Non-Additivity in Tsallis Statistics

• Non-additivity in Tsallis statistics is defined by a q-dependent deviation from standard entropy additivity, introducing pseudo-additive composition rules.
• The framework employs q-sum and q-exponential functions to model complex systems with long-range interactions, fractal geometries, and memory effects.
• Its thermodynamic implications include modified Legendre transforms, finite-size corrections, and experimentally observable signatures in diverse physical systems.

Non-additivity in Tsallis statistics refers to the violation of the ordinary additive law for entropy and related thermodynamic quantities, as observed in the Tsallis generalization of statistical mechanics. This framework plays a central role in the description of systems exhibiting long-range interactions, strong correlations, fractal phase-space structures, or fat-tailed distributions—cases where Boltzmann–Gibbs–Shannon (BGS) statistics fails to capture observed phenomenology.

1. Definition and Mathematical Structure of Tsallis Non-additivity

The Tsallis entropy for a normalized discrete probability distribution $\{p_i\}$ ($\sum_i p_i=1$) is defined by

$$S_q\left[\{p_i\}\right] = \frac{1 - \sum_i p_i^q}{q-1}, \quad q \in \mathbb{R}.$$

For $q \to 1$, this expression recovers the familiar BGS entropy $S_{BGS}= -\sum_i p_i \ln p_i$.

Non-additivity in this context is manifested through the composition law for two statistically independent subsystems, $A$ and $B$, (with $p_{ij}(A \cup B) = p_i(A) p_j(B)$), which reads: $$S_q(A\cup B) = S_q(A) + S_q(B) + (1-q) S_q(A) S_q(B).$$ This “pseudo-additivity” (or “generalized additivity”) is succinctly captured in the $q$-sum operation: $x \oplus_q y \equiv x + y + (1-q) x y,$ so that $S_q(A\cup B) = S_q(A) \oplus_q S_q(B)$.

When $q=1$, the additional term vanishes, recovering strict additivity. For $q \neq 1$, the entropy is non-extensive, and $q$ quantifies the deviation from ordinary additivity. The same non-additivity is built into the associated $q$-exponential ($e_q(x)=[1+(1-q)x]^{1/(1-q)}$) and $q$-logarithm ($\ln_q(x) = (x^{1-q}-1)/(1-q)$) structures, which underlie the entire Tsallis formalism (Kalogeropoulos, 2016, Ferri et al., 2015, Korbel, 2017, Krisut et al., 24 Nov 2024).

2. Thermodynamic Foundations and Formal Logarithms

A systematic foundation for non-additivity in Tsallis statistics is established by demanding that the zeroth law of thermodynamics (separability and transitivity of equilibrium) be maintained, even in the presence of non-additive composition rules for entropy and energy. For any composition law of the form $X_{12} = f^{-1}(f(X_1)+f(X_2))$, one can always define a “formal logarithm” $f$ that restores additivity in the transformed variable.

Imposing the further principle of universal thermostat independence—requiring the absence of finite-size corrections in the canonical limit—singles out an exponential form for the formal logarithm, thereby directly leading to the Tsallis entropy. The parameter $q$ is thus related to the generalized susceptibility of the thermostat,

$q = 1 + \frac{S''(X_0)}{[S'(X_0)]^2},$

where $S(X)$ denotes the entropy as a function of conserved quantity $X$ (Ván et al., 2012). This establishes $q$ not as a phenomenological constant but as a measure of reservoir-induced curvature in entropy.

3. Physical Interpretation of Non-extensivity Parameter $q$

The entropic index $q$ admits direct physical interpretations in several contexts:

• Long-range interactions: $q \neq 1$ signals the breakdown of independent-accessible-states additivity, pertinent in systems with nonlocal interactions or collective behavior.
• Fractality and self-similarity: Dynamical systems with multifractal phase-space structure or self-similar energy cascades naturally realize equilibrium distributions of Tsallis form. Explicit construction in field theories (e.g., QCD) yields $q$ as a function of group-theory parameters ($$q = 1 + [\frac{11}{3}N_c - \frac{4}{3}(N_f/2)]^{-1}$$ for QCD with $N_c$ colors and $N_f$ flavors), with experimental fits in high-energy collisions confirming the predicted non-additive parameter $q \simeq 1.14$ (Deppman et al., 2020).
• Finite system size and interaction with finite reservoirs: For an isolated ideal gas of $N$ particles, explicit calculation shows $q(N) = 1 + 2/(3N)$ (Lima et al., 2020), with nonadditivity (hence correlation) diminishing as $N\to\infty$.
• Memory, intermittency, and superstatistics: Fluctuations in intensive parameters (e.g., inverse temperature) or long-term correlations in stochastic processes yield marginal distributions that are $q$-exponentials, with $q-1$ quantifying the strength of non-Markovianity (Kalogeropoulos, 2016).

