N=4超对称Yang-Mills流体动力学中的等离子体吸引子研究

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本文主要探讨了在相对论流体力学背景下，特别是与夸克-胶子等离子体相关的物理现象。研究焦点是N=4超对称Yang-Mills理论（SYM）中的Bjorken流，这是一种在高能粒子碰撞实验中观察到的特征性流体动力学行为。随着系统远离局部平衡状态，非流体动力学模式逐渐倾向于一个称为吸引子的稳定状态。吸引子的存在意味着系统的长期行为趋向于一个共同的行为模式，即便初始条件和扰动各异。作者利用梯度扩展的Borel总和技术，对SYM中能量动量张量期望值的理论计算进行了深入分析。这种方法允许他们确定了流体动力学吸引子的具体形式，这对于理解高温强相互作用物质的行为至关重要。Borel总和是一种数学工具，用于处理无穷级数，通过有限的数值工作捕捉到原理论的精确行为。作者进一步将理论计算的结果与基于AdS/CFT（Anti-de Sitter空间/Conformal Field Theory对应关系）的数值模拟进行了对比。AdS/CFT对偶是弦理论中的一个关键概念，它提供了一种将量子引力理论（如SYM）与二维统计物理模型关联起来的方法。通过这种对偶，研究人员能够在相对容易控制的AdS空间中模拟复杂的QCD现象，如等离子体的行为。通过这种对比，研究者证实了他们在SYM中的流体动力吸引子理论提供了对实际等离子体行为非常准确且无歧义的近似。这意味着，即使在远离平衡的情况，也能用这个理论有效地预测等离子体的长期动力学行为，对于建立有效的水动力学理论具有重大意义，这有助于科学家们更深入地理解高能物理实验中的现象，并可能为未来的设计和解释实验结果提供有力工具。总结来说，这篇文章的核心贡献在于运用严谨的数学方法确定了N=4 SYM中Bjorken流的流体动力学吸引子，并通过与AdS/CFT的比较验证了其理论价值，这在理解和预测夸克-胶子等离子体的非平衡行为上迈出了重要一步。这一研究成果对于推动高温强相互作用领域的理论发展和实验应用具有深远的影响。

Physics Letters B 776 (2018) 468–472

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

On the hydrodynamic attractor of Yang–Mills plasma

Michał Spali

nski

a,b,∗

Physics Department, University of Białystok, 15-245 Białystok, Poland

National Centre for Nuclear Research, 00-681 Warsaw, Poland

a r t i c l e i n f o a b s t r a c t

Article history:

Received

2 November 2017

Accepted

24 November 2017

Available

online 2 December 2017

Editor:

N. Lambert

Keywords:

Quark–gluon

plasma

AdS/CFT

There is mounting evidence suggesting that relativistic hydrodynamics becomes relevant for the physics

of quark–gluon plasma as the result of nonhydrodynamic modes decaying to an attractor apparent

even when the system is far from local equilibrium. Here we determine this attractor for Bjorken ﬂow

in N = 4 supersymmetric Yang–Mills theory (SYM) using Borel summation of the gradient expansion

of the expectation value of the energy momentum tensor. By comparing the result to numerical

simulations of the ﬂow based on the AdS/CFT correspondence we show that it provides an accurate

and unambiguous approximation of the hydrodynamic attractor in this system. This development has

important implications for the formulation of effective theories of hydrodynamics.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

Heavy-ion collision experiments and their phenomenological

description have lead to the realization that relativistic hydrody-

namics

works very well rather far outside its traditionally under-

stood

domain of validity. Variants of Müller–Israel–Stewart (MIS)

theory [1–3] have successfully been applied in rather extreme con-

ditions,

which could hardly be assumed to be close to local equi-

librium.

Furthermore, model calculations exist where it is possible

to study the emergence of universal, hydrodynamic behaviour and

test to what extent an effective description in terms of hydrody-

namics

can match microscopic results [4]. Such calculations were

initially carried out in N = 4SYM using the AdS/CFT correspon-

dence [5–7],

but similar studies have since also been performed in

models of kinetic theory [8–10]. The conclusion from these inves-

tigations

is that the domain of validity of a hydrodynamic descrip-

tion

is delimited by the decay of nonhydrodynamic modes [5,6,11,

12].

The outcome of this transition to hydrodynamics (“hydroniza-

tion”)

is that the system reaches a hydrodynamic attractor [13]

which

governs its subsequent evolution toward equilibrium. This

attractor is a special solution to which generic histories decay ex-

ponentially,

and do so well before local equilibrium sets in. It in-

corporates

all orders of the hydrodynamic gradient expansion, and

at suﬃciently late times coincides with the predictions of relativis-

tic

Navier–Stokes theory. The existence of an attractor in this sense

Correspondence to: National Centre for Nuclear Research, 00-681 Warsaw,

Poland.

E-mail

address: michal.spalinski@ncbj.gov.pl.

is a critically important issue for hydrodynamics, because it deﬁnes

its very meaning. It has conceptual as well as practical implications

for the formulation of hydrodynamic theories in general as well as

for their application to the physics of quark–gluon plasma.

Attractor

behaviour was ﬁrst identiﬁed explicitly in the differ-

ential

equations of hydrodynamics [13,14]. An outstanding prob-

lem

is the determination of such attractors at the microscopic

level [15,4]. The ﬁrst calculations of this type were described by

Romatschke [15], who found approximate attractor solutions in the

context of kinetic theory and N = 4SYM by scanning for the cor-

responding

initial conditions. The purpose of this Letter is to argue

that the Borel sum of the hydrodynamic gradient expansion pro-

vides

a direct way of estimating the attractor. While at late times

this calculation clearly must give the correct result (which coin-

cides

with the prediction of Navier–Stokes hydrodynamics) it is

not obvious a priori that this calculation gives an accurate estimate

at earlier times. We will however show explicitly that the result

of Borel summation does indeed act as an attractor for histories

of Bjorken ﬂow simulated using techniques based on the AdS/CFT

correspondence. This should be viewed in the context of the idea

that higher orders of the gradient expansion may be relevant for

real-world physics [16–19].

important point is that the hydrodynamic gradient expan-

sion

is the leading element of a transseries [13], and in general

the higher order elements (“instanton sectors”) play an important

role in deﬁning the summation properly. These transseries sectors

involve integration constants which need to be ﬁxed. However,

their contributions are exponentially suppressed and it is tempt-

ing

to ignore them as a ﬁrst approximation. Such an approach will

https://doi.org/10.1016/j.physletb.2017.11.059

0370-2693/

SCOAP

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