Trinity of Varentropy: Finiteness, Fluctuations, and Stability in Power-Law Statistics

T rinit y of V aren trop y: Finiteness, Fluctuations, and Stabilit y in P o w er-La w Statistics Hiroki Suy ari Gr aduate Scho ol of Informatics, Chib a University, 1-33, Y ayoi-cho, Inage-ku, Chib a 263-8522, Jap an (Dated: Marc h 31, 2026) P ow er-la w distributions are widely observed in complex systems, yet establishing their thermo- dynamic consistency remains a theoretical challenge. In this pap er, we present a thermodynamic framew ork for p o w er-law statistics based on the r enormalize d entr opy s 2 − q . Derived from the asymp- totic scaling of the com binatorial q -factorial, this quan tity yields a stable thermo dynamic limit, remaining finite ( O ( N 0 )) for systems with strong correlations. F urthermore, w e clarify the physical origin of the nonlinearity parameter q through the concept of V ar entr opy (V ariance of Entrop y). By unifying the macroscopic v ariational principle with the microscopic Sup erstatistics framew ork, we deriv e the relation | q − 1 | ≃ 1 /C , where C is the heat capacity of the reserv oir. This result suggests that pow er-la w statistics provides a thermo dynamic description of finite systems, where the finite heat capacit y of the heat bath necessitates a generalization beyond the standard Boltzmann-Gibbs limit ( C → ∞ ). I. INTR ODUCTION P ow er-la w distributions are a common feature of com- plex systems. F rom high-energy particle collisions [1] and turbulen t flows [2] to financial markets [3] and biological net works, systems with strong correlations or long-range in teractions consisten tly exhibit hea vy-tailed statistics that deviate from the standard Boltzmann-Gibbs de- scription. T o capture these phenomena, non-extensive statistical mechanics, pioneered by Tsallis [4], has pro- vided a framework, generalizing the standard exp onen tial factor to the q -exp onen tial function. Despite its empirical success, a theoretical c hallenge remains: the consistent definition of the thermo dynamic limit . In standard statis- tical mec hanics, the extensivit y of entrop y (i.e., en trop y prop ortional to system size N ) is necessary for the exis- tence of stable thermodynamic p oten tials. How ever, for systems gov erned b y p ow er-law statistics, the standard non-extensiv e en trop y S q generally becomes non-additiv e and loses the extensiv e scaling prop ert y ( S q ∝ N 2 − q or similar anomalous scaling) as the system size diverges [5]. This b eha vior has led to a debate regarding the physical v alidity of the framework: Do es a consistent thermody- namics exist for finite systems that exhibit pow er-la w be- ha vior? In this pap er, w e address this issue by establishing a thermo dynamic framew ork mathematically grounded in the combinatorics of the q -factorial and physically in- terpreted through the concept of V ar entr opy (V ariance of Entrop y). W e sho w that the anomalous scaling is a consequence of the finite-size effects inheren t in corre- lated systems. Our approach introduces a generalized thermo dynamic state v ariable, the r enormalize d entr opy s 2 − q , which is constructed to remain finite ( O ( N 0 )) in the thermo dynamic limit, thereb y resolving the stability problem. F urthermore, w e inv estigate the ph ysical origin of the non-extensivit y parameter q . While q is often treated as a fitting parameter, w e show that it is a direct mea- sure of the finite he at c ap acity of the en vironmen t. By linking the macroscopic v ariational principle with the mi- croscopic theory of Sup erstatistics [6], w e deriv e a re- lation linking q to the heat capacity C of the reser- v oir: | q − 1 | ≃ 1 /C . This result implies that p o w er-law statistics describes the thermo dynamics of finite systems, where the fluctuations of in tensiv e parameters (temp er- ature) cannot be neglected. The pap er is organized as follows. In Sec. I I, we es- tablish the mathematical foundations, deriving the q - factorial gro wth rule and introducing the renormalized en tropy s 2 − q . In Sec. I II, w e explore the ph ysical mean- ing of q through the expansion of en trop y and the concept of V arentrop y . In Sec. IV, we demonstrate how the ne- cessit y of Gamma fluctuations relates to