# “Lagrangian Temperature”: Derivation and Physical Meaning for Systems Described by Kappa Distributions

## Abstract

**:**

**PACS Codes:**05.20.-y (Classical Statistical Mechanics); 05.20.Dd (Kinetic theory); 94.05.-a (Space plasma physics)

## 1. Introduction

^{q}probability distributions of energy ɛ, related by P(ɛ)~p(ɛ)

^{q}(normalized accordingly) [52]. The respective generalization of the Maxwellian distribution of velocities is derived by maximizing the Tsallis entropy (under the constraints of Canonical Ensemble [34]). The generalized distribution is a q-exponential function (see [34] and references therein). Such q-exponential distributions are observed quite frequently in nature and it is now widely accepted that these distributions constitute a suitable generalization of the Boltzmann-Gibbs exponential distribution. (For applications of q-exponential distributions, see [50,34] and refs therein.) [34] showed that the q-exponential distribution coincides precisely with the kappa distribution and that the entropic index q that characterizes the Tsallis entropy [49], is related to the κ-index of the kappa distribution by q = 1 + 1/κ [34]. Therefore, the q-exponential distribution is expressed in terms of the q-index and is widely used in the Statistical Physics community, while its equivalent, the kappa distribution, is expressed in terms of the κ-index and is more commonly used in the Space Physics community. The kappa distribution for a 3-dimensional system is given by the escort distribution [1,34,36],

_{B}is the Boltzmann constant), while the respective ordinary probability distribution is:

_{+}= y, if y ≥ 0 and [y]

_{+}= 0, if y ≤ 0, [50]). Indeed, Equation (2) becomes:

_{B}T̃

_{mx})], for κ→∞. However, the included temperature T̃

_{mx}does not constitute the actual temperature T (that is the real or measure one), but some function of the actual temperature T and the kappa index κ, T̃

_{mx}= T̃

_{mx}(T;κ), that is:

_{mx}is defined and determined by the temperature that the system would have if it was residing at thermal equilibrium and described by a Maxwell distribution (e.g., see [20,27,30,53]). However, such a definition is inconsistent:

_{B}T).

## 2. Derivation of the Canonical Probability Distribution

_{q}is the following probability functional:

_{u}is the smallest speed scale parameter characteristic of the system, so that the quantity:

_{u}

^{−}

^{f}, or, p(u⃗)·σ

_{u}

^{f}is dimensionless.)

^{2}is the particle kinetic energy, and <ɛ>=(1/2)m<(u⃗−<u⃗>)

^{2}> is the particle mean kinetic energy, that is, the internal energy U in the absence of a potential energy. Given the two constraints, the entropy is maximized using the Lagrange method, that is, by using the two Lagrange multipliers λ

_{1}and λ

_{2}to maximize the constructed probability functional G[p(u⃗)] ≡ S[p(u⃗)] + λ

_{1}·μ[p(u⃗)] + λ

_{2}·E[p(u⃗)], i.e.:

_{2}represents the negative inverse of the Lagrangian temperature T

_{L}, i.e., λ

_{2}≡ −β

_{L}with β

_{L}≡ 1/(k

_{B}T

_{L}), while the actual temperature T is given by β = β

_{L}/φ

_{q}and β = 1/(k

_{B}T). Hence:

_{b}≡ <u⃗>.

_{0}+d/2) [36]: The known kappa index κ is dependent on the correlated degrees of freedom f, and can be related to an invariant kappa index κ

_{0}by κ(f) = κ

_{0}+ (1/2)f. If N

_{C}is the number of correlated particles and d is the number of degrees of freedom per particle, then f = d·N

_{C}, and the dependent kappa index is κ(N

_{C}) = κ

_{0}+ (d/2)N

_{C}. Note that κ

_{0}is the actual kappa index that characterizes a stationary state, and it is invariant from the number of particles and degrees of freedom of the system [36]. The “thermalization” of the system—the system to reach thermal equilibrium—is realized for quite large values of the kappa index, i.e., 1 ≪ N

_{C}≪ κ

_{0}→ ∞, that is to approach infinity and the distribution to become Maxwellian. Using the invariant kappa index κ

_{0}, the kappa distribution is written as:

_{B}T), for f = 3 degrees of freedom, and various values of the kappa index. As the kappa index increases, the maximum of the distribution lowers, while the two “edges” (for small and large x) raise, so that the mean value of x (mean energy) to be preserved. Indeed, the mean kinetic energy does not depend on the kappa index, thus, it remains constant under variations of the kappa index (kinetic definition of temperature—see Section 3).

## 3. Kinetic/Thermodynamic Definitions of Temperature for Systems out of Thermal Equilibrium

_{B}T. This is the equipartition theorem applied to each of the (f) kinetic degrees of freedom of a particle system at thermal equilibrium. In other words, the mean kinetic energy per (half) degrees of freedom defines the kinetic energy k

_{B}T, or, the temperature (in units of J/k

_{B}). The temperature must be independent of other thermodynamic parameters, e.g., the kappa index. Indeed, the kinetic definition of temperature is applicable for both systems at or out of thermal equilibrium, and the equipartition theorem is identical for any kappa index [1,17,34]. While the mean kinetic energy defines the temperature, the correlations between the individual particle energies define the kappa index κ

_{0}[36]. In principle, the kappa index and temperature are independent thermodynamic parameters, meaning that the mean kinetic energy has no effect on the particle correlations. As it was correctly stated in [32], “… clearly, from the definition of temperature, all distributions with the same mean energy per particle have the same temperature”. It is worth noting that only the second statistical moment of velocities has this fundamental physical meaning of defining temperature, and thus, no other moment is independent of the kappa index. It can be shown that the α

^{th}moment, μ ≡ <x

^{α}

^{/2}>

^{1/}

^{α}(with x ≡ ɛ/(k

_{B}T) = (u⃗−u⃗

_{b})

^{2}/θ

^{2}), is $\mu {(\alpha ;{\kappa}_{0})}^{\alpha}\equiv {{\kappa}_{0}}^{{\scriptstyle \frac{\alpha}{2}}}\xb7\mathrm{\Gamma}({\kappa}_{0}+1-{\scriptstyle \frac{\alpha}{2}})/\mathrm{\Gamma}({\kappa}_{0}+1)$. Figure 2 demonstrates the derivative of the α

^{th}statistical moment with respect to the invariant kappa index κ

_{0}, showing that only the moments for α = 0 and α = 2 are independent of the kappa index.

^{−1}[55]; in the absence of a potential energy, the internal energy is given by the mean kinetic energy, i.e., the temperature. At thermal equilibrium, the two temperature definitions are equivalent, namely, they lead to the same temperature, $T\equiv {(\partial S/\partial U)}^{-1}={\scriptstyle \frac{2}{f\hspace{0.17em}{k}_{\text{B}}}}\xb7<\varepsilon >$.

^{−1}·[1−(1/κ)·S/k

_{B}]. [56] showed that this is the most generalized formulation of the temperature’s thermodynamic definition that can be consistent with the zero-th law of Thermodynamics. [34] showed the equivalence of these two different temperature definitions—the kinetic and thermodynamic—that produces a well-defined temperature for systems out of thermal equilibrium described by kappa distributions.

_{2}) that corresponds to the constraint of internal energy in the Canonical Ensemble, as it is shown in Equations (16) and (17). While all three temperature definitions (kinetic, thermodynamic, Lagrangian) coincide at thermal equilibrium, they are typically different out of thermal equilibrium. In particular, for systems out of thermal equilibrium described by kappa distributions, the kinetic and thermodynamic temperature definitions are still equivalent, leading to a well-defined temperature for stationary states out of thermal equilibrium. However, the Lagrangian definition gives the actual temperature only at thermal equilibrium, while for any state other than thermal equilibrium, it behaves like a function of the actual temperature. (Note: The Lagrangian definition gives the temperature used in the formulation of Maxwell distribution of velocities. Therefore, the Lagrangian temperature can be also referred to as Maxwellian temperature—see Equation 8.)

_{L}and the actual temperature T may not be a simple proportionality. The origin of the Lagrangian temperature is T

_{L}= (−λ

_{2})

^{−1}/k

_{B}or T

_{L}= (∂S/∂U)

^{−1}= T/[1−(1/κ)·S/k

_{B}] = T/ϕ

_{q}, that is Equation (20), which coincides with T at thermal equilibrium (κ→∞). The non-proportionality arises because the argument ϕ

_{q}depends also on the temperature, ϕ

_{q}= ϕ

_{q}(T). This dependence is inherited by the speed scale σ

_{u}= σ

_{u}(T) (Section 5).

