Intermittency in second-order phase transitions

Abstract

It is shown that self-similar behavior in multiplicity fluctuations exists in the Ginzburg-Landau description of second-order phase transitions. Furthermore, there exists a numerical exponent that characterizes the intermittency properties in the hadronic phase and is independent of the specific values of the coefficients in the Ginzburg-Landau potential. Current data on intermittency are only 2a away from the critical exponent.PACS numbers: 25.75.+r, 05.70.Fh, 24.60.Lz, 24.85.+p In a high-energy nuclear collision, which is the only feasible way in the laboratory to possibly create a thermalized quark phase, the remnants of the phase transition to hadrons would be copious and readily measurable. The usual connection between correlation functions and the Ginzburg-Landau description of phase transitions in conventional statistical physics [1] has been applied to the study of multiparticle final state a long time ago [2], and were revived more recently in consonance with the development of interest in multiparticle production [3][4][5]. The introduction of intermittency to particle physics [6] has at the same time stimulated considerable activities in the study of self-similarity behavior of multiplicity fluctuations in varying sizes of resolution cells [7,8]. In relating intermittency to the quark-hadron phase transition there is so far only a speculation on the behavior of the intermittency index [9], based partly on the results of studies of the Ising model [10,11]. In this paper we integrate the various concepts mentioned above and determine the properties of intermittency in the Ginzburg-Landau (GL) theory. What emerges is a critical exponent that is independent of the precise values of the coefficients in the GL potential, so long as they allow the hadronic phase to develop.To simplify our problem let us focus our attention on one small cell in phase space of size £, ignoring all other parts of the phase space. We shall take the variables to be the rapidity y and transverse momentum p^, and denote them collectively by z; thus 8 represents 8y8pr. Let the number of particles observed in 8 in an event be AZ, and let the multiplicity distribution in n after many events be P n (8). The scaled factorial moments [6] are defined bywhere ( • • • > denotes (vertical) averaging with weight P n (8). Intermittency refers to the power-law behavior F q (8)oc8(2) over a range of small 8. There exists abundant evidence for si [12].for such behavior in e + e , up, pp, pA, and AA collisions If there is no dynamical contribution to the multiplicity fluctuation, such as due to a phase transition or any other production mechanism, P n should be just the Poisson distribution, P®, reflecting statistical fluctuation only. In that case F q would have no dependence on («), and therefore on 8. Thus our interest is in the deviation of P n from P®. It suggests the use of the coherent-state representation, since the multiplicity distribution of a pure coherent state \</>) is Poissonian, i.e., \(n\<f>)\ 2: =Pn, withwhere the property a(z)...

KNO scaling 30 years later

“…In recent years considerable interest has been devoted to the Ginzburg-Landau theory of the phase transition from quark-gluon plasma to hadronic matter [13,14]. Instead of a strict powerlaw scaling F q ∝ δ −ϕq of the normalized factorial moments, a so-called F -scaling behavior…”

Factorial moments of continuous order

“…However, the more familiar arena for comparison involves the factorial moments, which are more closely related to the data. An interesting side remark that can be made here is that the feasibility of determining D(t) from the data makes possible an experimental look at such theoretical quantity as the Landau free energy if the data on hadronic multiplicity distribution correspond to quark-hadron phase transition [13], or if the data are on photon distribution at the threshold of lasing in quantum optics [14]. That is because in such problems…”

“…The multifractal analysis described above cannot be applied then. However, it has been found phenomenologically [17,18,19] as well as theoretically [13,20], not only in hadronic and nuclear collisions, but also in quantum optics [14], that F q satisfies a different scaling law…”

Preequilibrium evolution of a non-Abelian plasma

“…It is also generally believed that the transition is of the first order, although more recent lattice studies [3] and those of intermittency [4] suggest that this belief needs more substantiation.…”