Nonlinear eective dynamics of Brownian particle in magnetized plasma Yanyan Bu Biye Zhang and Jingbo Zhang

2025-05-02 0 0 947.08KB 33 页 10玖币

侵权投诉

Nonlinear eﬀective dynamics of Brownian particle in

magnetized plasma

Yanyan Bu ∗, Biye Zhang †, and Jingbo Zhang ‡

School of Physics, Harbin Institute of Technology, Harbin 150001, China

October 6, 2022

Abstract

An eﬀective description is presented for a Brownian particle in a magnetized plasma.

In order to systematically capture various corrections to linear Langevin equation, we con-

struct eﬀective action for the Brownian particle, to quartic order in its position. The

eﬀective action is ﬁrst derived within non-equilibrium eﬀective ﬁeld theory formalism, and

then conﬁrmed via a microscopic holographic model consisting of an open string prob-

ing magnetic AdS5black brane. For practical usage, the non-Gaussian eﬀective action is

converted into Fokker-Planck type equation, which is an Euclidean analog of Schr¨odinger

equation and describes time evolution of probability distribution for particle’s position and

velocity.

∗yybu@hit.edu.cn

†zhangbiye@hit.edu.cn (correspondence author)

‡jinux@hit.edu.cn

arXiv:2210.02274v1 [hep-th] 5 Oct 2022

Contents

1 Introduction 2

2 Eﬀective dynamics from symmetry principle 4

2.1 Construction of eﬀective action . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Fokker-Planck equation from non-Gaussian eﬀective action . . . . . . . . . . . 8

3 Study in a microscopic model 10

3.1 Magnetic AdS5black brane and its ﬁeld theory dual . . . . . . . . . . . . . . . 12

3.2 Dynamics of open string in magnetic brane . . . . . . . . . . . . . . . . . . . . 15

3.3 Eﬀective action for Brownian quark: quadratic order . . . . . . . . . . . . . . . 18

3.4 Eﬀective action for Brownian quark: quartic order . . . . . . . . . . . . . . . . 21

4 Summary and Discussion 24

A Subtlety due to non-commutativity of →0 versus ω→026

B KMS relations when (2.15)is relaxed 27

1 Introduction

Brownian motion is perhaps the simplest example of non-equilibrium phenomena, which, how-

ever, has played a profound role in the development of non-equilibrium statistical mechanics [1].

In the simplest case, a Brownian particle moving in a thermal medium is eﬀectively described

by linear Langevin equation

Md2

dt2q(t) + η0

dtq(t) = χ(t), (1.1)

where q(t) and Mare the position and eﬀective mass of the Brownian particle, η0is damping

coeﬃcient. A Gaussian white noise χ(t) could be characterized by one- and two-point functions,

hχ(t)i= 0, hχ(t)χ(t0)i= 2T η0δ(t−t0), (1.2)

where the coeﬃcient in the second relation is due to ﬂuctuation-dissipation theorem. Here, T

is the temperature of thermal medium.

For speciﬁc purpose, linear Langevin theory (1.1)-(1.2) would be recast into an alternative

formalism. For instance, in order to avoid repeatedly solving stochastic equation (1.1) with

(inﬁnitely) many diﬀerent samplings of noise, one could equivalently consider Fokker-Planck

equation, which is a deterministic diﬀerential equation for probability distribution function

P(q, ˙q,t), where a dot means time derivative. Moreover, for a Gaussian distribution of noise,

the Langevin equation (1.1) could be reformulated as a functional integral [2], with the weight

given by the Martin-Siggia-Rose-deDominicis-Janssen (MSRDJ) action. The functional integral

formalism based on MSRDJ action makes it natural to adopt modern ﬁeld theoretic methods

to analyze more general stochastic processes.

Indeed, linear Langevin theory (1.1)-(1.2) could be generalized in a number of ways [2–6].

First, linear Langevin equation (1.1) could be made nonlinear by adding general polynomial

terms such as f1(q, ˙q,χ), which contains self-interactions for dynamical variable qand nonlinear

interactions between dynamical variable qand noise ξ. One special case is multiplicative noise,

which amounts to making a replacement χ→f2(q, ˙q)χin (1.1). Second, the noise would

obey a non-Gaussian distribution, and might be coloured as well, which requires to go beyond

(1.2). Third, isotropy would be broken by an external ﬁeld, such as in magnetized thermal

medium [7,8]. Then, beyond linear level, dynamics of transverse and longitudinal modes (with

respect to external ﬁeld) would get mixed. These corrections may become relevant and/or

important for more realistic systems. A natural question arises: what is a more systematic

way of organizing these extensions? This will be pursued here through two complementary

approaches.

In this work we search for an eﬀective description for a Brownian particle in a magnetized

plasma, with potential applications in heavy-ion collisions in mind. The main purpose is to

reveal nonlinear corrections to linear Langevin theory (1.1)-(1.2) in a systematic way. This will

be achieved by non-equilibrium eﬀective ﬁeld theory (EFT) formalism for a quantum many-

body system at ﬁnite temperature [9–12] (see [13–15] for an alternative approach). Within

such a formalism, dynamics of Brownian particle is entirely dictated by an eﬀective action,

which will be constructed on a set of symmetries. The eﬀective action may be thought of

as generalization of the MSRDJ action for a linear theory (1.1)-(1.2). Moreover, the eﬀective

action contains “free parameters” representing UV physics and information of the state as well.

Generically, it is challenging to compute those free parameters from an underlying UV theory

(here, it is a closed system consisting of the Brownian particle and the magnetized plasma).

Given that quark-gluon plasma produced in heavy-ion collisions is strongly coupled, we turn

to a microscopic holographic model and derive the eﬀective action (including values of free

parameters).

Holography [16–18] is insightful in understanding symmetry principles underlying non-

equilibrium eﬀective action. Of particular importance is the dynamical Kubo-Martin-Schwinger

(KMS) symmetry [9–11] acting on dynamical variable of the eﬀective action, which guarantees

the generalized ﬂuctuation-dissipation theorem [19,20] at full level. In [9–11], dynamical KMS

symmetry is implemented in the classical statistical limit where ~→0, which corresponds to

neglecting quantum ﬂuctuations in the eﬀective theory. However, for a holographic theory, the

mean free path is ∼~/T , which implies that gradient expansion would generally inevitably

bear quantum ﬂuctuations [21]. Via the example of Brownian motion, we will elaborate on

this point from both non-equilibrium EFT approach and holographic calculation. Intriguingly,

imposition of a constant translational invariance (i.e., q→q+cwith ca constant) renders

resultant eﬀective theory to be of classical statistical nature.

While eﬀective action formalism is more systematic in covering nonlinear corrections alluded

above, it turns out to be inconvenient to convert non-Gaussian eﬀective action into Langevin

type equation [9]. The main obstacle stems from non-Gaussianity in a-variable (to be deﬁned

below), which prohibits from carrying out Hubbard-Stratonovich transformation [2]. Interest-

ingly, we are able to convert the non-Gaussian eﬀective action constructed in present work into

Fokker-Planck type equation, which will be useful in numerical study.

The rest of this paper will be structured as follows. In section 2, we clarify the set of

symmetries and construct eﬀective action for Brownian particle, which is further put into

Fokker-Planck type equation. In section 3, we derive eﬀective action for a Brownian parti-

cle moving in magnetized thermal plasma from a holographic perspective. In section 4we

summarize and outlook future directions. Appendices Aand Bprovide further calculational

details.

2 Eﬀective dynamics from symmetry principle

Dynamics of a closed system consisting of a Brownian particle and a thermal medium is pre-

sumably described by an action

SC=Sp[q] + Sth[Φ] + Sint[q, Φ], (2.1)

where Sp[q] is action for the Brownian particle, Sth[Φ] describes microscopic theory of the

constitutes (collectively denoted as Φ) for the thermal medium, and Sint[q, Φ] is the interaction

between Brownian particle and constituents of thermal medium. In principle, eﬀective action

for the Brownian particle would be obtained by integrating out degrees of freedom {Φ}for the

thermal medium, as illustrated below:

Z=Z[Dq][DΦ]eiSC=Z[Dq]eiI[q], (2.2)

where I[q] is the desired eﬀective action. For such a quantum many-body system, time evolution

of the state will eﬀectively go forward and backward along the Schwinger-Keldysh (SK) closed

time contour, see Figure 1. Apparently, when carrying out the “integrating out” procedure in

Figure 1: The SK closed time contour: ρ0is initial density matrix, and U(tf,ti) is the time-

evolution operator from initial time tito ﬁnal time tf.

