ARMA manuscript No.(will be inserted by the editor)

Cauchy-Born Rule and the Stability ofCrystalline Solids: Static Problems

W E1, PM2

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. The Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1. Simple and complex lattices . . . . . . . . . . . . . . . . . . . . . . . . . 62.2. Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3. Cauchy-Born rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4. Spectral analysis of the dynamical matrix . . . . . . . . . . . . . . . . . . 142.5. Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3. The Stability Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1. Stability condition for the continuum model . . . . . . . . . . . . . . . . . 193.2. Stability condition for the atomistic model . . . . . . . . . . . . . . . . . . 20

4. Local Minimizers for the Continuum Model . . . . . . . . . . . . . . . . . . . . 235. Asymptotic Analysis on Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.1. Asymptotic analysis on simple lattices . . . . . . . . . . . . . . . . . . . . 255.2. Asymptotic analysis on complex lattices . . . . . . . . . . . . . . . . . . . 29

6. Local Minimizer for the Atomistic Model . . . . . . . . . . . . . . . . . . . . . 34Appendix A. Detailed asymptotic analysis for one-dimensional model . . . . . . . . 47Appendix B. Elastic stiffness tensor for simple and complex lattices . . . . . . . . . 50

Abstract

We study the connection between atomistic and continuum models for the elas-tic deformation of crystalline solids at zero temperature. We prove, under certainsharp stability conditions, that the correct nonlinear elasticity model is given by theclassical Cauchy-Born rule in the sense that the elastically deformed states of theatomistic model are closely approximated by solutions of the continuum modelwith stored energy functionals obtained from the Cauchy-Born rule. The analy-sis is carried out for both simple and complex lattices, and for this purpose, wedevelop the necessary tools for performing asymptotic analysis on these lattices.

2 W. E, P. M

Our results are sharp and they also suggest criteria for the onset of instabilities ofcrystalline solids.

Key words. Atomistic models of solids, nonlinear elasticity, Cauchy-Born rule,stability of crystals

1. Introduction

This series of papers ([12], [28], [29] and [30]) are devoted to a mathematicalstudy of the connection between atomistic and continuum models of crystallinesolids at zero temperature. In the present paper, we study the simplest situationwhen classical potentials are used in the atomistic models, and when there are nodefects in the crystal. In this case the bridge between the atomistic and continuummodels is served by the classical Cauchy-Born rule [7]. Our main objective is toestablish the validity of the Cauchy-Born rule, for static problems in the presentpaper and for dynamic problems in the next paper [12]. In doing so, we also es-tablish a sharp criterion for the stability of crystalline solids under stress and thisallows us to study instabilities and defect formation in crystals [29].

The characteristics of crystalline solids can be summarized as follows:

1. Atoms in solids stick together due to the cohesive forces. Consequently theatoms in a crystal are arranged on a lattice. The origin of the cohesive forceand the choice of the lattice are determined by the electronic structures of theatoms. However, once the lattice is selected, its geometry has a profound influ-ence on the mechanical properties of the solid.

2. If the applied force is not too large, the crystal lattice deforms elastically torespond to the applied force. In this regime, the mechanical properties of thesolid are characterized mainly by its elastic parameters such as the elastic mod-uli.

3. Above a threshold, defects, such as dislocations, form in the crystal. The struc-ture of the defects are influenced largely by the geometry of the lattice. Butas we will see in subsequent papers [28] and [30], this is not always the case,more refined considerations about the nature of the bonding between atomsare sometimes necessary. In this regime, the mechanical properties of the solidare characterized by various barriers such as the Peierls stress for dislocationmotion.

This paper is concerned with the second point. In particular, we are interestedin how the atomistic and continuum models are related to each other in the elasticregime. Naturally there has been a long history of work on this topic, going back atleast to Cauchy who derived expressions for the linear elastic moduli from atom-istic pair potentials and the well-known Cauchy relations [21]. Modern treatmentbegan with the treatise of Born and Huang [7]. The basic result is the Cauchy-Born

Cauchy-Born Rule 3

rule (see Section 2 for details) which establishes a relation between atomistic andcontinuum models for the elastically deformed crystals. In the mathematics liter-ature, Braides, Dal Maso and Garroni studied atomistic models using the conceptof Γ-convergence [8], and proved that certain discrete functionals with pairwiseinteraction converge to a continuum model. One interesting aspect of their workis that their results allow for fractures to occur in the material (see also the workof Truskinovsky [25]). Blanc, Le Bris and Lions assumed that the microscopicdisplacement of the atoms follow a smooth macroscopic displacement field, andderived, in the continuum limit, both bulk and surface energy expressions fromatomistic models [6]. Their leading order bulk energy term is given by the Cauchy-Born rule. Friesecke and Theil examined a special lattice and spring model. Byextending the work on convexity for continuous functionals to discrete models,they succeeded quite remarkably in proving that in certain parameter regimes, theCauchy-Born rule does give the energy of the global minimizer in the thermody-namic limit. They also identified parameter regimes for which this statement failsand they interpreted this as being the failure of the Cauchy-Born rule [15].

This paper is devoted to the proof that the Cauchy-Born rule is always valid forelastically deformed crystals, as long as the right unit cell is used in formulatingthe Cauchy-Born rule. This statement is intuitively quite obvious. Indeed much ofthe work in this paper is devoted to the existence and characterization of elasticallydeformed states for the atomistic model, and this is where the stability conditions,which are the key conditions for our theorems, come in. However, to formulatethe right theorem, it is crucial to understand that elastically deformed states are ingeneral only local minimizers of the energy, not global minimizers. This observa-tion is not new (see for example [25]) and can be seen from the following simpleexample.

Consider a chain of N atoms on a line with positions x1, · · · , xN . Their totalpotential energy is given by

Ex1, · · · , xN =N∑

i=1

V0

( xi+1 − xi

ε

),

whereV0 = 4ε0(r−12 − r−6)

is the Lennard-Jones potential and ε is the equilibrium bond length. In the absenceof external loading, the equilibrium positions of the atoms are given by x j = jε. Wewill consider the case when the following condition of external loading is applied:The position of the left-most atom is kept fixed, the right-most atom is displacedby an amount that we denote as D0. To have a finite elastic strain, D0 should scaleas D0 ∼ L = Nε.

There are two obvious solutions to this problem: The first is a uniformly de-formed elastic state: x j = j(ε + d) where d = D0/N. The energy of this state is

4 W. E, P. M

approximatelyE1 ∼ NV0(1 + D0/L),

f f f f f f undeformed state

f f f f f f elastically deformed state

f f f f f f fractured state

Fig. 1. A schematic figure for the one-dimensional chain example

The second solution is a fractured state: x j ' jε for j ≤ N−1 and xN = Nε+D0.The energy of this state is approximately:

E2 ∼ V0(1 + D0/ε).

Obviously for large N, the fractured state has less energy than the elastically de-formed state. This example indicates that elastically deformed states might only belocal minimizers at zero temperature. Fractured states have less energy, this phe-nomenon is already noticed in the literature [8] [25]. The reason that crystals donot fracture spontaneously under loading is that the energy barrier for fracture istoo high for real systems. Nevertheless this example indicates that we have to thinkabout local minimizers of the atomistic model in order to study elastic deformationof crystals.

The fact that we have to deal with local minimizers simplifies the analysis atzero temperature, but complicates the situation at finite temperature. In the lattercase, the right strategy is to prove that elastically deformed states are metastablestates. At the present time, this is still a difficult problem.

With these remarks, we can put previous results as well as the results obtainedin this paper into perspective. First of all, we understand that the counterexamplesconstructed by Friesecke and Theil are due to the instabilities of the lattice whichhave caused either the onset of plastic deformation, phase transformation, or melt-ing of the lattice. Their positive result that the Cauchy-Born rule gives the energyof global minimizers in the thermodynamic limit is quite profound, and extendingthat result to off-lattice models may help us to understand why crystals are crystalsin the first place. The work of Braides et al. also analyzes global minimizers. Theyhave realized the analytical consequence of the example discussed above and al-lowed fracture states in their set-up. Their results in high dimensions require theatomistic potential to satisfy the super-linear growth condition, a condition whichis rarely met in real solids. The approach that is closest to ours is perhaps that

Cauchy-Born Rule 5

of Blanc, Le Bris and Lions. The difference is that they assumed that the atomicdisplacement follows that of a macroscopically smooth vector field, whereas weprove that this is indeed the case under certain stability conditions. This differenceis best seen from a simple example. Consider the Lennard-Jones potential withnext nearest neighbor interaction on square and triangular lattices. As we showbelow, the stability conditions are satisfied by the triangular lattice but violated bythe square lattice. Therefore, from Theorem 2.2, we are able to conclude that theCauchy-Born rule is valid on the triangular lattice but not on the square lattice.In [17], we show that the square lattice is indeed unstable and spontaneous phasetransformation occurs. In contrast, the results of [6] are equally valid for the trian-gular lattice and the square lattice. From a technical viewpoint, we can view thepassage from the atomistic models to the continuum models as the convergence ofsome nonlinear finite difference schemes. The work of Blanc et. al is concernedwith consistency. Our work proves convergence. The basic strategy is the same asthat of Strang [23] for proving convergence of finite difference methods for nonlin-ear problems. Besides stability of the linearized problem, the other key componentis asymptotic analysis of the atomistic model. Since we have at hand a highly un-usual finite difference scheme, we have to develop the necessary tools for carryingout asymptotic and stability analysis in this setting. Indeed, much of the presentpaper is devoted exactly to that.

Since this is the first in this series of papers, we will discuss briefly the contentsof subsequent papers. In the next paper, we will extend the results of the presentpaper to dynamic problems. This will allow us to formulate the sharp stability cri-teria for crystalline solids under stress. [29] is a natural growth of the present paperand [12], in which we carry out a systematic study of the onset of instability andplastic deformation of crystals, including a classification of linear instabilities andthe subsequent nonlinear and atomistic evolution of the crystal. [28] is devotedto the generalization of the classical Peierls-Nabarro model, which is a model ofdislocations that combines an atomistic description on the slip surface and a con-tinuum description of the linear elastic deformation away from the slip surface.The generalized Peierls-Nabarro model allows us to study the core structure anddynamics of dislocations and the influence of the underlying lattice. Finally [30]considers the generalization of the classical Cauchy-Born rule to low dimensionaland curved structures such as plates, sheets and rods, with applications to the me-chanical properties of carbon nanotubes.

One theme that we will emphasize throughout this series of papers is the in-terplay between the geometric and physical aspects of crystalline solids. As wesaid earlier, the geometry of the lattice has a profound influence on the physicalproperties of the crystal, such as the onset of plastic deformation, the core structureand the slip systems of dislocations, and the nature of the cracks. However, this isnot the whole story. There are also examples of properties of solids which are notreflected at the level of geometry and have to be understood at the level of physics,

6 W. E, P. M

e.g. the nature of the bonding between atoms. Some of these issues are discussedin [28].

Part results of this paper has been announced in [11].

2. The Generalities

We will begin with a brief discussion on atomic lattices and the atomistic po-tentials of solids.

2.1. Simple and complex lattices

Atoms in crystals are normally arranged on lattices. Common lattice struc-tures are body-centered cubic (BCC), face-centered cubic (FCC), diamond lattice,hexagonal closed packing (HCP), etc [3]. Under normal experimental conditions,i.e. room temperature and pressure, iron (Fe) exists in BCC lattice, aluminum (Al)exists in FCC lattice, and silicon (Si) exists in diamond lattice.

Lattices are divided into two types: simple lattices and complex lattices.Simple lattices. Simple lattices are also called Bravais lattices. They take the

form:

L(ei, o) = x | x =d∑

i=1

ν iei + o ν i are integers , (2.1)

where eidi=1 are the basis vectors, d is the dimension, and o is a particular latticesite, which can be taken as the origin, due to the translation invariance of lattices.The basis vectors are not unique.

Out of the examples listed above, BCC and FCC are simple lattices. For FCClattices, one set of basis vectors are

e1 =ε

2(0, 1, 1), e2 =

ε

2(1, 0, 1), e3 =

ε

2(1, 1, 0).

For BCC lattices, we may choose

e1 =ε

2(−1, 1, 1), e2 =

ε

2(1,−1, 1), e3 =

ε

2(1, 1,−1).

as the basis vectors. Here and in what follows, we use ε to denote the lattice con-stant.

Another example of a simple lattice is the two-dimensional triangular lattice.Its basis vectors can be chosen as:

e1 = ε(1, 0), e2 = ε(1/2,√

3/2).

Complex lattices. In principle, any lattices can be regarded as a union of con-gruent simple lattices [13], i.e. they can be expressed in the form:

L = L(ei, o) ∪ L(ei, o+ p1) ∪ · · · L(ei, o+ pk)

Cauchy-Born Rule 7

for certain integer k, p1, · · · , pk are the shift vectors. For example, the two di-mensional hexagonal lattice with lattice constant ε can be regarded as the unionof two triangular lattices with shift vector p1 = ε(−1/2,−

√3/6). The diamond

lattice is made up of two interpenetrating FCC lattices with shift vector p1 =

ε/4(1, 1, 1). The HCP lattice is obtained by stacking two simple hexagonal lat-tices with the shift vector p1 = ε(1/2,

√3/6,

√6/3). Some solids consist of more

than one species of atoms. Sodium-chloride (NaCl), for example, has equal num-ber of sodium ions and chloride ions placed at alternating sites of a simple cubiclattice. This can be viewed as the union of two FCC lattices: one for the sodiumions and one for the chloride ions.

