Weibull thermodynamics: Subexponential decay in the energy spectrum of cosmic-ray nuclei

Keywords

• Subexponential spectral decay;
• Entropy of stationary non-equilibrium ensembles;
• Weibull entropy and excess entropy;
• Multiply broken power-law densities with Weibull cutoff;
• Heat capacities of a non-equilibrated gas mixture;
• Thermodynamics of cosmic-ray nuclei

1. Introduction

We discuss the thermodynamics of the all-particle cosmic-ray spectrum which has recently been measured over an extended energy range by several ground-based, balloon- and space-borne experiments [1]; [2]; [3]; [4]; [5]; [6]; [7]; [8] ;  [9]. The spectrum comprises relativistic nuclei with energies ranging from 1GeV$1\text{GeV}$ to 10¹¹GeV$10^{11}\text{GeV}$, and the measurements are sufficiently precise to reveal the spectral fine structure. In double-logarithmic representation, the wideband spectral map consists of multiple power-law slopes with smooth transitions at the spectral breaks, terminating with a subexponential spectral cutoff above 10¹⁰GeV$10^{10}\text{GeV}$.

The purpose of this paper is to model the spectral fine structure. Ab initio modeling of the cosmic ray flux based on kinetic equations requires to quantify the driving forces sustaining the stationary non-equilibrium, that is the production sites and acceleration and attenuation mechanisms as well as the nuclear mass composition, which are only vaguely known [10] ;  [11]. Numerical simulations suggest that protons can be accelerated up to a few 10⁶GeV$10^6\text{GeV}$ by diffusive shock acceleration in supernova remnants, and iron nuclei up to 10⁸GeV$10^8\text{GeV}$. Direct evidence for that is still lacking, as there are no intensity peaks in anisotropy maps which can be associated with specific remnants. Another Galactic driving mechanism is acceleration in the electric fields of pulsars. Ultra-high energy cosmic rays above 10⁸GeV$10^8\text{GeV}$ are believed to be of extragalactic origin. If these rays are predominantly protons, the spectral cutoff could stem from energy dissipation due to scattering by background photons. Another possible attenuation effect at ultra-high energies is photo-disintegration of heavier nuclei. The cutoff could also be caused by acceleration limits at the production sites. As ab initio modeling of the wideband fine structure is presently not a viable option owing to the uncertain production and interaction mechanisms, we will content ourselves with a reconstruction of the spectral density from the available data sets. We will treat cosmic-ray nuclei as a relativistic gas in stationary non-equilibrium, as the cosmic-ray flux does not change on accessible time scales, apart from solar modulations at low energy which are already averaged out in the data sets. We will also dissect the entropy functional of the empirical spectral density to characterize the deviation from equilibrium, which can be done quantitatively without specification of the driving forces as outlined below.

The spectral flux density extending over eleven decades in energy can be fitted by a multiply broken power-law distribution with a subexponential spectral cutoff at ultra-high energies. The measured subexponential spectral decay means to replace the Boltzmann factor in the flux density by the Weibull factor exp(−(E∕(k_BT))^σ)$exp(-{(E∕\left(k_{\text{B}}T\right))}^{\sigma })$ with spectral index (shape parameter) 0<σ<1$0<\sigma <1$ [12] ;  [13], also known as stretched exponential or Kohlrausch function in relaxation theory  [14] ;  [15]. In Sections 2 ;  3, we develop the thermodynamic formalism of a classical stationary non-equilibrium system described by a power-law density admitting subexponential Weibull decay. In Sections 4 ;  5, we employ power-law densities with Weibull cutoff to perform a spectral fit to the all-particle cosmic-ray spectrum and to estimate the thermodynamic parameters of this relativistic nuclear gas.

Cosmic-ray nuclei can be treated as a classical gas, quantum corrections as well as the Coulomb interaction being negligible due to their low fugacity and density and high temperature. Relativistic statistical systems with short- and long-range interactions exhibiting quantum effects are studied in Refs. [16]; [17]; [18]; [19] ;  [20] and references therein, pertaining to conceptual topics like negative heat capacities [16] and negative entropy [17] as well as to concrete applications such as astrophysical electron plasmas [18]; [19] ;  [20]. Astro- and geophysical applications of sub- and superexponential Weibull factors (stretched and compressed exponentials with shape parameters σ<1$\sigma <1$ and σ>1$\sigma >1$, respectively) include asteroid fragmentation statistics [21], speed distributions of planetary surface winds [22]; [23] ;  [24] and earthquake interevent times [25]; [26] ;  [27]. Examples of network applications of the Weibull distribution can be found in Refs. [28]; [29]; [30] ;  [31], ranging from traffic flows to market volatility. The original use of the Weibull distribution as empirical fracture probability of brittle solids is discussed in Refs. [32] ;  [33]. A deformation of the Weibull density interpolating between the stretched exponential distribution and Pareto power laws [34] was used in Ref. [35] to model income distributions. Applications thereof in reliability modeling and weakest link theory are discussed in Refs. [36] ;  [37]. Multi-parameter exponential densities with Pareto tails were also employed in Ref. [38] to model wealth distributions etc.

