Measuring shear viscosity using correlations

a20v373a

Measuring shear viscosity using correlations

Sean Gavin^I; Mohamed Abdel-Aziz^II

^IDepartment of Physics and Astronomy, Wayne State University, Detroit, MI, 48202, USA

^IIInstitut für Theoretische Physik, J.W. Goethe Universität, 60438 Frankfurt am Main, Germany

ABSTRACT

Measurements of transverse momentum fluctuations can be used to determine the shear viscosity [1]. We use current data to estimate the viscosity-to-entropy ratio in the range from 0.08 to 0.3, and discuss how future measurements can reduce this uncertainty.

Key words: Relativistic Heavy Ions; Event-by-event fluctuations

I. INTRODUCTION

Elliptic flow measurements at RHIC are described by ideal viscosity-free hydrodynamics, indicating that the quark-gluon system produced in these collisions is a nearly perfect liquid [2-6]. In particular, the strong suppression of flow due to shear viscosity predicted by weak-coupling transport calculations is not observed [5]. This result is exciting because a small viscosity relative to the entropy density of the system may indicate that the system is more strongly coupled than expected: The collisional shear viscosity is proportional to the mean free path, which is shorter when the coupling is stronger.

But is the viscosity really small? Hirano et al. point out that color glass condensate formation may produce more elliptic flow than considered in refs. [4,5], requiring a larger viscosity for agreement with data [8]. We therefore seek experimental information on viscosity that does not rely on elliptic flow.

We propose that transverse momentum correlation measurements can be used to extract information on the kinematic viscosity,

where h is the shear viscosity, s is the entropy density and T is the temperature. This ratio characterizes the strength of the viscous force relative to the fluid's inertia and, consequently, determines the effect of h on the flow [6]. We argue that viscous diffusion broadens the rapidity dependence of transverse momentum correlations, and then show how these correlations can be extracted from measurements of event-by-event p_t fluctuations.

A number of experiments have studied transverse momentum fluctuations at SPS and RHIC [9,10]. Interestingly, the STAR collaboration reports a 60% increase of the relative rapidity width for p_t fluctuations when centrality is increased [15]. While the STAR analysis differs from the one we propose, model assumptions provide a tantalizing hint that the viscosity is small.

Any experimental information on the kinematic viscosity of high energy density matter is vital for understanding the strongly interacting quark gluon plasma. Theorists had long anticipated a large collisional viscosity based on weak coupling QCD [16] and hadronic computations [17], with values of h/s roughly of order unity for both phases near the crossover temperature ~ 170 MeV. Supersymmetic Yang Mills calculations give the significantly smaller ratio h/s = 1/4p in the strong coupling limit [18]. Lattice QCD calculations of the shear viscosity will eventually settle the question of the size of the viscosity near equilibrium [19]. However, the effective viscosity in the nonequilibrium ion-collision system may differ from these calculations. In particular, plasma-instability contributions can also explain the small viscosities in nuclear collisions [20].

We begin in the next section by formulating a simple model to illustrate how shear viscosity attenuates correlations due to fluctuations of the radial flow. In section III, we show how transverse momentum fluctuations can be used to measure these correlations. The magnitude of viscosity in heavy ion collisions at RHIC and SPS energies is discussed in section IV. In section V, we then demonstrate the impact of viscosity on the rapidity distribution of fluctuations. We explore the implications of current fluctuation data in section VI.

Before wading into the quark-gluon liquid, it is useful to recall how shear viscosity affects the flow of more common fluids. In a classic example of shear flow, a liquid is trapped between two parallel plates in the xy plane, while one plate moves at constant speed in the x direction as shown in fig. 1. The fluid is pulled along with the plate, so that v_x varies with the normal distance z. The viscous contribution to the stress energy tensor is then

see ref. [21] for a general treatment.

