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2022

Lloyd N. Trefethen

Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG,
United Kingdom

# Spectacularly large expansion coefficients in Müntz’s theorem

<EMAIL_ADDRESS>*

###### Abstract

Müntz’s theorem asserts, for example, that the even powers
$1,x^{2},x^{4},\dots$ are dense in $C([\kern 0.3pt0,1])$. We show that the
associated expansions are so inefficient as to have no conceivable relevance
to any actual computation. For example, approximating $f(x)=x$ to accuracy
$\varepsilon=10^{-6}$ in this basis requires powers larger than
$x^{280{,}000}$ and coefficients larger than $10^{107{,}000}$. We present a
theorem establishing exponential growth of coefficients with respect to
$1/\varepsilon$.

###### keywords:

Müntz approximation theorem, expansion coefficients

###### pacs:

[

MSC Classification]41A10, 42C30

##  1 Introduction

The Müntz approximation theorem, conjectured by Bernstein in 1912 [4] and
proved by Müntz in 1914 [12], is a beautiful result. Suppose we are interested
in continuous functions on $[\kern 0.3pt0,1]$, i.e., $f\in C([\kern
0.3pt0,1])$, and we want to approximate them as linear combinations of the
monomials

$x^{a_{1}},x^{a_{2}},\dots,$

where $\\{a_{k}\\}$ is a set of exponents (not necessarily integers)
satisfying

$0=a_{0}<a_{1}<a_{2}<\cdots.$

Certainly this is possible if $a_{k}=k$ for each $k$, by the Weierstrass
approximation theorem [14], but what if the set of powers is sparser? For
example, are the even powers

$1,x^{2},x^{4},\dots$ (1)

dense in $C([\kern 0.3pt0,1])$? The theorem characterizes all suitable sets of
exponents:

######  Theorem 1 (Müntz approximation theorem).

The family $\\{x^{a_{k}}\\}$ is dense in $C([0,1])$ if and only if

$\sum_{k=1}^{\infty}{1\over a_{k}}=\infty.$ (2)

Thus the set (1) easily qualifies, as do many other collections of exponents,
such as the primes:

$1,x^{2},x^{3},\dots,x^{89},x^{97},\dots.$ (3)

For discussions of the theorem with proofs, see [2], [6], and [11]. After
1914, Müntz’s theorem was generalized by Szász and others.

As a numerical analyst, I work with algorithms based on expanding functions in
nonorthogonal bases, a powerful technique in certain contexts [1, 10]. This
led me to consider Müntz’s theorem from a computational angle, and what
emerged is startling. To make the point, it is enough to consider a particular
case of what might be regarded as the most basic nontrivial Müntz
approximation. The name “E” alludes to the use of even powers.

Problem E. Given $\varepsilon>0$, find an integer $n\geq 0$ and coefficients
$c_{0},\dots,c_{n}$ such that

$\Bigl{|}\kern 1.0ptx-\sum_{k=0}^{n}c_{k}x^{2k}\Bigr{|}\leq\varepsilon,\qquad
x\in[\kern 0.3pt0,1].$ (4)

We shall prove:

######  Theorem 2.

If $\varepsilon<1/2$, then any solution of Problem E has

$n>{1\over 20\kern 0.5pt\varepsilon}$ (5)

and

$\max_{k}|c_{k}|>0.75\kern 0.7pt\varepsilon\kern 0.7pt2^{\kern 0.8pt1/(40\kern
0.4pt\varepsilon)}.$ (6)

Actually, I believe the following sharper bounds hold:

$n>{1\over 8\varepsilon},$ (7) $\max_{k}|c_{k}|>{(1+\sqrt{2}\kern
1.0pt)^{2\kern 0.4ptn}\over 16\kern 0.5ptn^{1.5}}.$ (8)

For accuracy $\varepsilon=10^{-6}$, my estimate is that one needs
$n>140{,}000$ and $\max_{k}|c_{k}|>10^{107{,}000}$. In such an expansion, the
enormous coefficients have oscillating signs, so that they cancel almost
exactly (namely to one part in $10^{107{,}000}$). On a computer in floating-
point arithmetic, all information will be lost unless one works in a precision
of more than $107{,}000$ digits. (The usual precision is 16 digits.)

##  2 Proof

Problem E is equivalent to the more familiar problem of approximation of $|x|$
on $[-1,1]$:

$\Bigl{|}|x|-\sum_{k=0}^{2\kern
0.3ptn}c_{k}x^{2k}\Bigr{|}\leq\varepsilon,\qquad x\in[-1,1].$ (9)

Since Lebesgue first used approximations of $|x|$ for a proof of the
Weierstrass approximation theorem at age 23 in 1898, a great deal has been
learned about this problem, as recounted in chapter 25 of [14]. In particular,
Bernstein’s 1914 paper [5] was a landmark contribution. Among many other
things, Bernstein proved that $\varepsilon$ satisfies

$\varepsilon>{1\over 4(1+\sqrt{2}\kern 1.0pt)(2\kern 0.3ptn-1)}$ (10)

for any $n\geq 1$, which implies, since $4(1+\sqrt{2}\kern 1.0pt)\approx
9.66$,

$n>{1\over 20\kern 0.5pt\varepsilon}.$ (11)

Since $\varepsilon<1/2$ in (9) implies $n\geq 1$, this establishes condition
(5) of Theorem 2.