In this paper, we focus on the case when k = 1. Generalization to high valuesof k is in principle straightforward but the technicalities can be quite tedious.

2.2. Potentials

We will restrict our attention to classical potentials of the form:

Ey1, · · · , yN = V(y1, · · · , yN) −N∑

j=1

f (x j)y j, (2.2)

where V is the interaction potential between the atoms, y j and x j are respectivelythe deformed and undeformed positions of the j-th atom. V often takes the form:

V(y1, · · · , yN) =∑

i, j

V2(yi/ε, y j/ε) +∑

i, j,k

V3(yi/ε, y j/ε, yk/ε) + · · · ,

where ε is the lattice constant as before.Examples of the potentials include:

1. Lennard-Jones potential:

V2(x, y) = V0(r) and V3 = V4 = · · · = 0,

where r = |x − y|, and

V0(r) = 4ε0(r−12 − r−6). (2.3)

Here ε0 is a constant.2. Embeded-atom methods: The Embeded-atom methods are introduced by Daw

and Baskes [9] [10] to model realistic metallic systems. The total energy con-sists of two parts: A function of the electron density and a term that accountsfor two-body interaction:

V =∑

i

F(ρi) +12

∑

i, j

V2(ri j/ε),

8 W. E, P. M

where ρi is the electron density around the i-th atom, and V2 is a pair potential,ri j = |x j − xi|. The density ρi is usually defined as

ρi =∑

j,i

f (ri j).

The functions f ,V2 and F are obtained empirically and calibrated by quantummechanical calculations.

3. Stillinger-Weber potential

V =12

∑

i, j

V2(ri j/ε) +13!

∑

i, j,k

V3(xi/ε, x j/ε, xk/ε),

where V2 is a pair potential and V3 is an angular term which usually takes theform:

V3(xi, x j, xk) = h(ri j, rik, θ jik),

whereh(ri j, rik, θ jik) = λe[γ(ri j−a)−1+γ(rik−a)−1](cos θ jik + 1/3)2

for some parameters λ and γ, θ jik is angle between x j − xi and xk − xi.4. Tersoff potential: Tersoff potential [24] is introduced to describe the open struc-

ture of covalently-bonded solids such as carbon and silicon. It takes the form:

V =12

∑

i, j

fC(ri j/ε)(fR(ri j/ε) + bi j fA(ri j/ε)

).

Here fC is a cut-off function and

fR(r) = A exp(−λ1r), fA(r) = −B exp(−λ2r).

The term bi j is a measure of local bond order,

bi j = (1 + βnξni j)(−1/2n),

where the function ξi j is given by

ξi j =∑

k,i, j

fC(rik/ε)g(θi jk) exp(λ33(ri j − rik)3)

with

g(θ) = 1 +c2

d2 −c2

d2 + (h − cos θ)2 .

The parameters A, B, λ1, λ2, λ3, β, n, c, d and h vary for different materials. Forexample, [24] recommended the following values for carbon:

A = 1393.6eV, B = 346.74 eV, λ1 = 3.4879Å, λ2 = 2.2119Å,β = 1.5724 × 10−7, n = 0.72751, c = 38049, d = 4.3484,h = −0.57058.

Cauchy-Born Rule 9

Clearly different potentials are required to model different materials. In thispaper, we will work with general atomistic models, and we will make the followingassumptions on the potential functions V.

1. V is translation invariant.2. V is smooth in a neighborhood of the equilibrium state.3. V is an even function.4. V has finite range and consequently we will consider only finite body interac-

tions.

To avoid complication in notations, we will sometimes only write out the three-body terms in the expressions for the potential. Extensions to general multi-bodyterms should be quite straightforward from the three-body terms.

At zero temperature, the atomistic model becomes a minimization problem:

miny1,··· ,yN

subject to certain boundary condition

Ey1, · · · , yN,

from which we determine the position of every atom. We define

u j = y j − x j

as the displacement of the j-th atom under the applied force.In the continuum model of solids, we describe the displacement by a vector

field u. Denote by Ω the domain occupied by the material in the undeformed state.The displacement field is determined by a variational problem:

∫

Ω

W(∇u(x)) − f (x) · u(x)

dx, (2.4)

subject to certain boundary conditions. Here W is the stored energy density, whichin general is a function of the deformation gradient ∇u. A very important questionis how to obtain W. In the continuum mechanics literature, W is often obtained em-pirically through fitting a few experimental parameters such as the elastic moduli.Here we will study how W can be obtained from the atomistic models.

For simplicity, we will concentrate on the case when the periodic boundarycondition is imposed over the material: The displacement is assumed to be thesum of a linear function and a periodic function, the linear part is assumed to befixed. Extending the analysis to non-periodic boundary conditions requires sub-stantial changes of the analysis, since new classes of instabilities may occur at theboundary.

There are two important length scales in this problem. One is the lattice con-stant. The other is the size of the material. Their ratio is a small parameter that wewill use in our estimates below.

10 W. E, P. M

2.3. Cauchy-Born rule

The Cauchy-Born (CB) rule is the standard recipe for computing the storedenergy function W from the atomistic potential V. We will denote such storedenergy function by WCB.

2.3.1. Cauchy-Born rule for simple lattice. First of all, let us fix the notations.We will fix one atom as the origin. All other atoms are viewed as translation of theorigin, and we denote the translation vector generically as s. In this way, we maywrite V2(s) = V2(0, s) and V3(s1, s2) = V3(0, s1, s2). We assume that V is zero ifone of the si is zero.

We let

D+` xi = xi+s` − xi, D−` xi = xi − xi−s` for ` = 1, 2, · · · , d,

where d is the dimension of the system. Clearly, D+`

and D−`

depend on s`. Butusing this simplified notations will not cause confusion.

For complex lattices, we need an additional notation. Assuming that the latticeis made up of two simple lattices, one with atoms of labelled by A and anotherwith atoms labelled by B, we let:

D+p xAi = xB

i − xAi .

The stored energy density WCB is a function of d × d matrices. Given a d × dmatrix A, WCB(A) is computed by first deforming an infinite crystal uniformly withdeformation gradient A, and then setting WCB(A) to be the energy of the deformedunit cell

WCB(A) = limm→∞

∑yi,y j,yk∈(I+A)L∩mD V(yi, y j, yk)

|mD| . (2.5)

Here D is an arbitrary open domain in Rd and L denotes the lattice L(ei, o) definedin (2.1).

The key point in (2.5) is that the lattice is uniformly deformed, i.e. no internalrelaxation is allowed for the atoms in mD. This is contrary to the definition ofenergy densities in Γ-limits (see [8] and [15]). It is easy to check that this definitionis independent of the choice of D.

The limit in (2.5) can be computed explicitly. For two-body potentials, we have

WCB(A) =1

2ϑ0

∑

sV2

((I + A)s

), (2.6)

where s runs over the ranges of the potential V2, ϑ0 is the volume of the unitcell. For simplicity of notations, we will omit the volume factor in subsequentpresentation.

Cauchy-Born Rule 11

In particular, if the atomistic model is a Lennard-Jones potential on a one-dimensional simple lattice, we have

WCB(A) =ζ2(6)ζ(12)

(14|1 + A|−12 − 1

2|1 + A|−6

), (2.7)

where ζ(·) is the Riemann-zeta function. See Fig. 2.For three-body potentials, we have

WCB(A) =∑

〈 s1,s2 〉

13!

V3((I + A)s1, (I + A)s2

). (2.8)

For general many-body potentials,

WCB(A) =∑

m=2

1m!

∑

〈 s1,··· ,sm−1 〉Vm((I + A)s1, · · · , (I + A)sm−1). (2.9)

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Fig. 2. Solid line: Stored energy density obtained from the Lennard-Jones potential via CBrule in terms of 1+A. Dotted line: the original Lennard-Jones potential function in terms ofr

The variational operator for WCB is:

div(DAWCB(∇u)

)=

∑

〈 s1,s2 〉

(∂2α1

V(s1 · ∇)2u + 2∂α1α2 V(s1 · ∇)(s2 · ∇)u

+ ∂2α2

V(s2 · ∇)2u), (2.10)

where ∂2α1

V, ∂α1α2 V and ∂2α2

V are all evaluated at(s1 + (s1 · ∇)u, s2 + (s2 · ∇)u

).

12 W. E, P. M

2.3.2. Cauchy-Born rule for complex lattice For a complex lattice, we first as-sociate with it a Bravais sublattice so that the unit cell generated by the basisvectors coincides with the unit cell of the complex lattice. The remaining latticepoints are treated as internal degrees of freedom, denoted by p. These are the shiftvectors. To compute WCB(A), we deform the Bravais sublattice uniformly with de-formation gradient A. We then relax the internal degrees of freedom keeping theposition of the deformed Bravais lattice fixed. This gives

WCB(A) = minp

W(A, p), (2.11)

where

W(A, p) = limm→∞

1|mD|

∑[V(yi + p, y j + p, yk + p) + V(yi, y j, yk)

]. (2.12)

Here the summation is carried out for yi, y j, yk ∈ (I + A)L ∩ mD.We will give two specific examples of WCB for complex lattices. First we con-

sider a one-dimensional chain with two alternating species of atoms A and B, withpairwise interactions. We assume that the interaction potential between A atomsis VAA, the interaction potential between B atoms is VBB, and the interaction po-tential between A and B atoms is VAB. Denote the shift of a B atom from its leftneighboring A atom by p. Then WCB(A) = minp W(A, p) with

W(A, p) =∑

j∈Z

(VAB

((1 + A)( j + 1)ε − p

)+ VAB

((1 + A) jε + p

))

+∑

j∈Z

(VAA

((1 + A) jε

)+ VBB

((1 + A) jε

)),

where ε is the lattice constant. Observe that for any A, W(A, p) is symmetric withrespect to (1+A)ε/2, therefore p = (1+A)ε/2 is either a local maximum or a localminimum of W(A, p). In the latter case, we have

WCB(A) =∑

j∈Z

(VAA

((1 + A) jε

)+ VBB

((1 + A) jε

)

+ 2VAB((1 + A)( j + 1/2)ε

))(2.13)

at that local minimum.Next we consider the hexagonal lattice. We again assume that there are two

species of atoms, A and B, located at the open and filled circles in Fig. 3, respec-tively. As in the one-dimensional case, there are three terms in W(A, p):

W(A, p) = WAA(A) +WAB(A, p) +WBB(A)

withWκκ(A) =

∑

〈 s1,s2 〉Vκκ((I + A)s1, (I + A)s2) κ = A or B,

Cauchy-Born Rule 13

A

B

Fig. 3. Hexagonal Lattice. Two species of atoms: Atom A and atom B.

and

WAB(A, p) =∑

〈 s1,s2 〉

[VAB((I + A)s1 + p, (I + A)s2 + p)

+ VAB((I + A)s1 + p, (I + A)s2)

+ VAB((I + A)s1, (I + A)s2 + p)].

A special case of this example is the graphite sheet for carbon. In that case, thereis only one specie of atoms. Hence VAA,VBB and VAB are all equal.

We next derive the Euler-Lagrange equations in this case. There are two setsof Euler-Lagrange equations. The first comes from the minimization with respectto the internal degree of freedom p, i.e. (2.11), which reads:

∂pWAB(A, p) = 0, (2.14)

namely,∑

〈 s1,s2 〉

[(∂α1 + ∂α2 )VAB((I + A)s1 + p, (I + A)s2 + p)

+ ∂α1 VAB((I + A)s1 + p, (I + A)s2)

+ ∂α2 VAB((I + A)s1, (I + A)s2 + p)]= 0. (2.15)

The second Euler-Lagrange equation comes from the minimization problem (2.4),

div(DAWCB(∇u)

)= f , (2.16)

where DAWCB(A) = DA(WAA(A) +WAB(A, p) +WBB(A)

), and for κ, κ′ = A or B,

div(DAWκκ′(∇u)

)=

∑

〈 s1,s2 〉

2∑

i, j=1

∂2αiα j

Vκκ′(si · ∇)(s j · ∇)u,

14 W. E, P. M

where ∂2αiα j

Vκκ′(κ, κ′ = A or B) is evaluated at(s1 + (s1 · ∇)u, s2 + (s2 · ∇)u

).

It follows from the Hellmann-Feynman theorem [27] that

div(DAWAB(∇u)

)

=∑

〈 s1,s2 〉

2∑

i, j=1

∂2αiα j

VAB(si · ∇)(s j · ∇)u

+∑

〈 s1,s2 〉

[(∂2α1

(V1AB + V2

AB) + ∂α1α2 (V1AB + V3

AB))(s1 · ∇)(DA p · ∇)u

+(∂α1α2 (V1

AB + V2AB) + ∂2

α2(V1

AB + V3AB)

)(s1 · ∇)(DA p · ∇)u

],

where VAB = V1AB + V2

AB + V3AB with

V1AB = VAB

(s1 + (s1 · ∇)u + p(∇u), s2 + (s2 · ∇)u + p(∇u)

),

V2AB = VAB

(s1 + (s1 · ∇)u + p(∇u), s2 + (s2 · ∇)u

),

V3AB = VAB

(s1 + (s1 · ∇)u, s2 + (s2 · ∇)u + p(∇u)

),

where p is obtained from the algebraic equations (2.14).

2.3.3. The elastic stiffness tensor For a given stored energy function WCB, theelastic stiffness tensor can be expressed as

Cαβγδ =∂2WCB

∂Aαβ∂Aγδ(0) 1 ≤ α, β, γ, δ ≤ d.

