# An Open-Ended Approach to Understanding Local, Emergent Conservation Laws in
Biological Evolution
††thanks: This work was funded by Cross Compass, LTD.

Alyssa M Adams  Cross Labs, Cross Compass
Kyoto, Japan
<EMAIL_ADDRESS>Eliott Jacopin  Center for Biosystems Dynamics
ResearchRIKEN
Kobe, Japan
http://orcid.org/0000-0002-4568-283X  Praful Gagrani  Wisconsin Institute for
Discovery
University of Wisconsin-Madison
Madison, WI, USA  Olaf Witkowski  Cross Labs, Cross Compass
Kyoto, Japan
<EMAIL_ADDRESS>
###### Abstract

While fields like Artificial Life have made huge strides in quantifying the
mechanisms that distinguish living systems from non-living ones, particular
mechanisms remain difficult to reproduce in silico. In particular, biology
seems to endlessly produce new, innovative states and new, innovative rules.
Known as open-endedness, we’ve been successful in finding mechanisms that
generate new states, but have been less successful in finding mechanisms that
generate new rules. In this discussion paper, we weigh whether or not
analyzing the effects of internal and external system constraints on a
system’s dynamics would be a fruitful avenue to understanding open-endedness
in future studies. We discuss the conceptual connection between physical
constraints and the ways that the system can physically reach possible states
while those constraints are present. It seems that the physical constraints
that define biological objects (and dynamics) are maintained by dynamics that
occur from within the system. This is in opposition to current modeling
approaches where system constraints are maintained by external forces. We
suggest that constraints can be characterized as variables whose values are
either completely conserved, quasi-conserved, or conditionally conserved.
Regardless of whether or not a constrained variable is a part of the
biological object or present in the object’s environment, we discuss how the
accessible system states under that constraint can lead to local, emergent
conservation laws (rules). We provide several examples of constraints based on
real, well-studied systems and highlight the differences in mechanisms that
constrain the variables being held constant. Finally, we discuss the possible
benefits of formally understanding how system constraints that emerge from
within a system lead to system dynamics that can be characterized as new,
emergent rules– particularly for artificial intelligence, hybrid life,
embodiment, astrobiology, and more. Understanding how new, local rules might
emerge from within the system is crucial for understanding how open-ended
systems continually discover new update rules, in addition to continually
discovering new states.

###### Index Terms:

Open-endedness, emergent rules, biological evolution, constraints, embodiment,
biological functions

##  I Introduction

Over the last few decades, researchers have made much progress in identifying
aspects of living systems that set them apart from non-living ones [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 54, 55], even if there is still much
more to be learned [15, 16, 17, 18, 19, 20, 21]. One popular approach,
particularly in Artificial Life (ALife), is to replicate some aspects of
living systems in a computer simulation. By doing so, it becomes possible to
tinker with the computer code and hardware to explore different system
behaviors. Based on these results, researchers can make connections between
their experiment’s design, and real, biological processes.

Many mechanisms seem to drive living systems. But they have been difficult to
quantify, measure, and understand well enough that they can be replicated in a
computer program. One aspect of life, in particular, continues to be difficult
to understand– its ability to continually find and select new, previously
unexplored states and dynamics without external intervention [20, 17, 16].
Known as open-endedness, this “behavior” continuously produces new, innovative
states and evolves under new, innovative dynamics without repeating itself
exactly over time [20, 17, 16, 25]. Theoretical approaches to exactly define
open-endedness have faced an inability to arrive at a field-wide consensus
[20, 17, 16]. So while it is understood in terms of broad, qualitative
definitions, an exact, mathematical definition that applies to any dynamic
system remains to be decided.

Computationally, engineering and quantifying open-ended dynamics in a closed
system has been particularly challenging [20, 26]. Because there isn’t a
field-wide standard mathematical definition to detect or quantify “how much
open-endedness” or “what type of open-endedness” is in a system [16, 26], it
is unclear how to interpret novel models that produce interesting dynamics
([27, 28, 29, 30], to name a just a few). But it is also unclear how open-
ended systems continue to produce new rules as a result of dynamics and
interactions from within a system, as opposed to constant, external
interventions being made on the system [1].

For bounded, deterministic computational approaches, implementing systems that
continually find new, innovative rules to evolve under is a challenge. As has
been suggested previously by others [31, 32], we suggest that a big part of
this challenge is due to the way we approach modeling. Current approaches
involve manually writing a set of rules (a program) with tunable variables
(parameters), running the program on a set of hardware, and characterizing the
resulting dynamics of the system. The program, parameters, and hardware are
either all manually designed, or can be set to change as some function of the
system’s state, dynamics, and/or external parameters [33, 34, 35].

There already exist a few explicit methods of changing the set of rules within
a system over time. In probabilistic computing, a data-prediction algorithm
can be edited and changed by a secondary “meta-program.” But even if the
lower-level program is continually changing, the meta-program that controls
doesn’t change directly as a result of “within-the-system” dynamics. The
resulting dynamics of the changing program are still a direct result of an
external force acting externally on the system.

So then, how do rules in systems that evolve open-endedly change as a result
of dynamics and interactions from within the system [1]? Moreover, how can we
understand biological rules like autopoesis, homeostasis, metabolism,
replication, etc. in terms of local, emergent conservation laws (or, from
another perspective, dynamics that could be described as new, emergent updates
rules) acting from within the system, even if not defined explicitly from
global update laws (such as the laws of physics)? Is it possible to do so
starting from global laws like physics and chemistry? How do internal and
external system constraints change the system dynamics in a way that they
appear to evolve under new, emergent rules?

Understanding how new, local “rules” might emerge from within the system is
crucial for understanding how open-ended systems continually discover new
update rules, in addition to continually discovering new states. The purpose
of this paper is to weigh whether or not analyzing the effects of internal and
external system constraints on a system’s dynamics would be a fruitful avenue
to understanding open-endedness in future studies. We start by outlining some
of the current challenges in ALife like open-endedness that are impacted by
these problems. We review current ALife approaches on how open-endedly
changing rules might build on each other to create the kind of “ratcheting
effect” we see in in the evolution of various systems (biological, cultural,
etc.) [36, 37]. Next, we briefly review the characterization of the most
important rules that distinguish living systems (“lyfe”) from abiotic ones
(“non-lyfe”) [2, 3]. Finally, we discuss future directions and possible
implications of a “constraint-centric” perspective of open-endedness for
ALife, artificial intelligence (AI), hybrid life, major transitions in
biological evolution, and astrobiology [20, 17, 16, 38].

##  II Challenges in Artificial Life

Understanding how biology evolves open-endedly, and what the mechanisms for
open-endedness are more broadly, remains an open challenge for ALife [39]. In
general, open-endedness can be classified into different types and
characterized in different kinds of systems [24, 13, 17, 18, 40, 41]. It is
generally agreed upon that any system that continually explores new states and
also continually explores new ways to find those states is open-ended to some
degree [20, 17, 16, 25]. But that degree can vary wildly from system to
system.

These challenges are at the intersection of ALife machine learning [22, 19,
15], astrobiology [25], economics [19, 15], mathematics [43, 42], and complex
systems [44]. The implications for fully understanding and being able to
engineer the necessary and sufficient mechanisms that produce open-ended
behavior in a system are profound.

###  II-A Emergence

The concept of emergence (such as the emergence of new system behavior, the
emergence of new kinds of objects, etc) is often at the center of many
problems in ALife [40]. The standard approach involves showing how new rules
(new, “effective laws of physics”) emerge as a result of lower-level, finer-
grained dynamics of “particles.”

Some examples of particles from physics include defects in a crystal lattice
and vortices in a fluid. The equations that describe their behavior, such as
potentials and motion, treat the objects as if they are fundamental particles.
But those equations are mediated by the collective action of many, many sub-
particles at a lower level. In these cases, “new physics” emerges from the
underlying dynamics, each with its own apparent rules and conservation laws
that are not explicitly part of the symmetries of the underlying physics.

In a crystal lattice, the ’defects’ are not conserved because they’re
invariants of the Hamiltonian of the universe. They’re conserved because
removing them would involve rearranging the entire crystal, and the crystal as
a whole is much bigger than the defect; it’s an event that happens with
(asymptotic with system size) probability zero. Therefore, the defect becomes
a new emergent conservation law of the coarse-grained system even if the
variable it tracks (defined on the coarse-grained macro-scale) doesn’t even
exist at the micro-scale.

###  II-B The continual creation of new, local rules

Overall, we have a good understanding of static mechanisms that continually
produce new states. For example, counting to infinity is a static mechanism
(always add 1 to the previous number) that always guarantees a new, never-
before-seen state. But understanding internal processes that continually
generate new, never-before-seen rules is much less understood [24, 13, 17, 18,
40, 41, 35].