## 4. The N-Particle Kappa Distribution

_{C}-particle kappa distribution gives the probability of N

_{C}correlated particles to have kinetic energies ɛ

_{1}, ɛ

_{2}, …, ɛ

_{N}[1], i.e.:

_{C}correlated particles have f = d·N

_{C}total kinetic degrees of freedom. Each velocity vector contributes equally to the summation in Equation (25), which can be rewritten by substituting the summation on the particles with the summation on the degrees of freedom, i.e.:

_{j}and u

_{b}

_{,}

_{j}denote components of the particle velocity and the bulk velocity of the plasma, respectively. The bulk velocity vector is identical for all the particles, so that the j-th component is the same with the mod(j, d) -th. Then, the summation ${\sum}_{j=1}^{d\xb7{N}_{\text{C}}}{({u}_{j}-{u}_{b,j})}^{2}$ can be considered as the velocity magnitude of a (f=d·N

_{C})-dimensional velocity vector u⃗. Hence, the N

_{C}-particle kappa distribution is written as Equation (24), but now the kinetic degrees of freedom for the velocity vector are f = d·N

_{C}instead of d.

_{q}. In general, the formalism of non-extensive statistical mechanics is not supporting the usage of the more simplified 1-particle kappa distribution, but rather the more complicated N-particle kappa distribution.

## 5. Relation between the Lagrangian and Actual Temperature

_{q}that connects the actual temperature T with the classical temperature T

_{L}. Then, substituting ϕ

_{q}in Equation (20), we end up to the desired relation.

_{q}is given in terms of a speed scale parameter σ

_{u}, or equivalently, in terms of a dimensionless scale parameter σ = σ

_{u}/θ. (The subscript u indicates that the constant σ

_{u}is a scale of speed.) Then, we have:

_{b}= 0, thus p(u⃗)=p(u) and:

^{f}

^{/2}/Γ(f/2) is the surface area of the f-dimensional sphere of unit radius. Then, by substituting p(u) to Equation (29), we obtain:

_{u}/θ.

_{q}is by using directly the escort distribution function, P(u⃗). This is more convenient method, because the distribution that describes the particles of the system and its statistical moments is the escort P(u⃗) and not the ordinary p(u⃗) distribution. The duality of ordinary/escort distributions is given by the following scheme:

_{b}= 0 for simplicity, so that P(u⃗) = P(u). Then, the argument ϕ

_{q}, given by Equation (30), is derived by substituting P(u) into:

_{q}, the connection between the actual temperature and the Lagrangian temperature is given by substituting Equation (30) into Equation (20), i.e.:

_{0}+ (1/2)f [36]. Using the invariant kappa index, κ

_{0}≡ κ − (1/2)f, Equation (30) is rewritten as:

_{C}and the degrees of freedom per particle d = 3, then f = 3N

_{C}:

_{C}≫1, the argument φ

_{q}is ~σ

^{2}/πe, i.e., it is independent of the kappa index. Such examples are the weakly coupled plasmas; these have large number of correlation particles within Debye spheres, N

_{D}≡ (4π/3)nλ

_{D}

^{3}≫ 1, where λ

_{D}is the Debye length. For smaller N

_{C}numbers, ϕ

_{q}increases slightly with the kappa index. When the kappa index exceeds the number of particles, κ

_{0}> (3/2)N

_{C}, ϕ

_{q}increases more abruptly with the kappa index, approaching ϕ

_{q}~ 1. The dependence of ϕ

_{q}on the invariant kappa index κ

_{0}, and for various N

_{C}, is shown in Figure 3. Next, we derive the dimensionless scale σ that can be related to the temperature, and other thermodynamic parameters, depending on the system and its correlation length.

## 6. The Scale Parameter

_{C}correlated particles is different from that of a classical system of uncorrelated particles (at thermal equilibrium). For classical systems, the number of microstates in an infinitesimal volume of N-particle phase-space is given by dΩ = m

^{3}

^{N}d⇉

_{1}du⃗

_{1}···d⇉

_{N}du⃗

_{N}/(N!h

^{3}

^{N}). For systems with local correlations, the microstate number is dΩ = m

^{3}

^{N}d⇉

_{1}du⃗

_{1}···d⇉

_{N}du⃗

_{N}/(c

_{N}h

^{3}

^{N}), where the combinations of correlated particles give c

_{N}≅ N!N

_{C}!