To establish condition (6), we make use of (11). Given $\varepsilon$, let $n$
and $\\{c_{k}\\}$ define a solution (4) of Problem E. If we split the series
into roughly the first quarter and the last three-quarters,

$\sum_{k=0}^{\lfloor 1/(80\kern
0.3pt\varepsilon)\rfloor}c_{k}x^{2k}+\sum_{k=1+\lfloor 1/(80\kern
0.3pt\varepsilon)\rfloor}^{n}c_{k}x^{2k},$ (12)

then by (11), the first part can approximate $|x|$ no more closely than
$4\kern 0.5pt\varepsilon$. More to our purpose, by a linear scaling, it can
approximate $|x|$ over the subinterval $[-1/2,1/2]$ no more closely than
$2\kern 0.5pt\varepsilon$. Therefore, since the sum of the two series in (12)
has accuracy better than $\varepsilon$, the second series must have maximal
size at least $\varepsilon$ over $[-1/2,1/2]$. Since $|x^{2k}|\leq 2^{-2k}$
for $x\in[-1/2,1/2]$, this implies that there must be some huge coefficients.
Specifically, summing a power series involving powers of $4$ shows that the
second series of (12) is bounded by

$\sum_{k=1+\lfloor 1/(80\kern
0.3pt\varepsilon)\rfloor}^{n}|c_{k}x^{2k}|\leq{4\over 3}\kern
1.0pt\max_{k}|c_{k}|\kern 1.0pt2^{-2(\lfloor 1/(80\kern
0.3pt\varepsilon)\rfloor+1)}.$ (13)

Therefore we must have

${4\over 3}\kern 1.0pt\max_{k}|c_{k}|\kern 1.0pt2^{-2(\lfloor 1/(80\kern
0.3pt\varepsilon)\rfloor+1)}>\varepsilon,$ (14)

that is,

$\max_{k}|c_{k}|>0.75\kern 0.7pt\varepsilon\kern 0.8pt2^{2(\lfloor 1/(80\kern
0.3pt\varepsilon)\rfloor+1)}.$ (15)

This implies (6), completing the proof of Theorem 2.

##  3 Numerical estimates

The theorem and proof just given were all about lower bounds, but now let us
look at more accurate (though unrigorous) estimates. Bernstein [5] also proved
that the best degree $2\kern 0.4ptn$ maximum-norm approximation errors
$\varepsilon$ satisfy

$\varepsilon\sim{\beta\over 2\kern 0.4ptn},\quad n\to\infty$ (16)

for some $\beta$, and in 1985 Varga and Carpenter [16] gave the numerical
estimate

$\beta\approx 0.28016949902386913303643649\dots.$ (17)

To achieve $\varepsilon\leq 10^{-1}$, $10^{-2}$, $10^{-3}$, and $10^{-4}$,
respectively, this suggests (rounding up to the next even numbers) that we
will need degrees $2\kern 0.4ptn$ of approximately $4$, $28$, $282$, and
$2802$. It turns out that the actual minimal degrees (as computed with the
Chebfun minimax command [7, 8]) are exactly these four numbers. For accuracy
$10^{-6}$, for example, though this is beyond Chebfun, it seems clear that the
required degree will be close to $n=280{,}170$.

Thus we see again that an approximation (4) requires degrees of order
$O(1/\varepsilon)$, but why are the coefficients so large? The explanation is
that the monomials $1,x^{2},x^{4},\dots,x^{2\kern 0.4ptn}$ are an
exponentially ill-behaved basis for the space of even degree $2\kern 0.4ptn$
polynomials on $[-1,1]$. Numerical analysts quantify this observation by
noting that the condition number of this set of functions is of the
approximate order

$\kappa_{2\kern 0.4ptn}\approx(1+\sqrt{2}\kern 1.0pt)^{2\kern 0.4ptn}\approx
10^{\kern 0.5pt0.766n}$ (18)

[3, 9]. With $2\kern 0.4ptn=280{,}170$ for accuracy $10^{-6}$, this suggests
the expansion coefficients will need to be of order about $10^{107{,}000}$.
Our best empirical approximation based on calculations for $n$ up to $300$ is

$\max_{k}|c_{k}|\approx 0.066\times{(1+\sqrt{2}\kern 1.0pt)^{2\kern
0.4ptn}\over n^{1.5}}.$ (19)

Table 1 summarizes our computations and estimates for accuracies
$\varepsilon=10^{-1},\dots,10^{-8}.$

Table 1: Numerical data for approximation of $x$ by even powers on $[\kern 0.3pt0,1]$, or equivalently, approximation of $|x|$ on $[-1,1]$. The numbers up to $2802$ are based on numerical computations, and the remaining ones are estimates. accuracy $\varepsilon$ | minimal degree $2\kern 0.4ptn$ | maximal coefficient $|c_{k}|$
---|---|---
$10^{-1}~{}~{}~{}$ | $4~{}~{}~{}~{}~{}$ | $~{}~{}~{}~{}~{}~{}~{}~{}1.93$
$10^{-2}~{}~{}~{}$ | $28~{}~{}~{}~{}~{}$ | $~{}~{}~{}~{}~{}~{}~{}~{}7.4\times 10^{7}$
$10^{-3}~{}~{}~{}$ | $282~{}~{}~{}~{}~{}$ | $~{}~{}~{}~{}~{}~{}~{}~{}3.5\times 10^{103}$
$10^{-4}~{}~{}~{}$ | $2802~{}~{}~{}~{}~{}$ | $~{}~{}~{}~{}~{}~{}~{}~{}10^{1068}$
$10^{-5}~{}~{}~{}$ | $28018~{}~{}~{}~{}~{}$ | $~{}~{}~{}~{}~{}~{}~{}~{}10^{10{,}700}$
$10^{-6}~{}~{}~{}$ | $280170~{}~{}~{}~{}~{}$ | $~{}~{}~{}~{}~{}~{}~{}~{}10^{107{,}000}$
$10^{-7}~{}~{}~{}$ | $2801696~{}~{}~{}~{}~{}$ | $~{}~{}~{}~{}~{}~{}~{}~{}10^{1{,}070{,}000}$
$10^{-8}~{}~{}~{}$ | $28016950~{}~{}~{}~{}~{}$ | $~{}~{}~{}~{}~{}~{}~{}~{}10^{10{,}700{,}000}$
|  |

Figure 1 illustrates graphically where the big coefficients come from. For
$2\kern 0.4ptn=28,56,\dots,140$, it plots the coefficients $|c_{k}|$ for
$k=0,1,\dots,n$ in a monomial expansion of the best approximations.

Figure 1: Monomial coefficients $|c_{k}|$ of best approximations of $x$ by
even powers on $[\kern 0.3pt0,1]$, or equivalently, of $|x|$ on $[-1,1]$, for
$2\kern 0.4ptn=28,56,\dots,140$. These values correspond to approximation
errors approximately $0.01,0.01/2,\dots,0.01/5$. The end of the curve is of
order $2^{2n}$, and the peak in the middle is of order $(1+\sqrt{2}\kern
1.0pt)^{2n}$.

##  4 A remark about mathematics

Theorem 2 is startling and interesting. From the usual mathematical point of
view, however, it is not much more than that. After all, Müntz’s theorem
remains valid and beautiful. From the usual mathematical perspective, Müntz’s
theorem expresses a fundamental truth, and Theorem 2, however interesting, is
an engineering footnote.

As I have discussed in the context of other problems [13, 15], I believe this
usual perspective is too comfortable. Theorem 2 implies that typical sets of
powers deemed useful by Müntz’s theorem would in fact be useless in any actual
application. If it was not the business of mathematicians to notice and
analyze this effect in the past 110 years, then whose business was it?

Acknowledgments

I am grateful for helpful suggestions to Michael Ganzburg, Daan Huybrechs,
Christian Lubich, Doron Lubinsky, Yuji Nakatsukasa, Allan Pinkus, and Endre
Süli.

Conflict of Interest

The author has no relevant financial or non-financial interests to disclose.

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# Initial perturbations dependence of non-equilibrium continuous and
discontinuous pattern transition

Rikuya Ishikawa and Rei Kurita  Department of Physics, Tokyo Metropolitan
University, 1-1 Minamioosawa, Hachiouji-shi, Tokyo 192-0397, Japan

###### Abstract

A phase separation in a spatially heterogeneous environment is closely related
to intracellular science and material science. For the phase separation,
initial heterogeneous perturbations play an important role in pattern
formations. In this study, a pattern transition from a lamellar pattern to a
columnar pattern is investigated in the presence of a slit pattern as the
initial perturbations. Here it is found that the transition behavior depends
on the initial slit width. When the initial slit width is close to the width
of the columnar pattern at the steady state, the pattern transition is the
second-order-like (continuous) transition. Meanwhile, the pattern transition
becomes the first-order-like (discontinuous) transition if the width of the
initial slit is much larger than that at the steady state. Then those
transition behaviors can be explained by the dynamical path during the pattern
formation. This finding will advance understanding of the initial perturbation
dependence of nonequilibrium phenomena.

Phase separation; Pattern formation; Nonequilibrium phenomenon; Spinodal
decomposition; Directional quenching

##  I Introduction

A phase separation (PS) in which two phases are separated from a mixture is
important in a wide range of fields such as material performance and
biological activities Cahn and Hilliard (1958); Hyman  _et al._ (2014); Cross
and Hohenberg (1993); Onuki (2002); Hamley (2007). For example, it is known
that SnPb alloy used as solder exhibits phase separation and the formation of
phase-separated domains may facilitate the formation of cracks at phase
boundaries, resulting mechanical failure Dreyer and Müller (2001). Moreover,
it has been reported that protein droplets formed by the liquid-liquid phase
separation (LLPS) in cells play important roles in transcription Hnisz  _et
al._ (2017), signal transduction Su  _et al._ (2016), and the pathogenesis of
neurodegenerative diseases Wegmann  _et al._ (2018); Alberti and Dormann
(2019).