thermodynamic stabilit y and the observed anomalous scaling. Finally , Sec. V summarizes our findings and discusses the impli- cations for finite-size thermodynamics. I I. THEORETICAL FRAMEWORK In this section, w e outline the mathematical founda- tions for pow er-la w statistics. Rather than p ostulating a macroscopic entrop y functional a priori, w e analyze ho w the num ber of microstates grows with the system size. A nonlinear generalization of the growth rule leads to the q -algebra and the associated combinatorial structure. A. Generalized Phase Space Growth Consider a ph ysical system with an effectiv e phase space v olume W ( N ) dep enden t on the system size N . F or standard extensiv e systems with short-range interactions, the phase space volume gro ws exp onen tially , go v erned b y the linear differen tial equation dW/dN = λW , where λ 2 is a gro wth constant. Ho wev er, for systems character- ized b y long-range correlations or strong entanglemen t, the growth rate may dep end non-linearly on the current v olume. W e generalize the gro wth law as: dW dN = λW q , (1) where the exp onen t q c haracterizes the degree of non- extensivit y and the nature of the phase space correla- tions. T o ensure thermo dynamic stabilit y , sp ecifically the conca vit y of the entrop y functionals S q and S 2 − q , the physical regime is strictly b ounded to 0 < q < 2 throughout this pap er. Within this range, the system exhibits distinct gro wth regimes: • q = 1: Standard extensive systems (Indep enden t gro wth). • 0 < q < 1: Restricted gro wth (Phase space is sup- pressed by correlations). • 1 < q < 2: Sup er-extensiv e growth (Phase space is expanded). T o analyze the solution W ( N ), it is useful to con- sider the transformation that linearizes the gro wth. Re- arranging Eq. (1) as W − q dW /dN = λ , we can express this non-linear gro wth equiv alently using the q -lo garithm (ln q x := x 1 − q − 1 1 − q ): d ln q W dN = λ. (2) Equation (2) shows that the q -logarithm transforms the p o wer-la w gro wth in to a constan t in tensiv e rate. Inte- grating this equation yields the linear relation: ln q W = λN + const . (3) The q -logarithm serves as a natural scaling function that maps the correlated phase space v olume W in to a linear extensiv e quantit y . T o maintain the extensivity of the macroscopic state v ariable, the entrop y of the system is defined as: S q := k B ln q W . (4) While the in v erse relation is given b y the q -exponential function ( W = exp q ( λN + const)), the linearit y of Eq. (3) underpins the generalized thermodynamics. B. Algebraic Structure: Linearization via q -Logarithm The nonlinear growth rule Eq. (1) implies that the comp osition of subsystems is no longer multiplicativ e. If w e consider a composite system formed by tw o subsys- tems A and B , the total n umber of microstates W A ∪ B in volv es a generalized pro duct W A ⊗ q W B due to the presence of correlations. W e define the comp osition rule ⊗ q b y the requirement that the generalized entropies are additiv e: ln q ( W A ⊗ q W B ) = ln q W A + ln q W B . (5) This relation asserts that, although the microstate vol- umes W in teract nonlinearly via the generalized q - pro duct [7, 8], the corresp onding v ariables ln q W follo w the standard addition rule. Thus, the q -logarithm trans- forms the nonlinear composition into a linear sup erpo- sition, serving as the appropriate v ariable for describing correlated systems. C. q -Stirling’s F orm ula F or large N , the direct analytical ev aluation of the gen- eralized factorial N ! q is in tractable. W e therefore emplo y the q -deformation of Stirling’s approximation [9], here- after referred to as the q -Stirling’s formula . Consisten t with the classification presented in Section I I-A, this ap- pro ximation is applicable across the gro wth regimes: re- stricted growth (0 < q < 1), standard extensive gro wth ( q = 1), and sup er-extensiv e gro wth (1 < q < 2). As- suming the parameter range 0 < q < 2, the q -Stirling’s form ula is giv en by: ln q ( N ! q ) ≃ N ln q N − N . (6) This expression reduces to the standard Stirling’s for- m ula, ln( N !) ≃ N ln N − N , in the limit q → 1. F or q  = 1, Eq. (6) captures the non-linear scaling of the phase space volume, pro viding a to ol for