Tsallis Non-additivity in Statistical Mechanics

• Tsallis non-additivity is the defining property of Tsallis entropy, characterizing the deviation from classic additive behavior through a q-dependent pseudo-additive law.
• It introduces q-deformed algebraic operations and modified thermodynamic potentials that enable new descriptions of finite, correlated, and complex systems.
• The framework generates power-law equilibrium distributions and finds applications in statistical mechanics, quantum systems, and gravitational models.

Tsallis non-additivity is the signature structural property of the Tsallis entropy, a one-parameter family of trace-form entropies generalizing the Boltzmann–Gibbs–Shannon (BGS) entropy. It manifests in the failure of strict additivity for the entropy of independent subsystems and underlies power-law equilibrium distributions, q-deformed thermodynamic structures, and generalized transport equations. The non-additivity parameter $q$ quantifies the degree to which entropy and associated ensemble quantities depart from extensivity, with $q=1$ recovering the classical (additive) theory. Tsallis non-additivity serves as a unifying concept across statistical mechanics, information theory, complex systems, and quantum frameworks.

1. Mathematical Definition and Pseudo-Additive Law

For a probability distribution $\{p_i\}$, the Tsallis entropy of order $q\in\mathbb{R}$, $q\neq1$, is given by

$S_q[\{p_i\}] = \frac{1 - \sum_i p_i^q}{q-1}$

and equivalently,

$S_q = - \sum_i p_i^q \,\ln_q(p_i)$

where the $q$-logarithm is $\ln_q(x) = (x^{1-q} - 1)/(1 - q)$ (Kalogeropoulos, 2016, Ferri et al., 2015).

For two independent subsystems $A$ and $B$ (joint probabilities $p_{ij} = p_i^A p_j^B$), Tsallis entropy satisfies the pseudo-additive (non-additive) composition law: $S_q(A+B) = S_q(A) + S_q(B) + (1-q) S_q(A) S_q(B)$ This quadratic correction term, $(1-q) S_q(A) S_q(B)$, is the hallmark of non-additivity (Ferri et al., 2015, Krisut et al., 24 Nov 2024, Megias et al., 2022, Ván et al., 2012).

Regimes:

• $q<1$: Super-additivity (the whole exceeds the sum of the parts).
• $q>1$: Sub-additivity.
• $q\to1$: Additivity, recovering Boltzmann–Gibbs entropy $S_{BG} = -\sum_i p_i \ln p_i$.

2. Algebraic Structures and q-Deformations

The Tsallis pseudo-additivity is naturally encoded in the $q$-sum,

$x\oplus_q y = x + y + (1-q)xy$

and the associated $q$-algebra includes deformed operations for addition, multiplication, exponentiation, and logarithms (Korbel, 2017, Krisut et al., 24 Nov 2024, Ferri et al., 2015). The $q$-exponential,

$\exp_q(x) = [1 + (1-q)x]^{1/(1-q)}$

satisfies

$\exp_q(x)\exp_q(y) = \exp_q(x\oplus_q y)$

demonstrating non-additivity in the kernel of canonical distributions.

A transformation group $T_\alpha$ acting on $q$,

$q_\alpha = \frac{q+\alpha-1}{\alpha}$

forms an abelian group under composition with important implications for the structure of $q$-deformed distributions and generalized distributivity in $q$-algebra (Korbel, 2017).

3. Physical and Thermodynamic Origin

The non-additivity parameter $q$ is not arbitrary but captures intrinsic properties of the reservoir or the system-environment interface. When a subsystem exchanges energy with a finite-capacity reservoir, universal thermostat-independence arguments (Ván et al., 2012) and phase-space topology analysis (Megias et al., 2022, Lima et al., 2020) both lead naturally to the Tsallis structure. For a finite ideal gas of $N$ particles, the effective $q$ is given by

$q = 1 + \frac{2}{3N}$

showing that non-additivity emerges from correlations induced by energy/volume constraints, vanishing as $N\to\infty$ (Lima et al., 2020, Megias et al., 2022).

The generalized composition law for extensive quantities can be made additive by mapping onto "formal logarithms," and the unique structure selected by demanding universal thermostat independence is Tsallis entropy (Ván et al., 2012).

Table: Limiting Cases of Non-Additivity

$q$ Regime	Correction Sign	Physical Regime / Example
$q = 1$	None (additive)	Classical thermodynamics
$q < 1$	Positive (super-additive)	Long-range correlations, finite systems
$q > 1$	Negative (sub-additive)	Correlation-induced phase-space truncation

4. Deformed Thermodynamic Potentials and Legendre Structure

In standard thermodynamics, Legendre transforms relate thermodynamic potentials via convex duality. Tsallis' $S_q$ lacks standard concavity; it is s-concave with $s=1/(1-q)$ (Kalogeropoulos, 2017). The correct Legendre transform is replaced by the "q–Legendre" transform: $$(\mathcal{L}_s f)(x) = \inf_{y: f(y)>0} \frac{(1-\langle x, y\rangle/s)_+^s}{f(y)}$$ As $q\to1$ ($s\to\infty$), the classical Legendre structure is recovered. All thermodynamic potentials—free energy, Massieu function, partition sums—are built by replacing exponentials and logarithms with their q-deformed counterparts (Kalogeropoulos, 2017).

