QCD Dynamical Entropy Insights

• QCD dynamical entropy is a family of information-theoretic probes that characterize nonperturbative QCD through measures like configurational, relative, and thermal entropies.
• It employs definitions such as Shannon-based configurational entropy, Kullback–Leibler divergence for gluon state evolution, and temperature derivatives from static color sources to analyze confinement and deconfinement.
• Applications of these entropy diagnostics include predicting meson spectra via entropic Regge trajectories, assessing gluon saturation effects, and tracking entropy peaks near the QCD phase transition.

Searching arXiv for recent and foundational papers on QCD dynamical entropy, configurational entropy in AdS/QCD, and related entropy notions in QCD. QCD dynamical entropy denotes a family of entropy constructions used to characterize nonperturbative QCD and QCD-like systems, rather than a single universal observable. In the literature, the term is applied to Shannon-based configurational or conditional entropy of holographic energy-density profiles, Kullback–Leibler–type relative entropy for rapidity evolution of dense gluon states, thermal entropy of static color sources across deconfinement, entanglement entropy in holographic phase diagrams, and information entropy diagnostics of the hadronic spectrum (Bernardini et al., 2016, Peschanski, 2012, Iatrakis et al., 2015, Dudal et al., 2018, Rocha et al., 16 Nov 2025). Across these uses, entropy quantifies how QCD dynamics organizes degrees of freedom: over holographic modes, over gluon transverse momenta, over heavy-quark screening states, or over spectral distributions.

1. Conceptual range and definitions

A common feature of this literature is that entropy is used as an information-theoretic or thermodynamic diagnostic of QCD structure, but the underlying random variable depends on context. In dynamical AdS/QCD, the relevant object is a localized energy density in the holographic coordinate; in small-$x$ physics it is a normalized transverse-momentum distribution extracted from the unintegrated gluon distribution; in thermal heavy-quark observables it is the temperature derivative of a free energy; in spectral analyses it is a Shannon entropy of digit or mass distributions (Bernardini et al., 2016, Ramos et al., 2022, Weber, 2016, Rocha et al., 16 Nov 2025).

Construction	Defining quantity	Typical interpretation
Configurational or conditional entropy	$S=-\int d\omega\,\rho(\omega)\log \rho(\omega)$	Complexity of localized holographic or hadronic configurations
Dynamical entropy of dense gluon states	$$\Sigma^{Y_0\rightarrow Y}=\int d^2k_\perp\,P(Y,k_\perp)\ln \frac{P(Y,k_\perp)}{P(Y_0,k_\perp)}$$	Entropy production under rapidity evolution
Heavy-quark thermal entropy	$S_{Q\bar Q}=-\partial F_{Q\bar Q}/\partial T$, $S_Q=-\partial F_Q/\partial T$	Medium reorganization around static color sources

In the configurational-entropy program, the central statement is that entropy measures the “shape complexity” of a localized physical system. Lower configurational entropy means that fewer momentum modes dominate the profile, so the configuration is more compressed in information-theoretic terms and is interpreted as more ordered, more stable, more abundant, or more likely to be observed (Bernardini et al., 2016, Barbosa-Cendejas et al., 2018). In the dense-gluon program initiated by Peschanski, the entropy is explicitly a relative entropy between two rapidity-dependent probability distributions and is constructed to satisfy positivity and irreversibility properties (Peschanski, 2012, Ramos et al., 2022).

This diversity of definitions rules out identifying QCD dynamical entropy with a single thermodynamic state function. The literature instead uses entropy as a family of probes of confinement, Regge organization, saturation, deconfinement, and spectral structure.

