Non-ergodic transitions in many-body Langevin systems: a method of dynamical system reduction

Abstract

We study a non-ergodic transition in a many-body Langevin system. 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 average density 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. Comment: 8 pages, 1 figure; ver.3: Calculation errors have been fixed

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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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Non-ergodic transitions in many-body Langevin systems
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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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Non-ergodic transitions in many-body Langevin systems
5
∞
?
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