To fix definitions, recall that relativistic hydrodynamic evolution satisfies the energy-momentum conservation law ¶_mT^mn = 0. The stress-energy tensor is T^mn = (e + P)U^mUⁿ - Pg^mn + t^mn, where e is the energy density, P is the pressure, U^µ = g(1, v) is the four-velocity of the fluid, and g = (1 - v²)^-1/2. The dissipative contribution is t^mn. Following Landau and Lifshitz, we define the four velocity to vanish when T⁰ⁱ = 0 [21]. Then the momentum current is T_0i = g²(e + P)v_i and t^mn depends only on the viscosity, with t⁰⁰ = 0 = t⁰ⁱ and

in a local rest frame. We mention that the bulk viscosity z resists compression or expansion in which Ñ · v ¹ 0; the classic example is a Hubble-like expansion. Typically z h [17]; see however [23].

II. VISCOSITY AND DIFFUSION

Central nuclear collisions produce a high energy density fluid that flows outward with an average radial velocity v_r. In the hydrodynamic description of these collisions, we typically assume that v_r varies smoothly with spacetime (t, x) and is the same for all collisions of a fixed impact parameter. For central collisions, v_r is cylindrically symmetric. More realistically, small deviations u(x) of the radial flow occur throughout the fluid, varying with each ion-collision event. Such deviations occur, e.g., because the number and location of nucleon-nucleon subcollisions varies in each event.

Viscous friction arises as neighboring fluid elements flow past each other as illustrated in fig. 2. This friction reduces u, driving the velocity toward the local average v_r. The final size of the velocity increment u depends on the magnitude of the viscosity and the lifetime of the fluid.

In order to illustrate how the damping of radial flow fluctuations depends on the viscosity of the fluid, we introduce a velocity increment in the radial direction u that depends only on the longitudinal coordinate z and t. Our aim is to determine the linear response of the fluid to this perturbation. For simplicity, we take the unperturbed flow as slowly varying, and work in a co-moving frame where v_r locally vanishes. As in (2), the flow of neighboring fluid elements at different radial speeds u(z) produces a shear stress

This stress changes the transverse momentum current of the fluid, which is generally

for energy density e, pressure p, and g = (1 - v²)^-1/2 [21]. The perturbation u results in the change

in the co-moving frame, while

which follows from the energy-momentum conservation equation ¶_mT^mn = 0.

We combine these results to obtain a diffusion equation for the momentum current

to linear order. The kinematic viscosity is given by

This quantity measures the relative strength of the viscous relative to the fluid's and inertia, as is most apparent in the nonrelativistic limit, where e + p ® r for r the mass density (c = 1). It follows that n » h/r [21]. In our relativistic system at small net baryochemical potential, µ 0, so that e + p » Ts. Then n is given by (1).

Observe that (8) applies for any fluctuation g_t for which Ñ · g_t = 0 [21]; our physically-motivated radial g_t(z,t) is a specific instance of such a flow. Such shear modes are related to sound waves (compression modes) but diffuse rather than propagate [22]. Note that the scale over which sound is attenuated G_s = (4h/3 + z)/Ts depends on both shear and bulk viscosity [21-23].

Viscosity tends to reduce fluctuations by distributing the excess momentum density g_t over the collision volume. This effect broadens the rapidity profile of fluctuations. We write (8) in terms of the spatial rapidity y = 1/2 ln(t + z)/(t - z) and proper time t = (t² - z²)^1/2 to find

A similar equation is used to study net charge diffusion in ref. [26], and we can translate many of those results to the present context. Defining

for áyñ = 0, we multiply both sides of (11) by y² and integrate to obtain

for a constant n, we compute the rapidity broadening

where DV º V- V(t_o) for t_o the formation time. Observe that V ® 2n/t_o as t ® ¥. Also, note that the diffusion equation (10) can be solved directly to find g_t(t,y) µ exp[-y²/2V(t)^1/2]. The ratio (11) remains finite despite the fact that g_t ® 0 in that limit.