2. Configurational entropy in dynamical AdS/QCD

In dynamical AdS/QCD, the starting point is a five-dimensional Einstein–dilaton model in which the background geometry is dynamically determined rather than imposed by hand. In one formulation, the Einstein-frame action is

$$\mathcal{S}=\int d^{5}x\sqrt{- g }\left[ \frac{1}{2}g^{AB}\nabla_{A} \phi\nabla_{B} \phi+V(\phi )-{R}\right],$$

with conformally AdS metric

$$g_{AB}=e^{-2A(z)}\eta _{AB},\qquad A(z)=\log(z)+\mathring{A}(z),$$

where $\mathring{A}(0)=0$ ensures AdS$_5$ behavior in the UV (Barbosa-Cendejas et al., 2018). A phenomenological dynamical soft-wall warp factor,

$$\mathring{A}(z,S)=f(S)\frac{z^2\Lambda^2}{e^{(1-z\Lambda)}+1},\qquad f(S)=\frac{\sqrt{3}+1}{\sqrt{3}-1+2S},$$

with $S=-\int d\omega\,\rho(\omega)\log \rho(\omega)$0, reproduces linear Regge behavior while matching QCD data (Barbosa-Cendejas et al., 2018).

The holographic meson spectrum is obtained from a Schrödinger-like equation,

$S=-\int d\omega\,\rho(\omega)\log \rho(\omega)$1

with

$S=-\int d\omega\,\rho(\omega)\log \rho(\omega)$2

leading to Regge-like behavior $S=-\int d\omega\,\rho(\omega)\log \rho(\omega)$3 (Barbosa-Cendejas et al., 2018). The entropy is then not computed from the meson wavefunction directly, but from the localized energy density of the background,

$S=-\int d\omega\,\rho(\omega)\log \rho(\omega)$4

Its Fourier transform defines a normalized modal fraction,

$S=-\int d\omega\,\rho(\omega)\log \rho(\omega)$5

and the configurational entropy

$S=-\int d\omega\,\rho(\omega)\log \rho(\omega)$6

In the conditional-entropy variant, the same structure is justified by a lattice-to-continuum limit in which the modal fraction is identified with the normalized structure factor of the warped energy density (Bernardini et al., 2016).

The physical interpretation is explicit. Configurational entropy “measures the entropy of shape of localized physical systems and configurations,” and modes with lower entropy “request less energy to be yielded, being also more abundant, more dominant, and also more probable to be detected and observed” (Barbosa-Cendejas et al., 2018). Applied to light-flavour mesons, the conditional entropy is minimal for low spins and increases monotonically with spin in both UV and IR regimes, providing an information-theoretic basis for the lower phenomenological occurrence of high-spin mesons and a quantitative tool for studying their instability (Bernardini et al., 2016).

A recurrent misconception in this branch of the literature is to read the entropy as a thermal or time-dependent quantity. The papers explicitly exclude that interpretation: no thermal ensemble is introduced, and the entropy is static. It is called “dynamical” because it is attached to dynamically generated Einstein–dilaton or Einstein–dilaton–tachyon backgrounds, and because it depends on dynamical parameters such as spin, asymptotic regime, and large-$S=-\int d\omega\,\rho(\omega)\log \rho(\omega)$7 corrections (Bernardini et al., 2016, Barbosa-Cendejas et al., 2018).

3. Entropic spectroscopy and holographic state classification

A major development is the use of configurational entropy as a spectroscopy tool. In the tachyonic extension of dynamical AdS/QCD, the Einstein–dilaton action is generalized by adding a closed-string tachyon field $S=-\int d\omega\,\rho(\omega)\log \rho(\omega)$8,

$S=-\int d\omega\,\rho(\omega)\log \rho(\omega)$9

For the solution

$$\Sigma^{Y_0\rightarrow Y}=\int d^2k_\perp\,P(Y,k_\perp)\ln \frac{P(Y,k_\perp)}{P(Y_0,k_\perp)}$$0

with $$\Sigma^{Y_0\rightarrow Y}=\int d^2k_\perp\,P(Y,k_\perp)\ln \frac{P(Y,k_\perp)}{P(Y_0,k_\perp)}$$1, the potential is chosen so that the theory remains ghost-free for suitable parameter choice and yields good IR physics (Barbosa-Cendejas et al., 2018). The central entropic result is qualitative and systematic: introducing the bulk tachyon always reduces the configurational entropy at fixed spin, in both UV and IR, for both the dynamical AdS/QCD and the Sakai–Sugimoto-type deformed soft-wall backgrounds. The authors summarize this as: tachyonic bulk corrections produce more dominant and abundant dual mesonic states on the four-dimensional boundary (Barbosa-Cendejas et al., 2018).

