# Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle

^{*}

## Abstract

**:**

_{n}(E − ω)/Ω

_{n}(E) = (1 − ω/E)

^{n}. The only parameters are 1/T = ⟨β⟩ = ⟨n⟩/E and q = 1−1/⟨n⟩ +Δn

^{2}/⟨n⟩

^{2}. For the binomial distribution of n one obtains q = 1−1/k, for the negative binomial q = 1+1/(k+1). These results also represent an approximation for general particle number distributions in the reservoir up to second order in the canonical expansion ω ≪ E. For general systems the average phase-space volume ratio ⟨e

^{S}

^{(}

^{E}

^{−}

^{ω}

^{)}/e

^{S}

^{(}

^{E}

^{)}⟩ to second order delivers q = 1−1/C +Δβ

^{2}/⟨β⟩

^{2}with β = S′(E) and C = dE/dT heat capacity. However, q ≠ 1 leads to non-additivity of the Boltzmann–Gibbs entropy, S. We demonstrate that a deformed entropy, K(S), can be constructed and used for demanding additivity, i.e., q

_{K}= 1. This requirement leads to a second order differential equation for K(S). Finally, the generalized q-entropy formula, K(S) = ∑p

_{i}K(− ln p

_{i}), contains the Tsallis, Rényi and Boltzmann–Gibbs–Shannon expressions as particular cases. For diverging variance, Δβ

^{2}we obtain a novel entropy formula.

## 1. Introduction

^{24}), considered in classical thermodynamics of atomic matter, complex networks, like, e.g., the human brain include about the square root of this number of elements O (10

^{12}). The internet contains approximately 10

^{7}hubs and 10

^{10}connections. On the other hand a relativistic heavy ion collision produces a fireball of several O (10

^{3}) new hadrons (strongly interacting particles), while in a more elementary pp collision about O (10) particles are detected[37–39]. Since one expects that the relative (scaled) fluctuations grow with the decreasing number of participants, it is evident that the high energy physics experiments are able to reveal finite reservoir effects quantitatively [21,22,26,27,33,40–45].

_{K}= 1? We note that q = 1 signalizes an additive composition rule, so the second question is equivalent for seeking an additive (“K-additive”) description in case of non-negligible finite size corrections on the classical thermodynamics [16,18,22].

## 2. Finite Heat Bath and Fluctuation Effects

^{−3}), but the number of produced particles, n, fluctuates appreciably. Its distribution will be considered first in terms of the simplest possible assumptions about combining occupied and unoccupied phase-space cells in a finite observed section of the available total phase-space. Following this analysis more general n distributions and finally a general heat bath, described by its equation of state, S(E), is considered. During this chain of models we seek answer for the question: What is the physics behind the parameter q ?

_{n}

_{+1}(E) is the total phase-space, while Ω

_{n}(E−ω) is the phase-space for the reservoir, missing one particle with energy ω. The number of particles, n, itself can have a distribution (based on the physical model of the reservoir and on the event by event detection of the spectra).

_{n,k}(f). The average statistical weight factor, w

_{E}(ω), with fixed E and the negative binomial distribution (NBD) of n becomes

_{T}spectra in the same experiments [37–39].

_{n}, not necessarily BD or NBD or Poissonian ones, the above result also applies, albeit only as an approximation. In the above detailed philosophy of the microcanonical approaching the canonical for large systems, we expand our formulas for ω ≪ E. The Tsallis–Pareto distribution as an approximation reads as

^{2}, our result is expressed via the heat capacity of the reservoir, defined as 1/C = dT/dE. In general we have opposite sign contributions from 〈S′

^{2}〉 − 〈S′〉

^{2}and from 〈S″〉. In the light of this result one realizes that

- q > 1 and q < 1 are both possible,
- for any relative variance $\mathrm{\Delta}\beta /\langle \beta \rangle =1/\sqrt{C}$ it is exactly q = 1,
- and for fixed E ∝ n/β we have Δβ/〈β〉 = Δn/〈n〉.

^{2}〉 = 〈n(n − 1)〉/E

^{2}, leading to

^{2}would have to be negative. It is impossible. This problem is also reflected in the fact that there is no guarantee that an inverse Laplace transformation results in an overall positive function. In this way the superstatistics due to n-fluctuations, P

_{n}(E), seems to be more general, than the approach with solely a β-distribution, γ(β). In particular a superstatistical β-distribution cannot ever match a q < 1 result.

## 3. Deformation of the Entropy

_{K}= 1?