In the following, we give an outline of this paper. In Section 2, we study the entropy of a stationary non-equilibrium gas (classical, relativistic) defined by a spectral number density composed of a multiply broken power law and a Weibull exponential. Comparing with the entropy function of an equilibrium system, there are two major differences. The contribution U∕T$U∕T$ of the internal energy to the entropy functional is replaced by the Weibull entropy accounting for the subexponential spectral decay, and the power-law factor in the number density describing the spectral breaks gives rise to an additional term in the entropy functional, the excess entropy which vanishes in an equilibrium system. Integral representations of Weibull and excess entropy suitable for numerical evaluation are also provided in Section 2. We also sketch the derivation of the spectral number density from the probability distribution in phase space, and derive excess entropy, Weibull entropy and total entropy as phase-space expectation values, the latter in −k_BPlogP$-k_{\text{B}}PlogP$ representation where P$P$ is the phase-space probability. We also briefly consider phase-space fluctuations of a stationary non-equilibrium gas, and show how to express the variances, covariances and higher-order correlations of the various thermodynamic variables by derivatives of the partition function. Like in an equilibrium system, the relative fluctuations vanish in the thermodynamic limit.

In Section 3, we make use of the entropy variable to define the effective temperature of a stationary non-equilibrium gas. As the Boltzmann factor is replaced by the Weibull exponential and also because of the power-law factor in the spectral number density, the cutoff temperature k_BT$k_{\text{B}}T$ in the Weibull exponential differs from the effective gas temperature defined as the reciprocal energy derivative of entropy, T_eff=1∕S_,U$T_{\text{eff}}=1∕S_{,U}$. The effective chemical potential is obtained in like manner as entropy derivative with respect to the number count, μ_eff=−S_,N∕β_eff${\mu }_{\text{eff}}=-S_{,N}∕{\beta }_{\text{eff}}$. We also find the explicit relation β_eff(β)${\beta }_{\text{eff}}\left(\beta \right)$ between the effective temperature and the cutoff temperature in the Weibull factor, in terms of the temperature parameters β_eff=1∕(k_BT_eff)${\beta }_{\text{eff}}=1∕\left(k_{\text{B}}{T}_{\text{eff}}\right)$ and β=1∕(k_BT)^σ$\beta =1∕{\left(k_{\text{B}}T\right)}^{\sigma }$, where σ$\sigma$ is the Weibull spectral index. Another basic difference compared to an equilibrium system is the pressure variable, which is unrelated to the partition function (unlike in equilibrium, where βPV=logZ$\beta PV=logZ$ holds) and has to be assembled from the spectral number density, the particle momentum and the particle speed. We relate the pressure variable to the effective temperature, to obtain the thermal equation of state, and also derive the caloric equation relating the internal energy to T_eff$T_{\text{eff}}$. The isochoric and isobaric heat capacities are derived from the entropy variable, and we show that the inequalities 0<C_V<C_P$0<C_V<C_P$ are equivalent to a positive effective temperature parameter with positive derivative ${\beta }_{\text{eff}}^{\prime }\left(\beta \right)$. Subject to the same conditions, the isobaric expansion coefficient is positive, and the isothermal and adiabatic compressibilities satisfy 0<κ_S<κ_Teff$0<{\kappa }_S<{\kappa }_{T_{\text{eff}}}$. To summarize Section 3, we study the entropy function of a stationary non-equilibrium gas in various parametrizations, derive numerically tame integral representations of the thermodynamic variables, and discuss positivity properties and thermodynamic inequalities relating to subexponential spectral decay.

In Section 4, we perform spectral fits to the all-particle cosmic-ray spectrum, employing the spectral densities discussed in Sections 2 ;  3. The nuclei constituting the all-particle flux can roughly be divided into a light component of protons and helium, an intermediate-mass component of nitrogen-like nuclei, and a heavy iron-like component. In the 1GeV−10⁴GeV$1\text{GeV}-10^4\text{GeV}$ range, the spectrum consists almost entirely, up to a fraction of one percent, of protons and helium nuclei. The proton and helium fluxes in this energy range have been measured by the Alpha Magnetic Spectrometer (AMS-02) on the International Space Station [1] ;  [2] and by the balloon-borne CREAM experiment [3]. In the 10⁵GeV−10¹¹GeV$10^5\text{GeV}-10^{11}\text{GeV}$ interval, the mass composition is not yet known, although there exist some preliminary qualitative studies [6] ;  [9]. This is the ultra-relativistic regime, where the nuclear mass becomes negligible compared to the particle energy. The all-particle spectrum, irrespectively of the nuclear mass composition, has been measured in this energy range by several ground-based experiments employing Cherenkov counters and fluorescence and scintillator detectors, such as Tibet-III [4], IceTop-73 [5] ;  [6], KASCADE-Grande [7], Auger [8] and Telescope Array [9].