We briefly remark on the effect of the mean radial flow v_r and transverse degrees of freedom on the growth of the rapidity width. In the comoving frame where v_r is small, we can generalize (8) as

We transform the diffusion equation (14) into (10) by replacing g_t by a quantity that is averaged over the transverse coordinate r_^. To see this, integrate both sides of (14) over r_^. The transverse contributions lead to "surface terms" that vanish to this linear order. We stress that this result is independent of the azimuthal anisotropy of the collision volume. This result follows because g_t is conserved - it doesn't matter where the momentum is in the transverse plane, as long as we add it up. Of course, that only applies if we limit our interest to the rapidity dependance. To study the azimuthal behavior of fluctuations we must solve the full eq. (14).

III. CORRELATIONS AND DIFFUSION

A. Transverse momentum correlations

We extend this discussion to address a more general ensemble of fluctuations by considering the correlation function

In local equilibrium, r_g has the value r_{g, eq}. The spatial rapidity dependence of Dr_g º r_g - r_{g, eq} is broadened by momentum diffusion. If the rapidity width of the one-body density follows (13), then the width of Dr_g in the relative rapidity y_r = y₁ - y₂ grows from an initial value s_o following

where t_f is the proper time at which freeze out occurs. This equation is entirely plausible, since diffusion spreads the rapidity of each particle in a given pair with a variance DV. We then take

where (16) gives the width in relative rapidity and the width in average rapidity y_a = (y₁ + y₂)/2 is S.

To compute fluctuation observables in the next sections, we identify spatial and momentum-space rapidity at a fixed freeze out proper time t_f. Observe that ISR and FNAL data on the rapidity dependence of multiplicity fluctuations [24] can be characterized as Gaussian near midrapidity. Moreover, these data show that charged particle correlations are functions of the relative rapidity y_r with only a weak dependence on the average rapidity y_a. Correspondingly, we take S s. In the next section we see that diffusion increases both s and S compared to their initial values s_o and S_o at the hadronization time t_o in accord with (25) and (26). For simplicity, we assume that S_o is sufficiently large that we can neglect the time dependence of S in (26).

B. Transport in fluctuating systems

Observe that (16) and (17) are exact for our diffusion model. To see that this is the case, we apply the theoretical framework developed by Van Kampen and others [25] to viscous diffusion. We have used this framework elsewhere to study the effect of diffusion on net charge fluctuations [26]. Since the key results of this section, (16) and (17), are "obvious," this section may be skipped on a first reading. However, the details are amusing.

The framework of ref. [25] generalizes the classic problem of Brownian motion, in which a dust particle wanders through a fluid with mean displacement d that satisfies d² ~ áx²ñ - áxñ²µ t on average. Microscopic molecular collisions cause the particle's motion, while friction of the macroscopic particle with the liquid dampens it. Here, we seek an evolution equation for r_g, in which g_t(x,t) replaces the Brownian x. Our equation for r_g must have the correct limit r_{g, eq} when the system is in equilibrium. Equation (33) in the next section implies that

where f(x,p) is the phase space distribution of particles, n is the density.

In the previous section we found that g_t obeys a diffusion equation (8) in the regime of small fluctuations, but this statement requires clarification. Consider an ensemble of systems with different initial g_t. At the macroscopic scales of interest, viscosity indeed dampens fluctuations, working to drive g_t(x,t) ® 0. Opposing this dampening are Brownian-like velocity and thermal fluctuations due to the microscopic motion and collisions of particles. These fluctuations are present at all times prior to freeze out. In equilibrium, fluctuations and viscous dissipation are balanced.

To describe the effect of these fluctuations on the momentum current, we treat g_t as a stochastic quantity. It is the average of g_t over our ensemble that satisfies (8); such an ensemble average is always implicit in hydrodynamics. Going further, we write a Langevin equation for the fluctuating g_t as

where j_g = -nÑg_t + k and k is a stochastic contribution to the current. If we divide the fluid into tiny cells with a macroscopic number of particles, then particle and momentum flow into and out of each cell gives rise to k. Observe that (19) conserves the transverse momentum current in each 'event' in the ensemble. The current k is a Langevin noise term satisfying ákñ = 0 and

see [25]. The coefficient K is determined by the requirement that fluctuations have the correct value in equilibrium (the fluctuation-dissipation theorem).

