Tsallis-Thermometer in Non-Extensive Thermodynamics

• Tsallis-Thermometer is a framework that defines an effective temperature based on Tsallis statistics, capturing both thermal and non-thermal contributions.
• It employs non-linear fitting of particle spectra to extract key parameters, including the non-extensivity parameter q and the effective temperature T.
• The method is validated in high-energy collision experiments, offering a practical tool for probing local thermal properties and potential QGP signals.

The Tsallis-thermometer is a generalized framework for extracting and interpreting temperature-like parameters in systems governed by Tsallis statistics, a non-extensive generalization of Boltzmann–Gibbs statistical mechanics. It provides a procedure for mapping the emergent features of particle spectra or occupation-level distributions to a well-defined "effective temperature" $T$ in systems exhibiting non-equilibrium phenomena, intrinsic correlations, or event-by-event fluctuations. The Tsallis-thermometer paradigm enables the uniform characterization of thermal and non-thermal contributions in relativistic nuclear collisions and other complex systems, serving as both a diagnostic and a phenomenological tool for probing local and global properties of the underlying statistical ensemble.

1. Foundations: Tsallis Statistics and Physical Temperature

Tsallis entropy, defined as $S_q = (\sum_i p_i^q - 1)/(1-q)$ for $q\neq1$, generalizes Boltzmann–Gibbs by introducing the non-extensivity parameter $q$. The Tsallis canonical ensemble uses constraints of normalization and the usual expectation value of energy to yield the probability distribution

$$P_i \propto \left[1 - (1-q)\beta(E_i-U_q)\right]^{1/(1-q)},$$

where $\beta$ is the Lagrange multiplier associated with the energy constraint and $U_q$ is the $q$-expectation value of energy (Ishihara, 3 Aug 2025). The physical (equilibrium) temperature is defined via the generalized thermodynamic relation

$$\frac{1}{T_{\text{phys}}} = \frac{1}{1+(1-q)S_q}\,\frac{\partial S_q}{\partial U_q} = \frac{\beta}{R},\quad R \equiv 1+(1-q)S_q.$$

This modified temperature definition ensures that a composite system in equilibrium admits a physical temperature which reduces to its Boltzmann–Gibbs counterpart in the $q\to1$ limit and restores standard relations such as $\langle E\rangle = CT$ with $C$ the heat capacity (Ishihara, 3 Aug 2025, Ishihara, 21 Jul 2025).

2. Construction of the Tsallis-Thermometer

The operational realization of the Tsallis-thermometer involves:

• Measurement of single-particle spectra (momentum or energy distributions).
• Fitting the measured distribution to the Tsallis form with free parameters $q$ and $\beta=1/T$:

$$f_q(p) \propto \left[1 - (1-q)\beta \frac{p^2}{2m}\right]^{1/(1-q)}\quad \text{(non-relativistic)},$$

or the relevant relativistic/hadronic form, e.g.,

$$\frac{d^2N}{dp_T\,dy} \Big|_{y=0} \propto p_T\,m_T \left[1+(q-1)\frac{m_T}{T}\right]^{-q/(q-1)}.$$

• Extraction of $T$ and $q$ via non-linear least squares or $\chi^2$-minimization.
• Calibration using global properties: for instance, the mean kinetic energy $\langle p^2/(2m)\rangle$ should return $(3/2)T$ (Ishihara, 3 Aug 2025, Bhattacharyya et al., 2016).
• Verification that the inferred $T$ is stable across particle species, event classes, and pseudorapidity intervals, supporting the interpretation of $T$ as a physical or "thermometer" temperature (Azmi et al., 2013, Cleymans, 2012, Marques et al., 2015, Cleymans et al., 2015).

This approach is applicable to quantum (bosonic/fermionic) as well as classical distributions and supports direct analytic computation of thermodynamic observables via Tsallis-modified partition functions (Hussein et al., 2022, Bhattacharyya et al., 2016).

3. Analytical Properties, Constraints, and Calibration

Analytical solutions exist for the normalization and moments of the Tsallis distribution. For a system of $N$ particles, the normalization factor and the convergence of moments demand restrictions on $q$: $1-\frac{1}{3N/2+1} < q < 1\ \text{(for %%%%21%%%% free particles)},$ and

$0 \le q-1 < \frac{1}{3}\ \text{(massive, finite %%%%22%%%%: convergence of thermodynamic potentials)} [2508.01609, 1608.08965].$

For massive systems and quantum corrections, explicit formulae for pressure, number density, and energy density in the Tsallis framework are provided, including closed expressions for both the massless and massive regimes (Hussein et al., 2022). In the high-energy context, the raw Tsallis temperature $T_q$ systematically underestimates the true thermodynamic temperature for $q>1$; thus, calibration against a reference (Boltzmann–Gibbs) ensemble or through matching to known equation-of-state benchmarks (e.g., energy density from lattice QCD) is essential (Hussein et al., 2022, Bhattacharyya et al., 2016).

