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Also at Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA.

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This paper discusses possible phenomenological implications for

High energy

Although there exist conflicting observations, there seems to be wide agreement that two complementary approaches have been quite successful in modeling and analyzing the prehydro (

AdS/CFT duality, in its simplest form, allows one to solve the dynamics of maximal supersymmetric

In these studies, modeling QCD plasma at experimentally relevant temperatures using large-

To be clear, this assumption applies to observables sensitive to typical thermal momentum scales in the plasma and not, for example, to measurements of high transverse momentum particle or jet production.

This evidence-based hypothesis forms the basis for the considerations in our present work.It is remarkable how well viscous hydrodynamics describes HICs. Various hydrodynamics codes achieve a close to perfect agreement with an enormous amount of experimental data in spite of uncomfortably large spatial gradient terms. Considerations that help explain this unexpected success include the distinction between “hydrodynamization” and genuine “thermalization,” and the fact that hydrodynamics has attractor properties which set in long before true local equilibration is reached

High energy heavy ion experiments have generated many surprising experimental observations which call for microscopic explanations. One is the degree of similarity between high multiplicity

Sketch of a peripheral HIC.

Hence typical anti–de Sitter (AdS) shock wave calculations involving smooth initial energy densities are too idealized. Even symmetric

A further phenomenologically relevant aspect is that AdS/CFT models of collisions, in the leading infinite coupling limit, tend to predict surprisingly short hydrodynamization times and equilibration times,

The here cited model treated in

In our earlier work

Over the years many different hydrodynamics codes have been developed, improved, and fine-tuned to describe the experimental data. Their relative advantages and disadvantages are the topic of specialized workshops. We do not want to enter this discussion here. Rather, we will focus on just one relatively recent study

In the following section we briefly review those results of Ref.

In Sec.

The arguments in favor of strong fluctuations in the initial state of HICs are manifold, both theoretical and experimental. In, e.g., Ref.

Different models vary in their assumptions, including those which concern fluctuations. We will follow Ref.

Extending the AdS treatment to include realistic fluctuations is somewhat subtle because it relates to basic questions of what is exactly meant by “decoherence” and thermalization. While the fundamental T invariance of QCD seems to imply the absence of any decoherence, this is no longer true if specific probes of only limited spatial extent are considered. All standard observables for high energy heavy ion collisions do exactly that, probing only transverse scales which are much smaller than the nuclear radii, be it

The description of detailed properties of collisions of highly nonuniform nuclei by viscous relativistic hydrodynamics has been the topic of many careful and interesting investigations, far too many to review in this short paper. Let us only mention Ref.

One of the standard procedures, also adopted here, is to describe all collisions by means of a “nuclear thickness function”

When we refer to pixels we mean independent transverse areas with a radius of order

The physical idea behind the thickness function is that due to length contraction and time dilation partons in the colliding nuclei are coherent in the longitudinal direction but incoherent in transverse distance beyond a characteristic length scale which is typically chosen as the inverse saturation scale

This dependence of the initial entropy density on the geometric mean of the thickness functions

Explicitly,

To connect this holographic result for hydrodynamization time to the model

The probability distribution

The median of the distribution for

This simple model predicts that fluctuations in the transverse plane lead to large non-Gaussian fluctuations in the hydrodynamization time. For an individual collision of two pixels it doubles, on average, the hydrodynamization time, thereby converting the typical timescale of

One original motivation for our investigation was the observation that, if hydrodynamization would occur significantly more slowly in the asymmetric fringe regions (the orange areas in Fig.

Sketch of a peripheral heavy ion collision.

The resulting geometric mean of the scale parameter is

The function

Let us add that it follows also from the geometric mean of the energy scales that the difference between smooth

It is possible to extend the motivation for the present contribution to a grander scale. Decoherence, entropy production, and hydrodynamization or thermalization are intensely discussed also in other fields like quantum gravity and quantum computing. AdS/CFT duality has the potential to connect all of these fields. It was established in recent years that there exists an intimate connection to quantum error correction schemes while by construction AdS/CFT combines quantum gravity and quantum field theory.

In principle, the connection to QCD and HICs opens the very attractive possibility for experimental tests of theoretical predictions because the number of final state hadrons per rapidity interval

As entropy production is equivalent to information loss, this discussion centers on the question in which sense information can get lost under unitary time evolution and how this potential information loss in the boundary theory is related to the generation of the Bekenstein-Hawking entropy of the formed large black branes in the (

There exists a fundamental difference with respect to ergodic properties in the relation between

Here we want to address only one aspect of this extensive discussion to which our calculations may add some insight. For classical ergodic theories the coarse-grained entropy grows linearly in time with a rate given by the Kolmogorov-Sinai entropy

Quantum chaos can be described quantitatively in terms of exponential growth of out-of-time-order correlators (OTOCs)

As pointed out by the authors of

It is indisputable that classical Yang-Mills theories show classical chaotic behavior, i.e., after an initial phase, which depends on the chosen initial conditions, a period of linear growth of the coarse-grained entropy sets in, followed by saturation at the thermal equilibrium value. Numerical studies of classical Yang-Mills theories showing this behavior can be found, e.g., in Ref.

There exist many more publications worth mentioning in this context. A very recent example dealing with QCD is, e.g.,

Numerical

The event horizon is not suitable as it depends on the entire future history.

Figure 3 in Ref.Shock wave collisions in

Strictly speaking this relation is only valid in the equilibrium case. As in e.g.,

The entropy

The analogous plot as shown in Fig.

Comparison of the symmetric (red, dashed, Fig.

For large enough times the growth of the area of the apparent horizon is close to linear, as expected, with a slight superimposed oscillation which probably averages out over sufficiently long time periods.

The small wiggling around linear growth seen in Figs.

Thus, at large times linear entropy growth does not only seem to connect classical and quantum chaos for quantum field theories but also their holographic dual description. Obviously, these similarities could well be accidental at our present stage of understanding, but they are sufficiently intriguing to warrant further research.

In this paper we have argued that detailed holographic calculations for asymmetric collisions are highly relevant for any quantitative description of realistic HICs. We have performed such calculations and have found the following:

The characteristic large fluctuations in transverse energy and entropy densities, which are required in hydrodynamic descriptions to explain the observed large event-by-event fluctuations of flow observables, delay hydrodynamization and equilibration in the holographic description so strongly that they are still significant at time of

In contrast, the effect of the transverse dependence of energy densities in peripheral collisions has only a minor impact on the hydrodynamization time, such that it is well motivated to initialize hydrodynamics for the entire system simultaneously.

The long time linear growth of the apparent horizon is very similar for symmetric and asymmetric collisions which supports its interpretation as entropy.

B. M. acknowledges support from the U. S. Department of Energy Award No. DE-FG02-05ER41367. L. Y. acknowledges support from the U.S. Department of Energy Award No. DE-SC0011637. S. W. acknowledges support from the Elite Network of Bavaria in form of a Research Scholarship Grant.