Competition Between Energy and Dynamics in Memory Formation Varda F. HaghzChloe W. LindemanzyChi Ian Ip and Sidney R. Nagel Department of Physics and The James Franck and Enrico Fermi Institutes

2025-04-29 0 0 917.23KB 6 页 10玖币

侵权投诉

Competition Between Energy and Dynamics in Memory Formation

Varda F. Hagh‡,∗Chloe W. Lindeman‡,†Chi Ian Ip, and Sidney R. Nagel

Department of Physics and The James Franck and Enrico Fermi Institutes

The University of Chicago, Chicago, IL 60637, USA.

‡Equal Contribution

(Dated: October 25, 2022)

Bi-stable objects that are pushed between states by an external ﬁeld are often used as a simple

model to study memory formation in disordered materials. Such systems, called hysterons, are

typically treated quasistatically. Here, we generalize hysterons to explore the eﬀect of dynamics

in a simple spring system with tunable bistability and study how the system chooses a minimum.

Changing the timescale of the forcing allows the system to transition between a situation where its

fate is determined by following the local energy minimum to one where it is trapped in a shallow

well determined by the path taken through conﬁguration space. Oscillatory forcing can lead to

transients lasting many cycles, a behavior not possible for a single quasistatic hysteron.

I. INTRODUCTION

The ability of a physical system to store information

about how it was prepared — memory — is now rec-

ognized as being crucial for the behavior in a large va-

riety of disordered materials [1]. Jammed packings of

soft spheres subjected to repeated cycles of shear, cycli-

cally crumpled sheets of paper, and interacting spins in

an oscillating magnetic ﬁeld all form memories of how

they were trained [2–7]. Memory in such systems hinges

on the ability to learn a pathway between metastable

states of the energy landscape. It has been likened to the

memory seen in a collection of bi-stable elements, called

hysterons, which ﬂip between states when an external

ﬁeld is raised above or below a critical value as shown

in Fig. 1(a) [8–10]. Although an enormous simpliﬁcation

from the original materials, ensembles of hysterons are

able to capture some features of the memory formation

seen in complex systems surprisingly well [1, 10, 11].

However, such hysterons fail to capture certain features

of real systems [10, 12–14]. As long as the hysterons are

independent, for example, the conﬁguration produced at

the end of the ﬁrst cycle is guaranteed to be the same

as that found after subsequent cycles of the same ampli-

tude (since each hysteron separately has this property).

By contrast, cyclically sheared packings can take many

cycles to train, and can even exhibit a multi-period re-

sponse [15] in which the periodicity of the response is

an integer multiple of the driving period as ﬁrst demon-

strated in systems with friction [16]. Recent work has

shown that generalizing the simple idea of a hysteron

as an independent two-state object by adding interac-

tions can result in long training times and multi-period

responses [12, 13].

Here, we generalize the behavior of hysterons in a dif-

∗vardahagh@uchicago.edu

†cwlindeman@uchicago.edu

ferent way: by studying the eﬀect of dynamics. Starting

from a two-spring conﬁguration that gives rise to a sym-

metric double-well potential, we add features one at a

time to uncover the criteria for landing in one basin or the

other. When a symmetry-breaking third spring is added,

the behavior is determined by a competition between the

timescale of applied forcing and the timescale of inherent

system dynamics that relaxes the system to lower en-

ergy. There is a crossover, depending on forcing velocity,

between energy-dominated and path-dominated selection

criteria. Following from this, for oscillatory driving we

ﬁnd a critical frequency which separates the two regimes.

Finally, we characterize the eﬀect of allowing the system

to age by slowly evolving the spring stiﬀnesses.

II. UNBUCKLED-TO-BUCKLED TRANSITION

FOR TWO SPRINGS

In two dimensions, two identical harmonic springs with

rest lengths `0and stiﬀnesses kcan be connected by a

single node to produce a bistable system as shown in

Fig. 1(b,c). The central-node location is the only variable

since the positions of the two outer nodes are explicitly

controlled. We restrict the motion of those outer nodes

to be symmetric about the x-axis so that the middle node

moves only in one dimension along x. The distance of a

given outer node from its position when the two outer

nodes are exactly 2`0apart is given by . The energy is

the sum of the spring energies:

E=k(px2+ (`0−)2−`0)2

which to ﬁrst order in and fourth order in xis:

E≈k1

4`2

x4−

x2.(1)

The energy is symmetric around x= 0, the center of

symmetry of the energy landscape. When  > 0, the

quartic and quadratic terms are of opposite sign and the

arXiv:2210.13341v1 [cond-mat.soft] 24 Oct 2022

energy is given by a bistable (double-well) potential. This

corresponds to a buckled conﬁguration. For  < 0, the

springs are stretched and there is only a single minimum.