If WCB is obtained from a pairwise potential, we have

Cαβγδ =∑

s

(V ′′(|s|)|s|−2 − V ′(|s|)|s|−3)sαsβsγsδ, (2.17)

where the summation is carried out for all s = (s1, · · · , sd). The above formulais proven in Lemma 3.2. Discussions for the more general cases are found in Ap-pendix B.

2.4. Spectral analysis of the dynamical matrix

A lot can be learned about the lattice statics and lattice dynamics from thephonon analysis, which is the discrete Fourier analysis of lattice waves at the equi-librium state. This is standard material in textbooks on solid state physics (see forexample [3]). Since we need some of the terminologies, we will briefly discuss asimple example of a one-dimensional chain.

Consider the following example:

Md2y j

dt2 = −∂V∂y j= V ′(y j+1 − y j) − V ′(y j − y j−1),

Cauchy-Born Rule 15

where M is the mass of the atom. Let y j = jε + y j, linearizing the above equation,we get

Md2y j

dt2 = V ′′(ε)(y j+1 − 2y j + y j−1). (2.18)

Let y j(k) = ei(k x j−ω t), we obtain

ω2(k) =4M

V ′′(ε) sin2 kε2,

where k = 2π`Nε with ` = −[N/2], · · · , [N/2].

For the more general case, it is useful to define the reciprocal lattice, which isthe lattice of points in the k-space that satisfy ei k· x j = 1 for all x j ∈ L. The firstBrillouin zone in the k-space is defined to be the subset of points that are closer tothe origin than to any other point on the reciprocal lattice.

For complex lattices, the phonon spectrum contains both acoustic and opticalbranches [3], which will be denoted by ωa and ωo respectively.

What we really need in the present work is the spectral analysis of the dynam-ical matrix, which is the matrix defined by the right-hand side of (2.18). For thesimple example discussed above, we consider the eigenvalue problem:

ω2y j = −∂V∂y j= V ′(y j+1 − y j) − V ′(y j − y j−1),

and

ω2(k) = 4V ′′(ε) sin2 kε2.

The difference between the phonon spectrum and the spectrum of the dynamicalmatrix lies in the mass matrix. If there are only one specie of atoms, the massmatrix is a scalar matrix. In this case, the two spectra are the same up to a scalingfactor. If there are more than one species of atoms, then the two spectra can bequite different. But they are still closely related [18]. In particular, the spectrumof the dynamical matrix will have acoustic and optical branches, which will bedenoted by ωa and ωc respectively.

2.5. Main results

Let Ω be a bounded cube. For any nonnegative integer m and positive integerk, we denote by Wk,p(Ω;Rm) the Sobolev space of mappings y: Ω → Rm suchthat ‖y‖Wk,p < ∞ (see [1] for the definition). In particular, Wk,p

# (Ω;Rm) denotes theSobolev space of periodic functions whose distributional derivatives of order lessthan k are in the space Lp(Ω). We write W1,p(Ω) for W1,p(Ω;R1) and H1(Ω) forW1,2(Ω).

Summation convention will be used. We will use | · | to denote the absolutevalue of a scalar quantity, the Euclidean norm of a vector and the volume of a set.

16 W. E, P. M

In several places we denote by ‖ · ‖`2 the `2 norm of a vector to avoid confusion.For a vector v, ∇v is the tensor with components (∇v)i j = ∂ jvi; for a tensor field S,div S is the vector with components ∂ jS i j. Given any function y: Rd×d → R, wedefine

DA y(A) =( ∂y∂Ai j

)and D2

A y(A) =( ∂2y∂Ai j∂Akl

),

where Rm×n denotes the set of real m × n matrices. We also define Rm×n+ as the set

of real m × n matrices with positive determinant. For a matrix A = ai j ∈ Rm×n,we define the norm ‖A‖: = (∑m

i=1∑n

j=1 a2i j)1/2.

For any p > d and m ≥ 0 define

X: = v ∈ Wm+2,p(Ω;Rd) ∩ W1,p# (Ω;Rd) |

∫

Ω

v = 0 ,

and Y: = Wm,p(Ω;Rd).Given the total energy functional

I(v): =∫

Ω

WCB(∇v(x)) − f (x) · v(x)

dx, (2.19)

where WCB(∇v) is given by (2.5) or (2.11) with A = ∇v, we seek a solution u − B ·x ∈ X such that

I(u) = minv−B·x∈X

I(v).

The Euler-Lagrange equation of the above minimization problem is:L(v): = − div

(DAWCB(∇v)

)= f in Ω,

v − B · x is periodic on ∂Ω.(2.20)

Our main assumption is the following:

Assumption A: There exist two constants Λ1 and Λ2, independent of ε, such thatthe acoustic and optical branches of the spectrum of the dynamical matrix satisfy

ωa(k) ≥ Λ1|k|, (2.21)

andωo(k) ≥ Λ2/ε, (2.22)

respectively, where k is any vector in the first Brillouin zone.In the next section, we will discuss where the scaling factor ε in (2.22) comes

from. In subsequent papers [17] and [29], we will show that if Assumption A isviolated, then results of the type (2.23) cease o be valid. Therefore, Assumption Ais not only sufficient for Theorems 2.1, 2.2 and 2.3, but also essentially necessary.

Our main results are:

Cauchy-Born Rule 17

Theorem 2.1. If Assumption A holds and p > d,m ≥ 0, then there exist threeconstants κ1, κ2 and δ such that for any B ∈ Rd×d

+ with ‖B‖ ≤ κ1 and for anyf ∈ Y with ‖ f ‖Wm,p ≤ κ2, the problem (2.20) has one and only one solution uCB

that satisfies ‖uCB −B · x‖Wm+2,p ≤ δ, and uCB is a W 1,∞-local minimizer of the totalenergy functional (2.19).

Theorem 2.2. If Assumption A holds and p > d,m ≥ 6, then there exist twoconstants M1 and M2 such that for any B ∈ Rd×d

+ that satisfies ‖B‖ ≤ M1 and forany f ∈ Y with ‖ f‖Wm,p ≤ M2, the atomistic model (2.2) has a local minimizer yε

that satisfies‖ yε − yCB ‖d ≤ Cε, (2.23)

where yCB = x + uCB(x). The norm ‖ · ‖d is defined for any z ∈ R2N×d as

‖ z ‖d: = εd/2(zT H0 z)1/2, (2.24)

where H0 is the Hessian matrix of the atomistic potential at the undeformed state.

We will see later that ‖ · ‖d is a discrete analog of the H1 norm (c.f. Lemma6.4).

Theorem 2.3. Under the same condition of Theorem 2.2, if the crystal lattice is asimple lattice, then (2.23) can be improved to

‖ yε − yCB ‖d ≤ Cε2. (2.25)

3. The Stability Condition

In this section, we will show that our Assumption A implies that W(A, p)satisfies a generalized Legendre-Hadamard condition. We will also discuss explicitexamples of the stability conditions. This allows us to appreciate the differencebetween the results of Blanc et al. and the results of the present paper. Basically,the results of Blanc et al. concern consistency between the atomistic model and theCauchy-Born rule in the continuum limit. Our results concern convergence.

Lemma 3.1. If Assumption A is valid, then W(A, p) satisfies the generalized Legendre-Hadamard condition at the undeformed configuration: There exist two constantsΛ1 and Λ2, independent of ε, such that for all ξ, η, ζ ∈ Rd ,

(ξ ⊗ η, ζ)

D2

AW(0, p0) DApW(0, p0)

DpAW(0, p0) D2pW(0, p0)

(ξ ⊗ ηζ

)≥ Λ1|ξ|2|η|2 + Λ2|ζ|2, (3.1)

where p0 is the shift vector at the undeformed configuration.

Proof. We first note the following equivalent form of (3.1), we call it AssumptionB.

18 W. E, P. M

– D2p W(0, p0) is positive definite.

– WCB satisfies the Legendre-Hadamard condition at the undeformed configura-tion:

D2AWCB(0)(ξ ⊗ η, ξ ⊗ η) ≥ Λ|ξ|2|η|2 for all ξ, η ∈ Rd.

The equivalence between (3.1) and Assumption B is a consequence of thefollowing simple calculation: At A = 0 and p = p0, we have

(I −DApW

[D2

pW]−1

0 I

) D2

AW DApW

DpAW D2pW

(I 0

−[D2pW

]−1DpAW I

)

=

(D2

AW − DApW[D2

pW]−1DpAW 0

0 D2pW

)=

(D2

AWCB 00 D2

pW

).

Next we prove Assumption A implies Assumption B.For any a, b, c ∈ Rd, define

g(λ): = WCB(a ⊗ bλ) and G(λ, µ) = W(a ⊗ bλ, cµ). (3.2)

Obviously, we only need to prove that if Assumption A is valid, then g(λ) is locallyconvex at λ = 0 and G(λ, µ) is locally convex at (λ, µ) = (0, 0). Without loss ofgenerality, we only consider the three-body potential.

For simple lattices, we have

g′′(0) =∑

〈 s1,s2 〉

∑

1≤i< j≤2

[ ∂2V3

∂αi∂α j(s1, s2)

](a, a)(b · si)(b · s j) = zT H0 z > 0

by Assumption A, where z = (b · s)a with s equals s1 or s2 which runs throughall pairs 〈 s1, s2 〉.

For complex lattices, we have

H =

∂2G∂λ2 (0, 0) ∂2G

∂λ∂µ(0, 0)

∂2G∂λ∂µ

(0, 0) ∂2G∂µ2 (0, 0)

with

H11 =∑

〈 s1,s2 〉

∑

1≤i≤ j≤2

[ ∂2V3

∂αi∂α j(s1, s2)

](a, a)b · sib · s j,

H12 = H21 =∑

〈 s1,s2 〉

2∑

i=1

[∂2V3

∂α2i

(s1, s2) +∂2V3

∂α1α2(s1, s2)

](a, c)b · si,

and

H22 =∑

〈 s1,s2 〉

∑

1≤i≤ j≤2

[ ∂2V3

∂αi∂α j(s1, s2)

](c, c).

Cauchy-Born Rule 19

It is seen thatH = zT H0 z > 0

by Assumption A, where z = (z, c, · · · , c︸ ︷︷ ︸N

).

In terms of the elastic stiffness tensor, the second condition of Assumption Bcan also be rewritten as:

C(ξ ⊗ η, ξ ⊗ η) ≥ Λ|ξ|2|η|2 for all ξ, η ∈ Rd . (3.3)

3.1. Stability condition for the continuum model

We shall prove that (3.3) is valid for the triangular lattice and fails for thesquare lattice with Lennard-Jones potential. We write Lennard-Jones potential as

V(r) = 4ε0[(σ

r

)12−

(σr

)6]. (3.4)

The following lemma simplifies the expression of the elastic stiffness tensorby exploiting the symmetry property of the underlying lattices.

Lemma 3.2. The elastic stiffness tensor C is of the form:

Cαβγδ =∑

s

(V ′′(|s|)|s|2 − V ′(|s|)|s|)|s|−4sαsβsγsδ. (3.5)

Proof. A direct calculation gives

Cαβγδ =∑

s

(V ′′(|s|)|s|2 − V ′(|s|)|s|))|s|−4sαsβsγsδ

+∑

sV ′(|s|)|s|−1δαγsβsδ.

UsingDAWCB(0) = 0, (3.6)

we have ∑

sV ′(|s|)|s|−1δαγsβsδ = δαγDAWCB(0) = 0.

We thus conclude (3.5).

It is well-known that the elastic modulus takes the form (3.5). However, this isnot the case for the elastic stiffness tensor unless (3.6) is valid [27].

Using (3.5), we have

C1111 = C2222 =∑

s

(V ′′(|s|)|s|2 − V ′(|s|)|s|)|s|−4|s1|4,

C1122 = C1212 =∑

s

(V ′′(|s|)|s|2 − V ′(|s|)|s|)|s|−4|s1|2|s2|2.

(3.7)

20 W. E, P. M

For the triangular lattice, using the above lemma, we obtain

C(ξ ⊗ η, ξ ⊗ η) = C1111(ξ21η21 + ξ

22η

22) + 2C1122ξ1ξ2η1η2

+ C1122(ξ1η2 + ξ2η1)2

= (C1111 − 2C1122)(ξ21η21 + ξ

22η

22)

+ C1122[2(ξ1η1 + ξ2η2)2 + (ξ21η22 + ξ

22η

21)].

Using the explicit form of V (3.4), a straightforward calculation gives that

C1111 − 2C1122 > 0 and C1122 > 0.

Therefore, we have

C(ξ ⊗ η, ξ ⊗ η) ≥ (C1111 − 2C1122)(ξ21η21 + ξ

22η

22) + C1122(ξ21η

22 + ξ

22η

21)

≥ Λ|ξ|2|η|2

with Λ = min(C1111 − 2C1122,C1122) > 0.For the square lattice, let ξ1 = η2 = 0, we have

C(ξ ⊗ η, ξ ⊗ η) = C1122ξ22η

21 < 0

for any ξ2, η1 , 0 since C1122 < 0 by (3.7)2 and a direct calculation.If we only consider the nearest neighborhood interaction, we have C1122 =

C1212 = 0, this means that the shear modulus of the macroscopic model is zero.We refer to [29] for discussions on the manifestation of this instability.