This framework also yields “entropic Regge-like trajectories,” namely smooth fits of configurational entropy versus spin. For the dynamical AdS/QCD model, cubic interpolations were obtained in distinct regimes, for example

$$\Sigma^{Y_0\rightarrow Y}=\int d^2k_\perp\,P(Y,k_\perp)\ln \frac{P(Y,k_\perp)}{P(Y_0,k_\perp)}$$2

in the tachyonless IR, and

$$\Sigma^{Y_0\rightarrow Y}=\int d^2k_\perp\,P(Y,k_\perp)\ln \frac{P(Y,k_\perp)}{P(Y_0,k_\perp)}$$3

in the tachyonic UV, with accuracies of about $$\Sigma^{Y_0\rightarrow Y}=\int d^2k_\perp\,P(Y,k_\perp)\ln \frac{P(Y,k_\perp)}{P(Y_0,k_\perp)}$$4–$$\Sigma^{Y_0\rightarrow Y}=\int d^2k_\perp\,P(Y,k_\perp)\ln \frac{P(Y,k_\perp)}{P(Y_0,k_\perp)}$$5 in the spin range considered (Barbosa-Cendejas et al., 2018). The analogy to ordinary Regge theory is explicit: just as $$\Sigma^{Y_0\rightarrow Y}=\int d^2k_\perp\,P(Y,k_\perp)\ln \frac{P(Y,k_\perp)}{P(Y_0,k_\perp)}$$6 organizes hadrons by spin and radial number, configurational entropy organizes them by informational complexity.

The same logic was extended to even-spin and odd-spin glueballs, including pomeron and odderon trajectories, in dynamical AdS/QCD with anomalous dimensions. There the energy density

$$\Sigma^{Y_0\rightarrow Y}=\int d^2k_\perp\,P(Y,k_\perp)\ln \frac{P(Y,k_\perp)}{P(Y_0,k_\perp)}$$7

or its odd-spin analog built from $$\Sigma^{Y_0\rightarrow Y}=\int d^2k_\perp\,P(Y,k_\perp)\ln \frac{P(Y,k_\perp)}{P(Y_0,k_\perp)}$$8 is Fourier transformed, and the resulting configurational entropy is fit as a function of spin and mass. These “configurational-entropic Regge trajectories” were used to predict higher-$$\Sigma^{Y_0\rightarrow Y}=\int d^2k_\perp\,P(Y,k_\perp)\ln \frac{P(Y,k_\perp)}{P(Y_0,k_\perp)}$$9 glueball and pomeron masses and to rank states by configurational stability (Rodrigues et al., 2020, Rodrigues et al., 2020). In the odd-spin sector, the exponential modified dilaton with logarithmic anomalous dimensions was identified as the most suitable choice compatible with lattice QCD, and a hybrid lattice-plus-AdS/QCD paradigm was used to predict masses up to $S_{Q\bar Q}=-\partial F_{Q\bar Q}/\partial T$0 (Rodrigues et al., 2020).

A related but distinct construction is the nuclear configurational entropy of holographic light-front wave functions. For the pion, a cross section $S_{Q\bar Q}=-\partial F_{Q\bar Q}/\partial T$1 derived from the spin-improved holographic light-front wave function is Fourier transformed,

$S_{Q\bar Q}=-\partial F_{Q\bar Q}/\partial T$2

with modal fraction

$S_{Q\bar Q}=-\partial F_{Q\bar Q}/\partial T$3

and entropy

$S_{Q\bar Q}=-\partial F_{Q\bar Q}/\partial T$4

Its critical point optimizes the spin-improvement parameters at $S_{Q\bar Q}=-\partial F_{Q\bar Q}/\partial T$5 and $S_{Q\bar Q}=-\partial F_{Q\bar Q}/\partial T$6, yielding $S_{Q\bar Q}=-\partial F_{Q\bar Q}/\partial T$7 fm, in agreement with the PDG value $S_{Q\bar Q}=-\partial F_{Q\bar Q}/\partial T$8 fm at the $S_{Q\bar Q}=-\partial F_{Q\bar Q}/\partial T$9 level (Karapetyan, 2018).

