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arXiv:cond-mat/0605049v3 [cond-mat.stat-mech] 31 Aug 2006

Non-ergodic transitions in many-body Langevin

systems: a method of dynamical system reduction

Mami Iwata

E-mail: iwata@jiro.c.u-tokyo.ac.jp

Shin-ichi Sasa

E-mail: sasa@jiro.c.u-tokyo.ac.jp

Department of Pure and Applied Sciences, University of Tokyo, Komaba, Tokyo

153-8902, Japan

Abstract.

We first derive an equation for the two-point time correlation function of density

fluctuations, ignoring the contributions of the third- and fourth-order cumulants. For

this equation, with the averagedensity fixed, we find that there is a critical temperature

at which the qualitative nature of the trajectories around the trivial solution changes.

Using a method of dynamical system reduction around the critical temperature, we

simplify the equation for the time correlation function into a two-dimensional ordinary

differential equation.Analyzing this differential equation, we demonstrate that a

non-ergodic transition occurs at some temperature slightly higher than the critical

temperature.

We study a non-ergodic transition in a many-body Langevin system.

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Non-ergodic transitions in many-body Langevin systems

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1. Introduction

Glassy systems exhibit distinctive phenomena, such as divergence of the viscosity [1],

a history-dependent response [2], and an extreme slowing down of the time correlation

of density fluctuations [3, 4]. These phenomena have been studied extensively through

laboratory experiments, numerical experiments, and theoretical analyses of models [5].

Although each of these phenomena has gradually come to be understood individually,

we are still lacking a unified picture that can account for all of them. In contrast to the

rich variety of physical phenomena displayed by these systems, the types of theoretical

methods that have been used in their investigation are few. Particularly, many analyses

are based on the mode coupling theory (MCT) [3]. However, the MCT is not justified

completely from a microscopic viewpoint, and also it is not regarded as the first-order

description of a systematic formulation. Therefore, it is important to formulate a new

method for analyzing the phenomena of glassy systems.

Recalling the historical development of the theory of glassy systems, we realize that

the MCT has been the most commonly used theory [6] because it describes various

phenomena on the basis of the singular behavior of the two-point time correlation

function of density fluctuations.This time correlation function exhibits a two-step

relaxation as a function of the time difference and a plateau regime appears between

the two steps in the relaxation. The MCT predicts the transition temperature for a fixed

density (or the transition density for a fixed temperature) at which the plateau regime

extends to infinity.This phenomenon is called a non-ergodic transition.

the singularity might be an artifact of the approximation employed in the theory

[6], there are evidences suggesting that the non-ergodic transition has been observed

experimentally and numerically with fairly good quantitative correspondence to the

theoretical prediction [4]. Noting such evidence, in the present paper, we theoretically

study the behavior of the time correlation of density fluctuations.

Let us briefly review the essence of the MCT. This theory provides a self-consistent

integral equation of the time correlation function and the response function, ignoring

vertex corrections [3]. Then, by analyzing this equation numerically, one can find a

non-ergodic transition. Here, it should be noted that the memory contribution to the

equation for the time correlation function is expressed in terms of the time correlation

function. It is believed that such non-linear memory plays a key role in the non-ergodic

transition.

Now, it is interesting that behavior similar to that exhibited in the non-ergodic

transition seen in glassy systems is easily obtained in ordinary differential equations

possessing no nonlinear memory. As an example, let us consider Newton’s equation of

motion for a point particle moving in a one-dimensional space. Let x(t) be its position

and V (x) be a potential function given by V (x) = −x2[(x − x∗)2+ ε], with ε ≥ 0. We

consider the particle motion under the condition that x → 0 as t → ∞. With this

condition, it is easily found that when ε is small, the particle starting from x(0) (> x∗)

climbs the potential, slowly passes the non-zero maximum position, and then converges

Although

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Non-ergodic transitions in many-body Langevin systems

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to x = 0. The graph of x as a function of t exhibits a two-step relaxation behavior with

a long plateau. Furthermore, if ε = 0, the particle converges to the position x∗ and

cannot approach the origin. This qualitative change in the behavior of the system as a

function of ε occurring at ε = 0 is analogous to that exhibited by glassy systems at the

non-ergodic transition.

