributions: From Observational Evidences via Controversial Predictions to a Consistent Theory of Suprathermal Space Plasmas, which triggered many fruitful discussions that were beneficial for the work presented here. We are also grateful for helpful comments by Christian Röken.
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