4. Relative entropy of dense gluon states and heavy-ion initial conditions

In the saturation and Color Glass Condensate literature, QCD dynamical entropy is formulated as a relative entropy in transverse-momentum space. Starting from the unintegrated gluon distribution $S_Q=-\partial F_Q/\partial T$0, one defines the normalized probability density

$S_Q=-\partial F_Q/\partial T$1

and then the entropy for evolution from $S_Q=-\partial F_Q/\partial T$2 to $S_Q=-\partial F_Q/\partial T$3,

$S_Q=-\partial F_Q/\partial T$4

This is mathematically a Kullback–Leibler divergence and is interpreted as entropy production under rapidity evolution of a dense gluon system (Peschanski, 2012, Ramos et al., 2022).

In the original Gaussian CGC model, the probability distribution is controlled by the saturation scale $S_Q=-\partial F_Q/\partial T$5, or equivalently the correlation length $S_Q=-\partial F_Q/\partial T$6, and rapidity evolution is interpreted as a compression process. For that model, the entropy can be computed analytically: $S_Q=-\partial F_Q/\partial T$7 which is positive and vanishes only when the distributions are identical (Ramos et al., 2022). Matching this microscopic expression to Kutak’s macroscopic CGC entropy leads to an effective number of gluonic degrees of freedom per transverse cell $S_Q=-\partial F_Q/\partial T$8 (Peschanski, 2012).

Subsequent work replaced the simple Gaussian ansatz by phenomenological UGDs, including the MPM model with Tsallis-like tails, a Levin–Tuchin-inspired form, and the KS nonlinear UGD. The dynamical entropy then remains a relative entropy between normalized $S_Q=-\partial F_Q/\partial T$9 distributions but becomes sensitive to saturation-region details. Realistic models such as MPM, Levin–Tuchin, and KS yield nearly identical entropy curves, whereas the Gaussian model overestimates the growth with $$\mathcal{S}=\int d^{5}x\sqrt{- g }\left[ \frac{1}{2}g^{AB}\nabla_{A} \phi\nabla_{B} \phi+V(\phi )-{R}\right],$$0 (Ramos et al., 2022).

The total entropy density per rapidity is obtained by multiplying the microscopic entropy by the effective number of gluonic degrees of freedom in the transverse area. For proton–proton collisions this takes the form

$$\mathcal{S}=\int d^{5}x\sqrt{- g }\left[ \frac{1}{2}g^{AB}\nabla_{A} \phi\nabla_{B} \phi+V(\phi )-{R}\right],$$1

with $$\mathcal{S}=\int d^{5}x\sqrt{- g }\left[ \frac{1}{2}g^{AB}\nabla_{A} \phi\nabla_{B} \phi+V(\phi )-{R}\right],$$2 (Ramos et al., 2022). In the rapidity interval relevant for LHC kinematics, realistic models give total dynamical entropy densities of order tens per unit rapidity, comparable to phenomenological final-state entropy estimates from pion spectra (Ramos et al., 2022).

This formalism was extended to nuclei by replacing the proton UGD by a nuclear UGD and using either geometric scaling or a Glauber–Gribov construction. The nuclear entropy is again

$$\mathcal{S}=\int d^{5}x\sqrt{- g }\left[ \frac{1}{2}g^{AB}\nabla_{A} \phi\nabla_{B} \phi+V(\phi )-{R}\right],$$3