Phase separation in binary mixtures has long been studied as a classical
problem, and the dynamics of phase separation appears to be well understood
both experimentally and theoretically Cahn and Hilliard (1958); Onuki (2002).
However, the universality of the dynamics holds for spatially uniform systems,
but not for systems with inhomogeneous temperatures or concentrations. For
example, in the case of an inhomogeneous system with large initial
concentration fluctuations, a concentric pattern is formed around high
concentration areas Furukawa (1994). In the case of non-stationary temperature
fields such as directional quenching (DQ) Furukawa (1992); Krekhov (2009);
Tsukada and Kurita (2020); Ishikawa  _et al._ (2022), a random droplet,
lamellar and columnar pattern are formed depending on a migration speed $V$ of
a quenching front. There are many other examples for showing specific pattern
formations in systems such as PS with radial quenching Kurita (2017); Tsukada
and Kurita (2019a, b), PS with temperature gradient Roy  _et al._ (2018); Roy
and Maciolek (2018), PS with lamination Ishikawa  _et al._ (2023), PS with
double quenching Tanaka and Araki (1998), PS with containing particles Tanaka
_et al._ (1994); Araki and Fukai (2015); Araki and Tanaka (2006); Krekhov  _et
al._ (2013), particles creation Adachi and Kawaguchi (2021), self-propelled
systems Cates and Tailleur (2015), and nonreciprocal interaction systems Saha
_et al._ (2020). Understanding of the phase separation phenomena under such
inhomogeneous conditions is an urgent issue since the pattern formation are
related with development of new functional materials and LLPS in cells occurs
in inhomogeneous concentration and temperature.

Although the importance of initial perturbations has been suggested for the
pattern formation, there are few quantitative studies on the effects of the
initial perturbations on a transition point and a transition behavior.
Understanding the effects of the initial perturbations that cause transitions
will not only facilitate the understanding of nonequilibrium systems, but also
control the transitions which will be useful for applications in various
fields. Therefore, we investigated the effect of the initial perturbations for
the transition from columnar pattern to lamellar pattern in DQ. In this study,
we prepared a slit pattern into the lamellar pattern as initial perturbations
and studied a process of the transition from the columnar pattern to the
lamellar pattern. It was found that the transition behavior depends on the
slits width $h$ and it means that the condition of the initial perturbation
plays a critical role for the pattern formation.

##  II Methods

In this study, we used modified-Cahn-Hilliard equation for understanding a
dynamics of two dimensional phase separation under inhomogeneous temperature
fields Cahn and Hilliard (1958); Jaiswal  _et al._ (2013). The normalized
equation of the modified-Cahn-Hilliard equation is given as

$\displaystyle\frac{\partial\phi}{\partial
t}=\nabla^{2}[\epsilon(x,t)\phi+\phi^{3}-\nabla^{2}\phi]$ (1)

where $\phi,t$ and $\epsilon$ are the normalized concentration, the time
normalized by the diffusion time and the temperature normalized by the quench
depth, respectively. The length is normalized by the correlation length. We
note that the effect of temperature gradient (Ludwig-Soret effect) was
neglected. When $\epsilon\geq 0$, the mixed state is stable; when
$\epsilon<0$, the mixture is separated into two phases. In this study, we
define $\phi=1$ and $\phi=-1$ which are the concentrations after phase
separation as A (white region) and B phase (black region), respectively. To
compute eq. 1, we used the Euler method and set a grid size to $\Delta x=1$
and a time increment to $\Delta t=0.01$. We used a free surface condition in
the $x$ directions and a periodic boundary condition in the $y$ directions.
The system size is $L_{x}:L_{y}=1000:1000$.

We initially put A phase in $x<40$ and then we put one or two slits of B phase
in A phase as the initial perturbations (see Fig. 2). The width of one slit or
the period of two slits is $h$. The region $x\geq 40$ is a symmetric
composition $\phi=0$. Then, the quenching front was set to $x$ = 40 and
annealed for $-10<t<0$ to smooth the slit boundary of $\phi$. Even after the
annealing, the width of the slit was confirmed to be $h$. At $t$ = 0, the
quenching front moved in the $x$ direction with a constant velocity $V$.
Therefore, the temperature $\epsilon(x,t)$ is as follows.

$\displaystyle\epsilon(x,t)=\left\\{\begin{array}[]{ll}1&x\geq Vt+40\\\
-1&x<Vt+40\end{array}\right.,\ \ \ \ x\in[0,L_{x}-1]$ (4)

##  III Results

Firstly, we describe the pattern formation when the quenching front is moved
at $V$ in the slit systems. We defined “columnar pattern” when columns
percolate to the right boundary of the simulation box, while “lamellar
pattern” is stable when the columns change into the lamellae during the
pattern formation. We set the transition velocity $V_{t}$ at the median of the
lowest velocity for the lamellar pattern and the highest velocity for the
columnar pattern. Figure 1 shows the $h$ dependence of $V_{t}$ in one or two
slits system. There are two types of the transitions; one is a discontinuous
transition (DT) and the other is a continuous transition (CT). We will
describe the difference between DT and CT in the next paragraph. Circles,
triangles, and diamonds correspond to $V_{t}$ with DT for one slit, with CT
for two slits, and with DT for two slits, respectively. For the one slit
system, the $V_{t}$ line is smooth with a peak at $h$ = 9 and the CT cannot be
observed. Meanwhile, for the two slits system, $V_{t}$ is larger for $h\leq 9$
(triangle symbols) than for one slit system. For $h\geq 10$ (diamond symbols),
$V_{t}$ with DT was almost the same as $V_{t}$ with DT in the one slit system.
In the following, we will reveal the reason for the large $V_{t}$ jump around
$h$ = 9 in the two slits case and we will discuss the difference from the one
slit case.