#### 3.1. The Additive Entropy K(S)

_{K}. In order to simplify the differential equation posed on K(S) by requiring a given value for q

_{K}we introduce the notations F = 1/K′ = T

_{K}/T and Δβ

^{2}/〈β〉

^{2}= λ/C. Then the q

_{K}parameter for the K(S) entropy is expressed as

_{K}= q and q

_{K}= 1. Since we seek for entropy deformations with the property K(0) = 0 and K′ (0) = 1, one fixes the condition F(0) = 1. In this case the only solution for q

_{K}= q is F = 1, K(S) = S. It is obvious that the other choice, q

_{K}= 1, is the only purposeful deformation for reaching K-additivity [16,18]. Equation (26) becomes then easily solvable. We call this form of the q

_{K}= 1 requirement the “Additivity Restoration Condition” (ARC):

#### 3.2. Classification by Fluctuation Models

_{K}= 1 also means a re-exponentialization of the ω-expansion of the statistical weight based on the deformed entropy phase-space, ${w}_{E}^{\mathrm{K}}(\omega )$. In this way the effective equilibrium condition, the common temperature, least depends on the one-particle subsystem energy, ω. In earlier publications we called this the “Universal Thermostat Independence” (UTI) principle [22].

^{2}= 0 and therefore λ = 0. Applying our previous general result for this value we have to solve

_{i}p

_{i}K(−ln p

_{i}) at the statistical entropy formulas of Tsallis and Rényi: [1–4,6]

_{Δ}= C + λ. Its first integral,

_{12}) = K(S

_{1}) + K(S

_{2}), is equivalent to

_{Δ}. Using the auxiliary function, h

_{C}(S) = C(e

^{S}

^{/}

^{C}−1), we have h

_{∞}(S) = S and the entropy deformation function can also be written as

- For λ = 1 it is obviously K
_{1}(S) = S. This is the Gaussian fluctuation model, considered in several textbooks, and also believed to lead to the smallest physically possible variance due to a “thermodynamical uncertainty” principle [55–58]. Since β = S′(E), the variances are related as Δβ = |S″(E)|ΔE = ΔE/CT^{2}. Then from Δβ · ΔE ≥ 1 it follow $\mathrm{\Delta}E\ge T\sqrt{C}$ and $\mathrm{\Delta}\beta \ge 1/T\sqrt{C}$. A straightforward consequence of this is λ/C = Δβ^{2}/〈β〉^{2}≥ 1/C and therefore λ ≥ 1. We note, that if this “uncertainty” principle were correct, then only q > 1 canonical distributions of ω would exist in Nature. - For no fluctuations λ = 0 and we get K
_{0}(S) = h_{C}(S). We regain the Tsallis and Rényi formulas presented above in Equation (31). - It is also very intriguing to inspect the following particular limit: C → ∞, λ → ∞ but $\mathrm{\lambda}/{C}_{\mathrm{\Delta}}\to \tilde{q}-1$ finite. In this non-extensive limit the fluctuations are much larger than the normal Gaussian ones, and we obtain a nontrivial entropy deformation:$${K}_{NE}(S)={h}_{1/(\tilde{q}-1)}^{-1}({h}_{\infty}(S))=\frac{1}{\tilde{q}-1}\mathrm{ln}(1+(\tilde{q}-1)S).$$
_{12}) = K(S_{1}) + K(S_{2}), in this case leads to the non-additivity formula ${S}_{12}={S}_{1}+{S}_{2}+(\tilde{q}-1){S}_{1}{S}_{2}$, – investigated formerly in depth by Tsallis and Abe [6,54,59–64].

_{i}p

_{i}K(− ln p

_{i}):

- For normal fluctuations K
_{1}(S) = −∑_{i}p_{i}ln p_{i}is exactly the Boltzmann entropy. - Without fluctuations ${K}_{0}(S)=C{\displaystyle {\sum}_{i}\left({p}_{i}^{1-1/C}-{p}_{i}\right)}$ is the Tsallis entropy with q = 1 − 1/C and S the corresponding Rényi entropy.
- Finally considering extreme large fluctuations and a finite heat capacity, C(S) which however may be an arbitrary function of the total entropy, S, we obtain the non-extensive result Equation (38) with $\tilde{q}=2$:$${K}_{\infty}(S)=\mathrm{ln}(1+S)={\displaystyle \sum _{i}{p}_{i}\mathrm{ln}(1-\mathrm{ln}{p}_{i}).}$$
_{i}distribution maximizing this parameterless deformed entropy is also expressed in terms of Lambert-W function, it shows tails like the Gompertz distribution [65–67], known from extreme value statistics and nonequlibrium growth models for demography and tumors.

_{∞}(S[p, 1 − p]) is shown on the Figure 1 by the full line, while the back–deformed “Rényi-type” entropy formula is plotted by the dashed line. These curves are close to each other for the very low (and very high) probability, and maximally differ at the equiprobability point p = 0.5. However, in the latter point S[p, 1−p] has the same value as the traditional Boltzmann–Gibbs curve, for which S and K(S) coincide due to λ = 1.