deriving the macroscopic thermo dynamic prop erties of the reserv oir. D. q -Multinomial Co efficien t and En tropy W e define the q -multinomial c o efficient as the effective n umber of microstates for a system partitioned in to k groups with sizes N 1 , . . . , N k ( N = P N i ). Using the q - logarithm and the generalized additivit y , this co efficient is given b y: ln q  N N 1 . . . N k  q := ln q N ! q − k X i =1 ln q N i ! q . (7) This quantit y generalizes the coun ting factor to corre- lated systems. Substituting the generalized Stirling’s for- m ula (Eq. 6) in to Eq. (7), we obtain the asymptotic re- lation linking the microstate count to the macroscopic en tropy: ln q  N N 1 . . . N k  q ≃ N 2 − q 2 − q S Tsallis 2 − q ( p 1 , . . . , p k ) , (8) where S Tsallis q ( p ) := 1 − P p q i q − 1 is the Tsallis entrop y . This establishes a mathematical link [9]: the com binatorial 3 structure of the q -m ultinomial co efficien t corresp onds to the Tsallis entrop y with index 2 − q in the thermo dynamic limit. E. The Generalized Thermo dynamic Limit An issue in establishing a statistical mec hanics for p o wer-la w systems is the thermo dynamic limit. Assum- ing a generic case where the parameter q is fixed (or v aries slo wly), the total en trop y S q ( N ) := ln q W exhibits a scal- ing b eha vior: S q ( N ) ∼ N 2 − q . (9) This scaling, derived from the combinatorial coun ting of microstates, implies that the standard entrop y density S q / N either diverges (for 0 < q < 1) or v anishes (for 1 < q < 2) as N → ∞ . A w ell-known approach to resolve this non-extensivity is to identify a unique, strictly tuned index q ent dic- tated b y the sp ecific phase-space geometry , for which S q ent ∝ N [10]. Ho wev er, treating q as an indep endent ph ysical parameter c haracterizing the finite heat capacity of the reservoir (Sec. I II) requires a broader thermo dy- namic framew ork. T o construct a consistent macroscopic theory v alid for generic q , w e prop ose a gener alize d ther- mo dynamic limit . W e define the r enormalize d entr opy s 2 − q b y normal- izing the total entrop y with the correct scaling factor deriv ed from the q -factorial: s 2 − q := lim N →∞ S q ( N ) N 2 − q . (10) This renormalized quantit y remains finite and indep en- den t of the system size N in the thermodynamic limit. T able I summarizes the distinction b et ween the unscaled total entrop y and the renormalized in tensiv e state v ari- able. T ABLE I. Definition of thermodynamic v ariables in p o wer- la w statistics. Note that the renormalized entrop y s 2 − q serv es as the intensiv e state v ariable go verning equilibrium. Quan tity Scaling ( N → ∞ ) Ph ysical Role T otal En trop y S q ( N ) ∼ N 2 − q Measure of microstates Renormalized En trop y s 2 − q ∼ N 0 (Finite) In tensive state v ariable By constructing thermo dynamics based on s 2 − q , we ensure a stable macroscopic description. The expansion of this renormalized en trop y s 2 − q around q = 1 relates to the physical origin of the parameter q , whic h will be the fo cus of the next section. F. V ariational Principle and the q -Canonical Distribution W e determine the equilibrium probability distribu- tion by applying the Maximum Entrop y principle to the renormalized entrop y established abov e. A feature of the non-extensiv e formalism is the dualit y betw een the en- trop y index and the resulting distribution index when standard constraints are employ ed. As sho wn by W ada and Scarfone [11], maximizing the entrop y with index 2 − q (our renormalized entrop y s 2 − q ) under the lin- ear mean energy constraint U = P i p i E i yields the q - exp onen tial distribution. The v ariational functional is giv en by: δ δ p i   s 2 − q − β X j p j E j − α X j p j   = 0 , (11) where β and α are Lagrange m ultipliers asso ciated with the in ternal energy and the normalization, resp ectiv ely . Solving this v ariational problem leads to the q -canonical distribution: p i = 1 Z q [1 − (1 − q ) β E i ] 1 1 − q , (12) where Z q is the generalized partition function. Maximiz- ing the renormalized en tropy s 2 − q under standard linear constrain ts yields the q -canonical distribution. F urther- more, as w e will see in Section IV, this distribution de- riv ed from the macroscopic v