Beyond the importance of recreating open-ended systems computationally,
knowing how new rules emerge (and persist) is crucial for understanding major
evolutionary transitions, including the transition from a non-living state to
a living one. Each major transition is marked by a new kind of dynamical rule
that persists across several, future biological objects [24, 13, 17, 22, 23].

Computationally we already implement systems that change a system’s program
(set of rules) as a result of a secondary meta-rule. In machine learning,
training a model involves iterating over different individual models (“rules”)
to finally settle on one target “rule”– one that might closely mimic a human
decision-making task, for example. Probabilistic computing methods like Gen
[34] and PyMC [33] change a program over time according to a secondary, meta-
program that is implicitly or explicitly consistent with Bayesian updating.
But in both of these cases, the rules are being updated by a secondary, static
program that is set externally. In the case of machine learning, the secondary
program is the training methods, in combination with the data being used to
train. In the case of probabilistic computing, the secondary program is the
Bayesian update method.

But for open-ended systems like living systems, it is unclear how rules should
change for it to meet the standard of each type of open-endedness. Can we
characterize secondary meta-programs such that they relate to the potential
kind of open-endedness they could contribute to? In the case of machine
learning, the secondary method (training a machine learning model) evolves as
well, but at a slower time scale. As the community of machine learning
researchers discovers new, efficient methods to train particular model
architectures for specific tasks, they introduce new methods for training
models such that a certain rule can be discovered faster. Is it necessary to
have a rule that changes a rule that changes a rule (three layers of rules) to
achieve stronger forms of open-endedness? At present, the answers to these
problems are unclear.

###  II-C Imposed constraints

In the quest to understand how rules for open-ended and living systems change
over time, some have explored the relationship between the physical
constraints imposed on a system and the dynamics that are allowed under those
constraints [1, 23]. Note the how a constraint is defined in [23]:

> What do we mean by constraints? In contrast to fundamental physical
> equations and their underlying symmetries, constraints are contingent
> causes, exerted by specific structures or dynamics, which reduce the degrees
> of freedom of the system on which they act. As additional causes, they
> simplify (or change) the description of the system, and enable an adequate
> explanation of its behaviour to be provided, an explanation which might
> otherwise be under-determined or wrongly determined. In describing physical
> and chemical systems, constraints are usually introduced as external
> determinations (boundary conditions, parameters, restrictions on the
> configuration space, etc.), which contribute to determining the behaviour
> and dynamics of a system, although their existence does not depend on the
> dynamics on which they act.

Here, we stress the important distinction between constraints applied to the
system externally and constraints that appear fixed as a result of staying
within some subset of the entire possibility state space. For example, the
oxygen percentage of our atmosphere acts as a constraint for life on Earth,
but that constraint is not upheld externally. Instead, the constraint appears
from within the system of biological evolution.

###  II-D The state space is a lie

The full possibility space for any real system is unthinkably large. But to
characterize or describe a system, we, as observers, either make a decision on
how to best describe a system or evolve the ability to sense aspects of a
system. By doing so, we construct a symbolic state space that characterizes
different aspects (including the temporal dynamics) of that system. This
involves selecting relevant variables (color, temperature, pressure, etc) to
capture important features necessary for our survival. In other words, the
physically embodied way that organisms use to interact with a system
determines the structure of the internal representation of that system.

For living structures, it seems like there is something special about these
coarse-grained representations. Biological evolution doesn’t seem to produce
embodiments that produce overly fine-grained representations of the system.
While including additional variables would increase the resolution of the
system representation, the cost of including these variables may not be worth
the amount of information gained by including it [14, 45, 46]. But still,
living systems seem to continually select embodiments that translate aspects
of an system into information that enable it to survive [14, 45, 46]. So,
there seems to be a balance between the need to capture enough information to
continue survival, but not too much such that it costs more energy than its
worth.

This is exactly the kind of process we wish to explore. We observers never see
the full state space of possibilities for any system, but as we interact with
real systems as observers, we assign labels to some system macrostates. By
doing so, observers construct an internal topology that represents how
internal coarse-grained representations of a system transition to other
internal coarse-grained representations of a system. An accurate sensory
mechanism is a physical embodiment that coarse-grains particular aspects of a
system into variables as that observer interacts with the system. Some work
suggests that this whole process acts like a compression algorithm on
description space for other systems [14, 45, 43]. What is the relationship
between resulting internal variables, the topology of a “representation” of a
system, and the physical constraints in an observer’s embodiment?

In the next section, we discuss different kinds of constraints based on the
kind of behavior a system evolves under while these variables are constrained.
We use a toy model to demonstrate how physical constraints result in different
representation state spaces. We don’t specify whether or not these constrained
variables are physically a part of a biological object or in the environment.
Instead, we focus on how systems evolve under these constraints. The resulting
constrained system dynamics have particular topologies that reflect local,
emergent rules. However, we don’t address how these variables are being
constrained– whether it’s from within the system in a homeostatic way, or by
an external force doing work on the system. We discuss future directions in
establishing a framework that characterizes the relationship between system
constraints, constrained system dynamics, and internal dynamics that uphold
those constraints from within the system.

##  III Examples of constraints as local rules

Figure 1: An example system of a lipid solution in a petri dish, where a lipid
bilayer membrane bubble could form (upper right) or not (upper left). While
the membrane bubble is present, it acts as a constraint on the number of
possible states the system can physically access. Under some level of
description for that system, such as the positions of individual lipids, the
dynamics of the real system can be represented as an observed state space with
a particular topology, or emergent, local rule (bottom left). The lipids in
the solution can be arranged into a lipid bilayer membrane (upper right),
either as an energetically favored macrostate or by putting external work onto
the system. For either case, doing so constrains the number of possible
observed states (bottom right) while the constraint (the membrane) is present.
These topologies are only used as a toy example and don’t reflect the dynamics
of an actual, lipid-solution system. Created with BioRender.com.

When the laws of physics lead to a system remaining in a smaller region of the
much larger possibility space, some observed variables may appear constant
(depending on how the variables represent the original system). At least for
scientists, the goal of research is to study a system well enough to present
its properties and dynamics with certain labels and variables. It often seems
like a good set of labels results in a representation of the system’s dynamics
that appears deterministic under those particular labels. This way, we can
write the resulting phase space topology111In physics, a phase space is a
theoretical tool used to describe all possible states a system can find itself
in, along with all the possible transitions between unique states. Phase
spaces are easily visualized in 2 or 3 dimensions, where each axis spans the
range of possible variable values to represent the dynamics of that system. as
a system of symbolic equations. Even if a system is proven difficult to
approximate symbolically, recent advances in machine learning make it possible
to approximate movement along that topology with a trained model. In either
case, the resulting representation is a direct reflection of being in the
smaller region of the large possibility space. The topology (number of states
and connections between them) of these smaller regions are local, emergent
rules. If the system were to explore states in another smaller region, the
resulting representation of observed system dynamics would be different.

In this section, we discuss constraints (variables that don’t change over
time) as properties of these smaller regions of a system’s possibility state
space. While there are other ways of characterizing these smaller spaces
(these apparent rules), we use constraints as a simple way to point to
different regions of the whole possibility state space. We use the word
“constraint” to describe variables that remain fixed in some region of the
state space. However, we note this may be misleading in the usual sense of the
word. This is because it does not necessarily imply whether or not that
variable is being held by external forces or generated and maintained by
internal ones. We also justify our interest in calling “variables that remain
fixed in some region of the whole state space of possibilities” constraints to
make an explicit connection to the various physical embodiments that
biological objects take to map aspects of their world into internal states.

Figure 2: Topological alteration of a state graph. a) A system being coarse-
grained and receiving arbitrary labels from an external observer. The example
macrostate ($u$) labels are vectors of three binary-valued microstate
variables denoted $(v_{1},v_{2},v_{3})$. Panels b to d display the topologies
of the physically-reachable state graph after constraining a variable of the
labeling vector. The nodes and edges that became inaccessible as a consequence
of applying a conservation law are displayed with lighter colors. The top row
corresponds to setting value 0, while the bottom row corresponds to setting
the variable to 1. b) Constraining variable $v_{1}$. On the one hand (b.1), a
conservation law for $v_{1}=0$ leads the system to state $u_{1}$. On the other
hand (b.2), with $v_{1}=1$, the system cannot escape from $u_{2}$. c)
Constraining variable $v_{1}$. In the case $v_{1}=1$ in c.2), the graph is
restricted to $u_{3}$ and has no possible transitions. The node does exist
under a conservation law setting $v_{1}$ to $1$ but there is no way to reach
this node due to the topology of the unconstrained graph. d) Constraining
variable $v_{3}$. This variable is constant and equal to $1$ in the
unconstrained graph in a), so a conservation law setting $v_{3}=0$ such as in
panel d.1 is meaningless for this case.