Consequently, all thermodynamic equations of state, Maxwell relations, and ensemble averages reflect the underlying pseudo-additive structure, and ensemble equivalence, convexity, and thermodynamic stability must be re-examined in this framework.

5. Structural and Stability Properties

The Tsallis entropy $S_q$ is:

• Concave for $q>0$; convex for $q<0$ (Kalogeropoulos, 2016).
• Lesche-stable for all $q>0$—small perturbations in probability distributions result in small changes in entropy (normalized by its maximum) (Kalogeropoulos, 2016).
• Thermodynamically consistent: Both super- and sub-additive regimes satisfy the third law when the full "completed" deformed logarithms and exponentials ($q$ and $2-q$ branches) are considered; escort averaging becomes unnecessary (Bagci et al., 2016).

The pseudo-additive chain rule, $S_q(A\cup B)=S_q(A)\oplus_q S_q(B\mid A)$, holds for both independent and dependent systems when conditional entropy is defined via escort averages. However, uniqueness of $S_q$ is only guaranteed upon fixing the conditional mean prescription; alternative functionals (Sharma–Mittal, Jizba–Arimitsu) may share the pseudo-additive law but differ in operational and stability properties (Jizba et al., 2017).

6. Physical and Statistical Manifestations

Pseudo-additivity, as encoded by the $q$-sum, induces explicit correlations in equilibrium and transport phenomena:

• Microscopic Justification: In small or finite systems the phase-space measure and energy/momentum constraints enforce non-product forms for marginal densities. This induces q-exponential distributions and non-trivial correlations, with the strength of non-additivity set by $N$ or generalized susceptibilities (e.g., reservoir heat capacity) (Megias et al., 2022).
• Transport Equations: The non-extensive Boltzmann equation acquires a modified source term reflecting Tsallis non-additivity, reducing to the classical collision kernel in the additive limit (Megias et al., 2022).
• Partition Function and Mean Energy: For a q-monoatomic gas, the partition function and $\langle U\rangle$ can develop divergences, negative heat capacities, or clustering (quartet) states, all arising as statistical artefacts of the non-additive regime (Plastino et al., 2017).
• Quantum Domains: Tsallis non-additivity propagates to quantum mechanics, affecting propagators, wavefunctions (power-law q-exponentials replace Gaussians), and uncertainty relations. Tsallis q-coherent states feature modified variances and uncertainty products that exceed the Heisenberg limit as $q$ departs from unity (Ferri et al., 2015, Bizet et al., 2019).

7. Applications, Generalizations, and Open Problems

Tsallis non-additivity has been invoked to model:

• Long-range or non-local interacting systems,
• Space–plasma velocity distributions, astrophysical phenomena (Kalogeropoulos, 2016),
• Black hole entropy and possible non-additive corrections to area laws,
• Emergent gravity scenarios via q-generalized Ricci curvature and optimal transport,
• Nonlinear quantum equations and memory effects in open quantum systems (Bizet et al., 2019),
• Complexity measures and generalized information theory (Jizba et al., 2017),

Open issues include:

• First-principles derivation of $q$ from microscopic physics in gravity and quantum field theory,
• Laws of "q-thermodynamics" compatible with diffeomorphism invariance,
• Consistent treatment of interacting/coupled systems with different $q$ values (e.g., via superstatistics),
• The role of pseudo-additivity in the holographic principle and black-hole microstate counting (Kalogeropoulos, 2016),
• Precise mapping between operational definitions (conditional entropies, escort distributions) and physical measurables (Jizba et al., 2017).

Tsallis non-additivity thus constitutes not only a generalization of entropy, but a comprehensive deformation of the probabilistic, thermodynamic, and dynamical structure of statistical mechanics, with ramifications across classical, quantum, and gravitational domains. Its mathematical and physical frameworks are still being refined, particularly concerning uniqueness, physical origin, and regime validity.

References:

• (Kalogeropoulos, 2016) Kalogeropoulos, "Non-additive entropies in ... gravity ?"
• (Ferri et al., 2015) Ferri et al., "New mathematics for the non additive Tsallis' scenario"
• (Krisut et al., 24 Nov 2024) "Deriving Tsallis entropy from non-extensive Hamiltonian within a statistical mechanics framework"
• (Megias et al., 2022) "Transport Equation for Small Systems and Nonadditive Entropy"
• (Ván et al., 2012) Biró et al., "Nonadditive thermostatistics and thermodynamics"
• (Kalogeropoulos, 2017) Kalogeropoulos, "The Legendre Transform in Non-additive Thermodynamics and Complexity"
• (Jizba et al., 2017) Jizba & Korbel, "On the uniqueness theorem for pseudo-additive entropies"
• (Korbel, 2017) Korbel, "Rescaling the nonadditivity parameter in Tsallis thermostatistics"
• (Bagci et al., 2016) Bagci & Oikonomou, "Validity of the third law of thermodynamics for the Tsallis entropy"
• (Plastino et al., 2017) Plastino & Rocca, "Hidden correlations entailed by q-non additivity render the q-monoatomic gas highly non trivial"
• (Lima et al., 2020) Lima et al., "Tsallis meets Boltzmann: q-index for a finite ideal gas and its thermodynamic limit"
• (Bizet et al., 2019) "Quantum implications of non-extensive statistics"