Clearly, this simple Newtonian system has no direct relation with glassy systems.

Nevertheless, it might be possible that there exists a variable representing the time

correlation function for a glassy system that obeys an equation similar to that in the

example discussed above. Such a variable would provide a characterization of the non-

ergodic transition in that glassy system. Beginning from this speculation, in this paper,

we determine such a variable and derive its evolution equation from a many-body

Langevin model, employing several assumptions. We demonstrate that the obtained

equation indeed does exhibit a non-ergodic transition.

2. Model

We investigate a system consisting of N identical Brownian particles suspended in a

liquid. We denote the volume of the system by V and the temperature by T, and

consider the thermodynamic limit, represented by V → ∞ and N → ∞, with the density

¯ ρ = N/V fixed. Let ri(t) be the position of the i-th particle, with i = 1,2,···N. We

assume that the i-th particle interacts with the j-th particle through some interaction

potential U(|ri−rj|). The motion of each particle is described by a Langevin equation

with a friction constant γ. Then, the evolution equation of the fine-grained density field,

ρ(r,t) ≡?N

∂ρ(r,t)

∂tγ

i=1δ(r − ri(t)), is given by

= ∇ ·

?1

+T

?

d3r′ρ(r′,t)ρ(r,t)∇U(|r − r′|)

γ∇ρ(r,t) +

?

2T

γρ(r,t)Ξ(r,t)

?

,(1)

where Ξ(= (Ξx,Ξy,Ξz)) is zero-mean Gaussian white noise that satisfies

?Ξα(r,t)Ξβ(r′,s)? = δαβδ(t − s)δ(r − r′).(2)

This evolution equation can be derived exactly from the Langevin equation describing

the motion of N particles [7]. Explicitly, we choose the form of the interaction potential

as

E

λ1(k2+ λ2

(k2+ λ2

where k = |k|. Throughout this paper,ˆf(k) is used to represent the Fourier transform

of f(r):

ˆf(k) =

?d3reik·rf(r).

that we analyze numerically, we consider the case in which σ = 8.0 and λ2= λ1. With

ˆU(k) =

1)−

2kEσ

2)2,(3)

All quantities in this system are converted into

dimensionless forms by setting λ1 = E = γ = 1. Further, for the specific example

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this choice, U(r) has a minimum value −1.18 at r = 0.09, while U(r) → ∞ as r → 0

and U(r) → 0 as r → ∞. Explicitly one can obtain

U(r) =e−r

π2

2 dr

where Ei(r) represents an integral exponential function defined as Ei(r) =?r

parameter.

For later convenience, we define the quantity Ψ(k,k′;t,s) ≡ˆδρ(k,t)ˆδρ(k′,s)/¯ ρ2,

where δρ(r,t) ≡ ρ(r,t) − ¯ ρ. The statistical average of Ψ(k,k′;t,s) provides the two-

point correlation functionˆC(k,t) according to the relation

?Ψ(k,k′;t,s)? = (2π)3ˆC(k,t − s)δ(k + k′),

4πr−4

?

1 +r

d

?e−rEi(r) − erEi(−r)

r

, (4)

−∞dtet/t.

We also fix the density ¯ ρ as ¯ ρ = 4.436. The temperature T is regarded as a control

(5)

where we have assumed that the statistical properties of density fluctuations are

symmetric with respect to spatial translations, rotations, and temporal translations.

Although the last assumption is not valid below the glass transition temperature, we

focus on the higher temperature regime, in which stationarity holds. Now, following

the motivation of this study, we wish to obtain a differential equation that determines

the time dependence of the correlation functionˆC(k,t). For this purpose, as a trial, let

us first consider the quantity ∂2ˆC(k,t)/∂t2. This quantity can be expressed in terms

ofˆC(k,t) and a four-point correlation function. Then, writing this equality, we obtain

an evolution equation forˆC(k,t) by making the simplest approximation of omitting

the third- and fourth-order cumulant terms of this four-point correlation function. The

equation obtained with this approximation represents a mean field theory in the sense

that fluctuation effects of Ψ are ignored when we consider its average value. As far

as we know, this naive approximation was not employed in previous studies. It might

be important to find some connection with more traditional approximations such as

a Kirkwood superposition approximation.