2ℓ0x

x = 0

(b)

ε = 0.10

(c)

(d)

ε = 0.05

ε = - 0.05

ε = 0.05

fw = - 0.01

𝛾

𝛾-𝛾+

(a)

FIG. 1. (a) Double-well model of a hysteron. Under an

applied strain, γ, the landscape changes: one well disappears

at low strain and the other disappears at high strain. (b)

Schematic of the two-spring system. Outer nodes are shown

as black dots. Middle node is shown as a green circle. (c)

Energy versus position of the middle node. For large , the

two wells are deep. For small , the wells become smaller and

the minima are closer together; they coalesce when →0. For

≤0, there is only one minimum. (d) Three-spring system

with fw6= 0 showing energy versus middle-node position. In

the energy diagram, the force is pulling to the left (in the

negative x-direction).

We study the behavior of this system with over-

damped dynamics. The x-velocity of the middle node

is given by vm=f/β where fis the x-component of the

total force from all springs and βis a damping coeﬃ-

cient. In addition to the spring forces, an x-velocity of

the outer nodes, vo,x, will also inﬂuence the motion of the

middle node. We choose a reference frame in which the

outer nodes are stationary in the x-direction. By its deﬁ-

nition the position x= 0 then also remains stationary so

that the middle node moves with respect to x= 0 with

an additional velocity: ∆vm=−vo,x. This movement,

due solely to this motion of the origin, is in addition to

the velocity caused by the springs themselves. With over-

damped dynamics, this can be treated as if there were an

additional eﬀective force in the x-direction feﬀ =−βvo,x:

Evx≈k1

4`2

x4−

x2+βvo,xx. (2)

We probe the transition from one to two minima by

bringing the two outer nodes together in the y-direction

at a ﬁxed velocity vy. Starting from the stretched state

with one minimum, changes from negative to positive as

the two nodes approach one another, causing the initial

minimum to separate into two distinct minima. If the

motion is only along the y-direction (co-linear motion

of the two outer nodes), the system chooses a minimum

randomly (if there is any noise) or remains stuck in the

unstable equilibrium position x= 0.

However, if the outer nodes move in the x- as well as

the y-direction, the path of the outer nodes dictates into

which minimum the system will eventually come to rest.

One can see already that this two-state system is diﬀer-

ent from the conventional hysteron; the path of applied

boundary motion, not just the resulting energy landscape,

determines the conﬁguration of the system.

III. ADDITION OF A WEAK THIRD SPRING

One can break the symmetry of the two-spring system

by attaching a third, weak spring to the middle node so

that it applies an additional force along the x-axis, as

shown in Fig. 1(d). If the other end of the third spring is

pinned so that the equilibrium position is far away, the

force will be approximately independent of position, and

the modiﬁed energy will be given by

E≈k1

4`2

x4−

x2−fwx, (3)

where fwis the small force due to the weak spring. The

form of this equation is identical to that of Eq. 2, so we

can include any boundary-node motion in the x-direction

as an eﬀective weak spring force: feﬀ =fw−βvo,x.

This simple model of a hysteron has two distinct mech-

anisms that can compete to determine the eﬀective force.

If the eﬀective force is dominated by the weak spring,

then the behavior will be “energy dominated,” with the

landscape changing slowly enough that the energetics de-

termine which well is chosen. If, on the other hand, the

velocity of the boundary nodes dominates, then the be-

havior will be “path dominated”: the energy landscape

will change too quickly for the system to keep up with

the local minimum and so it will become trapped in a

state determined by the path of the boundaries.

We can make the crossover between energy-dominated

and path-dominated outcomes explicit by ﬁnding the

critical x-velocity of the boundary nodes that leads to

zero eﬀective force: feﬀ =fw−βvo,x = 0. If we start

with  < 0 at x= 0 and bring the nodes together at a

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：

CompetitionBetweenEnergyandDynamicsinMemoryFormationVardaF.Haghz,ChloeW.Lindemanz,yChiIanIp,andSidneyR.NagelDepartmentofPhysicsandTheJamesFranckandEnricoFermiInstitutesTheUniversityofChicago,Chicago,IL60637,USA.zEqualContribution(Dated:October25,2022)Bi-stableobjectsthatarepushedbetweenstatesbyanex...

展开>> 收起<<

Competition Between Energy and Dynamics in Memory Formation Varda F. HaghzChloe W. LindemanzyChi Ian Ip and Sidney R. Nagel Department of Physics and The James Franck and Enrico Fermi Institutes.pdf

共6页,预览2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！

Competition Between Energy and Dynamics in Memory Formation Varda F. HaghzChloe W. LindemanzyChi Ian Ip and Sidney R. Nagel Department of Physics and The James Franck and Enrico Fermi Institutes

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: