A Mathematical Journal
Vol. 7, No 2, (69 - 79). August 2005.

Two-Phase Structures as Singular Limit of
a one-dimensional Discrete Model

Thomas Blesgen
Max-Planck-Institute for Mathematics in the Sciences

Inselstraße 22-26, D-04103 Leipzig, Germany
e-mail: blesgen@mis.mpg.de

ABSTRACT
A one-dimensional energy functional that models the elastic free energy of

a monatomic chain of atoms occupying a bounded real domain is discussed and
the Γ-limit of this functional when the number of particles becomes infinite is
derived. The particular ansatz allows for the first time the presence of two coex-
isting phases in the singular limit and thus can be used as a prototype towards
modeling three dimensional cases of physical relevance.

RESUMEN

Se discute una funcional de enerǵıa unidimensional que modela la enerǵıa
elástica liberada po una cadena monotómica de átomos ocupando un dominio real
acotado y se deduce el ĺımite Γ de esta funcional cuando el número de part́ıculas
llega a ser infinito. El particular ansatz permite por primera vez la presencia de
dos fases coexistentes en el ĺımite singular y de esta forma puede ser utilizado
como un prototipo para modelar casos tridimensionales de relevancia en f́ısica.

Key words and phrases: Singular Limits, Phase Transitions, Elasticity
Math. Subj. Class.: 35E99, 49J45, 74M25, 74N99



70 Thomas Blesgen
7, 2(2005)

1 Introduction

In this article we study the behaviour of a one-dimensional monatomic lattice com-
prising of (n + 1) particles that interact via three and four body potentials that
represent the interatomic forces. We are interested in the description of the Γ-limit
when n → ∞. This leads to a continuum theory of regular crystals under the ideal-
ising assumption that the interatomic distance vanishes in the limit. The notion of
Γ-limit goes back to work by de Giorgi, [5]. Γ-limits can be regarded as a convergence
related not only to one lattice, but provide a natural framework to formulate conver-
gence with respect to an entire family of lattices, depending on the parameter n. A
comprehensive discussion of this issue can be found in [1].
The derivation of a continuum theory relying on atomistic descriptions has only re-
cently become a focus of research after the article [2] studied softening in fracture
mechanics. In a similar direction goes [3]. In [6] a stochastic framework is consid-
ered, in [8] the topic of phase transitions is discussed by considering oscillations of
lattices. In this article, we can extend the analysis in these articles and will discuss
a prototype of a free energy functional whose minimisers give rise to two different
lattice structures. This energy functional consists of one part that accounts for the
local lattice symmetry, a second part that represents the surface energy and a third
contribution that stands for the elastic energy.
This work is organised in the following way. In Section 2, a one-dimensional discrete
free energy functional W n(un) for deformations un is introduced that describes the
elastic energy of the atomic chain. In Section 3, the Γ-limit n → ∞ of W n(un) is
identified. The article is ended by a short discussion and outlook.

2 The energy functional

Let Ω := (0, 1) ⊂ R be the domain that contains a regular monatomic chain.
We suppose that the undeformed discrete reference configuration of Ω is given by a
system of n + 1 atoms with equal distance located at points Rni ∈ R, where

Rni := ihn 0 ≤ i ≤ n.

Here, the setting hn := 1/n defines for given number n ∈ N the interatomic distance.
The limit n →∞ corresponds to hn ↘ 0. The index n is always used to indicate the
dependency on the number of subdivisions.
By R̂ni , 0 ≤ i ≤ n we denote the position of atom i after the deformation. Finally, by
uni , 0 ≤ i ≤ n we denote the two-dimensional displacement vector of atom i, given by
the relationship

uni = R̂
n
i −Rni , 0 ≤ i ≤ n.

For shortness we introduce the numbers s1 := 1, s2 := 2 and s3 := 12 and let

Dnuni :=
uni+1 −uni

hn
.



7, 2(2005)
Two-Phase Structures as Singular Limit of a ... 71

This is a forward difference quotient that approximates (uni )
′ for small hn.

We will study the behaviour of the following energy functional.

W n(un) :=
{

+∞ if uni+1 = uni for some i,∑3
k=1 W

n
k (u

n) else

where

W n1 (u
n) :=

n−2∑
i=0

nα
3∏

k=1

∣∣∣sk − u
n
i+2 −uni+1
uni+1 −uni

∣∣∣2,

W n2 (u
n) :=

n−3∑
i=0

∣∣∣1 − uni+3 −uni+2
uni+1 −uni

∣∣∣2,

W n3 (u
n) := hn

n−2∑
i=0

[(uni+2 −uni
2hn

−α1
)2
βni +

(uni+2 −uni
2hn

−α2
)2
γni

]

and

βni :=
[
1 −nα

∣∣∣1 − uni+2 −uni+1
uni+1 −uni

∣∣∣2
]
+
,

γni :=
[
1 −nα

∣∣∣2 − uni+2 −uni+1
uni+1 −uni

∣∣∣2
∣∣∣1
2
− u

n
i+2 −uni+1
uni+1 −uni

∣∣∣2
]
+
.

Here, 0 < α < 1 and [·]+ denotes the positive part, this is [x]+ = x for x ≥ 0 and
[x]+ = 0 for x < 0.
The concept behind the ansatz for W n is the following. A minimiser of W n1 either
fulfils Dnuni+1 	 Dnuni which specifies one lattice order that is in the sequel referred
to as Phase 1, or Dnuni+1 	 2Dnuni resp. Dnuni+1 	 12Dnuni which characterises
Phase 2.
W n2 represents a surface energy. It counts(and limits) the number of transitions
between the two phases, as within a phase one asymptotically has Dnuni+2 = D

nuni .
Finally, W n3 represents an elastic energy. We will show below that β

n
i converges in

L1(Ω) to the indicator function of Phase 1 and γni to the indicator function of Phase 2
as n →∞. αk corresponds to the elastic constant to Phase k.
The functional W n1 represents the electrostatic energy due to interatomic potentials
that force the atoms to positions of a certain given lattice order. The given example
of a doubling (halfing) of the period of Dnun is only a first simple example. Energy
functionals motivated by real physical applications are higher dimensional and much
more complicated.
For the analysis we extend the discrete deformation values {uni }i, to piecewise linear
functions un in L2(Ω)∩An, where An denotes the space of piecewise linear functions,
see [2]. In this article, discrete quantities are always specified by subscript i.



72 Thomas Blesgen
7, 2(2005)

3 Identification of the Γ-limit for W n

Now we can state the main result. It characterises the Γ-limit of W n as n tends to
infinity. We use the notation χ1 := χ, χ2 := 1−χ. For u ∈ H1,2(Ω), χ ∈ BV (Ω,{0, 1})
we define

E(u,χ) :=
1
4

∫
Ω

|∇χ| +
2∑

k=1

∫
Ω

χk (u
′ −αk)2.

Additionally we introduce W : L2(Ω) → R by

W(u) :=
{

infχ∈BV (Ω,{0,1}) E(u,χ) if u ∈ H1,2(Ω) is strictly monotone,
+∞ else.

Theorem 3.1 (Characterisation of the Γ-limit of W n)
The following statements are valid:
(i) The boundedness of the energy functional W n(un) implies the boundedness of(∫

Ω

|(un)′|2
)

n
uniformly in n.

(ii) W is the Γ-limit of W n as n →∞ with respect to the convergence in L2(Ω).

Proof of (i):
Step 1: Construction of the characteristic function χ:
By C we denote various positive constants that may change from line to line.
Let (un) ⊂ L2(Ω) be a sequence with W n(un) ≤ C. We set

dik :=
∣∣∣uni+2 −uni+1
uni+1 −uni

−sk
∣∣∣, ki0 := argmin

{
k �→ dik

∣∣ 1 ≤ k ≤ 3}.
The boundedness of W n1 (u

n) implies

n−1∑
i=0

nα
3∏

k=1

(
sk −

uni+2 −uni+1
uni+1 −uni

)2
≤ C.

Therefore there exists a constant C > 0 such that

sup
i
diki0
≤ Chα/2n . (1)

For sufficiently large n we can thus define an indicator function χn to Phase 1 by

χn(x) :=

⎧⎨
⎩

0 if x ∈ [ihn, (i + 1)hn), i ≤ n− 2, ki0 
= 1,
1 if x ∈ [ihn, (i + 1)hn), i ≤ n− 2, ki0 = 1,

χn(1 − 2hn) if x ∈ [1 −hn, 1].
Next we show that χn ∈ BV (Ω; {0, 1}), i.e.

