Physical meaning of temperature in superstatistics

Gravanis, Elias; Akylas, Evangelos; Livadiotis, G.

doi:10.1209/0295-5075/130/30005

Cited by 13 publications

(8 citation statements)

References 32 publications

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“…We shall restrict ourselves to kinetic degrees of freedom. The random temperature distribution p T is given as a function of the inverse random temperature β and satisfies the following conditions [22] ∞…”

Section: Random Walk In the Super-ensemble (Derivation I)mentioning

confidence: 99%

“…First, observe that in the classical equilibrium limit (κ 0 → ∞), we have that 1/τ → 1/T , recovering equation ( 1). Also, it was found in [22] that the mean energy per degree of freedom E /f (times 2) defines the fluctuating temperature β -1 in the thermodynamic limit f → ∞. Approximating u 2 = 2E ∼ β -1 f , equation (20) implies that 1/τ → β in that limit.…”

Section: Langevin Equation Of Many Degrees Of Freedommentioning

confidence: 99%

“…We explain that one simple reason for the discrepancy between the two derivations is that the notion of fluctuating temperature is lost in the Langevin equation of one degree of freedom (derivation II). In [22] it was shown that the fluctuating temperature is realized by the mean energy per degree of freedom in a system of virtually infinite degrees of freedom whose velocities are distributed and correlated according to the kappa distribution. Hence, we introduce a non-linear Langevin equation for the whole system, that is, an equation that generates kappa distributed (and correlated) velocities for all degrees of freedom in the system.…”

Section: Introductionmentioning

confidence: 99%

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Stochastic dynamics and superstatistics of the many-particle kappa distribution

Gravanis¹,

Akylas²,

Livadiotis³

2021

J. Stat. Mech.

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The diffusion of particles with kappa distributed velocities is strongly influenced by statistical correlations. We argue that the consistent way to deduce the diffusion laws of any one degree of freedom is to analyze the simultaneous diffusion of virtually infinite correlated degrees of freedom. This is done by deriving the diffusion laws (I) by utilizing the superstatistics interpretation of the kappa distribution and averaging the usual Brownian motions correlators over the super-ensemble of fluctuating temperatures, (II) through the one degree of freedom Langevin equation, (III) through the many degrees of freedom Langevin equation, calculating the diffusion of any one degree of freedom. It turns out that only the results (I) and (III) agree. The disagreement between (II) and (III) is a striking outcome of the strong correlations between kappa distributed degrees of freedom. The agreement between (I) and (III) shows that the superstatistics is a fundamental interpretation of the kappa distribution. The discrepancy of (II) shows that focusing on a single degree of freedom or particle is inconsistent with a superstatistics interpretation. Derivation (III) explicitly realizes the recent observation by the authors that the mean energy per degree of freedom is the superstatistical fluctuating temperature in a system with a large number of particles. We conclude that superstatistics is intimately related to a system of correlated degrees of freedom (in our case, kappa distributed); one cannot consistently reason with a single degree of freedom.

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Section: Random Walk In the Super-ensemble (Derivation I)mentioning

confidence: 99%

Section: Langevin Equation Of Many Degrees Of Freedommentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stochastic dynamics and superstatistics of the many-particle kappa distribution

Gravanis¹,

Akylas²,

Livadiotis³

2021

J. Stat. Mech.

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“…There are multiple mechanisms that can generate kappa distributions in particle systems. Some examples are superstatistics (i.e., the temperature is not fixed but has a special distribution;e.g., Beck & Cohen 2003;Schwadron et al 2010;Hanel et al 2011;Livadiotis 2019a;Gravanis et al 2020;Sánchez et al 2021), effect of shock waves (Zank et al 2006), turbulence (Yoon 2012;Bian et al 2014;Yoon 2014Yoon , 2020, pump acceleration mechanism (Fisk & Gloeckler 2014), colloidal particles (Peterson et al 2013), and polytropes (Meyer-Vernet et al 1995;Livadiotis 2019b).…”

Section: Introductionmentioning

confidence: 99%

Transport Equation of Kappa Distributions in the Heliosphere

Livadiotis,

McComas

2023

ApJ

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In this paper, we develop the transport equation of kappa, the fundamental thermodynamic parameter that labels kappa distributions of particle velocities. Using the recently developed concept of entropy defect, we are able to formulate the transport equation of kappa as a function of a general, positive or negative, rate of entropy change. Then, we derive the particular case of exchanging plasma ions with low-dimensionality, newly born pickup protons, which interact and decrease the entropy of the flow of otherwise kappa-distributed plasma protons. Finally, we apply the transport equation of kappa to the solar wind plasma protons, which leads to the radial profile of kappa values, as well as the evolution of the kappa distributions through the heliosphere. The results show that the solar wind kappa decreases with increasing heliocentric distance, corresponding to plasmas residing in stationary states far from classical thermal equilibrium. Moreover, in the outer heliosphere and the heliosheath, kappa reaches its lowest values and is spread across the far-equilibrium region of 1.5 < κ < 2.5, which coincides with independent observations provided by NASA’s Interstellar Boundary Explorer mission.

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“…These particle systems are characterized by local particle correlations and strong collective behavior, leading to a framework of statistical mechanics different from the classical one that is based on the entropy of Boltzmann-Gibbs and the distribution function of Maxwell-Boltzmann. Some examples are: superstatistics (Beck & Cohen 2003;Schwadron et al 2010;Hanel et al 2011;Livadiotis 2019b;Gravanis et al 2020), effect of shock waves (Zank et al 2006), weak turbulence (Yoon 2014(Yoon , 2019, turbulence with a diffusion coefficient inversely proportional to velocity (Bian et al 2014), effect of pickup ions (Livadiotis & McComas 2011a, 2011b, pump acceleration mechanism (Fisk & Gloeckler 2014), polytropic behavior (Meyer-Vernet et al 1995Livadiotis 2018aLivadiotis , 2019a; see also Livadiotis (2017, chapters 4-6, 8, 10, 15, 16). Common plasma processes, such as Debye shielding and magnetic coupling, can also play an important role in the generation of kappa distributions in plasmas (Livadiotis et al 2018).…”

Section: Introductionmentioning

confidence: 99%

Physical Correlations Lead to Kappa Distributions

Livadiotis

McComas

2022

ApJ

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The recently developed concept of “entropic defect” is important for understanding the foundations of thermodynamics in space plasma physics, and more generally for systems with physical correlations among their particles. Using this concept, this paper derives the basic formulation of the distribution function of velocities (or kinetic energies) in space plasma particle populations. Earlier analyses have shown how the formulation of kappa distributions is interwoven with the presence of correlations among the particles’ velocities. This paper shows, for the first time, that the reverse is true: the thermodynamics of particles’ physical correlations are consistent only with the existence of kappa distributions.

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Physical meaning of temperature in superstatistics

Cited by 13 publications

References 32 publications

Stochastic dynamics and superstatistics of the many-particle kappa distribution

Stochastic dynamics and superstatistics of the many-particle kappa distribution

Transport Equation of Kappa Distributions in the Heliosphere

Physical Correlations Lead to Kappa Distributions

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