3.2. Stability condition for the atomistic model

We will check Assumption A for the triangular and square lattices. Write

ω2(k) = 2λ(k). (3.8)

A straightforward calculation gives, in the case of triangular lattice with nearestneighbor interaction

λ(k) = a[α + β + γ − (

(α − β)2 + (β − γ)2 + (γ − α)2)1/2/√

2],

where a = V ′′(ε), and

α = sin2 π

Nk1, β = sin2 π

Nk2, γ = sin2 π

N(k1 − k2) (3.9)

with k = (k1, k2).If the next nearest neighbor interaction is taken into account, then

λ(k) = (a + b)(α + β + γ) + (c + d)(α + β + γ)

− 12([(a − b)(2α − β − γ) + (d − c)(2β − α − γ)]2

+ 3[(a − b)(β − γ) + (c − d)(α − γ)]2)1/2,

Cauchy-Born Rule 21

where α, β and γ are the same as (3.9) while

α = sin2 π

N(k1 + k2), β = sin2 π

N(−k1 + 2k2), γ = sin2 π

N(−2k1 + k2),

andb = V ′(ε)ε−1, c = V ′(

√3ε), d = V ′(

√3ε)(√

3ε)−1,

where

ε =(2

1 + 3−6

1 + 3−3

)1/6σ.

For the square lattice, if we only consider the nearest neighbor interaction, thenwe have

λ1(k) = 2aα, λ2(k) = 2aβ.

where α, β are defined earlier. Obviously, there does not exist a constant Λ suchthat

ω(k) ≥ Λ|k|.

If we take into account the next nearest neighbor interaction, then we have

λ(k) = (a + b)(α + β) + (e + f )(α + γ)

− ([(a − b)2(α − β)2 + (e − f )2(α − γ)2)1/2

,

wheree = V ′′(

√2ε), f = V ′(

√2ε)(√

2ε)−1,

and

ε =(2

1 + 2−6

1 + 2−3

)1/6σ.

From Fig. 4 we see Assumption A is satisfied by the triangular lattice butfails for the square lattice. Therefore, our results imply that the Cauchy-Born ruleis valid for the triangular lattice but not for the square lattice. Numerical resultsshow that this is indeed the case [17]. In contrast, the work of Blanc et al. does notdistinguish the two cases.

Next we turn to complex lattices. Again we will consider a one-dimensionalchain with two species of atoms A and B. We do not assume nearest neighborinteraction. The equilibrium equations for A and B are:

mAd2yA

i

dt2 = V ′AB(yBi − yA

i ) + V ′AA(yAi+1 − yA

i )

− V ′AB(yAi − yB

i−1) − V ′AA(yAi − yA

i−1),

mBd2yB

i

dt2 = V ′AB(yAi+1 − yB

i ) + V ′BB(yBi+1 − yB

i )

− V ′AB(yBi − yA

i−1) − V ′BB(yBi − yB

i−1).

22 W. E, P. M

010

2030

4050

0

10

20

30

40

500

5

10

15

20

25

30

35

010

2030

4050

0

10

20

30

40

50−5

0

5

10

15

20

25

30

35

40

Fig. 4. Spectrum of the dynamical matrix corresponding to the larger wave speed for trian-gular (left) and square (right) lattices with next nearest neighbor interaction.

Let yAi = iε + yA

i and yBi = iε + p + yB

i , linearizing the above equation, and usingthe Euler-Lagrange equation for optimizing with respect to the shift p, we obtain

mAd2yA

i

dt2 = V ′′AB(p)(yBi − yA

i ) − V ′′AB(ε − p)(yAi − yB

i−1)

+ V ′′AA(ε)(yAi+1 − 2yA

i + yAi−1),

mBd2yB

i

dt2 = V ′′AB(ε − p)(yAi+1 − yB

i ) − V ′′AB(p)(yBi − yA

i−1)

+ V ′′BB(ε)(yBi+1 − 2yB

i + yBi−1).

Let yAi = εAei(kiε−ω t) and yB

i = εBei(kiε−ω t), we get

M(k)(εA, εB)T = (0, 0)T

withM11 = mAω

2 − 4V ′′AA(ε) sin2 kε2 − V ′′AB(p) − V ′′AB(ε − p),

M12 = V ′′AB(ε − p)eik(ε−p) + V ′′AB(p)eikp,

M21 = V ′′AB(ε − p)e−ik(ε−p) + V ′′AB(p)e−ikp,

M22 = mBω2 − 4V ′′BB(ε) sin2 kε

2 − V ′′AB(p) − V ′′AB(ε − p).

Solving the equation det M(k) = 0 we get

ω2 = 2(V ′′AA(ε)

mA+

V ′′BB(ε)mB

)sin2 kε

2+

( 1mA+

1mB

)V ′′AB(p) + V ′′AB(ε − p)2

±[(

2(V ′′BB(ε)

mB−

V ′′AA(ε)mA

)sin2 kε

2

+( 1mB− 1

mA

)V ′′AB(p) + V ′′AB(ε − p)2

)2

+

(V ′′AB(p) + V ′′AB(ε − p)

)2 − 4V ′′AB(p)V ′′AB(ε − p) sin2 kε2

mAmB

]1/2.

Cauchy-Born Rule 23

As |k| → 0, the acoustic branch tends to

ω2a =

4mA + mB

(V ′′AA(ε) + V ′′BB(ε) +

V ′′AB(p)V ′′AB(ε − p)V ′′AB(p) + V ′′AB(ε − p)

)sin2 kε

2,

and the optical branch tends to

ω2o =

( 1mA+

1mB

)(V ′′AB(p) + V ′′AB(ε − p)

).

Let mA = mB = 1, we obtain

ω2a → 2

(V ′′AA(ε) + V ′′BB(ε) +

V ′′AB(p)V ′′AB(ε − p)V ′′AB(p) + V ′′AB(ε − p)

)sin2 kε

2,

ω2o → 2

(V ′′AB(p) + V ′′AB(ε − p)

)

as |k| → 0. This example clearly verifies Assumption A.In the general case, the factor 1/ε in ωo is a consequence of scaling: If we take

the lattice constant to be O(1), then ωo = O(1). If we take the lattice constant to beO(ε), then V ′′(ε) = O(ε−2), which gives ωo = O(ε−1).

As we will see in § 6, Assumption A enables us to prove that an intrinsicdefined with the help of the Hessian matrix associated with the atomistic potentialis equivalent to a discrete H1−norm (see (6.19) for the definition of this discreteH1−norm).

In our analysis, it is sufficient to impose stability conditions on rank-one de-formations only. This is due to the fact that we have fixed the linear part of thedeformation gradient tensor through boundary conditions. If we allow the linearpart to vary, we have to impose additional stability conditions with respect to de-formations of higher rank. In this case, we need to require that the elastic modulitensor is positive definite. We refer to [20] for a discussion.

4. Local Minimizers for the Continuum Model

In this section, we prove Theorem 2.1. The proof is quite standard. The maintool is the implicit function theorem.

The linearized operator of L at u is defined by:

Llin(u)v = − div(D2

AWCB(∇u)∇v)

for any v ∈ W1,p# (Ω;Rd)

Proof for Theorem 2.1. For any p > d, define the map

T : Y × X → Rd with T ( f , v): = L(v + B · x) − f .

24 W. E, P. M

Since Ω is a cube, we may extend v to v such that v is defined over Rd and v|Ω = v.By Lemma 3.1, we have Assumption B, which together with Fourier transformleads to

∫

Ω

∇v · D2AWCB(0) · ∇v dx =

∫

Rd

∇v · C · ∇v dx

=

∫

Rd

d∑

α,β,γ,δ=1

Cαβγδvαvβ(2iπξγ)(2iπξδ)

≥ 4π2Λ

∫

Rd

| ˆv|2|ξ|2 dξ = Λ∫

Rd

|∇v|2 dx

≥ Λ∫

Ω

|∇v|2 dx, (4.1)

where v is the Fourier transform of v.Since

∫

Ω

v = 0, using Poincare’s inequality, we get

∫

Ω

∇v · D2AWCB(0) · ∇v dx ≥ C1‖v‖21, (4.2)

where C1 depends on Λ, c.f. C1 = Λπ2/(1 + π2). Define κ = min(C1/(2M), 1) with

M = max0≤t≤1‖A‖≤1|D3

AWCB(tA)|. If ‖B‖ ≤ κ, then

|D2AWCB(B) − D2

AWCB(0)| ≤ M‖B‖ ≤ C1/2.

Therefore, ∫

Ω

∇v · D2AWCB(B) · ∇v dx ≥ (C1/2)‖v‖21. (4.3)

Notice that T (0, 0) = 0. Standard regularity theory for elliptic systems (see [2])allows us to conclude that DvT (0, 0) is a bijection from X onto Y. Since p > d, weknow that Wk,p(Ω;Rd) is a Banach algebra [1] for any k ≥ 1, therefore, it is easy toverify that DAWCB is a C2 function from Rd×d

+ to Rd×d. It follows from the implicitfunction theorem [26, Appendix I] that there exist two constants R and r such thatfor all f satisfying ‖ f ‖Wm,p ≤ r, there exists one and only one solution v( f ) ∈ Xthat satisfies

T ( f , v( f )) = 0, ‖v( f )‖m+2,p ≤ R, (4.4)

and v(0) = 0. Finally we let uCB = v( f ) + B · x. It is clear that uCB satisfiesequation (2.20), uCB − B · x is periodic over ∂Ω and

‖uCB − B · x‖Wm+2,p ≤ R. (4.5)

Cauchy-Born Rule 25

Next we show that uCB is actually a local minimizer. Taylor expansion arounduCB and using (2.20) give

I(v) − I(uCB) =∫

Ω

∇(v − uCB) ·∫ 1

0(1 − t)D2

AW(∇ut) dt · ∇(v − uCB) dx, (4.6)

where ut = tv + (1 − t)uCB. It is clear that

∇ut − B = t∇(v − uCB) + ∇v( f ).

Therefore, there exist two constant κ and δ such that if ‖ f‖Lp ≤ κ and ‖v−uCB‖1,∞ ≤δ, then

∫

Ω

∇(v − uCB) · D2AW(∇ut) · ∇(v − uCB) dx ≥ (C1/4)‖v − uCB‖21 (4.7)

for any 0 ≤ t ≤ 1. It follows from the above inequality and (4.6) that

I(v) − I(uCB) ≥ (C1/4)‖v − uCB‖21.

This proves that uCB is a W1,∞−local minimizer of I. ut

5. Asymptotic Analysis on Lattices

In this section, we carry out asymptotic analysis on lattices. The results in thissection not only serve as a preliminary step for proving Theorem 2.2, but also haveinterests of their own. We refer to [19, Chapter 20] for related asymptotic analysis.

5.1. Asymptotic analysis on simple lattices

We first discuss asymptotic analysis on simple lattices. As we said earlier, with-out loss of generality, we will restrict our attention to the case when the potentialV is a three-body potential. The equilibrium equation at the site i is of the form:

Lε(yi) = −∂V∂yi= f (xi), (5.1)

where

Lε(yi) =∑

〈 s1,s2 〉

[∂α1 V(D+1 yi,D

+2 yi) − ∂α1 V(D−1 yi,D

+2 yi−s1

)

+ ∂α2 V(D+1 yi,D+2 yi) − ∂α2 V(D+1 yi−s2

,D−2 yi)].

Here the summation runs over all 〈 s1, s2 〉 ∈ L × L. In writing this expression, wehave paired the interaction at s1 and s2 directions.

26 W. E, P. M

xix

i−s

1x

i+s

1

xi−s

12x

i+s

2

xi−s

2x

i+s

12

Fig. 5. A schematic example for the atomic interactions on simple lattices

The plan is to carry out the analysis in two steps. The first is to approxi-mate (5.1) by differential equations. The second step is to carry out asymptoticanalysis on these differential equations.

Assuming that yi = xi + u(xi) and substituting it into the above equilibriumequations, collecting terms of the same order, we may write

Lε(yi) = L0(u(xi)) + εL1(u(xi)) + ε2L2(u(xi)) + O(ε3). (5.2)

Lemma 5.1. The leading order operator L0 is the same as the variational opera-tor for WCB.

Moreover, L1 = 0 and L2 is an operator in divergence form.

Proof. We may rewrite the operatorLε(yi) into

Lε(yi) =∑

〈 s1,s2 〉

[∂α1 V(D+1 yi,D

+2 yi) − ∂α1 V(D+1 yi−s1

,D+2 yi−s1)

+ ∂α2 V(D+1 yi,D+2 yi) − ∂α2 V(D+1 yi−s2

,D+2 yi−s2)]

=∑

〈 s1,s2 〉

(D−1∂α1 V + D−2∂α2 V

)(D+1 yi,D

+2 yi).

Denote by ∂α j V(D+1 yi,D+2 yi) = ∂α j Vi for j = 1, 2. For any smooth function ϕ(x)

satisfying the periodic boundary condition, after summation by parts, we have

N∑

i=1

Lε(yi)ϕ(xi) = −N∑

i=1

∑

〈 s1,s2 〉∂α1 ViD+1ϕ(xi) + ∂α2 ViD+2ϕ(xi). (5.3)

Fix i, for j = 1, 2, Taylor expansion at xi gives

D+j ϕ(xi) = (s j · ∇)ϕ(xi) +12

(s j · ∇)2ϕ(xi) + O(ε3),

D+j yi = s j + (s j · ∇)u(xi) + a j + b j + O(ε4),

Cauchy-Born Rule 27

where

a j =12

(s j · ∇)2u(xi) and b j =16

(s j · ∇)3u(xi).

In what follows, we omit the argument of u and V since u is always evaluatedat xi and V is always evaluated at

((s1 + (s1 · ∇)u(xi), s2 + (s2 · ∇)u(xi)

).