We first perform spectral fits to the AMS-02 proton and helium spectra in the 1GeV$1\text{GeV}$–10⁴GeV$10^4\text{GeV}$ interval. The all-particle spectrum in this interval can be approximated by adding the proton and helium fluxes. We then analytically extend the low-energy proton and helium fluxes into the 10⁵GeV−10¹¹GeV$10^5\text{GeV}-10^{11}\text{GeV}$ interval and perform a separate fit to the all-particle spectrum in this energy range. At energies above 10⁴GeV$10^4\text{GeV}$, heavier nuclei can be accommodated in the extended proton and helium components, as the nuclear mass drops out of the flux density in this ultra-relativistic regime. The nuclear gas can thus be treated as a two-component mixture: In the 1GeV−10⁴GeV$1\text{GeV}-10^4\text{GeV}$ interval, the flux density depends on the proton and helium (alpha particle) mass, and it becomes mass independent in the 10⁴GeV−10¹¹GeV$10^4\text{GeV}-10^{11}\text{GeV}$ interval. The all-particle flux can therefore be represented, over the complete 1GeV−10¹¹GeV$1\text{GeV}-10^{11}\text{GeV}$ spectral range, as the sum of two partial fluxes (one depending on the proton mass and one on the helium mass), which are defined by multiply broken power-law densities with the same subexponential spectral cutoff exp(−(E∕(k_BT))^σ)$exp(-{(E∕\left(k_{\text{B}}T\right))}^{\sigma })$. The cutoff temperature inferred from the spectral fit is k_BT=(2.5±0.1)×10¹⁰GeV$k_{\text{B}}T=(2.5\pm 0.1)\times 10^{10}\text{GeV}$, and the Weibull index is σ=0.66±0.02$\sigma =0.66\pm 0.02$, so that the spectral decay substantially deviates from an exponential Boltzmann cutoff.

To obtain the thermodynamic parameters of the all-particle cosmic-ray spectrum, we need to generalize the formalism developed in Section 3 to cover a multi-component gas mixture, owing to the different particle masses. It suffices to consider a two-component mixture, as the flux density only depends on the hydrogen and helium mass at low energy and is mass independent in the ultra-relativistic regime. The thermodynamics of a two-component mixture in stationary non-equilibrium is elaborated in Section 5. First, we extract the spectral number densities of the gas components from the two partial fluxes constituting the all-particle flux. Then we assemble the entropy variable of this non-equilibrated mixture and calculate the effective gas temperature, k_BT_eff≈16GeV$k_{\text{B}}{T}_{\text{eff}}\approx 16\text{GeV}$, from the energy derivative of entropy. In an equilibrium system, this temperature would coincide with the cutoff temperature in the Boltzmann factor. Because of the multiply broken power-law function in the spectral densities and the subexponential spectral decay, the cutoff temperature k_BT≈2.5×10¹⁰GeV$k_{\text{B}}T\approx 2.5\times 10^{10}\text{GeV}$ in the Weibull factor greatly differs from the actual gas temperature T_eff$T_{\text{eff}}$. We also determine the effective chemical potentials of the gas components, derive the thermal and caloric equations of state, and give estimates of the internal energy and pressure of cosmic-ray nuclei. As done in Section 3 for a one-component gas, we calculate the heat capacities and compressibilities of a two-component mixture in stationary non-equilibrium, and verify the inequalities 0<C_V<C_P$0<C_V<C_P$ and 0<κ_S<κ_Teff$0<{\kappa }_S<{\kappa }_{T_{\text{eff}}}$ as well as the positivity of the expansion coefficient of the nuclear gas. In Section 6, we present our conclusions.

2. Entropy of a classical relativistic gas in stationary non-equilibrium

2.1. Subexponential spectral decay

We will study a relativistic gas in stationary non-equilibrium defined by the spectral number density [39] ;  [40]

equation(2.1)

$\text{d}\rho \left(p\right)=\frac{4\pi s}{{(2\pi ħc)}^3}exp[-\alpha -\beta {(p^2+m^2)}^{\sigma ∕2}]h\left(p\right)p^2\text{d}p,$

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$E=\sqrt{p^2+m^2}$

$-\hat{\alpha }-\hat{\beta }m{\upsilon }^2∕2$

$\hat{\alpha }=\alpha +\beta m^{\sigma }$

$\hat{\beta }=\beta \sigma m^{\sigma -1}$

2.2. Weibull and excess entropies

To obtain integral representations of the thermodynamic variables, we employ a more general distribution function than stated in (2.1),

equation(2.2)