^{N}

^{/}

^{NC}, or (c

_{N}/N!)

^{1/}

^{N}= N

_{C}!

^{−1/}

^{N}

^{C}≅ e/N

_{C}.

_{C}≡ (4π/3)nℓ

_{C}

^{3}counting the number of correlated particles. In a different approach, we can show this using the number of phase-space microstates of the correlated particles, that is:

_{C}:

_{*}~10

^{−22}J·s, that replaces the Planck’s constant ħ ~10

^{−34}J·s.

_{C}≈ 1, and the “correlation” length were reduced to the interparticle distance, ℓ

_{C}≅b≡n

^{−1/3}. (Precisely, it is ℓ

_{C}= e

^{1/3}b, for recovering the undistinguished particles combination, N!.) In this case, we have:

^{N}.

_{L}and the system’s (actual) temperature T. For κ

_{0}≪ N

_{C}→∞, all of these relations can be described by a power-law dependence, T

_{L}~T

^{α}, where α is hereby called thermal equilibrium index. Therefore, different correlation lengths induce different types of the exponent α. Consider the following five types of correlation lengths ℓ

_{C}: (i) Interparticle distance, b~n

^{−1/3}, that gives the lower limit of a correlation length, because below this limit, there is one particle the most, and thus, correlations are no effective; (ii) Debye length, ${\lambda}_{\text{D}}\propto \sqrt{T/n}$, that is the smallest length scale of local correlations in plasmas and is caused by Debye shielding electrostatic particle-particle interactions [39]; (iii) thermal wavelength, ${\lambda}_{\text{W}}~\sqrt{T}$, that is the particle thermal speed over the wave frequency, and is caused by wave-particle interactions [57]; (iv) thermal gyro-radius, ${\rho}_{g}~\sqrt{T}/B$, that is the ion thermal gyro-radius caused by an ambient magnetic field; (v) mean free path, L

_{m}~ θ/ν

_{col}~T

^{2}/n (ν

_{col}: collision rate), that gives the upper limit of a correlation length; beyond the mean free path, correlations are strongly damped by collisions.

_{0}≪ N

_{C}→ ∞, the relation between the Lagrangian temperature T

_{L}and the actual temperature T, and gives the various values of the index α that correspond to the above five correlation lengths ℓ

_{C}.

## 7. Application to the Solar Wind Throughout the Heliosphere

_{L}, where the involved scale is given by Equation (46):

_{e}is the elementary charge, and

**ɛ**is the plasma permittivity; (in Figure 4 we assume T

_{i}≅ T

_{e}, i.e., T

_{0}≅ T

_{i}/2).

_{L}and the (given) actual temperature T of solar wind throughout the heliosphere. While the values of the solar wind’s temperature span a range between a few thousands to a few million degrees, the derived values of the Lagrangian temperature span a larger range from 1 K to 10

^{10}K. For example, for T ~ 10

^{4}K the Lagrangian temperature ranges from T

_{L}~ 1 K to T

_{L}~ 10

^{4}K, while for T ~ 10

^{5}K the Lagrangian temperature ranges from T

_{L}~ 10

^{2}K to T

_{L}~ 10

^{6}K. The auxiliary line, given by log T

_{L}~ −8.6+2.5·logT, describes the average of the determined values of the Lagrangian temperature for the inner heliosphere (r up to 10 AU), that is for all the used datasets except IBEX. The slope (on a log-log scale) is α ~ 2.5 (see Table 1), corresponding to average polytropic index ν ~ 0.5 or γ ~ 3, that is the polytropic index that characterizes space plasmas (e.g., see [58]). The auxiliary line, given by logT

_{L}~−14.7+4·logT, describes the average of the determined values for the inner heliosheath using the IBEX datasets [22,24,26].) In this case, the slope (on a log-log scale) is α ~ 4, corresponding to average polytropic index ν ~ −1 or γ ~ 0, that is the value of polytropic index that was found to describe the inner heliosheath [8].