∂α j Vi = ∂α j V +((a1 + b1)∂α1 + (a2 + b2)∂α2

)∂α j V

+12((a1 + b1)∂α1 + (a2 + b2)∂α2

)2∂α j V + O(ε2).

Substituting the above four equations into (5.3) and gathering terms of thesame order, we obtain the expressions for the operators L0,L1 and L2:

N∑

i=1

L0(u(xi))ϕ(xi) = −N∑

i=1

∑

〈 s1,s2 〉

2∑

j=1

∂α j V(s j · ∇)ϕ(xi),

N∑

i=1

L1(u(xi))ϕ(xi) = −N∑

i=1

∑

〈 s1,s2 〉

2∑

j=1

[(a1∂α1 + a2∂α2 )∂α j V(s j · ∇)ϕ(xi)

+12∂α j V(s j · ∇)2ϕ(xi)

],

and

N∑

i=1

L2(u(xi))ϕ(xi)

= −N∑

i=1

∑

〈 s1,s2 〉

2∑

j=1

[(b1∂α1 + b2∂α2 )∂α j V(s j · ∇)ϕ(xi)

+12

(a1∂α1 + a2∂α2 )∂α j V(s j · ∇)2ϕ(xi)

+16∂α j V(s j · ∇)3ϕ(xi)

].

Passing to the limit, and integrating by parts, we have

∫

Ω

L0(u(x))ϕ(x) dx = −∫

Ω

∑

〈 s1,s2 〉

2∑

j=1

∂α j V(s j · ∇)ϕ dx

=∑

〈 s1,s2 〉

2∑

j=1

∫

Ω

div(∂α j V s j

)ϕ(x) dx,

28 W. E, P. M

which gives

L0(u) =∑

〈 s1,s2 〉

(∂2α1

V(s1 · ∇)2u + 2∂α1α2 V(s1 · ∇)(s2 · ∇)u

+ ∂2α2

V(s2 · ∇)2u).

We see that L0 is the same as the operator that appears in (2.10).The proof for the fact that the operatorL2 is of divergence form is similar.Since the lattice L is symmetric, i.e. if s ∈ L, then −s ∈ L, all first order deriva-

tives sum to zero. This gives L1 = 0. This can also be proved by a straightforwardbut tedious calculation.

Next we expand the solution

u = u0 + εu1 + ε2u2 + · · · .

Substituting it into (5.2), we obtain the equations for u0, u1 and u2. The equationfor u0 is simply the Euler-Lagrange equation (2.10), and u0 satisfies the sameboundary condition as for uCB. Therefore,

u0 = uCB.

For u1 and u2, we have

Lemma 5.2. u1 satisfiesLlin(u0)u1 = 0, (5.4)

and u2 satisfiesLlin(u0)u2 = −L2(u0). (5.5)

Moreover, if Assumption A holds, then u1 = 0 and there exists a function u2 ∈ Xthat satisfies (5.5).

Proof. A straightforward calculation gives

Llin(u0)u1 = −L1(u0) = 0.

Using Lemma 5.1, we get (5.4). Using (4.7) with t = 0, there exists a constant κsuch that if ‖ f‖Lp ≤ κ, then Llin is elliptic at uCB = u0. Therefore, u1 = 0.

A simple calculation gives

Llin(u0)u2 = −12

(δ2L0

δu2 (u0)u1

)u1 −

δL1

δu(u0)u1 − L2(u0) = −L2(u0),

which gives (5.5). It remains to prove that the right-hand side of (5.5) is orthogonalto a constant function, namely,

∫

Ω

L2(u0(x)) dx = 0. (5.6)

This is true since L2 is of divergence form, see Lemma 5.1.

Cauchy-Born Rule 29

As a direct consequence of Lemma 5.1 and Lemma 5.2, we have

Corollary 5.1. Definey = x + u0(x) + ε2u2(x). (5.7)

If f ∈ W6,p(Ω;Rd), then there exists a constant C such that

|Lε (y) − f | ≤ Cε3. (5.8)

Proof. Since f ∈ W6,p(Ω;Rd), using Theorem 2.1, we conclude that u0 ∈ W8,p(Ω;Rd).Therefore, u0 ∈ C7(Ω) by Sobolev imbedding theorem. This gives that u0 + ε

2u2 ∈C5(Ω). Therefore,

|Lε (y) − L0(u0 + ε2u2) − ε2L2(u0 + ε

2u2)| ≤ Cε3,

where the constant C depends on ‖u0‖C7(Ω). UsingL0(u0) = f and (5.5), we obtain

|L0(u0 + ε2u2) + ε2L2(u0 + ε

2u2) − f | ≤ Cε4,

where the constant C depends on ‖u0‖C4(Ω). A combination of the above two in-equalities leads to (5.8).

5.2. Asymptotic analysis on complex lattices

Assume that in equilibrium, the crystal consists of two types of atoms, A andB, each of which occupy a simple lattice. Let us express the equilibrium equationsfor atoms A and B in the form:

LAε (yA

i , yBi ) = f (xA

i ) and LBε (yA

i , yBi ) = f (xB

i ). (5.9)

We will make the following ansatz:

yAi = xA

i + u(xAi ),

yBi = yA

i + εv1(xAi ) + ε2v2(xA

i ) + ε3v3(xAi ) + ε4v4(xA

i ) + · · · .

Substituting this ansatz into (5.9), we obtain

LAε =

1εLA−1(u, v1) + LA

0 (u, v1, v2) + εLA1 (u, v1, v2, v3)

+ ε2LA2 (u, v1, v2, v3, v4) + O(ε3), (5.10)

LBε =

1εLB−1(u, v1) + LB

0 (u, v1, v2) + εLB1 (u, v1, v2, v3)

+ ε2LB2 (u, v1, v2, v3, v4) + O(ε3). (5.11)

Therefore,

LAε + LB

ε =1ε

(LA−1(u, v1) + LB

−1(u, v1)))+ LA

0 (u, v1, v2) + LB0 (u, v1, v2)

+ ε[LA

1 (u, v1, v2, v3) + LB1 (u, v1, v2, v3)

]

+ ε2[LA

2 (u, v1, v2, v3, v4) + LB2 (u, v1, v2, v3, v4)

]+ O(ε3).

We will show later that

30 W. E, P. M

1. LB−1 + LA

−1 = 0.2. vi+2 cancels out in the O(ε i) term for i ≥ 0.

Therefore we may write

12(LAε +LB

ε

)= L0(u, v1) + εL1(u, v1, v2)

+ ε2L2(u, v1, v2, v3) + O(ε3) (5.12)

with

L0(u, v1) =[LA

0 (u, v1, 0) + LB0 (u, v1, 0)

]/2,

L1(u, v1, v2) =[LA

1 (u, v1, v2, 0) + LB1 (u, v1, v2, 0)

]/2,

L2(u, v1, v2, v3) =[LA

2 (u, v1, v2, v3, 0) + LB2 (u, v1, v2, v3, 0)

]/2.

Next consider LAε − LB

ε , we obtain

LA−1(u, v1) − LB

−1(u, v1) = 0,

LA0 (u, v1, v2) − LB

0 (u, v1, v2) = −(p0 · ∇) f ,

LA1 (u, v1, v2, v3) − LB

1 (u, v1, v2, v3) = −12

(p0 · ∇)2 f ,

LA2 (u, v1, v2, v3, v4) − LB

2 (u, v1, v2, v3, v4) = −16

(p0 · ∇)3 f .

Observe that these are algebraic equations for v1, v2, v3 and v4 respectively. Theirsolvability will be proved in Lemma 5.3.

In the second step, we assume

u = u0 + εu1 + ε2u2 + · · · .

Substituting the above ansatz into (5.12), we obtain the equations satisfied byu0, u1 and u2:

L0(u0, v1) = 0, (5.13)

Llin(u0, v1)u1 = −L1(u0, v1, v2) +12

(p0 · ∇) f , (5.14)

Llin(u0, v1)u2 = −L2(u0, v1, v2, v3) − δL1

δA(u0, v1, v2)u1

− 12

(δ2L0

δA2 (u0, v1)u1

)u1 +

14

(p0 · ∇)2 f , (5.15)

where Llin(·, v1) is the linearized operator of L0 for fixed v1. We next relate theseequations with the Euler-Lagrange equations and show that they are solvable.

To carry out the details of this analysis, again we will work with the case whenV consists of three-body interactions only. It is easy to see how the argument canbe extended to the general case.

Cauchy-Born Rule 31

Depending on the type of atoms that participate in the interaction, we can groupthe terms of V into the following subsets: AAA, AAB, ABB and BBB. The AAA andBBB terms are treated in the same way as for simple lattices. Hence we will restrictour attention to the AAB and ABB terms.

xiA−s

1x

iA x

iA+s

1

xiA+p−s

1x

iA+p x

i+p+s

1

Fig. 6. A schematic illustration of the interactions between atoms on complex lattices forpair 〈 p, p− s 〉

Fix an A atom site xi. Consider its interaction with two B atoms at xi + pand xi + p − s1. As in the case of simple lattice, we pair this interaction with theinteractions between the atoms at AAB, and at ABB (see Fig. 6). Neglecting otherterms in Lε , we have

LAε (yA

i , yBi ):

=[∂α1 VAB(yB

i − yAi , y

Bi−s1− yA

i ) − ∂α1 VAB(yAi − yB

i , yAi+s1− yB

i )

+ ∂α2 VAB(yBi − yA

i , yBi−s1− yA

i ) − ∂α2 VAB(yAi−s1− yB

i−s1, yA

i − yBi−s1

)],

where the first and third terms come from the interaction of atoms at xi, xi+ p, xi+

p − s1, the second term comes from the interaction of atoms at xi, xi + p, xi + s1,and the last term comes from the interaction of atoms at xi, xi + p− s1, xi − x1.

Similarly for the B atom at the site xi + p, we have, corresponding to the inter-action pair shown in Fig. 6:

LBε (yA

i , yBi ):

=[∂α1 VAB(yA

i − yBi , y

Ai+s1− yB

i ) − ∂α1 VAB(yBi − yA

i , yBi−s1− yA

i )

+ ∂α2 VAB(yAi − yB

i , yAi+s1− yB

i ) − ∂α2 VAB(yBi+s1− yA

i+s1, yB

i − yAi+s1

)].

We may rewrite LAε and LB

ε into a more compact form as

LAε (yA

i , yBi ) = ∂α1 VAB(D+p yA

i ,D+p−s1

yAi ) + ∂α1 VAB(D+p yA

i ,D+p−s1

yAi+s1

)

+ ∂α2 VAB(D+p yAi ,D

+p−s1

yAi )

+ ∂α2 VAB(D+p yAi−s1,D+p−s1

yAi ),

and

LBε (yA

i , yBi ) = −∂α1 VAB(D+p yA

i ,D+p−s1

yAi ) − ∂α1 VAB(D+p yA

i ,D+p−s1

yAi+s1

)

− ∂α2 VAB(D+p yAi ,D

+p−s1

yAi+s1

)

− ∂α2 VAB(D+p yAi+s1,D+p−s1

yAi+s1

).

32 W. E, P. M

Let ϕ be a smooth periodic function, using summation by parts, we have

N∑

i=1

(LAε (yA

i , yBi ) +LB

ε (yAi , y

Bi )

)ϕi

=

N∑

i=1

(∂α1 VAB(D+p yA

i ,D+p−s1

yAi ) + ∂α2 VAB(D+p yA

i−s1,D+p−s1

yAi )

)D−s1ϕi.

Taylor expansion at xAi gives

D−s1ϕ = (s1 · ∇)ϕ − 1

2(s1 · ∇)2ϕ + O(ε3),

D+p yAi = εv1 + ε

2v2 + ε3v3 + ε

4v4 + O(ε5),

andD+p−s1

yAi = −(I + ∇u)s1 + εv1 + a1, D+p yA

i−s1= εv1 + b1,

where

a1 =12

(s1 · ∇)2u − ε(s1 · ∇)v1 + ε2v2 and b1 = −ε(s1 · ∇)v1 + ε

2v2.

The following lemma gives a characterization for the the differential operators LAi

and LBi .

Lemma 5.3. For i ≥ 0, the differential operators LAi (·, vi+2) and LB

i (·, vi+2) arealgebraic equations for the argument vi+2.

Moreover,

LA−1(u, v) + LB

−1(u, v) = 0 (5.16)

for any smooth functions u and v.

Proof. This lemma is a tedious but straightforward calculation. We will omit thedetails except to say that it is useful to note the following:

∂α j VAB(−x,−y) = −∂α j VAB(x, y) for j = 1, 2.

Lemma 5.4. The differential operator L0 is of the form:

L0(u, v1) = − div(∂α2 VAB(εv1,−s1 − (s1 · ∇)u + εv1)s1

). (5.17)

Moreover, it is the variational operator for WCB.

Proof. We omit the interaction between the same species. For the pair 〈 p, p− s1 〉,we consider the following term in WCB:

VAB(p,−(I + A)s1 + p) with A = ∇u.

Cauchy-Born Rule 33

Applying (2.16) to the pair 〈 p, p− s1 〉, we get the differential operator correspond-ing to this pair:

L(u, v1) = −∂2α2

VAB[(s1 · ∇)2u − (s1 · ∇)p

] − ∂α1α2 VAB(s1 · ∇)p

= − div(∂α2 VAB(p,−(I + A)s1 + p) · s1

),

which is the same as the corresponding term in the equation (5.17).

Lemma 5.5. All higher-order differential operators Li(i ≥ 1) are in divergenceform.