$\text{d}\rho \left(p\right)=\frac{4\pi s}{{\left(2\pi \right)}^3}\text{exp}[-\alpha -\beta {(p^2+m^2)}^{\sigma ∕2}-\gamma log\frac{1}{h\left(p\right)}-\delta \frac{1}{3}\frac{p^2}{\sqrt{p^2+m^2}}-\epsilon \sqrt{p^2+m^2}]p^2\text{d}p,$

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$logZ=V{\int }_0^{\infty }\text{d}\rho \left(p\right)$

equation(2.3)

$N=-{(logZ)}_{,\alpha }=V{\int }_0^{\infty }\text{d}\rho \left(p\right)=logZ,$

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equation(2.4)

$U=-{(logZ)}_{,\epsilon }=V{\int }_0^{\infty }\sqrt{p^2+m^2}\text{d}\rho \left(p\right),$

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equation(2.5)

$PV=-{(logZ)}_{,\delta }=\frac{1}{3}V{\int }_0^{\infty }\frac{p^2}{\sqrt{p^2+m^2}}\text{d}\rho \left(p\right).$

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equation(2.6)

S=logZ−α(logZ)_,α−β(logZ)_,β−(logZ)_,γ,$S=logZ-\alpha {(logZ)}_{,\alpha }-\beta {(logZ)}_{,\beta }-{(logZ)}_{,\gamma },$

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equation(2.7)

$\frac{W}{\beta }=-{(logZ)}_{,\beta }=V{\int }_0^{\infty }{(p^2+m^2)}^{\sigma ∕2}\text{d}\rho \left(p\right),$

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equation(2.8)

$H=-{(logZ)}_{,\gamma }=V{\int }_0^{\infty }log\frac{1}{h\left(p\right)}\text{d}\rho \left(p\right),$

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equation(2.9)

S=(1+α)N+W+H.$S=(1+\alpha )N+W+H.$

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2.3. Weibull probability distribution in phase space

We start with the n$n$-particle phase-space measure

equation(2.10)

${\text{d}}^{3n}\text{(}p\text{,}q\text{)}=\frac{1}{{\left(2\pi \right)}^{3n}}{\text{d}}^3{p}_1{\text{d}}^3{q}_1{\text{d}}^3{p}_2{\text{d}}^3{q}_2\cdots {\text{d}}^3{p}_n{\text{d}}^3{q}_n,$

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equation(2.11)

$P_n\left(p\right)=\frac{b_n\left(p\right)}{Z},b_n\left(p\right)=exp(-\alpha n-\beta w_n-\gamma h_n-\delta v_n-\epsilon u_n),$

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equation(2.12)

$w_n=\sum _{k=1}^n{(p^2+m^2)}^{\sigma ∕2},h_n=\sum _{k=1}^{n}log\frac{1}{h\left(p_k\right)},v_n=\frac{1}{3}\sum _{k=1}^n\frac{p_k^2}{\sqrt{p_k^2+m^2}},u_n=\sum _{k=1}^n\sqrt{p_k^2+m^2}.$

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equation(2.13)

$\sum _{n=0}^{\infty }\frac{s^n}{n!}\int P_n\left(p\right){\text{d}}^{3n}(p,q)=1,Z=\sum _{n=0}^{\infty }\frac{s^n}{n!}\int b_n\left(p\right){\text{d}}^{3n}(p,q),$

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$logZ=V{\int }_0^{\infty }\text{d}\rho \left(p\right)$

2.4. Excess entropy, Weibull entropy and total entropy as phase-space expectation values

Differentiation of logZ$logZ$ in (2.13) with respect to the parameters of the exponential b_n(p)$b_n\left(p\right)$ in (2.11) leads to representations of the various thermodynamic variables as phase-space averages. For instance, the Weibull entropy (2.7) is assembled in phase space as

equation(2.14)

$\frac{W}{\beta }=-{(logZ)}_{,\beta }=\sum _{n=0}^{\infty }\frac{s^n}{n!}\int w_n{P}_n\left(p\right){\text{d}}^{3n}(p,q)=〈w_n〉,$

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equation(2.15)

$S=-\sum _{n=0}^{\infty }\frac{s^n}{n!}\int P_n\left(p\right)log{P}_n\left(p\right){\text{d}}^{3n}(p,q),$

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equation(2.16)

s_n=−logP_n(p)=logZ+αn+βw_n+h_n,$s_n=-log{P}_n\left(p\right)=logZ+\alpha n+\beta w_n+h_n,$