∫
Ω

|∇χn| ≤ C. (2)



7, 2(2005)
Two-Phase Structures as Singular Limit of a ... 73

This follows from the boundedness of W n2 (u
n). Since for large n

uni+2 −uni+1
uni+1 −uni

= sk + o(1) for some k ∈{1, 2, 3},

we see that if χn(x) jumps in x = (i + 1)hn between 0 and 1, then
(
1 − u

n
i+3 −uni+2
uni+1 −uni

)2
≥ 1

4
+ o(1)

which shows W n2 (u
n) ≥

(
1
4

+ o(1)
)∫

Ω
|∇χn| and proves (2). Here we adapted the

Landau notation and denote by o(1) terms that tend to 0 as n →∞.
With (2), well-known compactness results imply the existence of a subsequence (again
denoted by) χn and a χ ∈ BV (Ω, {0, 1}) such that χn → χ in L1(Ω).
Step 2: Convergence of βn, γn in L1(Ω):
We extend the discrete quantities {βni }i, {γni }i to piecewise constant functions in
L1(Ω) by the definition

βn(x) :=
{
βni if x ∈ [ihn, (i + 1)hn) and i ≤ n− 2,
0 if x ∈ [1 −hn, 1].

In the same manner, the extension γn of {γni }i is defined.
Now we show that

βn → χ, γn → (1 −χ) in L1(Ω) for n →∞, (3)
where the function χ ∈ BV (Ω, {0, 1}) is the limit of χn found in Step 1. Without
loss of generality we may restrict to i ≤ n− 2.
Case 1: χn(ihn) = 1.
Fix a small ε > 0. From the boundedness of W n(un) together with (1) we see that
there exists a n0 ∈ N such that∣∣∣1 − uni+2 −uni+1

uni+1 −uni
∣∣∣2nα ≤ ε for all n ≥ n0.

So we find 1 ≥ βni ≥ 1 −ε for large n.
Similarly,

∣∣∣2 − uni+2 −uni+1
uni+1 −uni

∣∣∣2
∣∣∣1
2
− u

n
i+2 −uni+1
uni+1 −uni

∣∣∣2 ≥ hαn for all n ≥ n0
and thus γni = 0 for sufficiently large n.

Case 2: χn(ihn) = 0.
Analogous to Case 1 we see that for large n

∣∣∣2 − uni+2 −uni+1
uni+1 −uni

∣∣∣2
∣∣∣1
2
− u

n
i+2 −uni+1
uni+1 −uni

∣∣∣2nα ≤ ε,



74 Thomas Blesgen
7, 2(2005)

so 1 ≥ γni ≥ 1 −ε for large n.
Similarly, ∣∣∣1 − uni+2 −uni+1

uni+1 −uni
∣∣∣2 ≥ hαn

and βni = 0 for sufficiently large n.
The discussion of these two cases yields the pointwise convergence of βn to χ and of
γn to 1 −χ. Together with Lebesgue’s dominated convergence theorem the proof of
(3) is finished.

Step 3: Boundedness of
∫
Ω

|(un)′|2 uniformly in n:
We choose constants a ∈ R+, b ∈ R such that

min{(x−α1)2, (x−α2)2}≥ ax2 − b.
Due to the boundedness of W n3 (u

n) we thus find that there exist constants C1, C2 > 0
such that

C1 ≥ h2nC2
n−2∑
i=0

(uni+2 −uni
2hn

)2(
βni + γ

n
i

)
.

Since Dnuni+1 = skD
nuni + o(1) for one k ∈{1, 2, 3} and large n we find that

(uni+2 −uni
2hn

)2
≥
(
1 +

1
2

+ o(1)
)(uni+1 −uni

2hn

)2
.

The term
(
βni + γ

n
i

)
can for large n be estimated from below by a constant. So we

find the existence of a constant C > 0 with

C ≥ h2n
n−2∑
i=0

(uni+1 −uni
2hn

)2
. (4)

Due to the estimate (Dnunn−1)
2 ≤ (2 + o(1))Dnunn−2 the sum in (4) can be extended

to i = n− 1 and the estimate still holds.
The sum

∑
i(D

nuni )
2 is directly related to

∫
Ω
|(un)′|2 where un is the piecewise affine

linear extension of {uni }i. With (4) extended to i = n− 1 this finally yields

sup
n

∫
Ω

|(un)′|2 = sup
n

hn

n−1∑
i=0

(Dnuni )
2 ≤ C. (5)

Proof of (ii): We assume that the reader is familiar with the concept of Γ-convergence.
Step 4: Lower semicontinuity inequality along the sequence W n:
We have to show: for every sequence (un)n∈N with un → u in L2(Ω) there exists a
subsequence (unk )k∈N with

W(u) ≤ lim inf
k→∞

W nk (unk ).