###  III-A Lipid membrane

For example, a petri dish with a lipid solution may or may not form a lipid
bilayer membrane bubble (Fig. 1, top). This system is described as some set of
variables with particular values (Fig. 1, bottom). The descriptive state space
(Fig. 1, lower left) has particular topological properties that are
conceptually related to the physical constraints on the system. When the lipid
bilayer membrane is present in the system, some set of descriptive variables
results in a particular topology of the physically accessible states under
that constraint. In the bottom row of Fig. 1, the nodes of this topology are
labeled with vectors (see Fig. 2). The membrane constrains the number of
system possibilities into a small subset of that state space (Fig. 1, lower
right). The constraint of the membrane not only imposes a physical constraint
on the movement of the lipids, but it also represents a particular subset of
an observer’s representation of that system as a lipid-position possibility
state space. As long as the membrane remains intact, the physically realized
states of the system remain constrained.

In this example, three things are happening: (1) A set of variables describes
the dynamics of the system into some “observed” state space with some
particular topology. The states are each labeled by a set of variables that
represent aspects and properties of individual lipid coordinates. (2) When the
membrane constraint is present in the system, the number of physically
accessible states shrinks into a smaller subset. For some observed state
spaces, the nodes in the smaller subset might have one or more variables with
values that are constant for these particular states. For the bubble, these
variables might represent the number of adjacent lipids to any single lipid,
which remains fairly constant while they are arranged in a lipid bilayer
membrane. (3) The membrane constraint is only present as a result of the laws
of physics of the system (being energetically favorable), or by putting
external work into the system (physically holding the lipids together through
some mechanism). Note how for this example, the constraint being present is
not a result of the internal dynamics of the bubble.

###  III-B Membrane of a living cell

In more complex, living cells, whatever is contained in the cell also
maintains the very membrane that constrains the contents of the cell. The loss
of the “living state” for the cell results in the mechanisms that maintain the
membrane to cease, thus dissolving the membrane. So for living objects like
cells, it seems that constraints that define the object are also present via
mechanisms that are only dynamically possible while that constraint is
present. In other words, the cell membrane conditionally constrains mechanisms
that allow it to persist [47, 48].

The stability of a cell membrane acts like a conditional conservation law
because the membrane traps objects inside the cell. While the cell membrane
itself is a constraint on the positions of lipids, the contents contained
inside the lipid cell are constrained conditionally as a result of the
membrane. In other words, the membrane constraint suddenly implies a second,
conditionally constrained set of variables in a descriptive state space that
describes the location of the trapped objects. In addition, the state of the
membrane constraint is maintained by the dynamics of objects it has trapped.
For this example, this constraint is a direct result of dynamics from within
the system, which also conditionally constrains the contents of the cell.

###  III-C Autocatalytic sets

Autocatalytic sets are an example of chemical systems with self-creating
dynamics. The conditions needed to perform particular chemical dynamics are
produced as a result of those dynamics. Like the membrane of a living cell,
the physical conditions needed for autocatalysis are maintained and upheld as
a result of the dynamics of the constrained state space.

Often considered a primitive, pre-biotic form of autopoiesis, researchers have
studied autocatalytic reaction networks characterized by chemical reaction
dynamics that display a kind of biological functionality [51, 52]. Here, the
autocatalytic functionality is a reflection of the topology of the chemical
system evolving under particular constraints.

###  III-D Autocatalytic networks

A reaction network is autocatalytic in a set of species if it consumes the set
while also producing it at an even higher rate [49]. Here, the conditions
needed to perform a particular chemical dynamic are produced as a result of
those dynamics. Like the membrane of a living cell, the physical conditions
needed for autocatalysis are maintained and upheld as a result of the dynamics
of the constrained state space. In dynamics where the autocatalytic species
are no longer spontaneously produced by the environment, an extinction event
of a particular autocatalytic set permanently constrains the system to a
smaller subspace. On the other hand, if the food (waste) needed (produced) by
the autocatalysts [50] is provided (removed) at a sufficiently large rate, the
same system will be constrained to an attractor basin or a growth regime very
far from equilibrium.

Often considered a primitive, pre-biotic form of autopoiesis, researchers have
studied autocatalytic reaction networks characterized by chemical reaction
dynamics that display a kind of biological functionality [51, 53]. It is shown
that ecosystems of autocatalytic subnetworks can recover familiar ecological
dynamics. Moreover, it is argued that a population of interacting ecosystems
(compartmentalized or diffusing on a lattice) can show selection toward the
discovery of autocatalytic subnetworks that increase the growth rate of the
ecosystem they constitute. At both levels, the autocatalytic functionality is
a reflection of the topology of the chemical system evolving under particular
constraints due to its own dynamics.

###  III-E Feedback-based canalization

The way that cracks form and grow in solids is a non-biological example of how
the creation of constraints can create a feedback mechanism that reinforces
the growth of that crack. Given a solid, there are lots of potential cracks
that could exist, but once any initial crack forms, it ends up concentrating
all of the available stress on its tip. Due to this stress, that particular
crack continues to form, rather than just other cracks also randomly forming.
Because of the physical properties of an initial crack, the system is
energetically favored to reinforce that one particular crack, since crack
propagation is fundamentally a non-equilibrium process.

###  III-F Living systems

But unlike membranes of bubbles and autocatalytic sets, living systems seem to
maintain these constraints as a result of the internal system doing work to
physically maintain the constraints that define them (homeostatic processes,
for example). These kinds of biological “endo-physics” don’t physically
realize the other possible states in the subspace because they actively spend
energy to avoid them.

Yet, when we describe biological systems, we often refer to their ability to
constantly explore new, innovative states. But we suggest describing living
objects according to their ability to avoid certain states, as per the
constraints that define their physical embodiment. As with the case of
autocatalytic sets and membranes of living cells, the topology of the
constrained state space results in functionality. Given that perspective, we
seek to identify the relationship between different types of constraints and
resulting functionality that may be related to well-known functions of living
systems [4]. In the next section, we use constraints (fixed variables) on
graphs (nodes connected by directed edges) to demonstrate how the resulting,
constrained possibility state space depends on (1) the value of the fixed
variable and (2) an observed system’s initial state space, especially the
transitions between states (global laws).

##  IV The effects of constraints on a system’s possibility state space

Here, we see what happens to the physically-accessible state space for a
system that is somehow constrained. For these examples, we don’t specify
whether these variables are fixed due to internal system dynamics or due to
external forces acting on the system. We also don’t specify if a fixed
variable is a part of a biological object or is a part of the object’s
environment. Instead, we never define an object in any of these systems.

First, we start with a different example dynamical system than the lipid
bilayer membrane from Fig. 1. Given some description of a new dynamical
system, shown in Fig. 2.a, we can systematically explore the effects of
constraining all three microstate variables, regardless of how those variables
are physically set to remain in those values. By being able to see how the
physically accessible possibility state space (Fig. 2, top and bottom rows of
panels b to d) changes as a result of that constraint, we conceptually connect
topological properties with apparent functionality. These functions, which we
call local, emergent rules, become physically realized when particular
variables are constrained. While we don’t yet know how to make these
connections, we discuss various aspects of this approach more broadly.

Namely, we can already identify patterns in this simple example. Functions
that could emerge from being in a sole state are particularly represented in
Fig. 2 when constraining variable $v_{1}$ to either $0$ or $1$ (Fig. 2.b.1 and
Fig. 2.b.2, respectively), or variable $v_{2}$ to $1$. Note that in the case
$v_{1}=0$, the state $u_{1}$ becomes the final state after transiting through
$u_{3}$ first. We can imagine extending this structure to a longer state chain
in which case the relevancy of the emerging function for the system would lie
in the specific sequence of state stabilized by the constraint. In the case of
$v_{1}=1$, however, the function would be a one-to-one map between a
constraint and a unique state maybe suggesting the criticality of this state
for the biological system. Indeed, the system cannot waste energy roaming the
state space in such cases and must remain in place. The case of $v_{2}=1$ is
yet another possibility where the state does exist but no dynamic can be
observed related to it. This might suggest that the state is reachable in
another graph (system description) that would share variable $v_{2}$ but not
$v_{1}$ and $v_{3}$. On the contrary to the case $v_{1}=0$ (Fig. 2.b.1), the
case $v_{2}=0$ yields a constrained graph with a loop between two states. This
type of structure might indicate a lower criticality of the function of the
biological system or the need for a dual solution to the same problem.
Finally, there is the case of constant variables such as $v_{3}$ in the
observable graph (see in Fig. 2.a that the third component of the vector is
always equal to 1). In such cases, constraints on the variable have either the
effect to turn off the whole observed state space, or to turn it on,
unaltered.

We propose three different classes of constraints based on how quickly or
readily they can change over time (or more generally, over a walk along the
state space). Some variables, like Planck’s constant, are completely
conserved. These variables are a direct property of the laws of physics that
determine the paths between arbitrarily labeled states. More subtly, some
variables define the local dynamics of a system such that the time derivative
of other variables is zero. While these variables themselves are not
conserved, they conditionally conserve other variables in the system. To these
two types, we add the possibility to be quasi-conserved. This quality
indicates that the constraint does not perfectly implement a subset of states
where it is satisfied but, at least, it is only possible to escape from the
subset that perfectly respects the constraint very slowly over time relative
to other variables. This includes co-evolutive phenomena like nitrogen
fixation in the soil or the amount of oxygen in the atmosphere.

##  V Future directions and discussion

In this discussion paper, we have examined the relationship between
constraints and functionality over a system’s state space represented as a
directed graph. We argue that the resulting, accessible state space under a
constraint is akin to a local, emergent conservation law or apparent “rule”
that some part of the system evolves under. While we have identified different
effects of constrained variables (ones that produce rules that are completely
conserved, quasi-conserved, and conditionally conserved), there is still much
more that needs to be explored to translate constrained topologies into
functions important for living systems– particularly how emergent “rules”
contribute to open-ended system dynamics. Which types of constrained variables
are most likely to lead to open-ended system dynamics? Since these topologies
also depend on the unconstrained representation of a system, we also want to
understand the relationship between constrained topologies and non-constrained
ones. For example, how do system topologies differ between isolated systems
and systems where external work is being done on them? Furthermore, is there
an analytical approach to measuring possible constraints from within the
system?

Understanding how biological systems and other open-ended systems continually
generate new, innovative rules over time has deep implications for many fields
and problems, all of which center on understanding open-endedness. These
include Artificial Life, characterizing diverse intelligences, understanding
biological processes and major evolutionary transitions, designing machine
learning architectures capable of self-teaching, and more. We hope that by
deepening our understanding of how physical constraints are added, changed,
and removed via the global laws of physics leads to emergent, local rules, we
gain a small step closer to understanding mechanisms that drive open-
endedness. We hope this discussion serves as a springboard to explore
approaches in this direction, as has been previously highlighted by [1, 23].

## Acknowledgments

We would like to thank Nicholas Guttenberg for several detailed discussions
and brainstorming that led to this paper. We would also like to thank Martin
Biehl, Joseph Austerweil, Tim Taylor, Hector Zenil, Kenneth Stanley, Lisa
Soros, and Lana Sinapayen for their feedback, guidance, and discussions for
many of the ideas discussed throughout the paper. PG was supported by the
National Science Foundation, Division of Environmental Biology (Grant No:
DEB-2218817). Finally, we thank the reviewers for their helpful comments,
feedback, and suggestions.

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m¯#1 m|#1|^2 m⟨#1 ⟩ $\mathrm{Id}$ $\mathbb{D}$ $\overline{\mathbb{D}}$
m$\underline{#1}$ ††thanks: The authors acknowledge support from the Labex
CEMPI (ANR-11-LABX-0007-01).

# Schwarz-Pick type inequalities from an operator theoretical point of view

Catalin Badea  Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé,
F-59000 Lille, France<EMAIL_ADDRESS>Axel Renard  Univ. Lille,
CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France
<EMAIL_ADDRESS>
###### Abstract

We use (versions of) the von Neumann inequality for Hilbert space contractions
to prove several Schwarz-Pick type inequalities. Specifically, we derive an
alternate proof for a multi-point Schwarz-Pick inequality by Beardon and
Minda, along with a generalized version for operators. Connections with model
spaces and Peschl’s invariant derivatives are established. Finally, Schwarz-
Pick inequalities for analytic functions on polydisks and for higher order
derivatives are discussed. An enhanced version of the Schwarz-Pick lemma,
using the notion of distinguished variety, is obtained for the bidisk.

###### Key words and phrases:

von Neumann inequality, Schwarz-Pick estimates, Beardon-Minda inequality,
Peschl’s invariant derivatives

###### 1991 Mathematics Subject Classification:

47A25, 47A30, 47A08, 30A10, 30C80, 15A60


Keywords: dendrites, secondary branching, perturbation, phase transformations,
Gibbs-Thomson equation, phase-field method

# Induced side-branching in smooth and faceted dendrites: theory and Phase-
Field simulations

Gilles Demange1, Renaud Patte1, Helena Zapolsky1 1GPM, CNRS-UMR 6634,
University of Rouen Normandy, 76801 Saint Étienne Du Rouvray, France
<EMAIL_ADDRESS>
###### Abstract

The present work is devoted to the phenomenon of induced side branching
stemming from the disruption of free dendrite growth. Therein, we postulate
that the secondary branching instability can be triggered by the departure of
the morphology of the dendrite from its steady state shape. Thence, the
instability results from the thermodynamic trade-off between non monotonic
variations of interface temperature, surface energy, kinetic anisotropy and
interface velocity within the Gibbs Thomson equation. For purposes of
illustration, the toy model of capillary anisotropy modulation is prospected
both analytically and numerically by means of phase field simulations. It is
evidenced that side branching can befall both smooth and faceted dendrites, at
a normal angle from the front tip which is specific to the nature of the
capillary anisotropy shift applied.

##  1 Introduction

Dendrites linger as a flagship example of patterning in nature and material
science. These peculiar structures emerge during a myriad of phase and
structural transformations, including the solidification of alloys and
semiconductors [1], the growth of ice crystals in undercooled water or vapor
[2, 3], and the crystallization of lava at the boundary of the Earth’s core
[4, 5]. Deciphering their growth process and ensuing morphology hitherto
remains a stumbling block of non-linear physics, as well as a strong
prerequisite to taylor the microstructural features determining the mechanical
properties of industrial cast ingots [6]. In this regards, the secondary
branching process receives a growing attention, insofar as side-branches set
the length scales and patterns of microsegregation [7], and contribute to the
adjustment of the primary spacing during directional solidification [8]. From
a more putative perspective, the recurrent presence of secondary branched
dendrites in the prebiotic soup might provide a plausible origin from
primitive biological processes [9].

The fundamental understanding of dendrite growth relies on the pioneering
works of Ivantsov [10] which latches the Péclet number to the thermodynamical
driving force, and the microscopic solvablity theory [11, 12, 12] which
provides a selection criterion for the charactertistic wavelength of the tip
[13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. As for secondary branching, the
conventional theory [23, 24, 25, 26] postulates that side-branches stem from
selective (thermal) noise amplification close to the tip. An alternative
deterministic mechanism was proposed in [27], which surmises that capillary or
kinetic anisotropy induces time periodic non monotone temperature distribution
along the tip. This causes velocity variations to befall the tip, thereby
triggering side-branching.

Now, blind spots remain concerning the description of side branching induced
by the disruption of the dendrite growth process. And yet, dendrites grow
under transient of perturbed growth conditions in practical casting processes
[28] and in nature. For instance, a smooth decrease of the velocity pulling on
the directional solidification in Fe-C alloys was proved to mitigate the
coarsening of side branches in a PFM study [29]. Moreover, a strong finite and
localized disturbance caused by a speck of dust was experimentally evidenced
to cause the violent destabilization of the front tip, and the formation of
abnormally fast growing side-branches [30]. Finally, the successive decrement
and increment of the supersaturation around the faceted tip of a snowflake
branch entails the formation of induced side branching (ISB) [31], and
multiplies the possible morphologies for ice crystals. The latter example in
particular shades light on the possibility of induced side branching, in case
of non local disturbance of the growth conditions of the propagating dendrite.

In this context, the present work addresses the phenomenon of induced side
branching after non-local disturbance of the growth conditions. A simple
mechanism for the secondary branching instability is proposed. It postulates
that side-branching can be induced by an extended deviation of the dendrite
shape from its steady state morphology ensuing growth conditions change.
Therefrom, the destabilization of the tip is fueled by a non-monotonic
deviation of the interface temperature as prescribed by the Gibbs Thomson
relation, which is compensated by localized modulations of the interface
normal velocity. The studycase of capillary anisotropy modulation is selected
as an analytically tractable toy model for a mismatched dendrite shape-growth
condition couple. It is prospected for both smooth and faceted dendrite
growth, before it is challenged by phase field simulations. As a result,
branches are prone to form symmetrically on both sides of the dendrite, at a
given normal angle from the tip. Moreover, two distinct patterns are
identified whether the capillary anisotropy parameter is increased or
decreased at the transition on the one hand, and faceting steps in the process
on the other hand.

Figure 1: Schematic representation of a dendrite tip propagating along
$x$-axis, and having a parabolic dendrite shape with curvature radius $R$ and
tip velocity $V$.