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In contrast to an equilibrium system, the probability distribution P_n(p)$P_n\left(p\right)$ in (2.11) (taken at γ=1$\gamma =1$, δ=ε=0$\delta =\epsilon =0$) does not maximize the entropy functional (2.15) subject to the particle number and internal energy constraints N=〈n〉$N=〈n〉$ and U=〈u_n〉$U=〈u_n〉$, see after (2.14). Only the equilibrium distribution defined by h(p)=1$h\left(p\right)=1$ and σ=1$\sigma =1$ in (2.1) maximizes the entropy functional (2.15). To see this, we consider the equilibrium distribution $P_n^{\text{eq}}\left(p\right)=b_n^{\text{eq}}\left(p\right)∕Z_{\text{eq}},b_n^{\text{eq}}\left(p\right)=exp(-{\alpha }_{\text{eq}}n-{\beta }_{\text{eq}}{u}_n)$, where Z_eq$Z_{\text{eq}}$ is defined as in (2.13) (with b_n$b_n$ replaced by $b_n^{\text{eq}}$), so that $P_n^{\text{eq}}$ is normalized to one. The constraints prescribing N=〈n〉$N=〈n〉$ and U=〈u_n〉$U=〈u_n〉$ are defined as in (2.14) with P_n$P_n$ replaced by $P_n^{\text{eq}}$; the equilibrium entropy S_eq$S_{\text{eq}}$ is defined as in (2.15), also with this replacement $P_n\to P_n^{\text{eq}}$. As P_n$P_n$ and $P_n^{\text{eq}}$ satisfy the same constraints, namely N=〈n〉,U=〈u_n〉$N=〈n〉,U=〈u_n〉$ and the normalization (2.13), we can also average the equilibrium entropy S_eq$S_{\text{eq}}$ over P_n$P_n$ instead of $P_n^{\text{eq}}$. That is, S_eq$S_{\text{eq}}$ can be defined by replacing P_nlogP_n$P_{n}log{P}_n$ in (2.15) by $P_{n}log{P}_n^{\text{eq}}$ instead of $P_n^{\text{eq}}log{P}_n^{\text{eq}}$. Finally we consider the difference S−S_eq$S-S_{\text{eq}}$, with S$S$ in (2.15) and S_eq$S_{\text{eq}}$ defined by $P_{n}log{P}_n^{\text{eq}}$, and apply the general logarithmic inequality $log(P_n∕P_n^{\text{eq}})>1-P_n^{\text{eq}}∕P_n$ and the normalization (2.13) (valid for P_n$P_n$ as well as $P_n^{\text{eq}}$) to find S<S_eq$S<S_{\text{eq}}$.

2.5. Variances, covariances and higher-order correlations: fluctuations in stationary non-equilibrium ensembles

We calculate the variances of the particle number 〈(Δn)²〉$〈{\left(\Delta n\right)}^2〉$, Weibull entropy 〈(Δw_n)²〉$〈{\left(\Delta w_n\right)}^2〉$, excess entropy 〈(Δh_n)²〉$〈{\left(\Delta h_n\right)}^2〉$, pressure 〈(Δv_n)²〉$〈{\left(\Delta v_n\right)}^2〉$ and internal energy 〈(Δu_n)²〉$〈{\left(\Delta u_n\right)}^2〉$, using the phase-space random variables in (2.12) and the shortcuts Δn=n−〈n〉,Δw_n=w_n−〈w_n〉$\Delta n=n-〈n〉,\Delta w_n=w_n-〈w_n〉$, etc. The variances $〈{\left(\Delta n\right)}^2〉=〈n^2〉-{〈n〉}^2,〈{\left(\Delta w_n\right)}^2〉=〈w_n^2〉-{〈w_n〉}^2$, etc., are then obtained as second derivatives of logZ$logZ$. For instance, 〈(Δn)²〉=(logZ)_,α,α$〈{\left(\Delta n\right)}^2〉={(logZ)}_{,\alpha ,\alpha }$ and 〈(Δw_n)²〉=(logZ)_,β,β$〈{\left(\Delta w_n\right)}^2〉={(logZ)}_{,\beta ,\beta }$, taken at γ=1$\gamma =1$, δ=ε=0$\delta =\epsilon =0$. We note the pairs (Δn,α)$(\Delta n,\alpha )$, (Δw_n,β),(Δh_n,γ),(Δv_n,δ)$(\Delta w_n,\beta ),(\Delta h_n,\gamma ),(\Delta v_n,\delta )$ and (Δu_n,ε)$(\Delta u_n,\epsilon )$, cf. (2.11), so that the variances and covariances like 〈ΔnΔw_n〉=(logZ)_,α,β$〈\Delta n\Delta w_n〉={(logZ)}_{,\alpha ,\beta }$ and 〈Δw_nΔh_n〉=(logZ)_,β,γ$〈\Delta w_n\Delta h_n〉={(logZ)}_{,\beta ,\gamma }$ are obtained by differentiation with the respective parameters. Higher-order correlations are assembled analogously, e.g. the triple correlations 〈Δn(Δw_n)²〉=−(logZ)_,α,β,β$〈\Delta n{\left(\Delta w_n\right)}^2〉=-{(logZ)}_{,\alpha ,\beta ,\beta }$ and 〈Δu_nΔw_nΔh_n〉=−(logZ)_,ε,β,γ$〈\Delta u_n\Delta w_n\Delta h_n〉=-{(logZ)}_{,\epsilon ,\beta ,\gamma }$; the minus sign arises in case of an odd number of factors. Spectral representations of correlations suitable for numerical computation are thus obtained by multiple differentiation of the partition function $logZ=V{\int }_0^{\infty }\text{d}\rho \left(p\right)$ as in (2.3) –(2.8). The heat capacities and compressibilities of a gas in stationary non-equilibrium can be expressed by the expectation values in Section 2.4, their variances, covariances and triple correlations, see the remarks after (3.28).