7, 2(2005)
Two-Phase Structures as Singular Limit of a ... 75

If W n(un) is unbounded, there is nothing to show. Hence we may assume w.l.o.g.
W n(un) ≤ C for all n. From (5) follows un, u ∈ H1,2(Ω) for all n ∈ N. Because of
the reflexivity of the Hilbert space H1,2(Ω) we know that there exists a subsequence
(again denoted by) un such that un ⇀ u, in H1,2(Ω) for n →∞.
From Step 2 we know that χn → χ, βn → χ, γn → 1 −χ in L1(Ω) for n →∞.
Because of

uni+2−uni+1
un

i+1−uni
≥ 1

2
+o(1), for n ≥ n0 we find that un is monotone for sufficiently

large n.
Now we estimate W n(un) from below. We claim

lim inf
n→∞

W n(un) ≥ E(u,χ) ≥ W(u). (6)

In order to prove (6), let us estimate every component of W n(un) separately.
1. W n1 (u

n) ≥ 0.
2. Estimate of the surface energy:

lim inf
n→∞

W n2 (u
n) ≥ lim inf

n→∞
(1
4

+ o(1)
)∫

Ω

|∇χn| ≥ 1
4

∫
Ω

|∇χ|.

3. Estimate of the elastic energy:
For x ∈ Ω let ũn be given by

ũn(x) :=
{

un(x+2hn)−un(x)
2hn

if x ∈ (0, 1 − 2hn),
0 if x ∈ [1 − 2hn, 1),

Using this notation we may rewrite W n3 ,

W n3 (u
n) =

∫
Ω

[
(ũn −α1)2βn + (ũn −α2)2γn

]
.

Next we show
ũn → u′ in L2(Ω) for n →∞. (7)

We observe ∣∣∣un(x + 2hn) −un(x)
2hn

∣∣∣ ≤
1∫

0

|(un)′(x + shn)|ds (8)

and due to the boundedness of un in H1,2(Ω), an application of Hölder’s inequality
yields the boundedness of the left hand side of (8) uniformly in n.
With Lebesgue’s dominated convergence theorem, ũn → u′ in L2(Ω) follows. In the
same way, the other convergence results in (7) can be derived.
Exploiting (7), with the help of Theorem 3.4, p.74 in [4] we obtain

lim inf
n→∞

W n3 (u
n) ≥

2∑
k=1

∫
Ω

(u′ −αk)2χk.



76 Thomas Blesgen
7, 2(2005)

Combining the estimates for W nl (u
n), 1 ≤ l ≤ 3, (6) is shown.

Step 5: Existence of a ”recovery sequence”:
We have to find a sequence (un) ⊂ L2(Ω) converging to u in L2(Ω) with

W(u) ≥ lim sup
n→∞

W n(un).

If W(u) = +∞, there is nothing to show. Due to the monotonicity properties of u
demonstrated above we know that the functional χ �→ E(u,χ) is bounded from below
in the BV-norm. Using the compactness properties of BV (Ω) and the coercivity
of E, it is clear that E(u, ·) attains its minimum, i.e. W(u) = E(u,χ) for some
χ ∈ BV (Ω,{0, 1}).
Next we show that for piecewise affine, strictly monotone u there exists a sequence
un with un → u and W n(un) → E(u,χ). We start with special cases, then generalise.
Case 1: u′ ≡ a1 > 0, χ ≡ const in Ω:
(a) χ ≡ 1 in Ω:
We simply set un := u for all n.
(b) χ ≡ 0 in Ω:
For x > 0 choose un such that Dnuni is alternating between

2
3
a1 and 43a1. Furthermore

un satisfies un(x = 0) = u(x = 0).

Case 2: u′ ≡ a1 > 0, χ ≡ 1 for 0 ≤ x ≤ 12 , χ ≡ 0 for x > 12 .
The treatment of this case is more difficult. It is not possible to directly combine the
two ansatz functions for un of Case 1 because for one index i this would mean Dnuni =
a1hn and either Dnuni+1 =

2
3
a1hn or Dnuni+1 =

4
3
a1hn, leading to limn→∞ W n1 (u

n) =
∞.
Therefore we have to introduce a transition layer of width hsn between the two phases,
where s > 0 is a small constant to be chosen later.
For convenience we introduce

ϕn(x) :=

⎧⎨
⎩

a1 for 0 ≤ x ≤ 12,
a1 +

a1
3
ns(x− 1

2
) for 1

2
< x ≤ 1

2
+ hsn,

4
3
a1 for 12 + h

s
n < x ≤ 1.

We set un such that un(x = 0) = u(x = 0) and

Dnuni :=
{

ϕn(ihn) for ihn ≤ 12,
1
2
ϕn(ihn), ϕn(ihn) alternating for ihn > 12.

