Thermodynamically Consistent Tsallis Distribution

• Thermodynamically Consistent Tsallis Distribution is a nonextensive statistical framework ensuring normalized probabilities and invariant energy shifts while preserving the Legendre structure and Maxwell relations.
• It employs q-weighted expectations and multiple canonical formalisms (Tsallis-1, -3, q-dual versus Tsallis-2) to reconcile statistical mechanics with observed high-energy particle spectra.
• The formulation is validated through precise phenomenological fits in p–p, p–Pb, and heavy-ion collisions, providing robust thermodynamic parameters and consistent free energy transformations.

Thermodynamically consistent Tsallis distribution denotes a class of nonextensive equilibrium distributions for which the probabilistic and thermodynamic structures remain compatible with equilibrium statistical mechanics. In the canonical-ensemble literature, one decisive criterion is invariance of the equilibrium probabilities under a uniform translation of the energy spectrum, $E_i \to E_i+\Delta$, at fixed temperature; in high-energy phenomenology, the same expression usually refers to the $q$-weighted single-particle spectrum whose thermodynamic integrals satisfy Euler, Gibbs–Duhem, and Maxwell relations and therefore require the spectral exponent $-q/(q-1)$ rather than $-1/(q-1)$ (Parvan, 2021, Cleymans, 2012).

1. Entropic setting and competing canonical formalisms

The common starting point is the Tsallis entropy

$$S_q=\frac{1-\sum_i p_i^q}{q-1} =\frac{\sum_i p_i^q-1}{1-q}, \qquad \sum_i p_i=1.$$

A second central object is the escort probability used in one major formulation,

$P_i^{(q)}=\frac{p_i^q}{\sum_j p_j^q}.$

A related construction is the $q$-dual entropy,

$S=\frac{q}{q-1}\sum_i\left(p_i^{1/q}-p_i\right),$

obtained from Tsallis entropy under the multiplicative transformation $q\to 1/q$ (Parvan, 2021).

The canonical literature distinguishes several nonextensive prescriptions according to the expectation-value rule. Their thermodynamic status is not the same.

Formalism	Expectation values	Status under $E_i\to E_i+\Delta$
Boltzmann–Gibbs	$q$0	Invariant
Tsallis-1	$q$1	Invariant
Tsallis-2	$q$2	Not invariant
Tsallis-3	$q$3	Invariant
$q$4-dual	$q$5	Invariant

In the notation of one review, these appear as Type I, Type II, and Type III Tsallis statistics, with Type III using escort averages. That review concludes that only Type III is consistent with the fundamental hypothesis of equilibrium statistical mechanics derived from the microcanonical ensemble (Kapusta, 2021). A separate canonical analysis instead proves invariance for Tsallis-1, Tsallis-3, and $q$6-dual statistics, while excluding Tsallis-2 (Parvan, 2021). This suggests that the nomenclature and the operative definition of consistency are not uniform across the literature.

2. Uniform energy translation and the equilibrium criterion

In equilibrium thermodynamics, only energy differences are physically meaningful. Accordingly, if the entire spectrum is shifted as

$q$7

with $q$8 fixed, the normalized canonical probabilities should remain unchanged, while the free energy should transform as $q$9. This is the criterion used to test thermodynamic consistency in the canonical ensemble (Parvan, 2021).

For Boltzmann–Gibbs statistics,

$-q/(q-1)$0

and the shift produces $-q/(q-1)$1, so $-q/(q-1)$2 and $-q/(q-1)$3.

For Tsallis-1, extremization gives

$-q/(q-1)$4

with normalization fixing $-q/(q-1)$5. Under $-q/(q-1)$6, one has $-q/(q-1)$7, so the bracket is unchanged and therefore $-q/(q-1)$8. The same logic yields invariance for Tsallis-3 and for the $-q/(q-1)$9-dual statistics. In all of these cases the free energy transforms as $-1/(q-1)$0 (Parvan, 2021).

Tsallis-2 fails precisely at this point. Its expectation-value rule is

$-1/(q-1)$1

so even for the constant observable one finds

$-1/(q-1)$2

in general, despite $-1/(q-1)$3. If one assumes $-1/(q-1)$4 under $-1/(q-1)$5, then

$-1/(q-1)$6

and the shifted probabilities do not reduce to the unshifted ones because the extra factor $-1/(q-1)$7 cannot be absorbed at fixed $-1/(q-1)$8. The noninvariance is thus the thermodynamic manifestation of the inconsistency between the expectation-value postulate and probability normalization (Parvan, 2021).