ariational principle exactly coincides with the effective macroscopic distribution ob- tained from the microscopic Sup erstatistics of a finite reserv oir. I II. PHYSICAL INTERPRET A TION: V ARENTR OPY AND FINITE HEA T CAP A CITY In the previous section, w e established the mathemati- cal framework based on the q -logarithm and the general- ized thermo dynamic limit. In this section, w e show that q acts as a conjugate v ariable to the higher-order statis- tics of information, sp ecifically related to the V ar entr opy and the finite heat capacity of the reservoir. A. Expansion of Renormalized Entrop y and V arentrop y T o clarify the physical role of q , we analyze the b e- ha vior of the renormalized en tropy s 2 − q in the vicinity of the standard limit q → 1. The q -logarithm function ln q x = ( x 1 − q − 1) / (1 − q ) can be T a ylor expanded around q = 1 as: ln q x = ln x + 1 − q 2 (ln x ) 2 + O ((1 − q ) 2 ) . (13) 4 By substituting x = 1 /p i in to this equation and taking the exp ectation v alue on b oth sides, we obtain the exact expansion: X p i ln q 1 p i | {z } S q ( P ):Tsallis = X p i ln 1 p i | {z } S BG ( P ):Boltzmann-Gibbs + 1 − q 2      X p i  ln 1 p i − S B G ( P )  2 | {z } V ( P ):V arentropy +( S B G ( P )) 2      + O  (1 − q ) 2  . (14) T runcating the higher-order terms and writing the rela- tion more concisely in terms of exp ectation v alues using the duality iden tity ln q (1 /x ) = − ln 2 − q ( x ), we hav e the follo wing approximation: ⟨− ln 2 − q p i ⟩ | {z } S q :Tsallis ≈ ⟨− ln p i ⟩ | {z } S BG :Boltzmann-Gibbs + 1 − q 2    ⟨ ( − ln p i − S B G ) 2 ⟩ | {z } V :V arentrop y + S 2 B G    . (15) Equation (15) is a perturbative expansion v alid in the quasi-asymptotic regime | q − 1 | ≪ 1 (i.e., for small de- viations from the extensive Boltzmann-Gibbs limit). It separates the leading-order physical p erturbation into the exact v ariance of microscopic information, known as the V ar entr opy : V ( P ) := ⟨ ( − ln p i ) 2 ⟩ − ⟨− ln p i ⟩ 2 , (16) and the square of the macroscopic bac kground ( S B G ) 2 . The ( S B G ) 2 term app ears as an algebraic consequence of isolating the exact v ariance V ( P ) from the second mo- men t of information. While ( S B G ) 2 represen ts the global macroscopic background of the standard reference state, it is the V arentrop y V ( P ) that captures the local micro- scopic fluctuations. Ho wev er, this quadratic appro ximation has mathemat- ical limitations. As the system mo v es in to the strongly non-extensiv e regime (1 < q < 2), higher-order fluctua- tions of information (e.g., skewness and kurtosis) b ecome non-negligible, and the truncation at the second order breaks down. This breakdo wn pro vides the ph ysical justification for adopting the full non-extensive framew ork. In a finite heat bath, thermal fluctuations are fundamen tally c har- acterized by a single macroscopic scale: the in v erse heat capacit y 1 /C . Due to this single-scale nature, the infinite to wer of higher-order information fluctuations (sk ewness, kurtosis, etc.) are not independent free parameters. In- stead, within the algebraic structure of the q -generalized framew ork, they are uniquely determined by and com- pletely parameterized b y the leading-order v ariance (V ar- en tropy), as will b e explicitly demonstrated in Section IV. The renormalized en tropy s 2 − q , gov erned by the q -logarithm, acts as the exact, non-p erturbativ e math- ematical structure that effectively r esums this single- parameter family of fluctuations into a closed function. Because these higher-order terms are sub ordinate to the v ariance, they do not alter the ph ysical direction of the deformation even in the strongly non-extensive regime (1 < q < 2). The sign of the leading-order co- efficien t (1 − q ) / 2 determines ho w the macroscopic equi- librium is driv en b y these information fluctuations: • Case q = 1 (Standar d Limit): The fluctuation w eight v anishes. The equilibrium state is deter- mined by the mean information, leading to the standard exp onen tial family (Boltzmann-Gibbs). • Case 0 < q < 1 (Fluctuation Pr ophilic): The lead- ing co efficien t is positive. T o maximize S q , the sys- tem fa vors states with lar ge V aren tropy . T o ac hieve this large v ariance, the system heavily