4. Non-additivity in Dynamics, Correlations, and Kinetics

Non-additive Tsallis statistics alters the kinetic and transport properties of finite systems. In the Boltzmann–Gibbs case, Liouville's theorem and the additivity of entropy imply uncorrelated ($f_2 = f_1 f_1$) behavior (molecular chaos).

Tsallis nonadditivity leads, for finite systems, to explicit two-particle correlations which scale as

$$C_2(v_1,v_2) = f_2(v_1, v_2) - f_1(v_1) f_1(v_2) \sim O(q-1) \sim O(1/N),$$

disappearing in the thermodynamic limit but becoming significant in small systems (Lima et al., 2020). In transport theory, the collision term in the Boltzmann equation is modified to incorporate the nonadditive composition

$$h_q[f_A, f_B] = \left[f_A^{1-q_A} + f_B^{1-q_B} - 1\right]^{1/(1-q_C)},$$

thereby generating a non-additive source term $I_q[f]$ (Megias et al., 2022). This is a direct consequence of the phase-space topology and conservation laws in small-N systems.

5. Quantum and Thermodynamic Consequences

Non-additivity affects not only classical but also quantum systems. In quantum mechanics, replacing the ordinary exponentials in coherent state constructions with $q$-exponentials yields so-called Tsallis pseudo-coherent states, exhibiting modified uncertainties and momentum distributions that explicitly depend on the non-additivity parameter $q$ (Ferri et al., 2015). The uncertainty product $(\Delta x)_q (\Delta p)_q$ attains its minimum at $q=1$ but remains bounded from below for all $q$, with the $q$-dependence encapsulating the deformations induced by non-additivity.

Thermodynamically, the nonadditive Tsallis entropy mandates a deformation of the Legendre transform. The standard involutive Legendre–Fenchel duality is replaced by a transformed dual suitable for $s$-concave (power-law) functionals. The Massieu potential, free energy, and partition function are all expressed in terms of $q$-algebraic constructions, and the thermodynamic relations acquire $q$-dependent correction terms that vanish for $q\to1$ (Kalogeropoulos, 2017, Krisut et al., 24 Nov 2024).

6. Scaling, Algebra, and the Transformation Group of Non-additivity

The nonadditivity parameter $q$ enjoys a rich algebraic structure. Under the rescaling group $q \mapsto q_\alpha = (q+\alpha-1)/\alpha$, induced by physical processes such as changing bath size or temperature fluctuation strength, the Tsallis entropy transforms covariantly, and the $q$-algebraic operations (deformed sum, product, exponential, logarithm) satisfy generalized distributive and composition rules under this transformation (Korbel, 2017). This algebraic framework organizes families of deformed distributions and connects various entropy functionals (e.g., Jizba–Arimitsu hybrid entropy) within a single parametrized scheme.

7. Implications, Applications, and Critiques

Non-additivity in Tsallis statistics leads to a broad class of observable phenomena:

• Emergent correlations: Even for nominally independent particles, the nonadditive term $(1-q)S_q(A)S_q(B)$ encodes “hidden correlations,” apparent in negative specific heats, bounded temperatures, and effective gravitational behavior in certain parameter regimes (Plastino et al., 2017).
• Experimental signatures: Finite-N corrections to distributions—such as the Doppler broadening of spectral lines—open direct experimental tests of nonadditivity (Lima et al., 2020).
• Complex systems: In gravitational, plasma, or field theoretic systems, the nonadditivity framework provides tools for modeling long-range and hierarchical effects inadequately described by BGS statistics (Kalogeropoulos, 2016, Deppman et al., 2020).

However, it is also established that the maximization of nonadditive entropies such as $S_q$ can violate the Shore–Johnson system independence axiom, introducing spurious correlations in inferred distributions not warranted by the data (Pressé et al., 2013). This arises because, for $q\neq1$, the maximum entropy joint distribution of independent subsystems does not factorize, but contains terms of order $q-1$ mixing subsystem probabilities: $$p_{ij} = \left[u_i^{q-1} + v_j^{q-1} - 1\right]^{1/(q-1)}.$$ Thus, care is required when employing nonadditive entropy in inferential contexts lacking justified couplings between components.

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In summary, non-additivity in Tsallis statistics is rigorously anchored in the $q$-parametrized deviation from the additive entropy composition law. It arises from reservoir effects, phase-space geometry, correlation structure, and algebraic considerations, leading to modified distributions, kinetics, thermodynamic dualities, and algebraic frameworks. Its phenomenological and theoretical implications are observable in finite-scale systems, systems with fractal or hierarchical structure, and various domains where classical extensivity fails. The theory is both structurally rich and subject to foundational constraints, notably in inferential applications, guiding its effective and appropriate application throughout statistical mechanics and complex systems science.