##  2 Theory: perturbation of Gibbs-Thomson relation

###  2.1 Governing equations

The present system of equations is based on the results obtained in [32]. For
a dendrite growing in a pure undercooled liquid, the transport of heat in the
stagnant melt follows the diffusion equation:

$\frac{\partial T_{l}}{\partial t}=D_{T}\nabla^{(2)}T_{l},$ (1)

where $T_{l}$ is the temperature in liquid, and $D_{T}$ is the thermal
diffusivity. Second, the heat condition is fulfilled at the solid/liquid
interface:

$T_{Q}v_{n}=D_{T}\left(\nabla T_{s}-\nabla T_{l}\right)\cdot\bm{n}.$ (2)

where $T_{s}$ is the temperature in the solid, and $T_{Q}=Q/c_{p}$ is the
hypercooling (temperature for complete adiabatic solidification), $\bm{n}$ is
the unit normal to the interface, and $v_{n}=\left(\bm{v}\cdot\bm{n}\right)$
will be referred as the interface normal velocity herefrom. Third, the Gibbs-
Thomson condition holds at the solid/liquid boundary:

$T_{i}=T_{l}=T_{s}=T_{m}-T_{Q}d(\bm{n})\mathcal{K}(\bm{n})-T_{Q}\beta(\bm{n})v_{n},$
(3)

where $T_{i}$ is the phase transition temperature at the dendrite interface,
$T_{m}$ is the melting temperature for the pure system, $\mathcal{K}(\bm{n})$
is the curvature of the interface, and $d(\bm{n})$ and $\beta(\bm{n})$ are the
anisotropic capillary length and anisotropic kinetic coefficient, which root
the anisotropic growth of the crystal. The far field condition is written in
the form $T_{l}=T_{\infty}$ where $T_{\infty}$ is the fluid temperature far
from the interface of the dendrite.

###  2.2 Perturbation of the GT equation in two dimensions

In two dimensions, the anisotropic capillary length $d(\theta)$ and the
anisotropic kinetic coefficient $\beta(\theta)$ are specified for the case of
$n$-fold lattice symmetry order:

$\left\\{\begin{aligned}
d(\theta)&=d_{0}\left\\{A_{d}(\theta)+A_{d}^{\prime\prime}(\theta)\right\\}=d_{0}\left\\{1-\alpha_{d}\cos\left[n\left(\theta-\theta_{d}\right)\right]\right\\}\\\
\beta(\theta)&=\beta_{0}\left\\{A_{\beta}(\theta)+A_{\beta}^{\prime\prime}(\theta)\right\\}=\beta_{0}\left\\{1-\alpha_{\beta}\cos\left[n\left(\theta-\theta_{\beta}\right)\right]\right\\},\end{aligned}\right.$
(4)

where $d_{0}$ and $\beta_{0}$ are the capillary length and isotropic kinetic
coefficient respectively,
$A_{d,\beta}(\theta)=1+\alpha_{d,\beta}\cos\left[n\left(\theta-\theta_{d,\beta}\right)\right]$
are the capillary and kinetic anisotropy functions, $\alpha_{d}$ and
$\alpha_{\beta}$ are the small anisotropy parameters, and $\theta_{d}$ and
$\theta_{\beta}$ designate the angles between the growth direction and the
crystal orientations minimizing $d(\theta)$ and $\beta(\theta)$. The parameter
$\alpha_{d}$ is tied to the anisotropy parameter $\epsilon_{d}$ by the
relation $\alpha_{d}=(n^{2}-1)\epsilon_{d}$.

The propagating 2D dendrite tip is sketched in figure 1, whereinto the
interface normal velocity $v_{n}(\theta)$, the angle
$\theta=\arctan[n_{y}/n_{x}]$ between $x$-axis and the normal vector $\bm{n}$,
the polar angle $\alpha$, and the distance between the center of the
parabola’s curvature circle $r(\theta)$ are indicated.

In the present work, the growth dynamics of the dendrite is disrupted by
imposing a discontinuous shift of the growth conditions from the initial
configuration (configuration 1), to a second configuration (configuration 2).
All parameters and growth quantities are henceforth indexed accordingly
(parabolic dendrite with tip radius $R^{(1,2)}$, curvature
$\mathcal{K}^{(1,2)}$, anisotropy parameters $\alpha_{d}^{(1,2)}$ and
$\alpha_{\beta}^{(1,2)}$ setting the anisotropic capillary length and
anisotropic kinetic coefficient $d^{(1,2)}(\theta)$ and
$\beta^{(1,2)}(\theta)$ respectively).

This transition should result in a rebalancing of thermodynamical quantities
stepping in GT condition (equation 3), including the interface normal velocity
$v_{n}^{(1,2)}$ and tip velocity $V^{(1,2)}$, the curvature radius
$R^{(1,2)}$, and the interface temperature $T_{i}^{(1,2)}$. For purposes of
emphasizing the early perturbation of the transition on the interface velocity
in the vicinity of the tip, the interface temperature will be assumed constant
during the early stage of the transition. Upon equalizing GT equation 3 for
configuration 1 and 2, through the constant interface temperature $T_{i}$, the
perturbed GT condition can be derived:

$\delta
v_{n}(\theta)=\frac{1}{T_{Q}\beta^{(2)}(\theta)}[T_{m}-T_{i}-T_{Q}d^{(2)}(\theta)\mathcal{K}^{(2)}(\theta)]-\frac{1}{T_{Q}\beta^{(1)}(\theta)}[T_{m}-T_{i}-T_{Q}d^{(1)}(\theta)\mathcal{K}^{(1)}(\theta)],$
(5)

where $\delta v_{n}(\theta)=v_{n}^{(2)}-v_{n}^{(1)}$.

###  2.3 Simple analytical model for side-branching induced by capillary
anisotropy parameter shift in the case of the 6-fold symmetry

In the present work, the perturbed GT equation 5 is harnessed to elucidate how
the swift change of one thermodynamical parameter may induce the disruption of
the dendrite growth dynamics, one manifestation of which can be induced side
branching (ISB).

Among the thermodynamical parameters in equation 5, only the capillary
anisotropy parameter was varied in the present study, insofar as it provides
an analytically tractable expression for the resulting normal velocity shift
$\delta v_{n}$, as well as this parameter can be easily tuned in a numerical
model, while keeping all other model parameters fixed. Some conclusions of
this case should remain valid upon acting on other parameters, including
kinetic anisotropy and tip radius. One step further, the specific studycase of
the 6-fold symmetry for the dendrite was chosen in this work for numerical
applications ($n=6$ in equation 4), so that $\alpha_{d}=35\epsilon_{d}$. In
this context, the growth direction was aligned with the $\bm{x}$-axis
($\theta_{d}=0$ and $\theta_{\beta}=\frac{\pi}{12}$ in equation 4). Finally,
for the sake of simplicity, only the capillary contribution to the anisotropic
growth was considered, for the kinetic anisotropy parameter $\alpha_{\beta}$
was set to zero. Under such hypothesis, the following relations hold:
$\beta^{(1)}(\theta)=\beta^{(2)}(\theta)=\beta_{0}$, and $R^{(1)}=R^{(2)}$ so
that
$\mathcal{K}^{(2)}(\theta)=\mathcal{K}^{(1)}(\theta)=\mathcal{K}(\theta)$.
However, the capillary anisotropy constant varies:
$\epsilon_{d}^{(1)}\neq\epsilon_{d}^{(2)}$, so that $d^{(1)}(\theta)\neq
d^{(2)}(\theta)$. Accordingly, equation 5 becomes:

$\delta
v_{n}(\theta)=\frac{\mathcal{K}(\theta)}{\beta_{0}}[d^{(1)}(\theta)-d^{(2)}(\theta)].$
(6)

#### Smooth dendrite:

for a smooth (non-faceted) dendrite, the morphology of the tip can be
approximated by Ivantsov’s parabola of curvature radius $R$:

$\frac{y(x)}{R}=1-\frac{1}{2}\left(\frac{x}{R}\right)^{2},$ (7)

whence assigning the curvature $\mathcal{K}$ the following analytical
expression:

$\mathcal{K}(x)\equiv\frac{y^{\prime\prime}(x)}{(1+(y^{\prime}(x))^{2})^{3/2}}=\frac{1}{R\left(1+\left[\frac{x}{R}\right]^{2}\right)^{3/2}}=\frac{\cos(\theta)^{3}}{R}.$
(8)

Then, equation 4 is used for the capillary distance
$d(\theta)=d_{0}(A_{d}(\theta)+A_{d}^{\prime\prime}(\theta))$. Herein,
equation 6 simplifies to the following explicit expression for the interface
normal velocity shift thereby induced:

$\delta
v_{n}(\theta)=\frac{35d_{0}\cos^{3}(\theta)\cos\left(6\theta\right)}{\beta_{0}}\left[\epsilon_{d}^{(2)}-\epsilon_{d}^{(1)}\right]\\\
$ (9)

The corresponding profile of $\delta v_{n}(\theta)$ as a function of the
normal angle $\theta$ is displayed in figure 2(a).