Figure 1: $h$ dependence of $V_{t}$. Circles, triangles, and diamonds
correspond to $V_{t}$ with a discontinuous transition (DT) for one slit, with
a continuous transition (CT) for two slits, and with a DT for two slits,
respectively. We set the transition velocity $V_{t}$ at the median of the
lowest velocity for the lamellar pattern and the highest velocity for the
columnar pattern. An error bar represents the lowest velocity for the lamellar
pattern and the highest velocity for the columnar pattern.

Next, we describe $h$ dependence of the pattern formation in the two slits
case. Figure 2 shows the pattern formations for different $h$. For $h$ = 8 and
$V$ = 0.665 just above $V_{t}$, the number of the slit increases as columns,
but the columns break off in the middle and the lamellae are formed; the
column formation have a finite lifetime (or a finite distance). Meanwhile, for
$h$ = 8 and $V$ = 0.66 just below $V_{t}$, the columnar pattern percolates to
the right boundary and the number of the column gradually increases. For the
slits width $h$ = 20 and $V$ = 0.595 just above $V_{t}$, no column is
generated from the slits and only lamellae are formed immediately after the DQ
starts. Meanwhile, for $h$ = 20 and $V$ = 0.59 just below $V_{t}$, a thin
column is formed at $t$ = 200 and then the number of the thin column increases
and the columnar pattern percolates to the right boundary. Here we defined the
continuous transition (CT) when the columns and lamellae coexist with some
length like the system with $h$ = 8, while the discontinuous transition (DT)
when the lamellae are formed immediately.

Figure 2: Time evolution of patterns in two slits. The yellow dotted lines
represent the quenching front. (a) For $h$ = 8 and $V$ = 0.665 just above
$V_{t}$, the number of the slit increases as columns, but they break off in
the middle and lamellae are formed; the column formation has a finite lifetime
(or a finite distance). Meanwhile, for $h$ = 8 and $V$ = 0.66 just below
$V_{t}$, the columnar pattern percolate to the right boundary and the number
of the column gradually increases. (b) For the slits width $h$ = 20 and $V$ =
0.595 just above $V_{t}$, no column is generated from the slits and only
lamellae are formed immediately. Meanwhile, for $h$ = 20 and $V$ = 0.59 just
below $V_{t}$, a thin column is formed at $t$ = 200 and then the number of the
thin column increases and the columnar pattern percolates to the right
boundary.

In addition, to clarify the mode of the transition, we investigated the
growing behavior of the column pattern just below $V_{t}$. When the quenching
front migrates toward $x$ direction, the lamellae are basically formed, but
the column pattern also grows inside the lamellae (see Fig. 2). We defined a
lamella and a column as one pair of A and B phase, and the number of lamellae
and columns as $n_{l}$, $n_{c}$, respectively. Figure 3 (a) and (b) show the
$n_{c}$ dependence of $n_{l}$ for $h$ = 8 and 20, respectively. For $h$ = 8,
$n_{c}$ linearly increases at any $V$. When $V$ is small, the slope $a$ is
about 2, which indicates that two columns are formed each time one lamella
formed. As $V$ approaches $V_{t}$, the slope decreases. Meanwhile, for $h$ =
20, the slope changes when the initial thick slit changes to the thin column
near $V=V_{t}$. It is also found that the slopes in the later period are
almost constant. We investigate the $a$ dependence on the velocity difference
from $V_{t}$ $\delta(=V_{t}-V)$ shown as in Fig. 3(c). Circles and squares
correspond to $a$ in $h$ = 8 and 20, respectively. Error bars reflect the
errors in the linear fitting. For $h$ = 8, $a$ obeys with $\delta^{\alpha}$
and the power $\alpha$ is about 0.40. The physical meaning of $\alpha$ is
unclear, thus the theoretical derivation of this relationship is a future
work. For $h$ = 20, $a$ has a non-zero value even $\delta\rightarrow 0$. Thus,
the growth behaviors of the column pattern depend on $\delta$ and there is a
clear difference between CT ($h$ = 8) and DT ($h$ = 20).

