In this paper we develop a model for the anisotropic Maxwell–Jüttner distribution and examine its properties. First, we provide the characteristic conditions that the modeling of consistent and well-defined anisotropic Maxwell–Jüttner distributions needs to fulfill. Then, we examine several models, showing their possible advantages and/or failures in accordance to these conditions. We derive a consistent model, and examine its properties and its connection with thermodynamics. We show that the temperature equals the average of the directional temperature-like components, as it holds for the classical, anisotropic Maxwell distribution. We also derive the internal energy and Boltzmann–Gibbs entropy, where we show that both are maximized for zero anisotropy, that is, the isotropic Maxwell–Jüttner distribution.
The Maxwell–Boltzmann distribution describes classical non-interacting particles at thermal equilibrium. This distribution is generalized within the framework of special relativity, leading to the Maxwell–Jüttner distribution, named after Ferencz Jüttner (1911).
In the case of anisotropic temperature, the Maxwell distribution becomes anisotropic on the kinetic degrees of freedom, which contribute differently in the internal energy of the system (e.g., solar wind: Olsen and Leer, 1999; Feldman et al., 1975; Pilipp et al., 1987; Phillips and Gosling, 1990; Kasper et al., 2002; Matteini et al., 2007; Štverák et al., 2008 – magnetosphere: Pilipp and Morfil, 1976; Renuka and Viswanathan, 1978; Tsurutani et al., 1982; Sckopke et al., 1990; Gary, 1992; Bavassano Cattaneo et al., 2006; Nishino et al., 2007; Cai et al., 2008; Winglee and Harnett, 2016 – also see the corresponding formulations in Krall and Trivelpiece, 1973; Livadiotis and McComas, 2014a; Livadiotis, 2015).
Therefore, we have two main types of generalization of the Maxwell–Boltzmann distribution, that is, the relativistic and the anisotropic description. However, there is no consistent modeling for both the generalizations together, namely, the anisotropic relativistic case, and the formalism of the anisotropic Maxwell–Jüttner distribution remains unknown. It is important to note that such a model may not be unique. From our experience on kappa distributions, a well-defined and unique isotropic model may degenerate to several anisotropic models, each one, however, corresponding to different physical meaning (see Sect. 5 in Livadiotis, 2015).
The purpose of this paper is to develop the anisotropic Maxwell–Jüttner distribution and examine its properties. First, the paper provides the characteristic conditions that a consistent and well-defined anisotropic modeling of the Maxwell–Jüttner distribution needs to fulfill. Then, guided by these conditions, the paper derives such a consistent model and, finally, examines several basic properties related to thermodynamics, e.g., the behavior of the internal energy, temperature, and entropy.
Next, Sect. 2 shows briefly the derivation of the standard, isotropic Maxwell–Jüttner distribution. Then, Sect. 3 proceeds to model the anisotropic Maxwell–Jüttner distribution. First, we provide the characteristic conditions of such a consistent model, and then examine several models, showing their possible advantages and/or failures in accordance with these conditions; finally, we end up with the correct model. In Sect. 4, we examine the properties and thermodynamics of this model, e.g., the internal energy, the partition of temperature to its directional components, and the entropy. Finally, Sect. 5 summarizes the conclusions.
The Maxwell–Boltzmann (MB) distribution
The relativistic generalization of Eq. (1a), that is, the
Maxwell–Jüttner (MJ) distribution, is given by
According to the relativistic formalism for the particle momentum and
energy, we have
The Boltzmann distribution of a Hamiltonian is
The MJ distribution of the electron kinetic energy
In Fig. 1 we plot the MJ distribution of the kinetic energy
When the MB distribution clearly deviates from the MJ distribution of the
same temperature and dimensionality, then a different MB distribution can
give a good approximation to the MJ distribution. This new MB distribution
can be either (i) a convected MB distribution, that is, an MB distribution
with the same dimensionality, but with different temperature
Also useful is the expression
of the distribution in the velocity space (Dunkel et al., 2007). Given that
Note that there were several other attempts of further generalization of the
MJ distribution, for example, using a power-law of energy, that is,
The plot of an MJ distribution of temperature
The derived distribution must have the following characteristic conditions.