Proof. This claim is a straightforward consequence of the fact that LAε + LB

ε is indivergence form.

Next we consider the terms in LAε − LB

ε .

Lemma 5.6. If Assumption A holds, then for i = −1, 0, 1, · · · ,

LAi (u, v1, · · · , vi+2) = 0 and LB

i (u, v1, · · · , vi+2) = 0 (5.18)

are solvable in terms of vi+2.

Proof. We only consider the interactions shown in Fig. 6.First, we consider theO(1/ε) equations. Applying (2.15) to the pair 〈 p, p− s1 〉,

we obtain

∂α1 VAB(p,−(I + A)s1 + p) + ∂α2 VAB(p,−(I + A)s1 + p) = 0,

which is always solvable with respect to p due to Assumption A. Notice that

LA−1(uε , v1) = (∂α1 + ∂α2 )VAB(εv1,−(I + ∇u)s1 + εv1).

Therefore, the O(1/ε) equations for A atoms are also solvable with v1 = p. Us-ing (5.16), we see that the other O(1/ε) equations LB

−1(u, v1) = 0 for B atoms arealso solvable with respect to v1.

In case when i ≥ 0, a straightforward calculation gives that the coefficients ofthe argument vi+2 is:

(∂α1 + ∂α2 )2VAB(εv1,−s1 + (s1 · ∇)u + εv1),

which is positive definite since p is a local minimizer.

From the O(1) equations, it is straightforward to obtain the equations for u1

and u2.

Lemma 5.7. If Assumption A holds, then there exist u1, u2 ∈ X that satisfies theequations (5.14) and (5.15), respectively.

34 W. E, P. M

Proof. Using Lemma 5.5, we see that the right-hand side of (5.14) and (5.15)belong to Y. Next, using Assumption A, there exists a constant κ such that if‖ f ‖Lp ≤ κ, then Llin is elliptic at u0. Therefore, there exist u1, u2 ∈ X that satisfiesthe equations (5.14) and (5.15), respectively.

As a direct consequence of Lemma 5.4 and Lemma 5.7, we have

Corollary 5.2. Define

yA= x + u0(x) + εu1(x) + ε2u2(x),

yB= yA

+ εv1(x) + ε2v2(x) + ε3v3(x) + ε4v4(x).(5.19)

If f ∈ W6,p(Ω;Rd), then there exists a constant C such that

|LAε (yA, yB) − f | ≤ Cε3, |LB

ε (yA, yB) − f | ≤ Cε3. (5.20)

Proof. Since f ∈ W6,p(Ω;Rd), using Theorem 2.1, we conclude that u0 ∈ W8,p(Ω;Rd).Therefore, u0 ∈ C7(Ω) by the Sobolev imbedding theorem. This gives that yA

, yB ∈C5(Ω). Therefore, using (5.14), (5.15), (5.17) and (5.12), we get

∣∣∣∣∣12[LAε (yA, yB) + LB

ε (yA, yB)

] − f∣∣∣∣∣ ≤ Cε3,

where C depends on ‖u0‖C7(Ω).Using Lemma 5.6 and the equations satisfied by v2, v3 and v4, we obtain

|LAε (yA, yB) − LB

ε (yA, yB)| ≤ Cε3,

where C depends on ‖u0‖C5(Ω). A combination of the above two results give (5.20).

6. Local Minimizer for the Atomistic Model

In this section, we prove Theorem 2.2 and Theorem 2.3. We will deal directlywith complex lattices with two species of atoms. We assume that there are a totalof 2N atoms, N atoms of type A, and N atoms of type B.

The Euler-Lagrange equations of the minimization problem are

T (y) = 0, (6.1)

where T = (T1, · · · , T2N) with Ti : R2N×d → Rd defined by

Ti(y): = −∂V∂yi− f (xi) 1 ≤ i ≤ 2N. (6.2)

We assume that y− x−B · x is periodic. To guarantee uniqueness of solutions, werequire

2N∑

i=1

yi = 0.

Cauchy-Born Rule 35

Define the Hessian matrix of the potential energy V at y by

H(y) = Hαβ(i, j)(y): =∂2V

∂yi(α)∂y j(β)(y), (6.3)

where yi(α) denotes the α−th component of yi. Let H0 = H(x) be the Hessian atthe undeformed state. The Hessian matrix H0 takes a block form

H0 =

(HAA HAB

HBA HBB

),

where

Hκκ′αβ(i, j) =∂2V

∂xκi (α)∂xκ′j (β)(x)

for κ, κ′ = A or B. We define the dynamical matrix associated with each block as

Diκκ′[n]αβ =

N∑

j=1

Hκκ′(i, j)αβei(xκ′j −xκi )·kn (6.4)

for κ, κ′ = A or B. It is easy to see that Diκκ′[n]αβ is actually independent of i

due to the translation invariance of Hκκ′ (see [18]). Moreover, it is clear that forκ, κ′ = A or B, we have

Hκκ′(i, j)αβ =1N

N∑

j=1

Diκκ′[n]αβe−i(xκ′j −xκi )·kn . (6.5)

Using [18, equation (2.22)], we have that Dκκ′ is Hermitian.For any z ∈ RN×d , we have

z j =1N

N∑

m=1

z[m]e−ix j·km ,

where z[m] are the Fourier coefficients of z, defined as

z[m]: =N∑

n=1

zneixn·km ,

and km are the discrete wave vectors in the first Brillouin zone. Similarly, wemay define zA[m] and zB[m].

For z ∈ RN×d, we define the discrete H1-norm as

‖z‖1: =( 1

N2

N∑

n=1

|kn|2| z[n]|2)1/2.

Throughout this section, we will frequently refer to the identities:∑

xeix·k = Nδk,0, (6.6)

36 W. E, P. M

and∑

k

eix·k = Nδx,0, (6.7)

where x = xA or x = xB and k runs through all the sites in the first Brillouin zoneof lattice L. We refer to [3, Appendix F] for a proof.

We first establish several inequalities concerning z, which serve to give adescription of the norm ‖ · ‖d defined in (2.24).

For any z = (zA, zB) ∈ R2N×d , define yet another norm

‖z‖a: = εd/2−1( N∑

i=1

∑

|xAi j|=ε

|zAi − zA

j |2 +∑

|xBi j |=ε

|zBi − zB

j |)1/2,

where xi j = xi − x j.

Lemma 6.1. For any z = (zA, zB) ∈ R2N×d , there exists a constant C such that

‖z‖a ≤ C(‖zA‖1 + ‖zB‖1). (6.8)

Proof. We have

|zAi − zA

j |2 =1

N2

N∑

n,m=1

zA[n] zA[m]

[eixA

i ·knm − eixAi ·kn e−ixA

j ·km

− eixAj ·km e−ixA

j ·kn + eixAi ·knm

].

We will decompose εd−2 ∑Ni=1

∑|xA

i j |=ε |zAi − zA

j |2 into I1 + I2 + I3 + I4 according to theabove expression. Using (6.6), we obtain

I1 =Kεd−2

N2

N∑

n,m=1

zA[n] zA[m]

N∑

i=1

eixAi ·knm

=Kεd−2

N2

N∑

n,m=1

zA[n] zA[m]Nδnm =

Kεd−2

N

N∑

n=1

| zA[n]|2,

where K is the coordination number of the underlying lattice. Similarly, I4 = I1.Note that

I2 = −εd−2

N2

N∑

i=1

∑

|xAi j |=ε

N∑

n,m=1

zA[n] zA[m]eixA

i ·kn e−i(xAi −xA

i j)·km .

Cauchy-Born Rule 37

For any point xAi , xA

i j is the same, so we denote it byα j = xAi j. A direct manipulation

leads to

I2 = −εd−2

N2

N∑

i=1

∑

|xAi j |=ε

N∑

n,m=1

zA[n] zA[m]eixA

i ·kn e−i(xAi +α j)·km

= − εd−2

N2

N∑

n,m=1

zA[n] zA[m]

N∑

i=1

eixAi ·knm

K∑

j=1

e−iα j·km

= − εd−2

N

N∑

n=1

| zA[n]|2K∑

j=1

e−iα j·km .

Similarly,

I3 = −εd−2

N

N∑

n=1

| zA[n]|2K∑

j=1

eiα j·km .

Summing up the expression for I1, · · · , I4, we obtain

εd−2N∑

i=1

∑

|xAi j|=ε

|zAi − zA

j |2 =2εd−2

N

N∑

n=1

| zA[n]|2K∑

j=1

(1 − cos(α j · kn)

)

=4εd−2

N

N∑

n=1

| zA[n]|2K∑

j=1

sin2 α j · kn

2.

Similarly,

εd−2N∑

i=1

∑

|xAi j|=ε|zB

i − zBj |2 =

4εd−2

N

N∑

n=1

| zB[n]|2K∑

j=1

sin2 α j · kn

2.

From these two identities and the definition of the discrete H1-norm ‖z‖1, we have

‖z‖2a ≤εd−2

N

N∑

n=1

(| zA[n]|2 + | zB[n]|2)K∑

j=1

|α j · kn|2

≤ C(‖zA‖21 + ‖zB‖21),

which leads to the desired estimate (6.1).

The following simple fact is useful.

Lemma 6.2. Given a block matrix A ∈ R2d×2d

A =(A11 A12

A∗12 A22

),

where A11 ∈ Rd×d and A22 ∈ Rd×d are positive definite. If A is semi-positive defi-nite, then for any w, v ∈ Rd,

w∗A11w + v∗A22v ≥ 12

(w, v)∗A(w, v). (6.9)

38 W. E, P. M

Lemma 6.3. Under Assumption A, for any z = (zA, zB), there exist two constantsC1 and C2 such that

‖ z ‖d ≥ C1εd/2−1‖zA − zB‖`2 −C2(‖zA‖1 + ‖zB‖1). (6.10)

Proof. Using the translation invariance of H0, for any 1 ≤ j ≤ 2N, we have

2N∑

i=1

H0(i, j) = 0. (6.11)

Using the above identities, we get

zT H0 z = −12

2N∑

i, j=1

(zi − z j)H0(i, j)(zi − z j),

which can be expanded into

zT H0 z = −12

∑

κ=A,B

N∑

i, j=1

(zκi − zκj )Hκκ(i, j)(zκi − zκj )

− 12

N∑

i, j=1

(zAi − zB

j )HAB(i, j)(zAi − zB

j )

− 12

N∑

i, j=1

(zBi − zA

j )H∗AB(i, j)(zBi − zA

j ),

we rewrite the above equation into

zT H0 z = −12

N∑

i, j=1

(zAi − zA

j )(HAA + HAB)(i, j)(zAi − zA

j )

− 12

N∑

i, j=1

(zBi − zB

j )(HBB + H∗AB)(i, j)(zBi − zB

j )

− 12

N∑

i, j=1

(zAj − zB

j )HAB(i, j)(zAj − zB

j )

− 12

N∑

i, j=1

(zAj − zB

j )H∗AB(i, j)(zAj − zB

j ) + I3, (6.12)

Cauchy-Born Rule 39

where

I3 = −12

N∑

i, j=1

(zAi − zA

j )HAB(i, j)(zAj − zB

j )

− 12

N∑

i, j=1

(zAj − zB

j )HAB(i, j)(zAi − zA

j )

− 12

N∑

i, j=1

(zBi − zB

j )H∗AB(i, j)(zAj − zB

j )

− 12

N∑

i, j=1

(zAj − zB

j )H∗AB(i, j)(zBi − zB

j ).

We rewrite (6.11) into

N∑

i=1

[HAA(i, j) + H∗AB(i, j)] = 0,N∑

i=1

[HAB(i, j) + HBB(i, j)] = 0,

which implies

zT H0 z =12

N∑

i=1

(zAi − zB

i )

N∑

j=1

(HAA(i, j) + HBB(i, j)

)

(zAi − zB

i )

− 12

[ N∑

i, j=1

(zAi − zA

j )(HAA + HAB + H∗AB)(i, j)(zAi − zA

j )

−N∑

i, j=1

(zBi − zB

j )HBB(i, j)(zBi − zB

j )]+ I3 = :I1 + I2 + I3.

Using (6.5), we have

N∑

j=1

(HAA(i, j) + HBB( j, i)) =1N

N∑

n=1

(DAA + DBB)[n]. (6.13)

Using (6.9) with w = v = zAi − zB

i , for each i and n, we have

(zAi − zB

i )T · (DAA + DBB)[n] · (zAi − zB

i )

≥ 12

(zAi − zB

i , zAi − zB

i )T · D[n] · (zAi − zB

i , zAi − zB

i ).

For each fixed n, we let Q[n] be a 2d × 2d matrix consisting of the normalizedeigenvectors of the eigenvalue problem:

D[n]Qi[n] = ω2i [n]Qi[n] 1 ≤ i ≤ 2d,

40 W. E, P. M

where Qi[n] is the i−th column of Q[n]. Let Qi[n] = (QAi [n],QB

i [n])T . Combiningthe above two equations, we get

I1 ≥1N

N∑

n=1

N∑

i=1

2d∑

j=1

ω2j [n]|(QA

i [n] + QBi [n]) · (zA

i − zBi )|2. (6.14)

Invoking (6.9) again with w = zAi − zB

i and v = zBi − zA

i , for each i and n, we have

(zAi − zB

i ) · (DAA + DBB)[n] · (zAi − zB

i )

≥ 12

(zAi − zB

i , zBi − zA

i )T · D[n] · (zAi − zB

i , zBi − zA

i ).