Figure 3: $n_{c}$ dependence on $n_{l}$ for (a) $h$ = 8 and for (b) $h$ = 20.
Each symbols represent $n_{c}$ in a different $V$. For $h$ = 8, $n_{c}$
linearly increases at any $V$ and the slope decreases with increasing $V$. For
$h$ = 20, the slope change at the time of the changing from thick slits to
thin column when $V$ is close to $V_{t}$ and the slope in the later period are
almost constant near $V_{t}$. (c) $\delta(=V_{t}-V)$ dependence of the slope
$a$ in log-log scales. Circles and squares correspond to $h$ = 8 and 20,
respectively. For $h$ = 8, $a$ obeys with $\delta^{\alpha}$ and the power
$\alpha$ is about 0.40. For $h$ = 20, $a$ has a finite value even
$\delta\rightarrow 0$. The $a$ dependence of $\delta$ also suggests the
difference between CT and DT modes.

We also show the pattern formation behaviors above $V_{t}$. To clarify the
dynamics in CT, we investigated the coexistence length of the columns and the
lamellae for $h$ = 8. We define $\xi$ as the distance from the initial
position of the slit to the vanishing point of the columns (see Fig. 4 (a))
and investigate $\xi$ dependence on $|\delta|(=V-V_{t})$ shown as in Figure
4(b). $\xi$ diverges as $\xi\sim|\delta|^{\beta}$ with decreasing $|\delta|$
and $\beta$ is about -0.50. This behavior also suggests that the pattern
transition at $V=V_{t}$ should be continuous at $h$ = 8. Meanwhile, $\xi$ = 0
at $h$ = 20 as we noted above (see Fig. 2(b).

Figure 4: (a) We define $\xi$ as the distance from the initial position of the
slit to the vanishing point of the columns at $h$ = 8. (b) $|\delta|$
dependence of $\xi$. $\xi$ diverges as $\xi\sim|\delta|^{\beta}$ with
decreasing $|\delta|$ and $\beta$ is about -0.50. This behavior also suggests
that the pattern transition at $V=V_{t}$ should be continuous at $h$ = 8.

Here we investigated the time evolution of the concentration at a position
prior to the quenching front of the column in order to reveal the formation
dynamics in the continuous transition. Figure 5(a) shows the time evolution of
$\phi(x_{f}+3,y)$ at $h$ =8 and $V$ = 0.663, where $x_{f}$ is a position of
the quenching front at $t$. We also define the $y$-coordinate of the center of
the column as $y_{c}$ (indicated by yellow dotted line) and
$\phi(x_{f}+3,y_{c})$ is denoted by $\phi_{fc}$ in the following. Since the
concentration is conserved and A phase rich ahead of the column with B phase,
$\phi_{fc}$ is positive (A rich phase), while the column is composed by B
phase. As time passed, $\phi_{fc}$ periodically changes shown as in Fig. 5(b).
Circles, squares and triangles are $V$ = 0.66, 0.663, and 0.665, respectively.
When $V$ is smaller than $V_{t}$ = 0.6625 ($V$ = 0.66), the amplitude of
$\phi_{fc}$ oscillation is quite small at any $t$. Meanwhile, when $V$ is
larger than $V_{t}$ ($V$ = 0.663 and 0.665), the oscillations of $\phi_{fc}$
become larger with time and then the column finally transforms into lamellae.

Here, we considered the periodic fluctuations and divergence of $\phi_{fc}$.
Due to the conserved manner, A component accumulates in front of the B phase
columns ($\phi_{fc}>0$ in front of the B column). When B phase columns grow,
the accumulated A component is diffused in the lateral direction. When a
lamella of A phase is formed next to B phase column, a layer with $\phi<0$ is
formed in front of the quenching front. Then the accumulated A component in
front of the B phase columns tends to diffuse the laterally since the
difference of $\phi$ is large. Thus, $\phi_{fc}$ oscillates slightly with the
same period as the period of the lamella. Next, the quenching front movement
is faster than the characteristic time of the diffusion when $V\geq V_{t}$.
The A component is gradually accumulated in front of the B phase columns with
time. The amplitude of the oscillation gradually increases and the onset time,
when the amplitude of $\phi_{fc}$ oscillation becomes larger, becomes shorter
with increasing $V$. They are consistent that the accumulation of A component
is faster than the diffusion. Therefore, the dynamics in CT is determined by
the competition between the accumulation and the diffusion in front of the
column.

Figure 5: (a) The time evolution of $\phi(x_{f}+3,y)$ at $h$ =8 and $V$ =
0.663, where $x_{f}$ is a position of the quenching front at $t$. $x_{f}$ and
$y_{c}$ show the $x$-coordinate of the quenching front and the $y$-coordinate
of the column center (indicated by yellow dotted line), respectively. (b) Time
evolution of $\phi_{fc}$ at $h$ = 8. Circles, squares and triangles are $V$ =
0.66, 0.663, and 0.665, respectively. $V_{t}$ = 0.6625 at $h$ = 8. When $V$ is
smaller than $V_{t}$ = 0.6625 ($V$ = 0.66), the amplitude of $\phi_{fc}$
oscillation is quite small at any $t$. Meanwhile, when $V$ is larger than
$V_{t}$ ($V$ = 0.663 and 0.665), the oscillations of $\phi_{fc}$ become larger
with time and then the column finally transforms into lamellae. The dynamics
in CT is determined by the competition between the accumulation and the
diffusion in front of the column.