Correspondence at both classical and isotropic limits:
Correspondence for Correspondence for Non-symplectic energy–temperature components:
The inertial mass energy does not depend on kinetic terms and is isotropic. Each kinetic component is connected with the corresponding
temperature component. Temperature partition to its anisotropy components:
Internal energy must be additive to its components,
Internal energy and entropy is maximized for zero anisotropy.
Next, we are going to derive and criticize several models using the above
conditions.
The particle energy–momentum relation can be a motive for a certain model of
the anisotropic MJ distribution. Indeed, following the derivation of the
energy–momentum relation,
The problem with the above derivation is that we are violating the
energy–momentum relation of the free particle,
The kinetic degrees of freedom,
The anisotropy applies by considering different statistics for each component
(or group); hence, while the temperature
Namely, in the isotropic MB distribution the independence of the
probabilities
In the relativistic case, the probabilities are not independent, so their
product cannot lead to the joint distribution
Let
According to this, the energies
Again, we reiterate that the inertial energy
It is always possible to write the anisotropic relation in a symplectic way
so that both correspondence conditions hold (Sect. 3.1). In the following
example, the inertial energy is characterized by anisotropy, while there is
one symplectic kinetic component (
In another example, for any value of
We may write the energy–momentum relation in a way that the summation on the
energy component is linear, namely,
– Correspondence for
– Correspondence for
– Correspondence for
Correspondence of anisotropic Maxwell–Jüttner (MJ) distribution to
the isotropic Maxwell–Jüttner and anisotropic Maxwell distributions; we
use
We start with the isotropic MJ distribution. The internal energy,
In the anisotropic case, the corresponding integral cannot be solved
analytically,
The internal energy depends on the anisotropy
Finally, we show the partition of temperature
This shows the additivity of the temperature directional components characterizing each dimension, as required by the conditions in Sect. 3.1.
The Boltzmann–Gibbs entropic formulation, which is aligned with the
classical kinetic theory and the Maxwell distribution (Livadiotis and
McComas, 2009), is better known for the discrete probability distribution
Having introduced the momentum scale
the partition function as
the mean value of a momentum function
and the entropy as
The entropy is analytically derived, below. First, we derive this
for the isotropic case:
Then,
Here, we show four basic relations of thermodynamics using the isotropic
relations. Up to now, there is no consistent thermodynamic theory for
anisotropic temperature. Equation (51) gives the first thermodynamic
relation, from which we easily derive the second, the free-energy relation:
Note that the investigation to derive the thermodynamic definition of the temperature in the case of the anisotropic MJ, or even the classical MB, distributions is still in process.
The paper developed a model for the anisotropic Maxwell–Jüttner distribution, and examined its properties and thermodynamics. The Maxwell–Jüttner distribution is useful in space, geological, and other plasmas where high energy particles often reach the relativistic limits where the Maxwell–Boltzmann distribution is not applicable. On the other hand, these plasmas are known to be characterized by anisotropic Maxwell–Boltzmann distributions as long as particle energies are non-relativistic; thus, we have no reason to expect isotropic distributions at relativistic high energies. Therefore, it is necessary to deduce a consistent formulation of anisotropic Maxwell–Jüttner distributions.
First, it provided the characteristic conditions that a consistent and well-defined anisotropic model of Maxwell–Jüttner distributions needs to fulfill. Then, guided by these conditions, the paper derived a consistent model, and examined its properties and thermodynamics.
In particular, the paper showed and discussed the following analytical
developments:
provided the conditions for modeling consistent anisotropic
Maxwell–Jüttner distributions, examined several models, showing their possible advantages and
failures, derived a consistent anisotropic model that fulfills all the desired
characteristic conditions. For example, the correspondence with both the
classical and isotropic limits, for studied the properties and thermodynamics of this model, e.g., the internal
energy, the partition of temperature to its components, and the entropy. The
anisotropic internal energy and entropy are both depended on the anisotropy;
they are maximized for the isotropic case, while both become independent of
the anisotropy at the classical case of
It is now straightforward for space physics researchers to use the derived analytical model in applications. The next goals may be to (i) show the connection with thermodynamics for the anisotropic model, (ii) derive other different anisotropic model(s), and (iii) to establish both the isotropic/anisotropic models within the framework of kappa distributions for particle systems described by stationary states out of thermal equilibrium (e.g., Livadiotis, 2015).The topical editor, E. Roussos, thanks two anonymous referees for help in evaluating this paper.