Applying the methods in [25] to (19), we find that the correlation function (15) satisfies the diffusion equation

where Dr_g = r_g - r_{g, eq}. This equation means that over time r_g ® r_{g, eq} as small deviations in the momentum current diffuse over the collision volume. As for (8), we emphasize that (21) only applies in the linear regime where fluctuations are small.

To get a rough understanding of how (21) comes about, consider the change dg_t in the time increment dt. The change of the two-point equal-time function at points 1 º (x₁, t) and 2 º (x₂, t) is dág_t(1)g_t(2)ñ = ádg_t(1)g_t(2)ñ + ág_t(1)dg_t(2)ñ + ádg_t(1)dg_t(2)ñ, where the increment dg_t = (¶g_t/¶t)dt is given by (19). In the absence of noise, the term ádg_t(1)dg_t(2)ñ would vanish as dt². However, noise forces g_t to undergo a "random-walk" so that dg_t receives a stochastic increment µ dt^1/2. This increment contributes to ádg_t(1)dg_t(2)ñ µ dt. One finds

the right side is zero in the absence of noise. Choosing K so that r_g ® r_{g, eq} for a uniform system in equilibrium gives (21).

The correlation function Dr_g(y₁,y₂,t) then obeys

We write (23) in terms of the relative rapidity y_r º y₁ - y₂ and average rapidity y_a = (y₁ + y₂)/2:

the "2" follows from the transformation to relative rapidity y_r. To compute the widths of Dr_g(y_r, y_a,t) in relative or average rapidity, one multiplies (24) by or and integrates over both variables. We find

and

where DV(t) is calculated in the previous section.

IV. VISCOSITY IN ION COLLISIONS

Gyulassy and Hirano survey computations of the ratio of the shear viscosity to the entropy and find that both the hadron gas and the perturbative quark gluon plasma have h/s ~ 1, if one naively extrapolates these calculations near T_C [6]. These values correspond to n = h/Ts roughly of order 1 fm for T_C = 170 MeV. On the other hand, they argue that the entropy increase near T_C reduces h/s for a strongly interacting plasma, perhaps to the supersymmetric Yang-Mills value h/s = 1/4p.

Motivated by these estimates, we show in fig. 3 the increase of s given by (13) and (16) as a function of t. Calculations for two values n/t_o ~ 0.1 and 1 schematically exhibit the likely range of viscous broadening. For t_o ~ 1 fm, these values respectively correspond to h/s ~ 1/4p and 1. We provide these calculations as benchmarks; more realistically, n would effectively increase with t depending on the state of the fluid.

We stress that the rapidity width depends on the viscous diffusion coefficient integrated over the collision lifetime. Comparing the viscous and perfect scenarios in fig. 3, we see that the largest contribution to this width comes from the earliest times. Consequently, we expect measurements of this width to yield information on the viscosity when the evolution is dominated by partons.

V. FLUCTUATIONS AND CORRELATIONS

A. Transverse momentum covariance

Variation of the radial fluid velocity over the collision volume induces correlations in the transverse momenta p_t of particles [27]. To describe such correlations, we observe that an inhomogeneous fluid near local thermal equilibrium can be divided into cells that are small enough to be regarded as uniform, while containing a macroscopic number of particles. Particles emerging from cells of different radial velocity v_r are more likely to have different p_t than particles from the same cell. We describe the number of particles of momentum p in a cell at position x at the instant of freeze out as dn = f(x,p)dpdx, where dp º d³p/(2p)³ and dx º d³x. We take the phase space density f(x,p) to be a Boltzmann distribution corresponding to the fluid velocity v(x) and temperature T(x). The temperature and velocity profiles both vary from event to event. A similar formulation is used in [28] to compute nonequilibrium p_t fluctuations. Here, we focus on central collisions, where there is a body of RHIC data suggesting that local equilibrium is likely achieved.