When the width of the initial slit $h$ is large, the thinner column is
reformed during DQ. It means that the stable width of the column is mismatched
with the width of the initial slit. Thus, we compute the stable width of the
column here. According to the previous report in Ref. Ishikawa  _et al._
(2023), a stable thickness of the lamella $\xi^{\prime}$ can be expressed as
$\xi^{\prime}\sim V^{\prime-1/2}$ since the time scales of the diffusion and
the migration of the free surface are balanced. Similarly, the stable width of
the column is expected to be determined by the competition between the
velocity of the quenching front and the speed of the diffusion. We
investigated the $V$ dependence of the stable width of the column $\lambda$
just after the entire system was quenched. Figure 6 shows the $V$ dependence
of $\lambda$. Circles and triangles are $h$ = 8 and 20, respectively. Symbols
and error bars are the mean and the standard deviation of $\lambda$ for all
columns, respectively. It is found that $\lambda$ is independent of $h$ and
$\lambda\propto V^{-0.60}$. In the lamella pattern, the diffusion
perpendicular to the quenching front is only considered, while the diffusion
in both perpendicular and lateral direction should be related in the column
pattern. Thus we consider that the exponent in the column pattern is different
from that in the lamella pattern. When $h$ is large and $V<V_{t}$, the wide
initial slit branches into columns of the steady width $\lambda$. Meanwhile,
when $V\geq V_{t}$, the diffusion is slower than the migration of quenching
front. A layer of A component is formed in front of the B phase column and
then the layer becomes the stable lamellar pattern.

Figure 6: $V$ dependence of $\lambda$ just after the entire system is
quenched. Circles and triangles are $h$ = 8 and 20, respectively. Symbols and
error bars are the mean and standard deviation of $\lambda$ for all columns,
respectively. It is found that $\lambda$ is independent of $h$ and
$\lambda\propto V^{-0.60}$.

Finally, we explain the conditions under which CT occurs. Figure 7 is a
superimposed graph of Fig. 1 and Fig. 6 with $V$ on the vertical axis and $h$
on the horizontal axis. Filled circles and filled squares are the stable
column width $\lambda$ at the initial slit width $h$ = 8 and 20, respectively.
There is a critical-like point $(h,V)=(h_{c},V_{c})$, where
$\lambda(V_{c})=h_{c}$ and $V_{t}(h_{c})=V_{c}$. Figure 7 is a state diagram
and the transition modes between CT and DT can be explained by this diagram.
As an example, we consider the case $h$ = 15 and $V$ = 0.5 (black circle). In
this case, a misfit parameter between the initial slit and the stable column
$\Delta h=h-\lambda$ is large. When $V<V_{t}$, the slits for $h$ = 15 branch
to a column with $\lambda=8$ to eliminate this misfit. In the post-bifurcation
state, $\delta=V_{t}-V$ is also large and the ratio of column formation to
lamella $a$ is also large. Meanwhile, the slits immediately become a lamella
if $V>V_{t}$. Therefore, due to the bifurcation, the condition cannot be close
to the critical-like point and then the transition is discontinuous.
Meanwhile, $\lambda\sim h_{c}$ holds at $h$ = 8 and $V=V_{c}$. In this case,
the condition of DQ is close to the critical-like point and the transition
becomes continuous. We note here that if $h$ is slightly larger than
$\lambda$, the slits cannot branch. Therefore, we consider that the transition
is the continuous up to $h$ = 9.

In addition, we describe the difference of the transition mode between one
slit and two slits. One slit is more likely to be lamellae than two slits
because the upper and lower lamellae are closer together. Therefore, $V_{t}$
for one slit is smaller than for two slits. If $V<V_{t}$, two or more columns
are formed from one slit. When two or more columns are formed, the stability
increases and then $V_{t}$ becomes larger. Then, the condition becomes far
from the critical-like point. We note that the state diagram for three or more
slits is similar to that of two slits. Therefore, the stability of the pattern
is unchanged after two or more columns are formed.

These results suggest that the stability of the system changes with the time
dependent condition such as the number of the slit, the width of the slit or
the column. Then the discontinuous transition occurs since the re-established
transition point is farther away from the simulated state. Conversely, if the
initial inhomogeneous perturbations are close to the stable state, the
stability of the system is unchanged and then the continuous transition occurs
near the critical point. It can be concluded that the difference between the
initial inhomogeneous perturbations (the slit width $h$ in this study) and the
steady state (the column width $\lambda$ in this study) significantly affects
the mode of the pattern formation.

Figure 7: A superimposed graph of Fig. 1 and Fig. 6 with $V$ and $h$. The
vertical and horizontal axis of Fig. 6 are swapped. Filled circles and filled
squares are the column width $\lambda$ at $h$ = 8 and 20, respectively. There
exists a critical-like point $(h,V)=(h_{c},V_{c})$, where $\lambda$ and
$V_{t}$ curves cross. When the condition is close to the critical-like point,
the transition is continuous. In the case $h$ = 15 and $V$ = 0.5 (black
circle), a misfit parameter between the initial slit and the stable column
$\Delta h=h-\lambda$ is large. When $V<V_{t}$, the slits for $h$ = 15 branch
to a column with $\lambda=8$ to eliminate this misfit. In the post-bifurcation
state, $\delta=V_{t}-V$ is also large and then the transition is
discontinuous.

##  IV Discussion

Firstly, we discuss the relevance of this study with experimental studies on
directional pattern formation during eutectic growth and gelation process. In
metallic alloys, eutectic patterns have been observed to self-assemble
perpendicular to the crystal-growth direction (like columnar pattern) when it
solidifies in a certain direction Ginibre  _et al._ (1997); Jackson and Hunt
(1966); Kulkarni  _et al._ (2018). However, a horizontal pattern (like
lamellar pattern) is not formed with respect to the crystal-growth direction.
Similarly, in gelation of collagen induced by directional neutralization, a
pattern perpendicular to the gelation plane (like columnar pattern) is formed
Yonemoto  _et al._ (2021), but a horizontal pattern (like lamellar pattern) is
not formed. In this study, $V_{t}$ with the initial perturbations is about one
order of magnitude larger than $V_{t}$ in homogeneous DQ Tsukada and Kurita
(2020) and then the $V$ regime of the lamellar pattern becomes narrow.
Therefore, in nonequilibrium systems, the uncontrollable initial perturbations
exist and then it seems difficult to observe lamellar patterns in experiments.

Next, we discuss the relationship between CT in DQ and the general critical
phenomena. The results of the continuous transition in DQ are deterministic,
not taking into account thermal fluctuations. On the other hand, the critical
phenomena of the phase transition Onuki (2002) and a laminar turbulent
transition Sano and Tamai (2016) are stochastic with thermal fluctuations. In
fact, the columns do not disappear at the center of the columnar pattern,
which is different from a critical phenomenon. Therefore, in this stage, the
CT in DQ is different from these critical phenomena, and it is more
appropriate to consider the pattern transition as interfacial dynamics.

Finally, the transition of the pattern formation under inhomogeneous
conditions are not limited to phase separations, but also occur in other
nonequilibrium systems such as active matter Vicsek  _et al._ (1995); Ginelli
(2016); Chaté (2020), non-reciprocal phase transitions Fruchart  _et al._
(2021), reversible-irreversible transitions in jamming systems Nagasawa  _et
al._ (2019); Reichhardt  _et al._ (2023). It is interesting that
nonequilibrium continuous transition are observed in many systems such as the
laminar-turbulent transition in fluids Sano and Tamai (2016), two turbulent
states in liquid crystals Takeuchi  _et al._ (2007, 2009) and a reversible-
irreversible transition of fiber Franceschini  _et al._ (2011), vortices
Maegochi  _et al._ (2021); Mangan  _et al._ (2008) and granular materials Ness
and Cates (2020). Thus it is expected that the difference between the
condition at the steady state and the initial perturbations plays an important
role in those nonequilibrium transition.

##  V Summary

A phase separation is important in a wide range of fields such as material
science and biology. Those phase separations often exhibit in inhomogeneous
concentrations or non-stationary temperature fields. Understanding of phase
separation phenomena under such inhomogeneous conditions is an urgent issue
since LLPS in cells occurs in inhomogeneous concentration and temperature and
the pattern formation are related with development of new functional
materials.

Here, we investigated the columnar-lamellar pattern transition due to
directional quenching (DQ) in the presence of initial slits pattern to study
the effect of initial perturbations. In this study, we prepared one slit and
two slits which are the nuclei of the columnar pattern as initial
perturbations. Then, we studied the process of transition from columnar
pattern to lamellar pattern dependent on the initial perturbations.

It is found that the transition behavior depends on the slit width $h$ and the
number of the slits. The transition from the column to the lamella is
continuous when $h$ is comparable to the column width $\lambda$ and two slits.
Then, the growth of the columnar pattern is determined by the balance between
the diffusion and the accumulation of the concentration. Whereas, it is a
discontinuous transition when $h$ is larger than the column width $\lambda$.
The stability of the system changes with the time dependent condition such as
the number of the slit, the width of the slit or the column. Then the
discontinuous transition occurs since the re-established transition point is
farther away from the simulated state.

This study not only shows the empirically knowledge that inhomogeneous initial
perturbations can significantly change the transition point from a homogeneous
system. It also shows that in nonequilibrium systems, the transition behaviors
are determined by the difference between the initial perturbations and the
condition at the steady state. Therefore, it is important to design initial
perturbations comparing with the condition at the steady state in order to
control pattern formation and nonequilibrium transitions in experiments.

## Acknowledgements

R. I. was supported by JST SPRING, Grant Number JPMJSP2156. R. K. was
supported by JSPS KAKENHI Grant Number 20H01874.

## AUTHORS CONTRIBUTIONS

R. I. and R. K. conceived the project. R. I. performed the numerical
simulations and analyzed the data. R. I. and R. K. wrote the manuscript.

## COMPETING INTERESTS STATEMENT

The authors declare that they have no competing interests.

## CORRESPONDENCE

Correspondence and requests for materials should be addressed to R. I.
<EMAIL_ADDRESS>and R. K. (kurita@tmu.ac.jp).

## Availability of Data and Materials

All data generated or analyzed during this study are included in this
published article and its supplementary information files.

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