_{∞}(S

_{∞}), S

_{∞}and the Boltzmann–Gibbs formula K

_{1}(S

_{1}) = S

_{1}for the two-state system have to be inspected. This quantity is plotted on Figure 2 demonstrating that its value is negative in the whole range.

_{i}g(p

_{i}), its properties are determined by those of the function g(p). First we note that g(0) = 0 and g(1) = 0 rendering the totally ordered state, {p

_{i}} = {1, 0, 0, …}, to zero entropy. The canonical solution is obtained by its derivative:

^{1}

^{/C}

^{Δ}and ln p = −(1 + α + βω) as usual. In the non-extensive limit, λ ≫ C, it reduces to

^{L}= z. Replacing z = e

^{−}

^{x}and taking the logarithm of both sides leads to

_{Δ}≥ (1 − λ)w. But since w ≤ 1 and C

_{Δ}≥ C ≥ 1 for any reservoir containing at least one unit of heat capacity (at least one relativistic particle in at least one dimension or at least one massive particle in at least two dimensions) this requirement is fulfilled for all parameter values.

_{i}= 1/W for all i = 1, … W. Due to construction then we have

_{eqp}= ln W follows, whatever the K(S) function was. In this way the Rényi-entropy type expression S(p

_{i}) at equiprobability is additive for multiplicative numbers of states; it is by construction extensive. The same cannot be told about K(S), this deformed entropy may become non-extensive. On the other hand, assuming K(S)-additivity, as we did by prescribing the additivity restoration condition (ARC, Equation (27)), the product rule for the equipartioned probabilities is deformed—signalling statistical entanglement.

## 4. Conclusions and Outlook

^{2}/〈n〉

^{2}−1/〈n〉 reflects both the particle number variance and due to its expectation value the size of the reservoir. This formula also explains why both q > 1 and q < 1 cases can be observed in natural phenomena.

^{2}/〈β〉

^{2}− 1/C, with 1/C = dT/dE = −T

^{2}〈S″(E)〉.

_{1}+ E

_{2}) ≠ S(E

_{1}) + S(E

_{2}) for q ≠ 1 is a weakness of the classical thermodynamics which has to be cured. Our approach here was to look for a function, K(S), which restores additivity by leading to q

_{K}= 1. This requirement for such a function concludes in the additivity restoring condition, ARC, in a differential equation satisfied by K(S). Finally the usual canonical treatment must then be based on the additivity of K(S), applied to an ensemble of configurations, which in turn provides the general formula K(S) =∑p

_{i}K(−ln p

_{i}) (cf. Equation (40) and [20]).

^{L}= z. Recently, the Lambert-W function was met in relation to a study of scaling properties of a general two-parameter ansatz for non-additive entropy formulas by Hanel and Thurner [68] and in a non-extensive diffusion model by Andrade et al. [69]. Although these and ours are quite different contexts, a far analogy on the level of similar behavior at extreme low probabilities cannot be ab initio excluded.

^{2}/〈β〉

^{2}≫ C ≫ 1 with a finite limit for λ/C, shall deal with genuine non-extensivity of the K(S) entropy. The physical modelling of the reservoir environment, in particular with emphasis on the variable number of particles relevant for high energy physics, leads to more complex descriptions than presented here: a dependence like C(S) and λ(S) can be quite common. In such cases the ARC differential equation leads to further entropy formulas. Our approach provides a procedure to find the optimal entropy – probability relation from the viewpoint of the non-additive composition of two (or gradually more) subsystems. In addition, the superstatistics, originally conceptualized as a β-distribution behind non-Gibbsean factors in the statistics, may be extended to studies considering physical systems which cannot be described simply by an overall positive weight factor γ(β) under an integral.

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The general entropy K(S) (full line) and S (dashed line) are plotted for λ = ∞, meaning divergently large fluctuations with respect to the Gaussian model. For comparison the λ = 1 case, the traditional Boltzmann–Gibbs formula is indicated by the dotted line.

**Figure 2.**The second derivatives of the general entropy K(S) (full line) and S (dashed line) are plotted for λ = ∞. For comparison the same derivative for the λ = 1 (Boltzmann) case is indicated by the dotted line.

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**MDPI and ACS Style**

Biró, T.S.; Ván, P.; Barnaföldi, G.G.; Ürmössy, K.
Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle. *Entropy* **2014**, *16*, 6497-6514.
https://doi.org/10.3390/e16126497

**AMA Style**

Biró TS, Ván P, Barnaföldi GG, Ürmössy K.
Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle. *Entropy*. 2014; 16(12):6497-6514.
https://doi.org/10.3390/e16126497

**Chicago/Turabian Style**

Biró, Tamás Sándor, Péter Ván, Gergely Gábor Barnaföldi, and Károly Ürmössy.
2014. "Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle" *Entropy* 16, no. 12: 6497-6514.
https://doi.org/10.3390/e16126497