To characterize the dynamic correlations of p_t, we use the transverse momentum covariance

where i labels particles from each event and the brackets represent the event average. The average transverse momentum is

This quantity vanishes in local thermal equilibrium, where the momenta are uncorrelated and particle number fluctuations satisfy Poisson statistics, so that áN²ñ - áNñ² = áNñ. It follows that áp_t1p_t2ñ_eq = áN(N - 1)ñáp_tñ²/áNñ² = áp_tñ².

This covariance is related to the spatial correlations of the momentum current (15) by

To obtain this result, observe that near local equilibrium f(x,p) = áfñ + df, where the average distribution is áf(x,p)ñ and the event-wise deviation df is necessarily small. Then

The contribution of fluctuations to the momentum current,

vanishes on event averaging. Similarly, the unrestricted sum is

We find

the second equality follows from (27). In local equilibrium, C º 0 implies òr_{g, eq}dx₁dx₂ = áåñ. Subtracting this term from (33) gives (29).

B. Multiplicity variance

The correlation information probed by C differs from that found in the multiplicity variance

Multiplicity fluctuations and their relation to correlation functions are discussed in [29]. As before, and in accord with [29], we write

where Dr_n = r_n - r_{n, eq} and

The density correlation function (36) carries different information than (15) because particle number is not conserved. Density fluctuations evolve by the full hydrodynamic equations, while g_t follows diffusion. We mention that

which is analogous to (18) but somewhat easier to understand. Equation (37) implies that (36) vanishes when particle number fluctuations obey Poisson statistics, a property that is evident from (34).

C. Rapidity dependance of p_t covariance

Viscosity information can be obtained from C as follows. For simplicity, we identify spatial and momentum space rapidity. The broadening in rapidity of Dr_g depends on the shear viscosity via (16). Equation (29) implies that the rapidity dependence of Dr_g can be measured by studying the dependence of (27) on the rapidity window in which particles are measured.

We illustrate this acceptance dependence in fig. 2 for the n/t_o values from fig. 1 as follows. The covariance C is computed by integrating Dr_g over an interval -D/2 < y₁, y₂< D/2 corresponding to the experimental acceptance. We use (17), (29), and assume S D/2 to obtain

where the total number of charged particles is áNñ µ D and áNñC_¥ is the value obtained for a large rapidity window. We assume t_f / t_o ~ 20.

VI. CURRENT DATA

A. Rapidity dependence of p_t fluctuations

Transverse momentum fluctuations have been measured at STAR and at the CERN SPS. We now ask whether information from these measurements can provide any information on the viscosity. The covariance C we propose is sensitive to the variation of the p_t of particles as well as their number density, in that both quantities affect the momentum current. Other measures of p_t fluctuations in the literature are designed to minimize any density contribution, often termed "volume fluctuations" [31]. This difference is important for the following discussion.

The STAR analysis in ref. [15] incorporates some of these ideas and, intriguingly, finds a broadening in rapidity together with a narrowing in azimuth for p_t correlations in central compared to peripheral collisions. We will use the rapidity information to estimate the viscosity. However, the measured quantities differ sufficiently from C that this estimate requires significant model assumptions. We therefore regard the result only as a signal of our method's promise.

STAR employs the transverse momentum fluctuation observable to construct a correlation function as a function of rapidity and azimuthal angle. They find that near-side correlations in azimuth are broadened in relative rapidity, with a rapidity width s_* that increases from roughly 0.45 in the most peripheral collisions to 0.75 in central ones [15]. In our terms,

To relate this to our observables, we expand the right side and use (28) to find

since the average number of pairs is áN(N - 1)ñ. We then obtain

where C and R are respectively given by (27) and (34).

The quantity therefore depends on both momentum current and density correlation functions (15) and (36),

We can directly compare s_* to s in fig. 1 if Dr_g and Dr_n have the same widths. Equation (16) then implies that the widths in central and peripheral collisions satisfy

Observe that the dependence on t_o cancels. Taking the freeze out times in central and peripheral collisions to be t_{f, c} ~ 20 fm and t_{f, p} ~ 1 fm, we then find n ~ 0.09 fm. The value t_{f, p} ~ 1 fm is reasonable, since ref. [15] argues that the average participant path length is about 1 fm for these peripheral collisions. We use (1) to find h/s ~ 0.08.