The entropy variance $〈{\left(\Delta s_n\right)}^2〉=〈s_n^2〉-{〈s_n〉}^2$, with Δs_n=s_n−〈s_n〉$\Delta s_n=s_n-〈s_n〉$, is a linear combination of the above variances and covariances, since Δs_n=αΔn+βΔw_n+Δh_n$\Delta s_n=\alpha \Delta n+\beta \Delta w_n+\Delta h_n$, cf. (2.16). We square this to find

equation(2.17)

〈(Δs_n)²〉=α²(logZ)_,α,α+β²(logZ)_,β,β+(logZ)_,γ,γ+2αβ(logZ)_,α,β+2α(logZ)_,α,γ+2β(logZ)_,β,γ,$〈{\left(\Delta s_n\right)}^2〉={\alpha }^2{(logZ)}_{,\alpha ,\alpha }+{\beta }^2{(logZ)}_{\text{,}\beta ,\beta }+{(logZ)}_{,\gamma ,\gamma }+2\alpha \beta {(logZ)}_{,\alpha ,\beta }+2\alpha {(logZ)}_{,\alpha ,\gamma }+2\beta {(logZ)}_{,\beta ,\gamma },$

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$V\to \infty ,\sqrt{〈{\left(\Delta s_n\right)}^2〉}∕〈s_n〉\propto 1∕\sqrt{V}$

3. Effective temperature and Weibull spectral decay: Thermodynamics of a stationary non-equilibrium gas

3.1. (α,β,V)$(\alpha ,\beta ,V)$ parametrization of thermodynamic variables

We define the partition factor z$z$ by a rescaling of logZ$logZ$ in (2.3) with the volume factor and the fugacity f=e^−α$f={\text{e}}^{-\alpha }$,

$z(\beta ,\gamma ,\delta ,\epsilon )=\frac{logZ}{Vf}={\int }_0^{\infty }\text{d}\hat{\rho }\left(p\right),$

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equation(3.1)

$\text{d}\hat{\rho }\left(p\right)=\frac{4\pi s}{{\left(2\pi \right)}^3}\text{exp}[-\beta {(p^2+m^2)}^{\sigma ∕2}-\gamma log\frac{1}{h\left(p\right)}-\delta \frac{1}{3}\frac{p^2}{\sqrt{p^2+m^2}}-\epsilon \sqrt{p^2+m^2}]p^2\text{d}p,$

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$(log(1∕h\left(p\right)),\gamma ),(p^2∕\left(3\sqrt{p^2+m^2}\right),\delta )$

$(\sqrt{p^2+m^2},\epsilon )$

All thermodynamic variables can efficiently be expressed and calculated in terms of the partition factor and its derivatives. Particle number, internal energy and pressure read, cf. (2.3), (2.4) ;  (2.5),

equation(3.2)

$N=V{\text{e}}^{-\alpha }z,U=-V{\text{e}}^{-\alpha }{z}_{,\epsilon },P=-{\text{e}}^{-\alpha }{z}_{,\delta }.$

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equation(3.3)

$W∕\beta =-V{\text{e}}^{-\alpha }{z}_{,\beta },H=-V{\text{e}}^{-\alpha }{z}_{,\gamma },$

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equation(3.4)

S(α,β,V)=Ve^−α((1+α)z−βz_,β−z_,γ).$S(\alpha ,\beta ,V)=V{\text{e}}^{-\alpha }((1+\alpha )z-\beta z_{,\beta }-z_{,\gamma }).$

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3.2. (β,V,N)$(\beta ,V,N)$ parametrization of internal energy, pressure and entropy

We invert the first identity in (3.2), α=log(zV∕N)$\alpha =log(zV∕N)$, to find the (β,V,N)$(\beta ,V,N)$ representation of internal energy and pressure as

equation(3.5)

$U(\beta ,N)=Nu\left(\beta \right),u\left(\beta \right)=-\frac{z_{,\epsilon }}{z},$

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equation(3.6)

$P=\eta \left(\beta \right)\frac{N}{V},\eta \left(\beta \right)=-\frac{z_{,\delta }}{z},$

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equation(3.7)

$S(\beta ,V,N)=N\left(1-log\frac{N}{V}+s\left(\beta \right)\right),s\left(\beta \right)=logz-\beta \frac{z_{,\beta }}{z}-\frac{z_{,\gamma }}{z}.$

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3.3. Entropy in (U,V,N)$(U,V,N)$ parametrization: effective temperature and effective chemical potential

To eliminate the temperature variable β$\beta$ in the (β,V,N)$(\beta ,V,N)$ representation, we invert U=Nu(β)$U=Nu\left(\beta \right)$ in (3.5), β=u⁻¹(U∕N)$\beta =u^{-1}(U∕N)$, to find, cf. (3.7),

equation(3.8)

S(U,V,N)=N[1−log(N∕V)+s(u⁻¹(U∕N))].$S(U,V,N)=N[1-log(N∕V)+s\left(u^{-1}(U∕N)\right)].$

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equation(3.9)

$S_{,U}={\beta }_{\text{eff}},{\beta }_{\text{eff}}\left(\beta \right)=\frac{s^{\prime }\left(\beta \right)}{u^{\prime }\left(\beta \right)},$

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equation(3.10)

$S_{,N}={\alpha }_{\text{eff}},{\alpha }_{\text{eff}}=-log\frac{N}{V}+s\left(\beta \right)-\frac{U}{N}\frac{s^{\prime }\left(\beta \right)}{u^{\prime }\left(\beta \right)},$

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equation(3.11)

$u^{\prime }\left(\beta \right)=-z^{-2}(z{z}_{,\epsilon ,\beta }-z_{,\epsilon }{z}_{,\beta }),s^{\prime }\left(\beta \right)=-z^{-2}[\beta (z{z}_{,\beta ,\beta }-z_{,\beta }{z}_{,\beta })+(z{z}_{,\gamma ,\beta }-z_{,\gamma }{z}_{,\beta })].$

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equation(3.12)

${\mu }_{\text{eff}}=\frac{1}{{\beta }_{\text{eff}}}log\frac{N}{V}+u\left(\beta \right)-\frac{s\left(\beta \right)}{{\beta }_{\text{eff}}}.$

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In Section 5, we will obtain the effective temperature and the chemical potential from a spectral fit. Conceptually, it is also possible to infer these parameters by equilibration with a thermal system. For notational simplicity, we assume this reference system to be a free relativistic gas defined by number density (2.1) with h(p)=1$h\left(p\right)=1$ and σ=1$\sigma =1$. The entropy function and the variables of this thermometer system are labeled with a subscript of two: S₂(U₂,V₂,N₂)$S_2(U_2,V_2,N_2)$, so that S_2,U2=β₂,S_2,V2=β₂P₂$S_{2,U_2}={\beta }_2,S_{2,V_2}={\beta }_2{P}_2$ and S_2,N2=α₂$S_{2,N_2}={\alpha }_2$, with chemical potential μ₂=−α₂∕β₂${\mu }_2=-{\alpha }_2∕{\beta }_2$ and thermal equation β₂P₂=N₂∕V₂${\beta }_2{P}_2=N_2∕V_2$.

We keep the particle numbers N$N$ and N₂$N_2$ constant. When equilibrating S(U,V,N)$S(U,V,N)$ in (3.8) and S₂(U₂,V₂,N₂)$S_2(U_2,V_2,N_2)$, we use as contact variables β_eff=β₂≕β_c${\beta }_{\text{eff}}={\beta }_2≕{\beta }_{\text{c}}$ and P=P₂$P=P_2$. The total energy and volume are kept constant, U+U₂=U_tot,V+V₂=V_tot$U+U_2=U_{\text{tot}},V+V_2=V_{\text{tot}}$. We thus find the equations S_2,U(U,V,N)=β_c$S_{2,U}(U,V,N)={\beta }_{\text{c}}$ and S_2,U2(U₂,V₂,N₂)=β_c$S_{2,U_2}(U_2,V_2,N_2)={\beta }_{\text{c}}$, and the pressure identity gives η(β(β_c))N∕V=N₂∕(β_cV₂)$\eta \left(\beta \left({\beta }_{\text{c}}\right)\right)N∕V=N_2∕\left({\beta }_{\text{c}}{V}_2\right)$ by way of the thermal equations, cf. (3.6). (In the thermal scale factor η(β)$\eta \left(\beta \right)$, we have substituted β=β(β_eff)$\beta =\beta \left({\beta }_{\text{eff}}\right)$, which is the inversion of β_eff(β)${\beta }_{\text{eff}}\left(\beta \right)$, cf. (3.9).) Together with the conservation conditions U+U₂=U_tot$U+U_2=U_{\text{tot}}$ and V+V₂=V_tot$V+V_2=V_{\text{tot}}$, we obtain five equations to determine the variables U$U$, U₂$U_2$, V$V$, V₂$V_2$ and β_c${\beta }_{\text{c}}$; the latter is a Lagrange parameter used to maximize S(U,V,N)+S₂(U₂,V₂,N₂)$S(U,V,N)+S_2(U_2,V_2,N_2)$ with respect to the energy variables subject to the constraint U+U₂=U_tot$U+U_2=U_{\text{tot}}$. The total entropy S+S₂$S+S_2$ is not maximized with respect to the volume variables V$V$ and V₂$V_2$, as this would require to equate P∕η(β(β_c))=P₂β_c$P∕\eta \left(\beta \left({\beta }_{\text{c}}\right)\right)=P_2{\beta }_{\text{c}}$ instead of the pressure variables, P=P₂$P=P_2$. Only if η(β(β_c))=1∕β_c$\eta \left(\beta \left({\beta }_{\text{c}}\right)\right)=1∕{\beta }_{\text{c}}$, which happens if S(U,V,N)$S(U,V,N)$ is an equilibrium entropy, the contact identity P=P₂$P=P_2$ amounts to maximize the total entropy S+S₂$S+S_2$ also with respect to the volume variables subject to V+V₂=V_tot$V+V_2=V_{\text{tot}}$. This reflects the fact that the phase-space probability distribution P_n(p)$P_n\left(p\right)$ in (2.11) does not maximize the entropy functional (2.15); only the equilibrium distribution $P_n^{\text{eq}}\left(p\right)$ gives the maximum entropy, see after (2.16). In brief, the stationary non-equilibrium entropy (3.4) is always smaller than the equilibrium entropy if subjected to the same constraints.