w eights ex- treme v alues of microscopic information. Mathe- matically , this is realized b y driving the probabilit y to zero at a finite threshold, resulting in distribu- tions with compact support where the information ( − ln p i ) diverges at the boundaries. • Case 1 < q < 2 (Fluctuation Phobic): The leading co efficien t is negativ e. T o maximize S q , the sys- tem minimizes the V arentrop y . In standard exp o- nen tial distributions, tail states p ossess extremely small probabilities, causing their information con- ten t ( − ln p i ∝ E ) to gro w linearly and con tribute hea vily to the v ariance. T o suppress these extreme information v alues, the system forces the proba- bilit y to decay m uc h slo w er. Mathematically , this manifests as a heavy-tailed (p o wer-la w) distribu- tion where the information gro ws logarithmically ( − ln p i ∝ ln E ) rather than linearly , thereby com- pressing the fluctuations and av oiding the V aren- trop y p enalty inheren t in exponential cutoffs. Th us, q serv es as a conjugate field that couples to the fluctuations of the microscopic information conten t, with the V arentrop y acting as the leading macroscopic signa- ture of this coupling. B. Connection to Thermo dynamic Heat Capacity In statistical mechanics, macroscopic fluctuations in en tropy are link ed to the heat capacit y of the system. According to the Einstein fluctuation theory , the proba- bilit y distribution of entrop y fluctuations δS around the equilibrium is Gaussian: P ( δ S ) ∝ exp  − ( δ S ) 2 2 k B C  , (17) 5 where C is the heat capacit y of the system. This for- m ula implies that the macroscopic v ariance of en trop y is prop ortional to the heat capacit y: ⟨ ( δ S ) 2 ⟩ ∼ k B C . In our microscopic framework, this v ariance of information is captured b y the V aren trop y V ( P ). As established in the previous subsection, the co effi- cien t | 1 − q | serv es as the single leading parameter that go verns the scale of these information fluctuations. Com- paring this v ariance-con trolling parameter with the ther- mo dynamic stability condition of the Einstein theory sho ws that | 1 − q | acts as the generalized macroscopic cost of these fluctuations, corresp onding exactly to the in verse heat capacity . Ph ysically , the standard limit q → 1 corre- sp onds to an infinite heat bath ( C → ∞ ), where fluctua- tions v anish relative to the system size. Con v ersely , a de- viation from unit y ( q  = 1) implies a finite heat capacity , where the reservoir cannot fully absorb energy fluctua- tions. This in terpretation aligns with the result obtained b y Wilk and W lo darczyk [1], leading to the fundamen tal corresp ondence: | 1 − q | ≃ 1 C . (18) This relation pro vides a ph ysical interpretation of q in terms of the thermal environmen t: 1. Infinite He at Bath ( q → 1 ): When the system is coupled to an infinite reservoir ( C → ∞ ), the pa- rameter q approac hes unity . The temp erature is fixed, and the canonical ensemble is exact. 2. Finite He at Bath ( q  = 1 ): When the reserv oir is finite ( C < ∞ ), the exchange of energy induces non-negligible fluctuations in the in tensive param- eters (temp erature). In this regime, q deviates from unit y to accoun t for the finite-size effects of the en- vironmen t. Therefore, the non-extensiv e parameter q is a physical measure of the finite he at c ap acity of the heat bath. The anomalous scaling N 2 − q deriv ed in Section I I can be in- terpreted as a consequence of this finite-size effect, where global temperature fluctuations of the finite reservoir in- duce strong correlations among the system’s comp onen ts, effectiv ely reducing their independent degrees of freedom. This mec hanism pro vides a macroscopic form ulation for a fluctuating environmen t, which w e will microscopically demonstrate in the next section via Superstatistics. IV. MICR OSCOPIC F OUND A TIONS: SUPERST A TISTICS AND HEA T CAP ACITY In the previous section, we suggested the relation | 1 − q | ≃ 1 /C based on thermo dynamic stability ar- gumen ts. In this section, we pro vide the deriv ation of this relation and explore the “T rinit y of V arentrop y”: Mathematical Necessit y , Thermodynamic Stability , and Anomalous Scaling. W e demonstrate that the Sup er- statistics framew ork is