## 8. Conclusions

## Conflicts of Interest

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**Figure 1.**Invariant kappa distribution. (

**a**) The distribution is depicted in terms of the energy x = ɛ/(k

_{B}T) and for values of the invariant kappa index κ

_{0}= 0.01, 0.1, 1, and ∞. (

**b**) The same for κ

_{0}= 10

^{−2}, 10

^{−4}, 10

^{−6}, 10

^{−8}, and ∞.

**Figure 2.**The derivative of the α

^{th}statistical moment with respect to the invariant kappa index κ

_{0}, i.e., (∂/∂κ

_{0}) μ(α;κ

_{0})

^{α}. The derivative becomes zero only for α=0 and α=2, independently of the kappa index. Therefore, only these two statistical moments are independent of the kappa index.

**Figure 3.**Dependence of ϕ

_{q}= T/T

_{L}on the invariant kappa index κ

_{0}, for σ

^{2}= 0.5 and various N

_{C}(see Equation (39)).

**Figure 4.**Relation between the Lagrangian temperature T

_{L}and the actual temperature T of solar wind throughout the heliosphere. Datasets used: Helios 1 (daily averages from 1974 to 1981) for heliocentric distance r ~ 0.3–1 AU, ACE (4-min averages for the whole year of 2012) for r ~ 1AU, Ulysses (daily averages from 1990–2009) for r ~ 1–5AU, Voyager 1 (daily averages from 1977–1980) for r ~ 1–10AU, IBEX (datasets from [22,24,26] for the first year of operation, 2009–2010) for the inner heliosheath, that is roughly beyond r ~ 100 AU. Two extreme values of the kappa index were used, κ

_{0}=0.1 (upper points, e.g., grey, blue, red, light green) and κ

_{0}=100 (lower points, e.g., dark red, light blue, orange, deep green); exception in the case of IBEX where the values of the kappa index are given—all distributed in the interval 0<κ

_{0}<1. The two auxiliary black dash lines are given by log T

_{L}~ −8.6+2.5·logT for the inner heliosphere and logT

_{L}~ −14.7+4·logT (for the inner heliosheath).

ℓ_{C} | ${\sigma}_{u}\propto {\ell}_{\text{C}}^{-1}$ | γ = log n/logT[a] | T_{L}∝T^{2}ℓ_{C}^{2} | α = log T_{L}/logT[b] |
---|---|---|---|---|

b ~ n^{−1/3} | n^{1/3} | 5/3 | T^{2}n^{−2/3} | $2-\frac{2}{3}\nu =2-\frac{2}{3(\gamma -1)}$ |

${\lambda}_{\text{D}}\propto \sqrt{T/n}$ | $\sqrt{n/T}$ | 3/2 | T^{3}n^{−1} | $3-\nu =3-\frac{1}{\gamma -1}$ |

${\lambda}_{\text{W}}~\sqrt{T}$ | $1/\sqrt{T}$ | 1 | T^{3} | 3 |

${\rho}_{g}~\sqrt{T}/B$ | $B/\sqrt{T}$ | 1+b[c] | T^{3}n^{−2b} | $3-2b\nu =3-\frac{2b}{\gamma -1}$ |

L_{m} ~ θ/ν_{col} ~T^{2}n^{−1} | n/T^{2} | 7/5 | T^{6}n^{−2} | $6-2\nu =6-\frac{2}{\gamma -1}$ |

^{[a]}Polytropic index for a constant σ;

^{[b]}Given a polytropic relation n ~ T

^{ν}, γ ≡ 1+1/ν;

^{[c]}Given a relation of the type B ~ n

^{b}.

© 2014 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Livadiotis, G.
“Lagrangian Temperature”: Derivation and Physical Meaning for Systems Described by Kappa Distributions. *Entropy* **2014**, *16*, 4290-4308.
https://doi.org/10.3390/e16084290

**AMA Style**

Livadiotis G.
“Lagrangian Temperature”: Derivation and Physical Meaning for Systems Described by Kappa Distributions. *Entropy*. 2014; 16(8):4290-4308.
https://doi.org/10.3390/e16084290

**Chicago/Turabian Style**

Livadiotis, George.
2014. "“Lagrangian Temperature”: Derivation and Physical Meaning for Systems Described by Kappa Distributions" *Entropy* 16, no. 8: 4290-4308.
https://doi.org/10.3390/e16084290