All of these $-1/(q-1)$9-exponential forms impose positivity conditions on the bracketed argument. Depending on $$S_q=\frac{1-\sum_i p_i^q}{q-1} =\frac{\sum_i p_i^q-1}{1-q}, \qquad \sum_i p_i=1.$$0 and $$S_q=\frac{1-\sum_i p_i^q}{q-1} =\frac{\sum_i p_i^q-1}{1-q}, \qquad \sum_i p_i=1.$$1, these conditions may imply finite-energy cutoffs. The presence of cutoffs does not spoil the invariance proofs, but it restricts the admissible state space (Parvan, 2021).

3. Legendre structure, physical temperature, and quantum generalizations

In relativistic quantum formulations, thermodynamic consistency can be implemented directly at the level of occupation numbers. Maximizing Tsallis-Fermi–Dirac or Tsallis-Bose–Einstein entropies with $$S_q=\frac{1-\sum_i p_i^q}{q-1} =\frac{\sum_i p_i^q-1}{1-q}, \qquad \sum_i p_i=1.$$2-weighted constraints

$$S_q=\frac{1-\sum_i p_i^q}{q-1} =\frac{\sum_i p_i^q-1}{1-q}, \qquad \sum_i p_i=1.$$3

yields

$$S_q=\frac{1-\sum_i p_i^q}{q-1} =\frac{\sum_i p_i^q-1}{1-q}, \qquad \sum_i p_i=1.$$4

and, in the Boltzmann limit,

$$S_q=\frac{1-\sum_i p_i^q}{q-1} =\frac{\sum_i p_i^q-1}{1-q}, \qquad \sum_i p_i=1.$$5

Because particle number and energy are built from $$S_q=\frac{1-\sum_i p_i^q}{q-1} =\frac{\sum_i p_i^q-1}{1-q}, \qquad \sum_i p_i=1.$$6, the explicit checks

$$S_q=\frac{1-\sum_i p_i^q}{q-1} =\frac{\sum_i p_i^q-1}{1-q}, \qquad \sum_i p_i=1.$$7

can be carried out, establishing the relevant Maxwell relations and the Legendre structure (Cleymans et al., 2011).

A different but complementary route introduces the additive entropic variable

$$S_q=\frac{1-\sum_i p_i^q}{q-1} =\frac{\sum_i p_i^q-1}{1-q}, \qquad \sum_i p_i=1.$$8

which is equivalent to Rényi entropy. With $$S_q=\frac{1-\sum_i p_i^q}{q-1} =\frac{\sum_i p_i^q-1}{1-q}, \qquad \sum_i p_i=1.$$9 as the entropy-like state variable, the first law takes the standard form

$P_i^{(q)}=\frac{p_i^q}{\sum_j p_j^q}.$0

and the conventional thermodynamic potentials follow by ordinary Legendre transforms. Within that framework, the mean squares of the fluctuations of the physical quantities are the same as those in the conventional statistics, whereas the fluctuations of the Tsallis entropy and of the Tsallis temperature carry explicit $P_i^{(q)}=\frac{p_i^q}{\sum_j p_j^q}.$1-dependent terms (Ishihara, 2021).

These developments show that “thermodynamic consistency” is not reducible to one algebraic identity. It usually means preservation of the full equilibrium structure: normalized probabilities, well-defined intensive variables, correct derivative relations, and a coherent Legendre transform network.

4. The high-energy form: $P_i^{(q)}=\frac{p_i^q}{\sum_j p_j^q}.$2 integrals and the $P_i^{(q)}=\frac{p_i^q}{\sum_j p_j^q}.$3 spectrum

In nuclear and particle phenomenology, the thermodynamically consistent Tsallis distribution is usually written in terms of the basic factor

$P_i^{(q)}=\frac{p_i^q}{\sum_j p_j^q}.$4

The crucial point is that the thermodynamic quantities are not built from $P_i^{(q)}=\frac{p_i^q}{\sum_j p_j^q}.$5 but from $P_i^{(q)}=\frac{p_i^q}{\sum_j p_j^q}.$6:

$P_i^{(q)}=\frac{p_i^q}{\sum_j p_j^q}.$7

$P_i^{(q)}=\frac{p_i^q}{\sum_j p_j^q}.$8

$P_i^{(q)}=\frac{p_i^q}{\sum_j p_j^q}.$9

With these definitions one has

$q$0

as well as the Maxwell relations

$q$1

(Azmi et al., 2013).

Because $q$2 is an integral over $q$3, the observable momentum distribution carries an extra power of $q$4:

$q$5

In rapidity and transverse variables,

$q$6

and at mid-rapidity with $q$7,

$q$8

This is the form repeatedly used in LHC analyses (Cleymans, 2012).

The distinction from older “Tsallis-like” parametrizations is exact, not cosmetic. A commonly used alternative employs the exponent $q$9 directly in the spectrum. In the consistent construction, that choice is incompatible with the thermodynamic integrals because it drops the extra power of $S=\frac{q}{q-1}\sum_i\left(p_i^{1/q}-p_i\right),$0 generated by the $S=\frac{q}{q-1}\sum_i\left(p_i^{1/q}-p_i\right),$1 averages. The consistent form also contains the explicit $S=\frac{q}{q-1}\sum_i\left(p_i^{1/q}-p_i\right),$2 Jacobian factor. In the language of the LHC notes, “without the extra power of $S=\frac{q}{q-1}\sum_i\left(p_i^{1/q}-p_i\right),$3” thermodynamic consistency would not be achieved (Cleymans, 2012).

A further structural issue concerns $S=\frac{q}{q-1}\sum_i\left(p_i^{1/q}-p_i\right),$4 scaling. The consistent Tsallis form depends on $S=\frac{q}{q-1}\sum_i\left(p_i^{1/q}-p_i\right),$5 in its argument, whereas some widely used Lévy–Tsallis mappings introduce the rest mass into the scale parameter and thereby break the $S=\frac{q}{q-1}\sum_i\left(p_i^{1/q}-p_i\right),$6 scaling present in the thermodynamic construction (Cleymans, 2012).

5. Phenomenological performance in proton, proton–nucleus, and nucleus–nucleus data

The consistent form has been applied extensively to LHC transverse-momentum spectra. For identified hadrons in $S=\frac{q}{q-1}\sum_i\left(p_i^{1/q}-p_i\right),$7–$S=\frac{q}{q-1}\sum_i\left(p_i^{1/q}-p_i\right),$8 collisions at $S=\frac{q}{q-1}\sum_i\left(p_i^{1/q}-p_i\right),$9 GeV, a common Tsallis temperature and nonextensivity parameter describe several particle species with

$q\to 1/q$0

while the species-resolved fits remain compatible with that scale; for example, $q\to 1/q$1 gives $q\to 1/q$2 and $q\to 1/q$3 GeV (Cleymans, 2012).

System	Representative result	Phenomenological note
$q\to 1/q$4–$q\to 1/q$5, $q\to 1/q$6 GeV, identified hadrons	$q\to 1/q$7 MeV, $q\to 1/q$8	Same $q\to 1/q$9 and $E_i\to E_i+\Delta$0 describe several species
$E_i\to E_i+\Delta$1–Pb, $E_i\to E_i+\Delta$2 TeV, charged particles	$E_i\to E_i+\Delta$3, $E_i\to E_i+\Delta$4 MeV, $E_i\to E_i+\Delta$5 fm for $E_i\to E_i+\Delta$6	No significant pseudorapidity dependence of $E_i\to E_i+\Delta$7, $E_i\to E_i+\Delta$8, or $E_i\to E_i+\Delta$9
ATLAS/CMS charged particles, 0.9 and 7 TeV	$q$00–$q$01, $q$02–$q$03 MeV, $q$04–$q$05 fm	Fits extend up to $q$06 GeV/c over 14 orders of magnitude

For $q$07–Pb collisions at $q$08 TeV, the same consistent distribution gives an excellent description of the ALICE spectra in all measured pseudorapidity intervals. The extracted parameters remain nearly constant: $q$09, $q$10 MeV, $q$11 fm for $q$12; $q$13, $q$14 MeV, $q$15 fm for $q$16; and $q$17, $q$18 MeV, $q$19 fm for $q$20 (Azmi et al., 2013).

For inclusive charged particles in $q$21–$q$22 collisions, fits to ATLAS and CMS data at $q$23 and $q$24 TeV report

$q$25

for ATLAS at $q$26 TeV,

$q$27

for ATLAS at $q$28 TeV,

$q$29

for CMS at $q$30 TeV, and

$q$31

for CMS at $q$32 TeV, with fits extending up to $q$33 GeV/c and spanning 14 orders of magnitude in yield (Azmi et al., 2015).

A broader system-size analysis using $q$34+$q$35, $q$36+Pb, Xe+Xe, and Pb+Pb spectra at the LHC uses the same consistent form to extract energy density, pressure, particle density, entropy density, mean free path, Knudsen number, heat capacity, isothermal compressibility, expansion coefficient, and speed of sound at kinetic freeze-out. The rate of increase or decrease of these thermodynamic variables is found to be more rapid in small systems such as $q$37+$q$38 and $q$39+Pb collisions than in large systems such as Xe+Xe and Pb+Pb collisions. The same study reports a nearly linear $q$40 relation with slope

$q$41

and an average

$q$42

together with the observation that high-multiplicity $q$43+$q$44 collisions produce similar thermodynamic conditions as peripheral heavy-ion collisions at kinetic freeze-out (Sharma et al., 2024).

6. First-principles derivations, controversies, and preferred usages

A first-principles route derives generalized equilibrium distributions from Liouville’s theorem, self-similarity of correlations, and the thermodynamic limit. Choosing the compositional correlation map to be the $q$45-product yields the Tsallis distribution as a special case:

$q$46

In that framework the parameter $q$47 characterizes the deformation of the probability-composition law, and the Tsallis distribution is valid for additive Hamiltonians with nontrivial correlations rather than for arbitrary nonadditive energies (Saadatmand et al., 2019).

Another line of work shows that the phenomenological transverse-momentum distribution widely used in high-energy collisions is exactly the zeroth-term approximation of the $q$48-dual statistics in the Maxwell–Boltzmann case. The $q$49-dual formalism uses normalized probabilities and standard averages, so it provides a thermodynamically consistent interpretation of the phenomenological Tsallis fit function without relying on the inconsistent Tsallis-2 expectation rule (Parvan, 2019).

The major controversy concerns which criterion should be decisive. One review argues that only Type III, derived from the microcanonical hypothesis with escort averages, is consistent with the fundamental hypothesis of equilibrium statistical mechanics (Kapusta, 2021). A separate canonical analysis instead identifies invariance under uniform energy translation as the necessary condition and concludes that Boltzmann–Gibbs, Tsallis-1, Tsallis-3, and $q$50-dual statistics satisfy it, whereas Tsallis-2 does not (Parvan, 2021). The difference is substantive: one criterion emphasizes derivability from the microcanonical ensemble, the other emphasizes preservation of canonical Legendre structure under energy shifts.

Within the invariant canonical formulations, Tsallis-1 and $q$51-dual are singled out as the most structurally satisfactory because they reproduce a Boltzmann–Gibbs-like dependence on energy differences and use expectation values consistent with $q$52. Tsallis-3 is invariant and thermodynamically admissible, but its normalization and self-consistency relations entangle $q$53, $q$54, and $q$55, so it does not reproduce the simple Boltzmann–Gibbs structural dependence on $q$56. Tsallis-2 is not recommended for equilibrium canonical ensembles because $q$57 and its probabilities are not invariant under uniform energy shifts (Parvan, 2021).

Specialized many-body applications reinforce the same point from another direction. In the Hamiltonian Mean Field model, maximizing the Tsallis functional at fixed mass and energy yields thermodynamically consistent polytropic distributions, explicit free-energy and entropy functionals, and a canonical tricritical index $q$58 separating first- and second-order phase transitions (Chavanis et al., 2010).

Taken together, these results establish that the thermodynamically consistent Tsallis distribution is not a single universally agreed formula but a family of closely related constructions. Across that family, the durable requirements are normalized probabilities, a valid Legendre structure, correct transformation under $q$59, and, in the high-energy phenomenology literature, the $q$60-based spectrum with exponent $q$61.