a mathematical consequence of the V aren trop y-driven algebra established in our frame- w ork. A. Mathematical Requiremen t of Gamma Fluctuations The sup erstatistics framework [6] p osits that the macroscopic distribution P ( E ) is a sup erposition of Boltzmann factors weigh ted by a fluctuation distribution f ( β ): P ( E ) = Z ∞ 0 f ( β ) e − β E dβ . (19) Mathematically , Eq. (19) represents the Laplace trans- form of the fluctuation distribution f ( β ). In order to reco ver the macroscopic q -canonical distribution deriv ed in Eq. (12): P ( E ) ∝ e − β 0 E q := [1 − (1 − q ) β 0 E ] 1 1 − q , (20) the determination of f ( β ) is reduced to an in verse Laplace transform problem. According to Lerc h’s theo- rem, the in verse Laplace transform is unique. Therefore, the distribution f ( β ) that yields Eq. (20) is determined to b e the Gamma distribution: f ( β ) = 1 Γ( α ) θ α β α − 1 e − β /θ , (21) where α is the shap e parameter and θ is the scale pa- rameter. Due to the uniqueness of the in v erse Laplace transform, this establishes a one-to-one corresp ondence b et ween the macroscopic q -canonical distribution and the microscopic Gamma-distributed fluctuations. This mathematical uniqueness pro vides the micro- scopic pro of for the ph ysical claim made in Section I I I. F or a thermal reserv oir with a fixed mean inv erse temper- ature ⟨ β ⟩ = αθ , the Gamma distribution is completely c haracterized b y a single free parameter: its v ariance σ 2 β = αθ 2 . Consequently , all higher-order moments of the fluctuations, such as the sk ewness (2 / √ α ) and the excess kurtosis (6 /α ), are strictly algebraic functions of the v ari- ance. This pro v es that the infinite to w er of higher-order thermal fluctuations do es not p ossess indep enden t free parameters; instead, it is exactly resummed and uniquely parameterized b y the v ariance, whic h corresponds to the macroscopic V aren tropy . W e v erify this corresp ondence by performing the in te- gration: P ( E ) ∝ Z ∞ 0 β α − 1 e − β ( E +1 /θ ) dβ ∝ (1 + θ E ) − α . (22) 6 Comparing Eq. (22) with the definition in Eq. (20), we find that the deriv ed distribution is mathematically iden- tical to the q -exp onen tial function if and only if the pa- rameters satisfy: α = 1 q − 1 , θ = ( q − 1) β 0 . (23) Th us, the Gamma fluctuation is a mathematical require- men t to reco ver Tsallis statistics. This result ph ysically corresp onds to the lo cal temp erature fluctuations aris- ing from a sum of independent Gaussian v ariables (e.g., kinetic energies) in the heat bath. Based on the physi- cal mo del of the reservoir discussed in the previous sub- section, w e iden tify the shape parameter α with half its effectiv e degrees of freedom ( n/ 2). Under this physical iden tification, the distribution f ( β ) exactly corresponds to a sc ale d χ 2 distribution. Using the properties of the Gamma distribution, to ensure that the mean inv erse temp erature remains constan t ( ⟨ β ⟩ = αθ = β 0 ) regard- less of n , the scale parameter m ust scale as: θ = β 0 α = 2 β 0 n . (24) Consequen tly , the v ariance of the in v erse temp erature ( σ 2 β = αθ 2 ) b eha ves as: σ 2 β = αθ 2 = n 2  2 β 0 n  2 = 2 β 2 0 n . (25) This shows that the thermal fluctuations v anish in the thermo dynamic limit ( n → ∞ ). Substituting α = n/ 2 in to Eq. (23), w e obtain the relation betw een the non- extensivit y parameter q and the finite size of the reser- v oir: q = 1 + 2 n . (26) Ph ysically , this confirms that the deviation from Boltzmann-Gibbs statistics ( q  = 1) arises from the fi- nite heat capacity of the environmen t. Finally , in the thermo dynamic limit ( n → ∞ ), we hav e q → 1 and σ 2 β → 0. The Gamma distribution f ( β ) conv erges to the Dirac delta function δ ( β − β 0 ), and the superp osi- tion integral reduces to the standard Boltzmann factor ( P ( E ) → e − β 0 E ), ensuring that Boltzmann-Gibbs statis- tics is reco vered. B. Thermo dynamic Stability and Heat Capacit y W e can quantitativ ely link q to the heat capacity C . In the standard Sup erstatistics framework, the non- extensivit y parameter is related to the v ariance of the in- v erse temperature b y q = 1 + σ 2 β /β 2 0 . Since the v ariance is inheren tly positive ( σ 2 β > 0), q is strictly b ounded from b elo w b y 1. This fundamen tal p ositivit y , combined with 0.0 0.5 1.0 1.5 2.0 2.5 I n v e r s e T e m p e r a t u r e β 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 P r o b a b i l i t y D e n s i t y f ( β ) n = 5 n = 2 0 n = 1 0 0  β ® = β 0 FIG. 1. Microscopic temperature fluctuations describ ed b y the Gamma distribution f ( β ) for v arying degrees of freedom n of the heat bath. The mean inv erse temperature is fixed at ⟨ β ⟩ = β 0 . As n increases ( n → ∞ ), the distribution sys- tematically narro ws and conv erges to a Dirac delta function δ ( β − β 0 ), physically corresp onding to an exact canonical en- sem ble coupled to an infinite heat bath. the normalizabilit y condition of the probabilit y distribu- tion, naturally restricts the framework to the 1 < q < 2 regime. In this sp ecific fluctuation-induced regime, the absolute v alue is resolved as q − 1 > 0. F or a general fluc- tuating environmen t f ( β ), the non-extensivity parameter q is related to the relative v ariance of β as [6]: q − 1 = ⟨ β 2 ⟩ − ⟨ β ⟩ 2 ⟨ β ⟩ 2 = σ 2 β ⟨ β ⟩ 2 . (27) In standard statistical mec hanics, the temperature fluctuations of a system with finite heat capacity C are giv en by the Landau fluctuation form ula: σ 2 T T 2 = k B C , (28) where C is the thermodynamic heat capacity . In the fol- lo wing, we adopt natural units ( k B = 1) or consider C as the dimensionless heat capacity for simplicity . Since β = 1 /T (in natural units), the fluctuation δβ is related to the temperature fluctuation δ T by the first-order ex- pansion around the mean temp erature ⟨ T ⟩ : δ β ≈ dβ dT     ⟨ T ⟩ δ T = − 1 ⟨ T ⟩ 2 δ T = − ⟨ β ⟩ ⟨ T ⟩ δ T . (29) Squaring and a veraging this linear relation leads to the equalit y of the relativ e v ariances: ⟨ ( δ β ) 2 ⟩ ⟨ β ⟩ 2 ≈ ⟨ ( δ T ) 2 ⟩ ⟨ T ⟩ 2 = ⇒ σ 2 β ⟨ β ⟩ 2 ≈ σ 2 T ⟨ T ⟩ 2 . (30) Com bining Eq. (27) and Eq. (28), we deriv e the relation: q − 1 = 1 C . (31) 7 This equation indicates that q  = 1 is a signature of a finite heat capacity . In the thermo dynamic limit C → ∞ (infinite heat bath), we recov er q → 1. How ev er, for fi- nite systems or non-extensiv e systems where C do es not div erge linearly , the term 1 /C remains significant. The V aren trop y term (parameterized b y q ) acts as a thermo- dynamic stabilizer , incorp orating the finite-size thermal fluctuations in to the state definition to preven t thermo- dynamic singularities. 0 2 4 6 8 10 H e a t C a p a c i t y C 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 N o n - e x t e n s i v i t y p a r a m e t e r q q = 1 + 1 / C q = 1 ( B G l i m i t ) FIG. 2. Relationship betw een the non-extensivit y parameter q and the dimensionless heat capacit y C of the reservoir. As the heat capacit y div erges in the standard thermo dynamic limit ( C → ∞ ), the fluctuations v anish and the system reco vers Boltzmann-Gibbs statistics ( q → 1). F or finite systems with b ounded C , the parameter q > 1 acts as a thermodynamic stabilizer absorbing the macroscopic fluctuations. C. Anomalous Scaling The relation q − 1 ≃ 1 /C has implications for the ther- mo dynamic limit. If the parameter q remains different from unit y as N → ∞ , it implies that the total he at c a- p acity C ( N ) saturates to a finite v alue: C ( N ) = 1 q − 1 ∼ const . (32) This indicates a strongly non-extensive regime where the sp ecific heat v anishes ( C ( N ) / N → 0). Suc h b eha vior is consistent with systems constrained b y global correla- tions, suc h as those with long-range in teractions where standard extensivity is violated [12]. F urthermore, for systems ob eying area laws, such as blac k hole thermo dy- namics, it has been argued that the en tropy scales non- extensiv ely [13]. In our framew ork, this corresp onds to an effective heat capacit y scaling as C ∝ N γ ( γ < 1), implying that q approaches unity slo wly as q − 1 ∼ N − γ . This suggests that our result q − 1 ≃ 1 /C provides a con- sisten t thermo dynamic basis for these known anomalous scalings. V. CONCLUSION AND PERSPECTIVES In this paper, we hav e established a consistent thermo- dynamic framework for non-extensive systems, grounded in the scaling properties of the q -factorial and the physi- cal concept of V arentrop y . W e addressed the thermody- namic limit problem for p o w er-law statistics by introduc- ing the renormalized en tropy s 2 − q . Unlike the original Tsallis entrop y , where extensivit y is reco vered only for a sp ecific unique index q dep ending on the system’s cor- relations, the renormalized entrop y s 2 − q is constructed to remain finite ( O ( N 0 )) in the thermo dynamic limit for an arbitrary parameter q , serving as the correct in tensiv e state v ariable for systems with strong correlations. The Physic al Me aning of q .