#### Faceted dendrite:

for a faceted dendrite, equation 4 should be amended. For (rough) faceted
growth, a highly anisotropic surface energy
($\epsilon_{d}>1/35\equiv\epsilon_{m}$ for the 6-fold symmetry) should be
used. In this case, the surface energy anisotropy function $A(\theta)$
displays high energy directions in the interval $[-\theta_{m},\theta_{m}]$
(modulo $\pi/3$ for the 6-fold symmetry), which generate unstable orientations
in the equilibrium shape of the crystal (ECS). These unstable orientations
correspond to sharp corners in the ECS, associated with a negative stiffness
$A_{d}+A_{d}^{\prime\prime}<0$ [33], and a negative capillary distance.

In this case, the mathematical problem of equations 1, 2 and 3 becomes ill-
posed, and $A(\theta)$ must be regularized. In this work, the inverse polar
plot of $A$ was convexified, following the procedure of Eggleston [34].
Unstable orientations can be determined by the tangent construction as per
which the critical angle $\theta_{m}$ satisfies the following conditions:

$\frac{6\epsilon_{d}\sin(6\theta_{m})}{\sin(\theta_{m})}=\frac{1+\epsilon_{d}\cos(6\theta_{m})}{\cos(\theta_{m})}.$
(10)

In the interval $[-\theta_{m},\theta_{m}]$, the expression of the anisotropy
function $A(\theta)$ becomes:

$A_{d}(\theta)=A_{1}+B_{1}\cos(\theta),\quad\text{where:}\quad\left\\{\begin{aligned}
&A_{1}=1+\epsilon_{d}\cos(6\theta_{m})-B_{1}\cos(\theta_{m})\\\
&B_{1}=\frac{6\epsilon_{d}\sin(6\theta_{m})}{\sin(\theta_{m})},\end{aligned}\right.$
(11)

thereby ensuring the continuity of $A_{d}(\theta)$ and its first derivative
$A_{d}^{\prime}(\theta)$, and a positive value of the capillary distance
$d_{0}(A_{d}(\theta)+A_{d}^{\prime\prime}(\theta))\geq 0$ defined piece-wise
thereafter by:

$d(\theta)=\left\\{\begin{aligned}
&d_{0}\left\\{1-\alpha_{d}\cos\left[n\left(\theta-\theta_{d}\right)\right]\right\\}\quad\text{for}\quad\theta\in[-\theta_{m},\theta_{m}]\\\
&1+\epsilon_{d}\cos(6\theta_{m})-\frac{6\epsilon_{d}\sin(6\theta_{m})}{\sin(\theta_{m})}\cos(\theta_{m})\quad\text{otherwise}.\end{aligned}\right.$
(12)

The corresponding profiles of $\delta v_{n}(\theta)$ provided by equation 6
are displayed in figures 2(b) and 2(c) for smooth$\leftrightarrow$ facet and
facet$\leftrightarrow$ facet transitions respectively, under the approximation
$\mathcal{K}(\theta)\simeq\cos(\theta)^{3}/R_{\text{moy}}$, where
$R_{\text{moy}}$ is an effective curvature radius for the faceted tip. The
implications of this simplification will be discussed hereafter.

(a) smooth$\leftrightarrow$ smooth

(b) smooth$\leftrightarrow$ facet

(c) facet$\leftrightarrow$ facet

Figure 2: Variation of normal interface velocity $\delta v_{n}(\theta)$ as a
function of the normal angle $\theta$ (in rad). Red line: decrease of
capillary anisotropy parameter $\epsilon_{d}$. Green dashed line: increase of
$\epsilon_{d}$. a) transition from smooth to smooth dendrite. b) Transition
from faceted (smooth) to smooth (faceted) dendrite. c) Transition from faceted
to faceted dendrite. Special angles $\theta_{0}$, $\theta_{1}$ ans
$\theta_{2}$ correspond to local extrema of $\delta v_{n}$ where side-
branching or carving is more likely.

In figure 2(a), the decrease of the anisotropy parameter $\epsilon_{d}$ (red
curve, $\epsilon_{d}^{(1)}=0.02$, $\epsilon_{d}^{(2)}=0.01$) leads to the
shrinking of tip velocity $V=v_{n}(\theta=0)$, the rise of the normal velocity
$v_{n}(\theta)$ in the vicinity of angle $\theta_{1}\simeq 0.48$ for the
present setting of $\epsilon_{d}^{(1)}$ and $\epsilon_{d}^{(2)}$ (red dot,
$\delta v_{n}(\theta_{1})>0$), and the small deceleration of the interface
close to the angle $\theta_{2}\simeq 0.94$ (red dot, $\delta
v_{n}(\theta_{2})<0$).

On the contrary, when anisotropy parameter $\epsilon_{d}$ is incremented
(green dashed curve, $\epsilon_{d}^{(1)}=0.01$, $\epsilon_{d}^{(2)}=0.02$),
the situation is inversed, as $\delta v_{n}(\theta_{1})<0$ and $\delta
v_{n}(\theta_{2})>0$. In this work, we surmise that the anisotropic variation
of the normal velocity envisioned in figure 2(a), which stems from the step
variation of the capillary anisotropy parameter, might as well enable the
induced secondary branching of the dendrite.

In addition, we suggest that hiking or lessening the anisotropy parameter
$\epsilon_{d}$ leads to two distinct side branching configurations. Whenever
$\epsilon_{d}$ is decremented, the acceleration of the interface normal
displacement is highest ($\delta v_{n}(\theta_{1})>0$) at the normal angle
$\theta_{1}$, versus $\theta_{2}\simeq 2\theta_{1}$ when $\epsilon_{d}$ is
decreased. As a consequence, the initially induced secondary branching sprout
is formed closer to the tip in the first case, before it propagates outward
from the tip. Now, in the so-called linear region spanning a few tip radius
from the dendrite tip [24], the amplitude of the secondary branch grows
exponentially from the tip [30], so that the closer to the tip the side brand
is generated, the more it will thrive hereafter, and the stronger the ISB.

For this reason, the decrease of $\epsilon_{d}$ is less likely to induce
secondary branching. However, the decrease of $\epsilon_{d}$ also causes the
strong deceleration of interface normal displacement in the normal direction
$\theta=\theta_{1}$, resulting in a hollow region that precedes the $\theta$
region favorable to the formation of a side branch. We suggest that this
hollow portion might eventually curve undercooling isolines toward a slightly
stronger undercooling in the $\theta_{2}$, whence fostering side-branching.

A similar behavior can be expected for the smooth$\leftrightarrow$ facet
dendrite transition (figure 2(b)), notwithstanding two significant
differences. First, the second extremum $\theta_{2}=\pi/3-\theta_{m}\simeq
0.65$ for the present setting of $\epsilon_{d}^{(1)}$ and $\epsilon_{d}^{(2)}$
corresponds to the lower bond of unstable orientations centered in $\pi/3$. In
this regards, it is located closer to the dendrite tip than for the
smooth$\leftrightarrow$ smooth transition ($\theta_{2}\simeq 0.94$). As a
consequence, it is more likely to result in side-branching when $\epsilon_{d}$
is increased (faceted $\to$ smooth dendrite). Second, an additional extremum
in $\theta_{0}=\theta_{m}\simeq 0.4$ corresponding to the higher bound of the
unstable orientations in the vicinity of the $\theta$ direction is located,
which might hinder the secondary branching process when $\epsilon_{d}$ is
decreased.

The latter observations remain true for the facet$\leftrightarrow$ facet
dendrite transition (figure 2(c)).

##  3 Phase-Field simulations

###  3.1 Phase-Field model

In this work, we simplified the solidification model introduced in [35] for
strongly anisotropic dendrite growth in three dimensions, to the two-
dimensional case:

$\displaystyle A(\theta)^{2}\partial_{t}\psi$
$\displaystyle=-f^{\prime}(\psi)-\lambda g^{\prime}(\psi)\Theta$ (13)
$\displaystyle+\frac{1}{2}\nabla\cdot\left(|\nabla\psi|^{2}\frac{\partial\left[A(\theta)^{2}\right]}{\partial\nabla\psi}+A(\theta)^{2}\nabla\psi\right),$
$\displaystyle\partial_{t}\Theta$
$\displaystyle=\widetilde{D}\nabla^{2}\Theta+\frac{L}{2c_{p}}\partial_{t}\psi.$
(14)

Here, $\psi$ is the order parameter ($=1$ for the solid phase, and -1 for the
liquid), and $\Theta=(T-T_{m})/T_{Q}$ is the reduced undercooling. In equation
(13), the double well potential $f(\psi)=-\psi^{2}/2+\psi^{4}/4$ is the free
energy density of the system, at melting temperature $T_{m}$, and
$g^{\prime}(\psi)=(1-\psi^{2})^{2}$ is an interpolation function. The model is
equipped with an anisotropic function $A(\theta)=1+\epsilon_{d}\cos(6\theta)$
for smooth dendrite growth, the normal angle $\theta$ can be calculated from
the unit normal vector $\bm{n}=-\nabla\psi/|\nabla\psi|$ of $\psi$, as
$\theta=\arctan[n_{y}/n_{x}]$.

For faceted growth, the regularization algorithm introduced in the previous
section was used. The result of the regularization procedure is summarized in
figure3 for $\epsilon_{d}=0.1>\epsilon_{m}$. Before regularization, the 2D
Wulff shape (WS) for the ECS displays ears corresponding to unstable
orientations (figure 3(a)). The associated inverse 2D polar plot is concave in
these directions (red curve in figure 3(c)). After regularization, the ears
are removed (figure 3(b)), and the inverse polar plot becomes convex (blue
curve in figure 3(c)). Besides that, the resulting 2D polar plot is regular
and coincides with the original 2D polar plot where the curvature was already
positive before the regularization procedure (red curve in figure 3(c)).

(a) WS

(b) ECS

(c) $1/A$

Figure 3: Regularization procedure for $\epsilon_{d}=0.1$. a) Wulff shape (WS)
before regularization. b) Equilibrium crystal shape (ECS) after
regularization. c) Original and confexified inverse polar plot of $A_{d}$.
Figres are taken from a previous publication [36].

Equations 13 and 14 were solved in reduced units using periodic boundary
conditions. Simulations were performed using the grid spacing $\Delta x=0.8$,
and the time step $\Delta t=0.05$, in the bosom of a 3000 $\times$ 800
simulation box. This choice guarantees that the spacing between the dendrite
tip and the boundary is several times the diffusion length during the
simulation. The initial condition was a spherical germ of radius $4\Delta x$.
The coupling constant was set to $\lambda=3.0$ [37]. Simulations displayed in
this work were performed at reduced undercooling $\Theta_{\infty}=0.65$,
corresponding to the diffusion limited growth regime. All results are given in
reduced space and time units.

###  3.2 PFM simulations: the studycase of capillary effects perturbation

(a) $\epsilon_{d}^{(1)}=0.02\searrow\epsilon_{d}^{(2)}=0.01$

(b) $\epsilon_{d}^{(1)}=0.01\nearrow\epsilon_{d}^{(2)}=0.02$

Figure 4: Dendrite morphological evolution from smooth tip to smooth tip after
discontinuous change of anisotropy parameter $\epsilon_{d}$ at reduced time
$t=400$. a) From $\epsilon_{d}^{(1)}=0.02$ corresponding to a smooth parabolic
dendrite with steady state tip radius $R=9.6$ (red curve, $t=400$) to
$\epsilon_{d}^{(2)}=0.01$ corresponding to a smooth parabolic dendrite with
steady state tip radius $R=13.2$ (blue curves from $t=400$ to $t=1000$). b)
From $\epsilon_{d}^{(1)}=0.01$ (red curve, $t=400$, $R=13.2$) to
$\epsilon_{d}^{(2)}=0.01$ (blue curves from $t=400$ to $t=1000$, $R=9.6$).
a-b) Insert: initial variation of radius $r(\theta)$ between $t=400$ and
$t=402.5$.

(a) $\epsilon_{d}^{(1)}=0.1\searrow\epsilon_{d}^{(2)}=0.01$

(b) $\epsilon_{d}^{(1)}=0.01\nearrow\epsilon_{d}^{(2)}=0.1$

Figure 5: Dendrite morphological evolution from faceted (smooth) tip to smooth
(faceted) tip after discontinuous change of anisotropy parameter
$\epsilon_{d}$ at reduced time $t=400$. a) Decrease of $\epsilon_{d}$ from
$\epsilon_{d}^{(1)}=0.1$ corresponding to a faceted dendrite with steady state
tip radius $R=2.9$ (red curve, $t=400$) to $\epsilon_{d}^{(2)}=0.01$ (blue
curves from $t=400$ to $t=1000$, $R=13.2$). b) Increase of $\epsilon_{d}$ from
$\epsilon_{d}^{(1)}=0.01$ (red curve, $t=400$, $R=13.2$) to
$\epsilon_{d}^{(2)}=0.1$ (blue curves from $t=400$ to $t=1000$, faceted
dendrite with $R=2.9$). a-b) Insert: initial variation of radius $r(\theta)$
between $t=400$ and $t=402.5$.

(a) $\epsilon_{d}^{(1)}=0.1\searrow\epsilon_{d}^{(2)}=0.05$

(b) $\epsilon_{d}^{(1)}=0.05\nearrow\epsilon_{d}^{(2)}=0.1$

Figure 6: Dendrite morphological evolution from faceted tip to faceted tip
after discontinuous change of anisotropy parameter $\epsilon_{d}$ at reduced
time $t=400$. a) Decrease of $\epsilon_{d}$ from $\epsilon_{d}^{(1)}=0.1$ (red
curve, $t=400$, faceted dendrite with $R=2.9$) to $\epsilon_{d}^{(2)}=0.05$
(blue curves from $t=400$ to $t=1000$, faceted dendrite). b) Increase of
$\epsilon_{d}$ from $\epsilon_{d}^{(1)}=0.05$ (red curve, $t=400$, faceted
dendrite) to $\epsilon_{d}^{(2)}=0.1$ (blue curves from $t=400$ to $t=1000$,
faceted dendrite). a-b) Insert: initial variation of radius $r(\theta)$
between $t=400$ and $t=402.5$.

In order to assess the impact of capillary anisotropy shift on the
morphological stability of the growing dendrite, the following procedure was
implemented. The dendrite was first grown from a small germ, using a first
value $\epsilon_{d}^{(1)}$ of the capillary anisotropy parameter, until
$t=400$. This duration was sufficient to reach the steady state regime. Then,
the capillary anisotropy parameter $\epsilon_{d}^{(1)}$ was updated to a
second value $\epsilon_{d}^{(2)}$, while all other parameters were kept
constant, as well as the temperature field, including at the interface
($T_{i}^{(1)}=T_{i}^{(2)}$). Three representative cases were addressed in this
work:

#### Case 1:

transition from a smooth parabolic dendrite tip with underlying capillary
anisotropy parameter $\epsilon_{d}^{(1)}<\epsilon_{m}$, to a second smooth
dendrite tip $\epsilon_{d}^{(2)}<\epsilon_{m}$. The different stages of the
process are displayed in figure 4. Two subcases are prospected. $\epsilon_{d}$
is decremented from 0.02 to 0.01 on the one hand (see figure 4(a)), and
incremented from 0.01 to 0.02 on the other hand (see figure 4(b)).

#### Case 2:

transition from a faceted dendrite tip with underlying capillary anisotropy
parameter $\epsilon_{d}^{(1)}>\epsilon_{m}$, to a smooth dendrite tip
$\epsilon_{d}^{(2)}<\epsilon_{m}$ (see figure 3.2), and the opposite situation
wherein $\epsilon_{d}$ is raised from 0.01 to 0.1 (see figure 3.2).

#### Case 3:

transition from a faceted dendrite tip with underlying capillary anisotropy
parameter $\epsilon_{d}^{(1)}>\epsilon_{m}$, to a second faceted dendrite tip
$\epsilon_{d}^{(2)}>\epsilon_{m}$ (see figure 6). $\epsilon_{d}$ is
alternately decreased from 0.1 to 0.05 (figure 6(a)) and increased from 0.05
to 0.1 (figure 6(b)).

In figures 4, 5 and 6, only the last step ($t=400$) of the first configuration
dendrite contour ($\epsilon_{d}=\epsilon_{d}^{(1)}$) is depicted (red curve).
Herein, different growth stages from $t=400$ to $t=1000$ of the dendrite after
shift of capillary anisotropy parameter to $\epsilon_{d}^{(2)}$ are shown
(blue curves). On these curves, the red and green dots indicate the tip of the
secondary branch and the center of the secondary hollow respectively. These
dots are used as a guide for the eye, to trace back the initial variation of
the initial state of configuration 2 from the last step of configuration 1
($t=400$, red curve). The underlying idea is to connect the early
morphological dynamics of the dendrite after capillary anisotropy disruption,
to the outcoming ISB process, in order to assess the ISB mechanism summarized
in equation 9 and figure 2.