following evolution equation forˆC(k,t):

∂2ˆC(k,t)

∂t2

γγ

+¯ ρ2

γ2

?ˆU(k1)2(k · k1)2+ˆU(k1)ˆU(|k − k1|)(k · k1)(k · (k − k1))

with t > 0.Here, we impose the boundary conditions limt→∞ˆC(k,t) = 0 and

∂tˆC(k,t)|t=0+= −k2T ¯ ρ/γ, which can also be derived from (1).

Here, it is expected thatˆC(k,t) decays rapidly to zero in the limit k → ∞

and exhibits no singularity as a function of k for any t. Therefore, we can expand

ˆC(k,t) in terms of the set of Hermite functions φn(k), with n = 0,1,··· asˆC(k,t) =

?∞

?

With this approximation, we obtain the

=

?¯ ρ

ˆU(k) +T

?2

k4ˆC(k,t)

?

d3k1

(2π)3ˆC(k1,t)ˆC(|k − k1|,t)

?

,(6)

n=0Cn(t)φn(k). Then, (6) can be rewritten as

∞

GnmdCm(t)

m=0

dt

= Bn(t),(7)

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∞

?

m=0

GnmdBm(t)

dt

=

∞

?

m=0

Cm(t)Lnm+

∞

?

m,l=0

MnmlCm(t)Cl(t). (8)

Here, we have introduced the new variable Bn(t) in order to obtain a first-order

differential equation.The coefficients Lnm and Mnml in (8) are determined by the

system parameters. For example, Lnmis calculated as

Lnm=¯ ρ2

γ2

?∞

−∞

dkφn(k)φm(k)

?

ˆU(k) +T

¯ ρ

?2

, (9)

while Gnmin (7) and (8) is given by

Gnm= −

?∞

−∞

dk

1

k2+ Λ2φn(k)φm(k),(10)

where we have introduced a cutoff parameter Λ in order to allow expansion of the

inverse of the Laplacian in terms of the Hermite functions. We take the limit Λ → 0

after the transition temperature is calculated for a system with Λ fixed. In the actual

computations, checking the temperature in the cases Λ2= 10−2, 10−3, and 10−4, we

found that Λ2= 10−3is a sufficiently small value. The numerical values appearing in

the analysis hereafter were obtained using Λ2= 10−3.

Now we describe a further approximation. Setting

u = (C0,C1,···,CK−1,B0,···,BK−1), (11)

we approximate (7) and (8) by the following ordinary differential equation:

ˆGdu

dt=ˆΣu +ˆΘ(u,u). (12)

Here, the 2K × 2K matricesˆG andˆΣ are defined asˆGnm =ˆGK+n,K+m= Gnmand

ˆΣn,K+m= δnm,ˆΣK+n,m= Lnm, for 1 ≤ n,m ≤ K (The other components are zero).

Also,ˆΘ is determined from Mnmland represents a map from two copies of u to a vector.

We chose K = 100 for the analysis reported below. (We checked the K dependence of

Lnm, Gnmand Mnml, and confirmed that K = 100 is sufficiently large.)

3. Analysis

We investigate the system behavior as we change the temperature T. We first focus

on the behavior of solution trajectories of (12) near the origin, u = 0. When |u| is

sufficiently small, this behavior is approximated by the linear equation obtained by

omitting the nonlinear term in (12). Then, calculating the eigenvalues and eigenvectors

ofˆΣ, we can classify the solution trajectories. In the high temperature limit, from

the form of Lnm, it is easily found that all trajectories either approach or move away

from the origin exponentially, because the eigenvalues consist of K pairs of positive and

negative numbers. We found in the numerical computations that when the temperature

is decreasing there exists a temperature T0below which one pair of eigenvalues becomes

zero. (We have T0=19.535 in the example we study.) Thus, the solution trajectories

in the 2K-dimensional phase space undergo qualitative change when the temperature