but after normalization and geometric scaling the dependence on the nuclear mass number $$\mathcal{S}=\int d^{5}x\sqrt{- g }\left[ \frac{1}{2}g^{AB}\nabla_{A} \phi\nabla_{B} \phi+V(\phi )-{R}\right],$$4 nearly cancels. The result is that the dynamical entropy per saturation cell is almost $$\mathcal{S}=\int d^{5}x\sqrt{- g }\left[ \frac{1}{2}g^{AB}\nabla_{A} \phi\nabla_{B} \phi+V(\phi )-{R}\right],$$5-independent, while the total entropy density scales with the transverse size $$\mathcal{S}=\int d^{5}x\sqrt{- g }\left[ \frac{1}{2}g^{AB}\nabla_{A} \phi\nabla_{B} \phi+V(\phi )-{R}\right],$$6 (Ramos et al., 12 Jul 2025). This is used as a weak-coupling estimate of the initial entropy density in ultra-relativistic heavy-ion collisions and, in the original proposal, as input to AdS/CFT-based thermalization analyses of the glasma-to-hydrodynamics transition (Peschanski, 2012, Ramos et al., 12 Jul 2025).

5. Thermal entropy of static color sources

A thermodynamic notion of QCD dynamical entropy arises in the entropy carried by static heavy color sources in a thermal medium. For a heavy quark–antiquark pair of separation $$\mathcal{S}=\int d^{5}x\sqrt{- g }\left[ \frac{1}{2}g^{AB}\nabla_{A} \phi\nabla_{B} \phi+V(\phi )-{R}\right],$$7, the entropy is defined by

$$\mathcal{S}=\int d^{5}x\sqrt{- g }\left[ \frac{1}{2}g^{AB}\nabla_{A} \phi\nabla_{B} \phi+V(\phi )-{R}\right],$$8

Lattice QCD shows that this entropy has a sharp peak near the deconfinement temperature, suggesting that deconfinement is driven by the increase of the entropy of bound states and that quarkonium dissociation can be understood through an entropic force

$$\mathcal{S}=\int d^{5}x\sqrt{- g }\left[ \frac{1}{2}g^{AB}\nabla_{A} \phi\nabla_{B} \phi+V(\phi )-{R}\right],$$9

when $$g_{AB}=e^{-2A(z)}\eta _{AB},\qquad A(z)=\log(z)+\mathring{A}(z),$$0 grows with separation (Iatrakis et al., 2015).

In Improved Holographic QCD, the free energy of a sufficiently separated heavy pair is represented by a string stretching from the boundary to the horizon, with entropy

$$g_{AB}=e^{-2A(z)}\eta _{AB},\qquad A(z)=\log(z)+\mathring{A}(z),$$1

This entropy vanishes in the confined phase, jumps at deconfinement, peaks near $$g_{AB}=e^{-2A(z)}\eta _{AB},\qquad A(z)=\log(z)+\mathring{A}(z),$$2, and agrees well with lattice data for $$g_{AB}=e^{-2A(z)}\eta _{AB},\qquad A(z)=\log(z)+\mathring{A}(z),$$3 (Iatrakis et al., 2015). Real-time dynamics of the falling string gives an exponential approach to the horizon,

$$g_{AB}=e^{-2A(z)}\eta _{AB},\qquad A(z)=\log(z)+\mathring{A}(z),$$4

and dissociation times below a fermi near deconfinement, supporting entropy-driven destruction as a dominant mechanism in that temperature range (Iatrakis et al., 2015).

A related lattice observable is the entropy of a single static quark,

$$g_{AB}=e^{-2A(z)}\eta _{AB},\qquad A(z)=\log(z)+\mathring{A}(z),$$5

computed in $$g_{AB}=e^{-2A(z)}\eta _{AB},\qquad A(z)=\log(z)+\mathring{A}(z),$$6-flavor QCD with the HISQ action. This entropy is renormalization-scheme independent in the continuum and exhibits a peak at

$$g_{AB}=e^{-2A(z)}\eta _{AB},\qquad A(z)=\log(z)+\mathring{A}(z),$$7

with systematic variations placing it in the range $$g_{AB}=e^{-2A(z)}\eta _{AB},\qquad A(z)=\log(z)+\mathring{A}(z),$$8 (Weber, 2016). The entropy peak lies close to the chiral crossover, whereas the inflection point of the renormalized Polyakov loop is scheme-dependent and occurs at a higher temperature. In this sense, $$g_{AB}=e^{-2A(z)}\eta _{AB},\qquad A(z)=\log(z)+\mathring{A}(z),$$9 provides a cleaner deconfinement diagnostic than the Polyakov-loop inflection point (Weber, 2016).

Dynamical holographic QCD models based on Einstein–Maxwell–dilaton gravity refine the heavy-pair analysis by introducing a temperature-dependent confined-like phase through a small black hole. In that “specious-confined” phase, the heavy-quark entropy builds up strongly toward the critical temperature from below as well as above, and the string tension does not vanish at $\mathring{A}(0)=0$0, in qualitative agreement with lattice QCD (Dudal et al., 2017). This suggests that heavy-quark entropy probes both confinement-like physics and the thermodynamic softening of the medium near deconfinement.

6. Vacuum, entanglement, and spectral entropy

Zero-temperature QCD also supports entropy notions tied to disorder, solitons, and entanglement. A broad review identified four manifestations: entropy associated with disordered condensates such as $\mathring{A}(0)=0$1, vacuum entropy of solitonic ensembles such as center vortices, entanglement entropy of the vacuum when a region is traced out, and configurational entropy of light-particle world-lines and flux tubes (Cornwall, 2012). In this picture, zero-temperature entropy counts the large degeneracy of nonperturbative field configurations compatible with coarse variables, and plays a central role in confinement, dynamical gluon mass generation, and chiral symmetry breaking.

In holographic QCD phase diagrams, entanglement entropy of a strip region is computed by the Ryu–Takayanagi prescription,

$\mathring{A}(0)=0$2

In Einstein–Maxwell–dilaton models this quantity reproduces the standard connected-to-disconnected entanglement transition in a thermal-AdS confining phase, shows no such transition in the deconfined black-hole phase, and exhibits a connected-to-connected swallow-tail transition in the temperature-dependent “specious-confined” phase (Dudal et al., 2018). The associated entropic $\mathring{A}(0)=0$3-function drops sharply at a critical strip length $\mathring{A}(0)=0$4, allowing a phase diagram in the $\mathring{A}(0)=0$5 plane with a critical end point at the deconfinement temperature (Dudal et al., 2018). This does not define a time-dependent dynamical entropy, but it gives an information-theoretic measure that tracks thermodynamic transitions and the scale dependence of effective degrees of freedom.

A distinct spectral entropy notion appears in the analysis of leading digits in the hadronic mass spectrum. If the spectrum were scale invariant, the leading digits would follow Benford’s law,

$\mathring{A}(0)=0$6

with discrete Shannon entropy

$\mathring{A}(0)=0$7

Using PDG hadron masses, the observed digit entropy is smaller, and the entropy deficit

$\mathring{A}(0)=0$8

was found to be about $\mathring{A}(0)=0$9 nat for mesons, $_5$0 nat for baryons, and $_5$1 nat for the combined sample, corresponding to reductions of about $_5$2–$_5$3 relative to the Benford baseline (Rocha et al., 16 Nov 2025). The interpretation is that $_5$4 measures the information-entropy cost associated with the breaking of scale invariance by the emergent dynamical scale $_5$5 (Rocha et al., 16 Nov 2025).

Finally, the confined phase of pure-gauge SU(3) displays a thermodynamic entropy density that is accurately described by a glueball gas only when the known low-lying spectrum is extended by a Hagedorn density of states,

$_5$6

with $_5$7 extracted from the vanishing of the flux-loop mass (0905.4229). This result shows that even standard thermodynamic entropy in confined QCD is controlled by the organization of an exponentially growing spectrum of string-like states.

Taken together, these developments show that QCD dynamical entropy is best understood as a family of entropy-based diagnostics tailored to different sectors of QCD. In holographic spectroscopy it measures configurational complexity of bound states; in dense-gluon physics it measures irreversible reshaping of the small-$_5$8 gluon distribution; in thermal heavy-quark observables it measures medium reorganization near deconfinement; and in vacuum, entanglement, and spectral studies it measures disorder, correlation, or scale breaking. This suggests that entropy in QCD is not a single quantity but a structured language for describing how nonperturbative dynamics organizes information across scales, states, and phases.