For that purpose, the early variations $\delta r$ of the radius $r$ as defined
in figure 1 between dendrite contours at $t=400$ and $t=402.5$, is plotted as
a function of the normal angle $\theta$ in the insert on the right. This curve
serves as a reliable indicator of the variation of the normal interface
velocity $\delta v_{n}(\theta)$ resulting from capillary effects disruption,
after integration on an infinitesimal time.

The value of the tip radius $R$ in the parabolic dendrite tip assumption is
chosen case-wise. When the dendrite tip of configuration 1 is smooth
($\epsilon_{d}^{(1)}<\epsilon_{m}$), it is reasonably chosen the curvature
radius of steady state configuration 1: $R=R^{(1)}$ (figures 4(a), 4(b) and
3.2). When the dendrite tip of configuration 1 is faceted, but the capillary
anisotropy coefficient of configuration 2 leads to a smooth parabolic dendrite
tip, the curvature of the freshly smoothed faceted tip at $t=402.5$ is used
(figure 3.2). Finally, when both the dendrite tips of configurations 1 and 2
are faceted (figures 6(a) and 6(b)), we used the effective curvature radius
$R_{\text{moy}}=10$.

In addition, the initial variation of the position of the secondary branch tip
and the secondary hollow, which is framed on the initial dendrite contour (red
curve), is indicated in the insert. It was obtained by the local average of
the early variation of radius $\delta r$ (red curve in the insert). Finally,
the theoretical value of angles $\theta_{1}$ and $\theta_{2}$ (and
$\theta_{0}$ when relevant) are carried over the abscissa axis.

On the one hand, the decrease of $\epsilon_{d}$ induces the formation of one
side branch at a normal angle $\theta_{1}$ spotted by a red dot on the
$\theta$-axis in cases 1 and 2 (see insert in figures 4(a) and ). This
direction corresponds to the local maximum of the $(\theta,\delta r)$ curve in
the same insert. The value of $\theta_{1}$ is close to the theoretical local
maximum $\theta_{1}=0.48$ for Case 1, and $\theta_{1}=0.51$ for Case 2.
However, the initial bump at $\theta\simeq\theta_{1}$ fails to produce side
branching in case 3 (see 6(a)). On explanation might be the presence of two
closely located deep minima in $\theta_{0}$ and $\theta_{2}$ in figure 2(c)
(red curve), which could preclude the development of the initial interface
variation in $\theta_{1}$.

On the other hand, the increment of $\epsilon_{d}$ in figures 4(b), and 6(b)
entails the curbing of the interface for normal directions close to a maximal
curbing normal angle $\theta_{1}$. This direction is indicated by a green dot
in the insert, pertaining to the local minimum of the $(\theta,\delta r)$
curve. Also, the increase of $\epsilon_{d}$ additionally begets the formation
of a side-branch bump at a second angle $\theta_{2}>\theta_{1}$ corresponding
to the local maximum of the $(\theta,\delta r)$ curve (red dot in the insert).
This direction $\theta_{2}$ is close to its theoretical counterpart
$\theta_{2}=0.94$ for smooth dendrites (figure 4(b)). When faceting is
involved (figures 3.2 and 6(b)), a significantly smaller value is obtained
$\theta_{2}\simeq 0.75$ drawing near to its theoretical counterpart
$\theta_{2}=0.65$. The smaller value of $\theta_{2}$ in case 2 and 3 compare
to case 1 ($\theta_{2}=0.65$ in theory, and $\theta_{2}\simeq 0.75$ in cases 2
and 3 vs. $\theta_{2}=0.94$ in case 1) is indeed likely to foster the enhanced
sprouting of side-branches, insofar as the initial side-branch variation lies
closer to the tip in this configuration.

Noteworthy, the local minima $\theta_{0}$ and $\theta_{2}$ of $\delta v_{n}$
in figure 2(b) does not feed through the early deformation of the dendrite
shape whenever $\epsilon_{d}$ is shrunk in case 2. One explanation might be
the negligible amplitude of the corresponding two minima in figure 4 (green
curve) as compared to the maximum $\theta_{1}$.

One remarkable observation is that the location ($\theta_{1}$ or $\theta_{2}$)
of the initial variation inducing side branching is to some extent independent
of the initial morphology for the dendrite. This is particularly striking when
the dendrite tip is faceted. Contrary to what was surmised in [35], the ISB is
not nucleated at the tip front facet corner, but at the fixed direction
$\theta_{1,2}$. As for the initial bump at the tip front facet corner, it
flattens and vanishes (see figure 3.2).

##  4 Discussion

The side branching observed in this work differs from noise ISB, wherein
thermal fluctuations at the tip are converted in a train of periodic interface
fluctuations whose period is set by the Mullins-Sekerka spectrum. These
fluctuations are amplified as they propagate backward from the tip, until side
branches are formed. Side branches hereafter compete, until the surviving ones
reach the coarsening regime. In the end, side branches seldom reach the free
growth regime. This characteristic behavior is envisioned in most phase-field
studies equipped with noise induced side branching [38, 39]. Things obviously
go very differently in case of induced side branching. Only one side branch
couple is formed, so that no competition between side branches occur, and the
free growth regime can be reached. Besides, depending on the growth conditions
change applied, side-branches can sprout precociously in the vicinity of the
tip, thence exacerbating the subsequent development of the freely growing
secondary branches. Albeit we did not check this point, one natural
consequence of prematurely free growing side branches should be a growth law
deviating from [26].

A second observation relates to the influence of the choice of regularization
algorithm for unstable orientations on presently envisioned ISB in case of
faceted growth. Indeed, alternative regularization procedures can be found in
the literature for both capillary and kinetic faceting (see for instance [40]
for a short review), including cusp removal [41], and the addition of higher
derivative order terms to the surface energy function [42]. In this regards,
one can thus argue that the present conclusions on ISB for faceted growth may
reflect the choice of the selected regularization procedure, more than the
underlying physics of the process. On this point, we suggest that numerical
observations (figures 5 and 6) and conclusions might not be radically altered,
as we believe that it is the transition from stable to unstable crystal
orientations at $|\theta|=\theta_{m}$ where the stiffness vanishes, that is
responsible for the strong departure of the interface equilibrium temperature
between configuration 1 and 2, which itself triggers side-branching. In
particular, choosing to remove cusps in place of convexifying the inverse
polar plot will only result in a slight variation of $\theta_{0,1,2}$.
However, analytical curves displayed in figures 2(b) and 2(c) rely upon the
parabolic tip approximation for the interface curvature. This simplification
might be responsible for the discrepancy between expected values of
$\theta_{1}$ and $\theta_{2}$ and the numerical values spotted in the insert
of 5 and 6. Moreover, this simplification might become even more pregnant with
implications whenever the curvature should be added to the expression of the
surface energy [43, 44, 45]. This question should be addressed in another
study. More generally, the present work should also be extended to kinetic
anisotropy in a first time, and then to a general variation of the dendrite
morphology in a second time.

With all due caution, namely the difference of Péclet number and growth regime
(diffusion limited growth vs. slow growth), origin of growth anisotropy
(capillary vs. kinetic), as well as the peculiarities of smooth and rough
faceting [46], the presently described mechanism for ISB might provide an
alternative scenario for the sprouting of side branches in snowflake as was
experimentally observed upon changing growth conditions in [31]. In his book,
Libbrecht ascribes ISB to a purely morphological origin. It is surmised that
upon switching from a smooth to a faceted and then again to a smooth dendrite
by means of supersaturation tuning would cause side-branches to emerge at the
freshly formed tip, in virtue of Berg’s effect [47]. Now, the present work
suggests that under such disturbances, induced side-branches should form
closer to the tip, and only in a second time migrate backwards from the tip.
In addition, secondary branching might be achieved by alternative growth
condition shifts, including certain transitions from smooth to smooth, faceted
to faceted, and smooth to faceted dendrite tip.

##  5 Conclusion

In this work, the process of induced side branching caused by a long range
disturbance of the growth conditions was prospected. An alternative mechanism
for the secondary branching instability was proposed. It is suggested that
side-branching can be the final outcome of a global deviation of the dendrite
shape from its steady state morphology. Therefrom, the destabilization of the
tip reflects the trade-off between non monotonic variations of interface
temperature, surface energy, kinetic anisotropy and interface velocity to
comply with the Gibbs Thomson equation. In this regards, the toy model of side
branching induced by the tuning of capillary anisotropy effects was used for
both smooth and faceted dendrite growth, as it can be easily addressed
analytically and numerically through faceted phase-field simulations. In this
manner, it was envisioned that the induced secondary branching instability
could ensue a host of capillary anisotropy effects changes, with specific
signatures. The present study should now be extended to more realistic growth
condition changes, and meshed within the more general framework of bifurcation
in a side-branch surface [48, 49].

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