Statistical models in heavy-ion physics

The tradition to regard a system of energetic hadrons as a statistical system at a given temperature dates back to Rolf Hagedorn (to the mid sixties). There are, however, also deviations from the classically expected thermodynamics of hot hadronic or quark matter. One view is to claim that equilibrated and non-equilibrated components coexist and their production mechanisms and sources are uncorrelated. In our theoretical investigations we challenge this view: We investigate whether there is a global statistical behavior describing both low- and high-momentum parts of particle spectra.

Interpolating fits between exponential-looking and power-law tailed behavior are known for long. Recently a suggestion occured, that this formula may be derived from a generalization of Boltzmann's entropy formula S = k log w. It is called a "deformed" logarithm. We investigate which dynamical equations may lead to such a stationary state, and how general the required circumstances could be.

Generalized kinetic approach

Most generalizations of classical approaches to the dynamics behind the thermal equilibrium utilize a nonlinear approach, dropping the product formula of probabilities, i.e. not assuming statistical independency for interacting subsystems. Both the Fokker-Planck equation describing a general diffusion process and the Boltzmann equation describing particle collisions have been generalized this way. Traditionally this effect is expected to loose its relative importance for large size systems.

We investigate an other mechanism: dropping the dynamical independence assumption, general corrections to the asymptotic free particle picture are considered. While detecting particle energies according to the free dispersion relation, the physical pair-energy inside a medium may be a complicated function of these values. Instead of the sum of these energies, a more general expression h(E1,E2) may be conserved by the microscopic interactions. This modifies the Boltzmann equation by generalizing the summation of simple energy expressions. In the Langevin or Fokker-Planck approach one likewise considers energy-dependent diffusion and drift coefficients. The result in both cases is a "deformed" exponential for the stationary distribution.

Interesting is, that for an associative composition rule, h(x,h(y,z))=h(h(x,y),z), a mapping to the addition exists: X(h(x,y))=X(x)+X(y). With this the stationary distribution is like f~exp(-X(E)/T), which can be anything. Of course, for h(x,y)=x+y one gets X(t)=t and f~exp(-E/T) as in the classical case. An amusing fact is, that the low-argument expansion of a general composition rule (but satisfying h(x,0)=x and h(0,y)=y) leads to h(x,y)=x+y+axy+... Exactly this rule leads to the cut power-law distribution: X(t)=(1/a)ln(1+at) and therefore f~(1+aE)^{-1/aT}!

Our approach, however, does not suggest to use any non-extensive entropy formula for arriving at such a stationary distribution. The H-theorem is for the integral of -f ln f valid. It is another question, that by inverting the mapping X(t) a non-extensive entropy-density formula can be derived, which is using the above leading order result coincides with the one promoted (but not invented) by C. Tsallis. Only in the non-linear approach to the Boltzmann-equation can an expression other than -f ln f be used (G. Kaniadakis).

Non-extensive thermodynamics?

This question is sometimes treated emotionally even among scientists: does a thermodynamics, based on a non-extensive definition of entropy, make any sense? Are there real physical systems in a long term stationary state behaving unlike the classical thermodynamics teaches us? How should one combine non-extensive quantities when composing bigger systems out of subsystems?

Power-law tailed distributions instead of the well-known Gaussian, or Boltzmann-Gibbs exponential in the energy variable, are surprisingly common in nature. Financial market data, turbulence or high-energy particle spectra in cosmic rays show such behavior. It seems to be a universal class of finite size (or finite time) effects on the well-known canonical distribution. But the usefulness of a non-extensive entropy formula is still questionable. Using a non-extensive entropy and an extensive energy in the S-beta E = max principle is also inconsistent. The Tsallis parameter, q=1-aT, does not seem to be a fundamental quantity: it is rather a property of the piece of matter under investigation. It is an open question what happens when two cut power-law distributions with different power become into contact. We belive that to all such questions only a genuine microcanonical approach has a chance to answer.