This result is remarkably close to the supersymmetric Yang Mills value 1/4p, and is consistent with some hydrodynamic comparisons to elliptic flow data [5]. However, we must be cautious: If Dr_g and Dr_n have different rapidity widths s and s_n then their relation to s_* depends on the relative strength of these contributions. Data in ref. [32] may indicate that s_n is roughly twice s_*. As we argue shortly, s is bounded by s_n and s_*. At the maximum value s = 2s_*, our dynamic assumptions yield h/s = 0.3. Together, our estimates constitute an uncertainty range for the viscosity-to-entropy ratio, 0.08 < h/s < 0.3. We also indicate the range of implied by the STAR data in fig. 3 as a gray band corresponding to s_* < s < 2s_*.

To show s is bounded by s_n and s_*, we observe that (42) implies that s_* follows from a distribution in relative rapidity

It follows that

where G = òDr_g and N = òDr_n. We then find

where b = áp_tñ²N/G. Although b is not measured, the width cannot exceed s_n ~ 2s_*. (We restrict our integrals to a rapidity and p_t region where a Gaussian parameterization of Dr_g and Dr_n makes sense; hence b > 0).

We briefly comment on the narrowing in azimuth for p_t correlations observed in ref. [15] in central compared to peripheral collisions. The diffusion broadening of azimuthal correlations is compensated by a narrowing due to the mean radial flow v_r, which hurls particles outward, and surface bias, which hides particles that flow in the wrong direction. Opening-angle effects in resonance decays lead to further narrowing. These effects are familiar in HBT. To address the azimuthal dependence of correlations, an equation analogous to (24) can be derived from (14). One must also account for the azimuthal anisotropy of the collision volume.

B. Other experimental information

As an aside, we now note some of the information from other experimental p_t fluctuation studies. The observable ádp_t1dp_t2ñ studied in [28] and measured in [9], [10] and [11] satisfies

where R = (áN²ñ - áNñ² - áNñ)/áNñ² measures multiplicity fluctuations [29]. The energy dependence of this quantity is shown in fig. 5.

The energy independence of these measurements supports the hypothesis that the largest contribution to p_t fluctuations is from the collective behavior of soft and semi-hard particles rather than jets as sometimes noted [34]. New PHENIX energy-dependence data further supports the soft origin of fluctuations [35]. Observe that (47) implies that will increase with energy in proportion to the multiplicity if ádp_t1dp_t2ñ is energy dependent. All this concerns the overall scale of the fluctuations. Little can be said about the energy dependence of the rapidity distribution of these fluctuations at present.

Alternative observables , , and proposed in [30], [13], and [14] satisfy » /s » áNñádp_t1dp_t2ñ/2s² for s² = áñ - áp_tñ², see e.g. [31]. Broadly, the differences in these various quantities is important only in peripheral collisions where the multiplicity is small. The different quantities have depend differently on the experimental efficiency, e.g., ádp_t1dp_t2ñ is designed to minimize efficiency dependence. Other differences may reflect the different experimental acceptance and analysis issues.

VII. SUMMARY

In summary, we find that shear viscosity can broaden the rapidity correlations of the momentum current. This broadening can be observed by measuring the transverse momentum covariance (27) as a function of rapidity acceptance. Our rough estimate from current data, h/s ~ 0.08 - 0.3, is small compared to perturbative computations [6]. To reduce the experimental uncertainty, we suggest measuring C to allow more direct access to the momentum density correlation function.

We stress that there is additional theoretical uncertainty in this estimate, mainly due to our freeze out model. In practice, » , since t_{f, p} t_{f, c}. The freeze out time in peripheral collisions t_{f, p} is not plausibly smaller than our value 1 fm (the nucleon radius), but may be twice as large. This would double the upper limit of our uncertainty band. That added uncertainty can eventually be reduced by measuring t_{f, p} as in [33]. HBT and resonance effects omitted here may contribute only at the 10% and 15% levels, respectively [10]. Minijets, color glass, and other particle production effects modify s_o in (16). We assume that any modification cancels in studying the centrality dependence at fixed beam energy. As to our hydrodynamic treatment, we have not considered alternative transport formulations that enforce causality [36]. Furthermore, our linearized diffusion model of flow fluctuations is physically reasonable but highly idealized. A more refined hydrodynamic description will be necessary to confront the measurements we suggest.