— Through the ”T rin- it y of V aren trop y”—Mathematical Necessity , Thermo dy- namic Stability , and Microscopic Fluctuations—w e hav e clarified the ph ysical origin of the non-extensivity param- eter q . W e demonstrated that q is a measure of the finite he at c ap acity of the en vironment. The deriv ed relation: | q − 1 | ≃ 1 C , (33) pro vides a bridge b et w een non-extensiv e statistics and the ph ysics of finite reserv oirs. Standard Boltzmann- Gibbs statistics ( q = 1) emerges only in the limit of an infinite heat bath ( C → ∞ ). Conv ersely , for finite sys- tems where the heat capacit y is limited, the V arentrop y (fluctuation of information) b ecomes non-negligible, ne- cessitating the q -generalized description (1 < q < 2) to main tain thermo dynamic stability . Origin in Information The ory.— The concept of V ar- en tropy originates in Information Theory , sp ecifically in the analysis of finite blo ck-length c o ding [14, 15]. Just as the v ariance of information b ecomes non-negligible when the data blo c k le ngth is finite, the thermo dynamic fluctuations become significant when the heat capacity is finite. Our result | q − 1 | ≃ 1 /C establishes an iso- morphism: Tsallis statistics is to Boltzmann-Gibbs ther- mo dynamics what finite-length co ding is to asymptotic Shannon theory . Both frameworks represent the general- ization needed to describe systems constrained by finite r esour c es (i.e., finite blo ck-length and finite he at c ap ac- ity) . This thermo dynamic persp ectiv e is mathematically complemen ted b y the generalized limit theorems [16] and the rigorous algebraic form ulation of finite block-length p enalties in information theory [17]. F utur e Outlo ok.— The framew ork presented here op ens av enues for applying non-extensive statistics to small-scale systems where fluctuations are inheren t. Ap- plications include nanothermo dynamics, the statistics of single-molecule exp erimen ts, and complex netw orks where the effectiv e degrees of freedom are limited. 8 [1] G. Wilk and Z. W lodarczyk, Physical Review Letters 84 , 2770 (2000). [2] C. Beck, Ph ysical Review Letters 87 , 180601 (2001). [3] L. Borland, Physical Review Letters 89 , 098701 (2002). [4] C. Tsallis, Journal of Statistical Physics 52 , 479 (1988). [5] C. Tsallis, Intr o duction to Nonextensive Statistic al Me- chanics: Appr o aching a Complex World (Springer, New Y ork, 2009). [6] C. Beck and E. G. D. Cohen, Physica A: Statistical Me- c hanics and its Applications 322 , 267 (2003). [7] L. Niv anen, A. L. M´ ehaut ´ e, and Q. A. W ang, Rep orts on Mathematical Ph ysics 52 , 437 (2003). [8] E. P . Borges, Ph ysica A: Statistical Mec hanics and its Applications 340 , 95 (2004). [9] H. Suyari, Physica A: Statistical Mec hanics and its Ap- plications 368 , 63 (2006). [10] C. Tsallis, M. Gell-Mann, and Y. Sato, Pro ceedings of the National Academy of Sciences 102 , 15377 (2005). [11] T. W ada and A. M. Scarfone, Ph ysics Letters A 335 , 351 (2005). [12] A. Campa, T. Dauxois, and S. Ruffo, Ph ysics Rep orts 480 , 57 (2009). [13] C. Tsallis and L. J. L. Cirto, The European Ph ysical Journal C 73 , 2487 (2013). [14] V. Strassen, in T r ansactions of the Thir d Pr ague Confer- enc e on Information The ory, Statistic al De cision F unc- tions, R andom Pr o c esses (Czec hoslov ak Academy of Sci- ences, Prague, 1962) pp. 689–723. [15] Y. Poly anskiy , H. V. P oor, and S. V erd ´ u, IEEE T ransac- tions on Information Theory 56 , 2307 (2010). [16] H. Suy ari and A. M. Scarfone, A constructiv e ap- proac h to q -Gaussian distributions: α -div ergence as rate function and generalized de Moivre-Laplace theorem, arXiv:2603.21391 (2026). [17] H. Suyari, Unified algebraic absorption of finite- blo c klength p enalties via generalized logarithmic map- ping, arXiv:2603.22358 (2026).