Let us now analyse what this means for the limit of W n(un). We notice

uni+2 −uni+1
uni+1 −uni

= sk if (i + 1)hn ≤
1
2

or ihn >
1
2
. (9)

If 1
2
< (i + 1)hn ≤ 12 + hsn and ihn ≤ 12 we have
uni+2 −uni+1
uni+1 −uni

= sk
ϕn((i + 1)hn)
ϕn(ihn)

= sk
a1 +

a1
3
h1−sn

a1
= sk

(
1 +

1
3
h1−sn

)
(10)



7, 2(2005)
Two-Phase Structures as Singular Limit of a ... 77

with sk = 1 or sk = 12 . To guarantee the convergence to 0 of the corresponding
expressions in W n1 which are weighted with a factor n

α we require 0 < s < 1−α
2

.

If 1
2
< ihn ≤ 12 + hsn and (i + 1)hn > 12 + hsn we have

uni+2 −uni+1
uni+1 −uni

= sk
ϕn((i + 1)hn)
ϕn(ihn)

= sk
4
3
a1

a1 +
a1
3
h−sn hsn

= sk (11)

where sk = 2 or sk = 12 .

(9), (10) and (11) cover all possible cases and demonstrate the convergence to 0 of
the Dnuni -terms in W

n
1 . Hence we find W

n
1 (u

n) → 0 as n →∞.
For the estimation of the functional W n2 (u

n) we have

∣∣∣1 − uni+3 −uni+2
uni+1 −uni

∣∣∣2 =
∣∣∣1 − 1

2
ϕn((i + 2)hn)
ϕn(ihn)

∣∣∣2

=
∣∣∣1
2
− 1

2
ϕn(ihn) −ϕn((i + 2)hn)

ϕn(ihn)

∣∣∣2.

For I := ϕ
n(ihn)−ϕn((i+2)hn)

ϕn(ihn)
simple computations yield

I =

⎧⎪⎪⎨
⎪⎪⎩

0 if (ihn > 12 ) or ((i + 1)hn ≤ 12 )
or ( 1

2
< ihn ≤ 12 + hsn and (i + 2)hn > 12 + hsn),

−skh1−sn if (ihn > 12 and (i + 2)hn ≤ 12 + hsn)
or (ihn ≤ 12 and 12 < (i + 1)hn ≤ 12 + hsn).

and for 0 < s < 1 the convergence of W n2 (u
n) to 1

4
can be assured.

For the estimation of W n3 (u
n), it is clear that outside the strip of width hsn the

summands in W n3 (u
n) are exactly equal to

hsn

[
χ(a1 −α1)2 + (1 −χ)(a1 −α2)2

]
.

Inside the strip, we have approximately hs−1n summands, where each summand is of
the form hnC. Thus, for s > 0 the part inside the strip tends to 0 for n →∞.

Case 3: General χ ∈ BV (Ω; {0, 1}), u piecewise affine, monotone, continuous:
The construction of un can be done by iteratively applying the construction given in
Case 2.



78 Thomas Blesgen
7, 2(2005)

Case 4: General monotone u ∈ H1,2(Ω):
Let u be a generic monotone function in H1,2(Ω) and let {un} be a sequence in An
such that un → u in H1,2(Ω). For every n we can apply Case 3 to find a sequence
{wnl }l such that wnl → un in L2(Ω) as n → ∞ and lim supl W l(wnl ) ≤ W(un). Then
we have

lim sup
n→∞

lim sup
l→∞

W l(wnl ) ≤ lim sup
n→∞

W(un) = W(u), (12)

where (12) holds because of the strong convergence of un to u in H1,2(Ω). By
diagonalisation, we find a sequence ũn := wn

l(n)
such that ũn → u in L2(Ω) and

lim supn→∞ W
n(ũn) ≤ W(u,v). �

4 Discussion and Outlook

The present article analysed the Γ-limit of a one-dimensional lattice as the number of
particles tends to infinity in a particular case. The discussed free energy functional
represents a first example that gives rise to two different lattice orders. It seems very
likely that this concept can be generalised to more than two phases and higher space
dimensions although the energy is even more artificial in this case. Physically relevant
formulas for the elastic energy replacing (u′ − αk)2 can for instance be found in [7]
and are mostly non-linear.

Finally it is important to realize that the concept of Γ-convergence is only partly
suitable for the understanding of the static behaviour of solids. This is because the
Γ-limit is bound to global minimisers whereas the evolution in nature may get stuck
in a local minimum as the activation energy needed to pass an energy barrier is not
available. Unfortunately, at this point no appropriate mathematical instruments seem
to be at hand to deal with formulations that arise from this consideration.

Received: June 2004. Revised: August 2004.

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7, 2(2005)
Two-Phase Structures as Singular Limit of a ... 79

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