Repeating the above procedure, then we get another lower-bound for I1 as

I1 ≥1N

N∑

n=1

N∑

i=1

2d∑

j=1

ω2j [n]|(QA

i [n] − QBi [n]) · (zA

i − zBi )|2. (6.15)

A combination of (6.14) and (6.15) we have

I1 ≥1N

N∑

n=1

N∑

i=1

2d∑

j=1

ω2j[n]

(|QAj [n] · (zA

i − zBi )|2 + |QB

j [n] · (zAi − zB

i )|2).

For a fixed n, the optical branch, for example, is j = 1, · · · , d. For each fixedj, QA

j [n] and QBj [n] cannot be orthogonal to zA

i − zBi simultaneously, otherwise,

zAi − zB

i = 0. Therefore, there exists a constant C, independent of N and ε, suchthat

I1 ≥C1

N

N∑

n=1

N∑

i=1

d∑

j=1

ω2j [n]|zA

i − zBi |2.

Using Assumption A, there exists a constant C such that

I1 ≥ C1ε−2‖zA − zB‖2`2 . (6.16)

Using Lemma 6.1 and the fact that the atomistic potential has finite range, we have

|I2| ≤ C2ε−d(‖zA‖21 + ‖zB‖21). (6.17)

Similarly, we get

|I3| ≤ C3ε−d/2−1‖zA − zB‖`2 (‖zA‖1 + ‖zB‖1). (6.18)

A combination of (6.16), (6.17) and (6.18) gives (6.10).

Lemma 6.4. Under Assumption A, there exists a constant C such that

‖ z ‖d ≥ C(‖zA‖1 + ‖zB‖1 + εd/2−1‖zA − zB‖`2

). (6.19)

Cauchy-Born Rule 41

Proof. It is easy to see that

zT H0 z = (HAA zA, zA) + (HABzB, zA) + (H∗ABzA, zB) + (HBBzB, zB).

Using (6.5), we express each item in terms of the dynamical matrix D.

(HAA zA, zA)

=1

N3

N∑

i, j=1

N∑

m=1

zA[m]e−ixAi ·km

N∑

n=1

DiAA[n]e−i(xA

j −xAi )·kn

N∑

p=1

zA[p]eixA

j ·kp

=1

N3

N∑

m,n,p=1

zA[m]( N∑

i=1

e−ixAi ·kmnDi

AA[n])z

A[p]

N∑

j=1

e−ixAj ·knp .

We let DAA[n] = DiAA[n] since Di

AA[n] is independent of i. Using (6.6), we rewritethe above identity into

(HAA zA, zA)

=1

N3

N∑

m,n,p=1

( N∑

i=1

e−ixAi ·kmn

)zA[m] · DAA[n] · zA

[p]N∑

j=1

e−ixAj ·knp

=1

N3

N∑

m,n,p=1

zA[m]NδnmDAA[n] zA[p]Nδnp

=1N

N∑

n=1

zA[n]DAA[n] zA[n].

Proceeding along the same line, we obtain

(Hκκ′ zκ, zκ′) =

1N

N∑

n=1

zκ[n]Dκκ′[n] zκ′

[n]

for κ, κ′ = A, B. We thus write zT H0 z as

zT H0 z =1N

N∑

n=1

( zA[n], zB[n])D[n]( zA[n], zB[n]).

As in Lemma 6.3 and using

ω2a(n) ≥ Λ2

1|kn|2 and ω2o(n) ≥ ω2

a(n) ≥ Λ21|kn|2,

we have

z[n]T D[n] z[n] =d∑

i=1

((ω2

a(n))i + (ω2o(n))i

)|Q[n] z[n]|2 (6.20)

≥ 2Λ21|kn|2|Q z[n]|2 = 2Λ2

1|kn|2| z[n]|2,

42 W. E, P. M

where we have used the fact that Q is an orthogonal matrix. Therefore, we obtain

zT H0 z ≥2Λ2

1

N

N∑

n=1

|kn|2(| zA[n]|2 + | zB[n]|2) ≥ 2Λ2

1N(‖zA‖21 + ‖zB‖21)

≥ Cε−d(‖zA‖21 + ‖zB‖21),

which leads to

‖ z ‖2d ≥ C(‖zA‖21 + ‖zB‖21). (6.21)

A convex combination of (6.21) and (6.10) leads to (6.19).

The identity (6.20) give an alternative characterization of ‖ · ‖d norm. In nextlemma, we will show that the right-hand side of (6.19) is actually an equivalentnorm of ‖ z ‖d.

Lemma 6.5. If there exists a constant C independent of ε such that the opticalbranch of the dynamical matrix satisfying ωo(k) ≤ C/ε, then for any z ∈ R2N×d ,there exists a constant C1 such that

‖ z ‖d ≤ C1(‖zA‖1 + ‖zB‖1 + εd/2−1‖zA − zB‖`2 ). (6.22)

Proof. We start with the identity (6.13) in Lemma 6.3. Using the fact that (ωa(n))i ≤(ωo(n))i ≤ C/ε for 1 ≤ i ≤ d, we have

I1 ≤ C4ε−2‖zA − zB‖2`2 ,

which together with (6.17) and (6.18) leads to

‖ z ‖2d ≤ C4εd−2‖zA − zB‖2`2 + C2(‖zA‖21 + ‖zB‖21)

+C3εd/2−1‖zA − zB‖`2 (‖zA‖1 + ‖zB‖1)

≤ max(C4,C2,C3/2)(‖zA‖1 + ‖zB‖1 + εd/2−1‖zA − zB‖`2 )2,

which gives the left hand-side of (6.22) with C1 =√

max(C4,C2,C3/2).

Though we have used the fact that the optical branch is bounded from aboveby C/ε with certain constant C, we never use this upper bound for ‖ · ‖d in oursubsequent analysis. This lemma fully characterizes of the seeming mysteriousnorm ‖ · ‖d.

Next we establish a discrete Poincare inequality.

Lemma 6.6. For any z ∈ RN×d that satisfies∑N

j=1 z j = 0, there exists a constant Csuch that

‖z‖`2 ≤ Cε−d/2‖z‖1. (6.23)

Cauchy-Born Rule 43

Proof. Since∑N

j=1 z j = 0, then z[1] = 0 since k1 = 0. Therefore, by definition,

‖z‖2`2 =1

N2

N∑

j=1

N∑

m=2

z[m]e−ix j·km

N∑

m=2

z[m]eix j·km .

By the Cauchy-Schwartz inequality, we have

N∑

m=2

z[m]e−ix j·km ≤( N∑

m=2

| z[m]|2|km|2)1/2( N∑

m=2

|km|−2)1/2.

Combining the above two equations, we obtain

‖z‖2`2 ≤1N

N∑

m=2

| z[m]|2|km|2N∑

m=2

|km|−2 = N‖z‖21N∑

m=2

|km|−2.

Since km =∑d

j=1m j

N jb j, where b j is the basis of the reciprocal lattice, we get

|km| = 2πε

(∑dj=1|m j/N j|2

)1/2. A direct calculation gives

N∑

m=2

|km|−2 ≤ ( ε2π

)2N∑

m=2

( d∑

j=1

|m j/N j|2)−1≤ d−2

(2π)2

d∑

j=1

N j∑

m j=1

(ε N j/m j)2

≤ Cd∑

j=1

N j∑

m j=1

m−2j ≤ C,

where we have used ε N j ≤ C for 1 ≤ j ≤ d. Combining the above two inequalitiesand noting that N = O(ε−d), we obtain (6.23).

For any z ∈ R2N×d , we define a discrete W1,∞−norm as

|z|1,∞: = ε−1 max1≤i≤2N

max|xi j |=ε|zi − z j|. (6.24)

Lemma 6.7. If there exists a constant κ such that

εd zT H(y1)z ≥ κ‖z‖2d for all z = (zA, zB) ∈ R2N×d , (6.25)

then there exists a constant δ such that for any y2 that satisfies |y1 − y2|1,∞ ≤ δ, wehave

εd zT H(y2)z ≥ κ2‖z‖2d for all z = (zA, zB) ∈ R2N×d (6.26)

for sufficiently small ε.

44 W. E, P. M

Proof. Using translation invariance, we have for any 1 ≤ i ≤ 2N,

2N∑

j=1

H(i, j) = 0,

which leads to

zT H(y)z = −12

2N∑

i, j=1

(zi − z j)H(i, j)(y)(zi − z j).

Therefore, using the fact that the potential V is of finite range, and noting the basicinequality

|zAi − zB

j |2 ≤ 2|zAi − zB

i |2 + 2|zBi − zB

j |2,

we get

|2N∑

i, j=1

(zi − z j)(H(i, j)(y1) − H(i, j)(y2)

)(zi − z j)|

≤ Cδε−2N∑

i=1

∑

|xi j|=ε

(|zAi − zA

j |2 + |zBi − zB

j |2 + |zAi − zB

j |2)

≤ Cδε−2N∑

i=1

∑

|Ri j |=ε

(|zAi − zA

j |2 + |zBi − zB

j |2)+CLδε−2

N∑

i=1

|zAi − zB

i |2.

It follows from Lemma 6.4 that

εd |zT [H(y2) − H(y1)]z| ≤ Cδ‖z‖2a + CLδεd−2‖ zA − zB ‖2`2 ≤ Cδ‖ z ‖2d,

which yields

εd zT H(y2)z = εd zT H(y1)z + εd zT [H(y2) − H(y1)]z

≥ κ‖z‖2d − Cδ‖z‖2d≥ (κ/2)‖ z ‖2d

for δ = κ/(2C). This gives (6.26).

Lemma 6.8. Assume that y satisfies:

1. There exists a constant κ such that

εd zT H(y)z ≥ κ‖z‖2d for all z ∈ R2N×d .

2. There exists a constant p > 2 such that εd/2‖ T (y) ‖`2 ≤ K1εp.

3. y − x − B · x is periodic.

Then there exists a unique solution y that satisfies (6.1) such that

‖y − y‖d ≤ Cε p. (6.27)

Cauchy-Born Rule 45

Proof. Write

T (y) = T (y) +∫ 1

0(1 − t)H(yt) dt · (y − y),

where yt = ty + (1 − t)y. Hence y is a solution of (6.1) if and only if∫ 1

0(1 − t)H(yt) dt · (y − y) = −T (y). (6.28)

Define a set B by

B: = y ∈ R2N×d and y − x − B · x is periodic | ‖y − y‖d < ε2 .

We define a map F : B→ B as follows. For any y ∈ B, let F(y) be the solutionof the linear system

∫ 1

0(1 − t)H(yt) dt · (F(y) − y

)= −T (y). (6.29)

We first show that F is well-defined. Since ‖yt − y‖d ≤ t‖y − y‖d ≤ ε2, we have|yt − y|1,∞ ≤ Cε2−d/2 < δ, if ε is sufficiently small, given that d ≤ 3. Using the firstassumption on y and Lemma 6.7, we conclude that there exists a constant κ suchthat

εd zT H(yt)z ≥ κ2‖z‖2d.

Therefore, the linear system (6.29) is solvable and F is well-defined. Moreover,F(y) − x − B · x is periodic.

F is also continuous since V is smooth. Using Lemma 6.6 and notice that∑2Ni=1[F(y)i − yi] = 0, we obtain

κ ε−d‖F(y) − y‖2d ≤ ‖ F(y) − y ‖`2‖ T (y) ‖`2≤ CK1ε

p−d‖F(y) − y‖d. (6.30)

If ‖ F(y) − y ‖d = 0, we have F(B) ⊂ B. Otherwise, the above inequality gives

‖F(y) − y‖d ≤ Cε p,

which in turn implies F(B) ⊂ B for sufficiently small ε since p > 2. Now the exis-tence of y follows from the Brouwer fixed point theorem. Moreover, we concludethat y satisfies (6.27), and the solution y is locally unique since the Hessian at y isnon-degenerate.

Lemma 6.9. There exist two constants M1 and M2 such that if ‖B‖ ≤ M1 and‖ f ‖W6,p ≤ M2, then there exists y that satisfies the the second and third conditionsof Lemma 6.8 for p = 2.

This is a direct consequence of Lemma 5.1 and Corollary 5.2.

46 W. E, P. M

Lemma 6.10. There exist two constants M1 and M2 such that if ‖B‖ ≤ M1 and‖ f ‖W6,p ≤ M2, then there exist a constant κ1 and y such that

εd zT H(y)z ≥ κ1‖ z ‖2d for all z ∈ R2N×d . (6.31)

Proof. Define M1 = M2 = δ/(4C), where C is the constant that appears in theright-hand side of the following inequality. For simple lattice,

|y − x|1,∞ ≤ C‖∇uCB + B‖1,∞ +Cε‖∇u1‖1,∞ + Cε2‖∇u2‖1,∞≤ C‖uCB‖2,p +Cε‖u1‖2,p + Cε2‖u2‖2,p +C‖B‖≤ C‖ f ‖6,p +C‖B‖ ≤ C(M1 + M2) = δ/2 < δ.

As to complex lattice, the estimate for |yA − xA|1,∞ is the same as above, while

|yB − xB| ≤ |yA − xA|1,∞ +C(ε2‖∇v2‖1,∞ + ε3‖∇v3‖1,∞ + ε4‖∇v4‖1,∞)

≤ |yA − xA|1,∞ +C‖u‖6,∞≤ C(M1 + M2) +C‖ f ‖5,p≤ C(2M1 + M2) = 3δ/4 < δ.

Note that zT H0 z = ε−d‖ z ‖2d, therefore (6.31) follows from Lemma 6.7.