How can we reduce the theoretical uncertainty? The viscosity of a common fluid can be measured by applying a known pressure and observing the resulting flow in a fixed geometry, e.g., a pipe. Alternatively, one can study the attenuation of high frequency sound waves from a calibrated source. Efforts to compare flow measurements to viscous hydrodynamic calculations are analogous to the first method [5]. Our observable C is in the spirit of ultrasonic attenuation. The early dynamics produces a spectrum of fluctuations analogous to sound waves that are attenuated by viscosity [37]. We suggest that experimenters pursue both approaches to extract quantitative viscosity information from ion collisions, since the initial conditions and model parameters are all unknown.

Acknowledgements

We thank J. Dunlop for kindly bringing the data in ref. [15] to our attention, and R. Bellwied, M. Gyulassy, G. Moschelli, and C. Pruneau for useful discussions. This work was supported in part by a U.S. National Science Foundation PECASE/CAREER award under grant PHY-0348559 (S.G.) and BMBF, GSI and DAAD (M.A-A.).

[1] S. Gavin and M. Abdel-Aziz, Phys. Rev. Lett. 97, 162302 (2006).

[2] I. Arsene et al. [BRAHMS Collaboration], Nucl. Phys. A 757, 1 (2005); K. Adcox et al. [PHENIX Collaboration], ibid. 184; B. B. Back et al. [PHOBOS Collaboration], ibid. 28; J. Adams et al. [STAR Collaboration], ibid. 102.

[3] K. H. Ackermann et al. [STAR Collaboration], Phys. Rev. Lett. 86, 402 (2001); C. Adler et al. [STAR Collaboration], Phys. Rev. Lett. 87, 182301 (2001); Phys. Rev. Lett. 89, 132301 (2002); Phys. Rev. C 66, 034904 (2002); J. Adams et al. [STAR Collaboration], Phys. Rev. C 72, 014904 (2005); K. Adcox et al. [PHENIX Collaboration], Phys. Rev. Lett. 89, 212301 (2002); S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 91, 182301 (2003); B. B. Back et al. [PHOBOS Collaboration], Phys. Rev. Lett. 89, 222301 (2002); Phys. Rev. Lett. 94, 122303 (2005); Phys. Rev. C 72, 051901 (2005).

[4] P. F. Kolb, P. Huovinen, U. W. Heinz, and H. Heiselberg, Phys. Lett. B 500, 232 (2001); P. Huovinen and P. V. Ruuskanen, arXiv:nucl-th/0605008.

[5] D. Teaney, Phys. Rev. C 68, 034913 (2003); J. Phys. G 30, S1247 (2004).

[6] T. Hirano and M. Gyulassy, Nucl. Phys. A 769, 71 (2006).

[7] L. P. Csernai, J. I. Kapusta, and L. D. McLerran, Phys. Rev. Lett. 97, 152303 (2006).

[8] T. Hirano, U. W. Heinz, D. Kharzeev, R. Lacey, and Y. Nara, Phys. Lett. B 636, 299 (2006).

[9] C. Pruneau, Proc. Quark Matter 2005, in press; J. T. Mitchell, J. Phys. G 30, S819 (2004).

[10] J. Adams et al. [STAR Collaboration], Phys. Rev. C 72, 044902 (2005).

[11] D. Adamova et al. [CERES Collaboration], Nucl. Phys. A 727, 97 (2003).

[12] T. Anticic et al. [NA49 Collaboration], Phys. Rev. C 70, 034902 (2004).

[13] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 93, 092301 (2004).

[14] J. Adams et al. [STAR Collaboration], Phys. Rev. C 71, 064906 (2005).

[15] J. Adams et al. [STAR Collaboration], J. Phys. G 32, L37, (2006); arXiv:nucl-ex/0509030.

[16] A. Hosoya and K. Kajantie, Nucl. Phys. B 250, 666 (1985); P. Danielewicz and M. Gyulassy, Phys. Rev. D 31, 53 (1985); G. Baym, H. Monien, C. J. Pethick, and D. G. Ravenhall, Phys. Rev. Lett. 64, 1867 (1990); H. Heiselberg, Phys. Rev. D 49, 4739 (1994); P. Arnold, G. D. Moore, and L. G. Yaffe, JHEP 0011, 001 (2000); 0305, 051 (2003).

[17] S. Gavin, Nucl. Phys. A 435, 826 (1985); M. Prakash, M. Prakash, R. Venugopalan, and G. Welke, Phys. Rept. 227, 321 (1993); A. Muronga, Phys. Rev. C 69, 044901 (2004); S. Muroya and N. Sasaki, Prog. Theor. Phys. 113, 457 (2005).

[18] P. Kovtun, D. T. Son, and A. O. Starinets, Phys. Rev. Lett. 94, 111601 (2005); 87, 081601 (2001).

[19] A. Nakamura and S. Sakai, arXiv:hep-lat/0510039.

[20] M. Asakawa, S. A. Bass, and B. Muller, Phys. Rev. Lett. 96, 252301 (2006).

[21] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1959).

[22] D. M. Forster, Hydrodynamic fluctuations, Broken symmetry, and Correlation Functions, (Benjamin, New York, 1975).

[23] K. Paech and S. Pratt, arXiv:nucl-th/0604055.

[24] J. Whitmore, Phys. Rept. 27, 187 (1976); L. Foa, Physics Reports, 22, 1 (1975); H. Boggild and T. Ferbel, Ann. Rev. Nucl. Part. Sci. 24, 451 (l974).

[25] N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, (Elsevier Science, Amsterdam, 1997); C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, (Springer, Berlin, 2002).

[26] M. Abdel-Aziz and S. Gavin, Phys. Rev. C 70, 034905 (2004).

[27] S. A. Voloshin, Phys. Lett. B 632, 490 (2006).

[28] S. Gavin, Phys. Rev. Lett. 92, 162301 (2004); J. Phys. G 30, S1385 (2004).

[29] C. Pruneau, S. Gavin and S. Voloshin, Phys. Rev. C 66, 044904 (2002).

[30] M. Gazdzicki and S. Mrowczynski, Z. Phys. C 54, 127 (1992).

[31] S. A. Voloshin, V. Koch, and H. G. Ritter, Phys. Rev. C 60, 024901 (1999).

[32] J. Adams et al. [STAR Collaboration], Phys. Rev. C 73, 064907 (2006).

[33] M. Lisa, Acta Phys. Polon. B 35, 37 (2004); C. Markert, J. Phys. G 31, S169 (2005).

[34] M. Abdel-Aziz and S. Gavin, Nucl. Phys. A 774, 623 (2006).

[35] J. T. Mitchell, Proc. Quark Matter 2006.

[36] A. Muronga, Phys. Rev. Lett. 88, 062302 (2002); Phys. Rev. C 69, 034903 (2004); A. Muronga and D. H. Rischke, arXiv:nucl-th/0407114; U. W. Heinz, H. Song, and A. K. Chaudhuri, Phys. Rev. C 73, 034904 (2006); A. K. Chaudhuri, Phys. Rev. C 74, 044904 (2006); R. Baier, P. Romatschke, and U. A. Wiedemann, Phys. Rev. C 73, 064903 (2006); R. Baier and P. Romatschke, arXiv:nucl-th/0610108.

[37] The fluctuations probed by our p_t observable are shear modes, which are not strictly sound waves; see the discussion following eq. (1).

Received on 15 December, 2006

Publication Dates

• Publication in this collection
  24 Sept 2007
• Date of issue
  Sept 2007

History

• Received
  15 Dec 2006