3.4. Entropy in (β_eff,V,N)$({\beta }_{\text{eff}},V,N)$ parametrization, thermal and caloric equations of state, and isochoric heat capacity

We use the effective temperature parameter β_eff=s^′(β)∕u^′(β)${\beta }_{\text{eff}}=s^{\prime }\left(\beta \right)∕u^{\prime }\left(\beta \right)$ in (3.9) as independent variable, and substitute the inversion β=β(β_eff)$\beta =\beta \left({\beta }_{\text{eff}}\right)$ into the (β,V,N)$(\beta ,V,N)$ representation in Section 3.2. The existence of this inversion is assured by the positivity conditions β_eff(β)>0${\beta }_{\text{eff}}\left(\beta \right)>0$ and ${\beta }_{\text{eff}}^{\prime }\left(\beta \right)>0$, see Section 3.8, but we will not need to perform it explicitly. The caloric and thermal equations of state read U=Nu(β(β_eff))$U=Nu\left(\beta \left({\beta }_{\text{eff}}\right)\right)$ and P=η(β(β_eff))N∕V$P=\eta \left(\beta \left({\beta }_{\text{eff}}\right)\right)N∕V$, cf. (3.5) ;  (3.6), and the (β_eff,V,N)$({\beta }_{\text{eff}},V,N)$ representation of entropy is, cf. (3.7),

equation(3.13)

S(β_eff,V,N)=N[1−log(N∕V)+s(β(β_eff))].$S({\beta }_{\text{eff}},V,N)=N[1-log(N∕V)+s\left(\beta \left({\beta }_{\text{eff}}\right)\right)].$

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equation(3.14)

$C_V=-{\beta }_{\text{eff}}{S}_{,{\beta }_{\text{eff}}}({\beta }_{\text{eff}},V,N)=-N{\beta }_{\text{eff}}^2\frac{u^{\prime }\left(\beta \right)}{{\beta }_{\text{eff}}^{\prime }\left(\beta \right)}.$

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${\beta }_{\text{eff}}^{\prime }\left(\beta \right)=(s^{″}-u^{″}{\beta }_{\text{eff}})∕u^{\prime }$

u^″(β)=z⁻³[z(z_,εz_,β,β−zz_,ε,β,β)+2z_,β(zz_,ε,β−z_,εz_,β)],$u^{″}\left(\beta \right)=z^{-3}[z(z_{,\epsilon }{z}_{,\beta ,\beta }-z{z}_{,\epsilon ,\beta ,\beta })+2z_{,\beta }(z{z}_{,\epsilon ,\beta }-z_{,\epsilon }{z}_{,\beta })],$

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equation(3.15)

s^″(β)=z⁻³[βz(z_,βz_,β,β−zz_,β,β,β)+2β_effz_,β(zz_,ε,β−z_,εz_,β)+z(z_,γz_,β,β−zz_,γ,β,β)−z(zz_,β,β−z_,βz_,β)],$s^{″}\left(\beta \right)=z^{-3}[\beta z(z_{,\beta }{z}_{,\beta ,\beta }-z{z}_{,\beta ,\beta ,\beta })+2{\beta }_{\text{eff}}{z}_{,\beta }(z{z}_{,\epsilon ,\beta }-z_{,\epsilon }{z}_{,\beta })+z(z_{,\gamma }{z}_{,\beta ,\beta }-z{z}_{,\gamma ,\beta ,\beta })-z(z{z}_{,\beta ,\beta }-z_{,\beta }{z}_{,\beta })],$

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$F_{,{\beta }_{\text{eff}}}=S∕{\beta }_{\text{eff}}^2$