Proof of Theorem 2.2 It follows from Lemma 6.8 that there exists a yε thatsatisfies (6.1) and ‖ yε − y ‖d ≤ Cε3. Therefore,

‖ yε − yCB ‖d ≤ ‖ yε − y ‖d + ‖ y − yCB ‖d ≤ Cε3 +Cε ≤ Cε.

This gives (2.23).For any y ∈ R2N×d with |y− yε |1,∞ ≤ δ/2, where δ is the same as in Lemma 6.7,

we write

E(y) − E(yε) = (y − yε) ·∫ 1

0(1 − t)H(ty + (1 − t)yε) dt · (y − yε).

Note that

|ty + (1 − t)yε − y|1,∞ ≤ t|y − yε |1,∞ + |yε − y|1,∞ ≤ δ/2 +Cε3−d/2 ≤ δ

for sufficiently small ε. Using Lemma 6.7, there exists a constant C such that

E(y) − E(yε) ≥ Cε−d‖ y − yε ‖2d > 0.

Therefore yε is a local minimizer. utProof of Theorem 2.3 The proof is essentially the same as that of Theorem 2.2.

The only difference is

‖ yε − yCB ‖d ≤ ‖ yε − y ‖d + ‖ y − yCB ‖d ≤ Cε3 +Cε2 ≤ Cε2,

since u1 = 0 due to Lemma 5.2. This gives (2.25). ut

Cauchy-Born Rule 47

Appendix A. Detailed asymptotic analysis for one-dimensional model

In this appendix, we will give a detailed asymptotic analysis of one dimen-sional models. We will consider a complex lattice with two species of atoms A andB.

Considering the equilibrium equations for atom A and B respectively

LAε (yA

i , yBi ) = f (xA

i ) and LBε (yA

i , yBi ) = f (xB

i ),

where

LAε (yA

i , yBi ) = V ′AB(yB

i − yAi ) − V ′AB(yA

i − yBi−1)

+ V ′AA(yAi+1 − yA

i ) − V ′AA(yAi − yA

i−1),

LBε (yA

i , yBi ) = V ′AB(yA

i+1 − yBi ) − V ′AB(yB

i − yAi )

+ V ′BB(yBi+1 − yB

i ) − V ′BB(yBi − yB

i−1),

which can be rewritten as

LAε (yA

i , yBi ) = V ′AB(D+pyA

i ) + V ′AB(D+p−εyAi ) + D−ε VAA(D+ε yA

i ),

LBε (yA

i , yBi ) = −V ′AB(D+p−εy

Ai+1) − V ′AB(D+ε yA

i ) + D−ε VBB(D+ε yBi ).

Proceeding as in subsection 5.2, we get

L−1(u, v1) = 0, L0(u, v1, v2) = − f ′,

L1(u, v1, v2, v3) = −12

f ′′, L2(u, v1, v2, v3, v4) = −16

f (3),(A.1)

where

L−1(u, v1) = 2V ′AB(v1) + 2V ′AB(v1 − 1 − ux),

L0(u, v1, v2) = 2[V ′′AB(v1) + V ′′AB(v1 − 1 − ux)

]v2

− V ′′AB(v1 − 1 − ux)v1x

+ ∂x(V ′AA(1 + ux) − V ′BB(1 + ux)

),

L1(u, v1, v2, v3) = 2V ′′AB(v1) + V ′′AB(v1 − 1 − ux)(b1 + b3)

+ V (3)AB(v1)v2

2 +12

V (3)AB(v1 − 1 − ux)(a2

1 + a23)

+ ∂x(V ′′BB(1 + ux)uxx

),

where

a1 = v2 − v1x +12

uxx, b1 = v3 − v2x +12

v1xx −16

uxxx,

a2 = v1x +12

uxx, b2 = v2x +12

v1xx +16

uxxx,

a3 = v2 −12

uxx, b3 = v3 −16

uxxx.

48 W. E, P. M

We omit the expression for L2 since we do not need it to solve v4.In what follows, we give the explicit expressions of L0,L1 and L2. The argu-

ments of VAB,VAA and VBB will be omitted unless otherwise stated.

L0(u, v1) = −∂x(V ′AB − V ′AA − V ′BB

),

L1(u, v1, v2) = −12∂2

x(V ′AB + V ′AA + V ′BB

)

− ∂x(V ′′ABa1 −

12

V ′′AAuxx − V ′′BBa2),

L2(u, v1, v2, v3) = −16∂3

x(V ′AB − V ′AA − V ′BB

)

− 12∂2

x(V ′′ABa1 +

12

V ′′AAuxx + V ′′BBa2)

− ∂x[V ′′ABb1 +

12

V (3)ABa2

1 −16

V ′′AAuxxx −18

V (3)AAu2

xx

− V ′′BBb2 −12

V (3)BBa2

2].

Defineα = V ′′AB(v1 − 1 − ux), β = V ′′AB(v1).

It is easy to see that

L1(u, v1, v2) = −∂x((a1 + v1x/2 − uxx/2)V ′′AB + (−a2 + uxx/2)V ′′BB

).

Differentiating the equation L−1(u, v1) = 0, we have

(α + β)v1x = α uxx.

Solving L0(u, v1, v2) = 0, we get

v2 =α

α + β

v1x

2− [

(V ′′AA − V ′′BB − f ′)uxx]/(2(α + β)).

A combination of the above three equations gives

L1 = ∂x

[ V ′′AB(v1)V ′′AB(v1 − 1 − ux) + V ′′AB(v1)

(V ′′AB + V ′′AA + V ′′BB + f ′/2

)],

where

V ′′AB =V ′′AB(v1 − 1 − ux)V ′′AB(v1)

V ′′AB(v1 − 1 − ux) + V ′′AB(v1).

Solving L1(u, v1, v2, v3) = − f ′′/2, we obtain

v3 =α

α + β(v2x/2 − v1xx/4 + uxxx/6)

−V (3)

AB(v1)α + β

v22

2−

V (3)AB(v1 − 1 − ux)α + β

a21 + a2

3

4−G,

Cauchy-Born Rule 49

where G =(∂x(V ′′BB(1 + ux)ux) + f ′′/4

)/(α + β). Substituting v3, v2, a1 and a3 into

L2, we obtain

L2 = L12 +L2

2 + L32,

where

L12 = ∂

2x

(uxx

6(V ′′AA + V ′′BB) +

αβ

α + βuxxV ′′BB

)

+1

24∂x

(u2

xx(V(3)AA + V (3)

BB) − V ′′BBG),

and

L22 = ∂x

[ αβ

4(α + β)

( αβ

(α + β)2 uxx

)x+ α

( 112− 1

8α2

(α + β)2

)( βα + β

)xuxx

− αβ

12(α + β)uxxx +

αβ

2(α + β)Gx

],

and

L32 = ∂x

[( 124

β2

(α + β)2 −αβ3

8(α + β)4

)V (3)

AB(v1 − 1 − ux)u2xx

− 12

V (3)AB(v1 − 1 − ux)

( αβ2

(α + β)3 uxx +β

α + βG)G

+12

V (3)AB(v1)

( α3

(α + β)3 uxx −α

α + βG)G].

Next let u = u0 + ε u1 + ε2 u2 + · · · , and substituting this ansatz intoL0,L1 and

L2, we get the equations for u0 as

−∂x(V ′AB(v1 − 1 − u0x) − V ′AA(1 + u0x) − V ′BB(1 + u0x)

)= f (x).

A straightforward calculation gives

Llin(u0, v1)u1 = ∂x

((V ′′AB(v1 − 1 − u0x) + V ′′AA(1 + u0x) + V ′′BB(1 + u0x)

)u1x

).

Therefore, we obtain the equations for u1 and u2:

Llin(u0, v1)u1 = −L1 + f ′/2,

Llin(u0, v1)u2 = −L2 −δL1

δu0− 1

2

(δ2L0

δu20

)u1 + f ′′/4.

50 W. E, P. M

Appendix B. Elastic stiffness tensor for simple and complex lattices

Elastic stiffness tensor and the elastic modulus tensor are two different con-cepts. They coincide while the internal stress vanishes [27], i.e. DAWCB(0) = 0.

Equation (2.17) is an explicit expression for the elastic stiffness tensor in thecase of two-body potentials. Here we generalize this formula to many-body poten-tials. In the case of simple lattice, we have, for any ξ, η ∈ Rd,

C(ξ ⊗ η, ξ ⊗ η) =∑

m=2

∑

〈 s1,··· ,sm−1 〉

∑

〈 si,s j 〉1≤i< j≤m−1

(si · η∂αi + s j · η∂α j )2Vm(ξ, ξ).

where Vm = Vm(s1, · · · , sm−1). For example, if V contains only three-body poten-tial V3, then we have, for any ξ, η ∈ Rd,

C(ξ ⊗ η, ξ ⊗ η) =∑

〈 s1,s2 〉

[∂2α1

V3(ξ, ξ)|s1 · η|2 + ∂2α2

V3(ξ, ξ)|s2 · η|2

+ ∂α1α2 V3(ξ, ξ)s1 · ηs2 · η],

where V3 = V3(s1, s2).Next we consider complex lattices. We first consider the one-dimensional case.

Minimizing W(A, p) with respect to p, we obtain

DA p =∑

s(V ′′AB((1 + A)s − p) − V ′′AB((1 + A)s + p)

)s

∑s V ′′AB((1 + A)s − p) +

∑s V ′′AB((1 + A)s + p)

.

Note that

D2AWCB(A) = D2

AW(A, p(A)) + D2ApW(A, p(A))

=∑

s

(V ′′AA((1 + A)s) + V ′′BB((1 + A)s)

+ V ′′AB((1 + A)s + p) + V ′′AB((1 + A)s − p))s2

+∑

s

(V ′′AB((1 + A)s + p(A)) − V ′′AB((1 + A)s − p(A))

)sDA p.

A combination of the above two identities leads to:

D2AWCB(A) =

∑

s

(V ′′AA((1 + A)s) + V ′′BB((1 + A)s)

+ V ′′AB((1 + A)s + p(A)) + V ′′AB((1 + A)s − p(A)))s2

−

(∑s(V ′′AB((1 + A)s − p(A)) − V ′′AB((1 + A)s + p(A))

)s)2

∑s V ′′AB((1 + A)s − p(A)) +

∑s V ′′AB((1 + A)s + p(A))

.

Cauchy-Born Rule 51

Therefore, for any k ∈ R, let p0 = p(0), we have

C k2 =∑

s

[V ′′AA(s) + V ′′BB(s) + V ′′AB(s + p0) + V ′′AB(s − p0)

]|sk|2

−

(∑s[V ′′AB(s − p0) − V ′′AB(s + p0)

]sk

)2

∑s V ′′AB(s − p0) +

∑s V ′′AB(s + p0)

. (B.1)

Next we consider high dimensional case. We only consider the case when V is athree-body potential, other cases can be dealt with similarly. Solving the algebraicequations (2.15), we obtain p = p(A). Differentiating (2.15) with respect to A, weget

DA p = −S(A)−1∑

〈 s1,s2 〉

2∑

i=1

Ki(A)si

with

S(A) =∑

〈 s1,s2 〉(∂α1 + ∂α2 )2V1

AB(A) + ∂2α2

V2AB(A) + ∂2

α1V3

AB(A),

Ki(A) =2∑

j=1

∂2αiα j

V1AB(A) +

2∑

j=1

∂2α jα2

V2AB(A) +

2∑

j=1

∂2α jα1

V3AB(A),

where

V1AB(A) = VAB((I + A)s1 + p(A), (I + A)s2 + p(A)),

V2AB(A) = VAB((I + A)s1 + p(A), (I + A)s2),

V3AB(A) = VAB((I + A)s1, (I + A)s2 + p(A)).

Therefore, we get

D2A WAB =

∑

〈 s1,s2 〉

2∑

i, j=1

3∑

k=1

∂2αiα j

VkAB(A)(si, s j)

−∑

〈 s1,s2 〉

2∑

i=1

Ki(A)(si,

∑

〈 s1,s2 〉

2∑

j=1

S(A)−1K j(A)s j

).

Define V(A) = VAA(A) + VBB(A) + 2∑3

i=1 V iAB(A). Then, we have

C(ξ ⊗ η, ξ ⊗ η)

=∑

〈 s1,s2 〉

[(∂α1 s1 · η + ∂α2 s2 · η)2V(0)

](ξ, ξ) (B.2)

−( ∑

〈 s1,s2 〉

2∑

i=1

[Ki(0)

](ξ, ξ)si · η,

∑

〈 s1,s2 〉

2∑

i=1

[S(0)−1Ki(0)

](ξ, ξ)si · η

).

Compare this formula with simple lattices, we see that (B.2) has the form of aSchur complement when the terms involving p are eliminated from the full dy-namical matrix.

52 W. E, P. M

Acknowledgements. The work of E is supported in part by an NSF grant, with additionalsupport from the project “Research Team on Complex Systems” of the Chinese Academy ofScience. The work of Ming is partially supported by the National Natural Science Founda-tion of China under the grant 10571172 and also supported by the National Basic ResearchProgram under the grant 2005CB321704.

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for publication, 2005

Department of Mathematics and PACM,Princeton University

e-mail: weinan@Princeton.EDU

and

Institute of Computational Mathematicsand Scientific/Engineering Computing, AMSS,

Chinese Academy of Sciences,No. 55, Zhong-Guan-Cun East Road,

Beijing, 100080, Chinae-mail: mpb@lsec.cc.ac.cn