4. Phenomenological Applications in High-Energy Collisions

The Tsallis-thermometer is widely deployed in high-energy nuclear and hadronic collisions (pp, pPb, AA) for reconstructing effective temperatures from measured transverse momentum ($p_T$) spectra of identified hadrons. Fit results consistently demonstrate that:

• A universal Tsallis temperature, typically $T\simeq 68$–$75$ MeV for LHC pp and pPb collisions, is obtained across species and energies, validating the thermometer-like role of $T$ (Marques et al., 2015, Azmi et al., 2015, Cleymans et al., 2015, Cleymans, 2012, Azmi et al., 2013, Gyulai et al., 7 Jan 2026).
• The non-extensivity parameter $q$ encodes deviations from equilibrium, with $q\in[1.10,1.17]$ characteristic of LHC hadronic systems.
• The temperature parameter rises monotonically with event multiplicity and, for given particle species, is proportional to the average transverse momentum, $T\simeq 4.26\,\langle p_T\rangle$ (Gyulai et al., 7 Jan 2026).
• $q$ is particularly sensitive to event-shape selection (jet-like vs. isotropic events), allowing the Tsallis-thermometer to serve as a probe of the interplay between soft and hard QCD processes (Gyulai et al., 7 Jan 2026).

5. Thermodynamic Consistency and Fluctuations

Tsallis-thermodynamics requires rigorous verification of thermodynamic identities:

• The generalization of the first law, $dU_q = T_q\,dS' + \sum_i X_i dY_i$, where $S'$ is an additive entropic variable derived from $S_q$ (Ishihara, 2021).
• The heat capacity and fluctuations in $T_q$ are characterized by $q$-dependent corrections:

$$\left\langle (\delta T_q)^2 \right\rangle = T_q^2 \left\{ \frac{1}{C_{qV} + 2(q-1) + (q-1)^2 C_{qP}} \right\},$$

where $C_{qV}, C_{qP}$ are heat capacities at constant volume and pressure. For $q>1$, temperature fluctuations increase, imposing practical considerations for thermometer material selection and calibration (Ishihara, 2021).

6. Role as a QGP Indicator and Systematic Features

The "Tsallis-thermometer" maps phases of QCD matter. By plotting $(T, q)$ for various collision systems on the $T$–$(q-1)$ plane, collisions lying within the window

$$T_{\rm eq}=0.144\pm0.010~{\rm GeV}, \quad q_{\rm eq}=1.156\pm0.007,$$

are identified as most likely to have produced a deconfined quark–gluon plasma (QGP) (Bíró et al., 2020). The migration of $(T, q)$ into this window, even for high-multiplicity small systems, signals QGP-like behavior and provides a phenomenological criterion for deconfinement. Benchmark multiplicity thresholds for reaching this region are reported for pp, pPb, and PbPb collisions (Bíró et al., 2020).

7. Practical Caveats, Limitations, and Calibration

Several caveats must be emphasized:

• The Tsallis effective temperature $T$, while robustly extractable, incorporates contributions from collective flow and intrinsic correlations as encoded in $q$, and cannot be naively equated with the local kinetic or chemical freeze-out temperature of equilibrium models (Zhang et al., 31 May 2025).
• Empirically, $T$ and standard effective temperatures (from Boltzmann, Bose–Einstein, Fermi–Dirac fits) are linearly related but differ in normalization and interpretation. The Tsallis fit provides a compact two-parameter thermometer, but disentangling purely kinetic from non-thermal effects requires additional cross-comparisons (Zhang et al., 31 May 2025).
• Systematic uncertainties arise from fit procedure, acceptance, feed-down corrections, and the finite $p_T$ range (Bhattacharyya et al., 2016). Analytical convergence and thermodynamic consistency are maintained only within specified $q$-intervals—typically $q<4/3$ for convergence of thermodynamic integrals (Hussein et al., 2022, Bhattacharyya et al., 2016).

References Table: Representative Tsallis-Thermometer Results

Paper (arXiv ID)	System	$T$ (MeV)	$q$ (~)	Additional Notes
(Cleymans, 2012)	LHC (pp, pPb)	68–75	1.11–1.16	Universal $T$ across species
(Marques et al., 2015)	LHC (pp)	68±5	1.146	43 spectra: $T$ universal
(Azmi et al., 2015)	LHC (pp)	73–76	1.13–1.15	Stable $T$ over $p_T$ and energy
(Azmi et al., 2013)	ALICE (pPb)	112	1.139	$T$, $q$ stable across $\eta$
(Gyulai et al., 7 Jan 2026)	ALICE (pp, 13TeV)	90–350 (species)	1.09–1.19	$T\sim 4.26\,\langle p_T\rangle$
(Bíró et al., 2020)	Multi-system	144$\pm$10	1.156	QGP indicator window in $T$–$q$ space
(Zhang et al., 31 May 2025)	RHIC (dAu, pp)	95–240	1.06–1.17	Linear map to quantum-statistical $T$
(Ishihara, 3 Aug 2025)	Free particles	$T$ from fit	$q<1$	Strict $q$ bound, energy–$T$ relation

Summary

The Tsallis-thermometer unifies the extraction and interpretation of temperature-like parameters in non-extensive systems through robust fitting of power-law distributions, analytic calibration, and connection to thermodynamic consistency. Its phenomenological power is demonstrated in high-energy collision systems, both as an operational tool for extracting effective temperatures and as a QGP phase indicator. Careful theoretical and empirical calibration is required to anchor the Tsallis temperature to first-principles quantities, but the framework offers a coherent and widely adopted extension of traditional thermodynamics to complex, correlated, and fluctuating media.