(2.2), one shall place the closed system on the SK closed time contour of Figure 1. Resultantly,

the degrees of freedom are doubled, q→(q1,q2), where the subscripts 1, 2 correspond to the

upper and lower branches of Figure 1.

Except for a few simple models [3,4,12,22–24], it is very challenging to implement the

“integrating out” procedure illustrated in (2.2). It is thus natural to construct the eﬀective

action based on symmetry principle, which will be pursued here.

2.1 Construction of eﬀective action

The eﬀective action I[q1;q2] is usually presented in (r,a)-basis:

qr≡1

2(q1+q2), qa≡q1−q2. (2.3)

where qris the physical variable and qais an auxiliary variable (conjugate to noise ξ(t)). We

summarize various symmetries and constraints obeyed by the eﬀective action I[q1;q2] = I[qr;qa]

for a Brownian particle moving in a magnetized plasma.

•Z2-reﬂection symmetry

Take the complex conjugate of partition function (2.2), we ﬁnd the reﬂection conditions,

I∗[q1;q2] = −I[q2;q1]⇐⇒ I∗[qr;qa] = −I[qr;−qa]. (2.4)

•Normalization condition

If we set the two coordinates to be the same q1=q2=q, we ﬁnd

Tr U(+∞,−∞;q)ρ0U†(+∞,−∞;q)= Trρ0= 1, (2.5)

which lead to the normalization condition,

I[qr;qa= 0] = 0. (2.6)

So, it will be convenient to present the eﬀective action as an expansion in number of qa-variable.

Moreover, for the path integral based on I[qr;qa] to be well-deﬁned, imaginary part of

I[qr;qa] should be non-negative:

Im (I[qr;qa]) ≥0, (2.7)

which will constrain some parameters in the eﬀective action.

•Dynamical KMS symmetry

I[q1;q2] = I[˜q1; ˜q2], (2.8)

where

˜q1(−t) = q1(t), ˜q2(−t) = q2(t−iβ). (2.9)

Here, β= 1/T is the inverse temperature of plasma medium. The dynamical KMS symmetry

is crucial in formulating an EFT for quantum many-body system at ﬁnite temperature. It

guarantees the generalized nonlinear ﬂuctuation-dissipation theorem (FDT) at full quantum

level [19], which originates from time-reversal invariance of underlying microscopic theory and

relies on the fact that initially the system is in a thermal state.

Intriguingly, it is possible to take classical statistical limit of dynamical KMS symmetry so

that only thermal ﬂuctuations in EFT will survive [9]. Let us properly restore Planck constant

β→~β,qr→qr,qa→~qa. (2.10)

Then, the classical statistical limit is achieved by taking ~→0 in the eﬀective action. Conse-

quently, the classical statistical limit of (2.9) becomes

˜qr(−t) = qr(t), ˜qa(−t) = qa(t)+iβ∂tqr(t). (2.11)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

NonlineareectivedynamicsofBrownianparticleinmagnetizedplasmaYanyanBu*,BiyeZhang,andJingboZhangSchoolofPhysics,HarbinInstituteofTechnology,Harbin150001,ChinaOctober6,2022AbstractAneectivedescriptionispresentedforaBrownianparticleinamagnetizedplasma.Inordertosystematicallycapturevariouscorrections...

展开>> 收起<<

Nonlinear eective dynamics of Brownian particle in magnetized plasma Yanyan Bu Biye Zhang and Jingbo Zhang.pdf

共33页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Nonlinear eective dynamics of Brownian particle in magnetized plasma Yanyan Bu Biye Zhang and Jingbo Zhang

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: