# Unifying self-organization and evolution principles in material and biological discrete systems

François Nicot, Antoine Wautier, Richard Wan, Felix Darve

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François Nicot, Antoine Wautier, Richard Wan, Felix Darve. Unifying self-organization and evolution principles in material and biological discrete systems. Granular Matter, 2025, 27 (4), pp.88. 10.1007/s10035-025-01565-0. hal-05318688

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# Unifying self-organization and evolution principles in material and biological discrete systems

François Nicot¹, Antoine Wautier², Richard Wan³, and Felix Darve⁴

(1) Université Savoie Mont Blanc, ISTerre, Chambéry (France)

(2) Aix-Marseille Université, INRAE, Unité de Recherche RECOVER, Aix-en-Provence (France)

(3) University of Calgary, Calgary (Canada)

(4) Université Grenoble-Alpes, Laboratoire L3SR, Grenoble (France)

Corresponding author: francois.nicot@univ-smb.fr

## ABSTRACT:

The post-Darwinian era has been marked by a long-term effort to lay the foundations for a generalized theory of evolution in the broad sense. We suggest throughout this article that most of biological systems, including living species, could stand as multiscale complex systems due to microscopic or mesoscopic properties of the entity interacting with its environment. Intriguing commonalties which exist between the living and non-living species as complex systems give a strong hint that a unified approach could be developed. The paper explores this hypothesis by analyzing how complex systems, such as granular matter, evolve and adapt when brought out of equilibrium. The inherent disorder in most of granular materials gives way to a wide spectrum of structural patterns that can transform according to the external conditions applied. When brought out of equilibrium, phase transitions can occur spontaneously, leading to profound configurational reorganizations where new and unexpected structures can emerge. Using most of the fundamentals derived for granular systems, a material approach of evolution is proposed, whereby living and non-living architectures can be brought together within a rational framework whereby key concepts such as self-organization, emergence, scale effects, fluctuations and memory storage are at the very forefront.

## KEYWORDS:

Granular solids, Complex systems, Continuum thermodynamics, Statistical mechanics, Configurational transformation, Critical state regime, out-of-equilibrium thermodynamics, Bifurcation, Natural selection, Genetic fluctuations, Heredity, Adaptations, Emerging structures.
# 1. INTRODUCTION

Granular materials are all around us: sand, gravel, snow, coffee, salt, rice, powders, pharmaceutical pills, among others. They are large aggregates of small scale individual solid particles interacting with each other through apparently simple deterministic laws. This simplicity is defied by the complex behaviour displayed by granular systems as they segregate when vibrated to form patterns which resemble vortices in fluids. In the quasi-static regime, another striking phenomenon in these materials is the appearance of localized deformation patterns such as shear bands when they are continuously loaded, even when starting from an initially homogeneous state. For example, tectonic deformations often concentrate preferentially along narrow, linear regions to form shear bands that trigger a range of geophysical phenomena such as earthquakes. Basically, particulate materials under specific loadings are prone to transition suddenly from a homogeneous deformation regime to a heterogeneous one, signaling the occurrence of a bifurcation (Parisi et al., 2017).

## Bifurcation and out-of-equilibrium

Many examples of bifurcation can be reported on Earth within both living or inanimate systems as in essence these systems are always open, exchanging energy or matter with the outside environment. Unlike isolated systems which converge irresistibly towards a thermodynamic equilibrium state (thus, of maximum disorder), open systems are most likely to be out-of-equilibrium which may lead to very clear structural reorganization. In essence, natural processes evolve into more and more efficient energy conversion by optimizing dissipation (Dewar, 2005). To wit, new efficient systems with more order emerge by minimizing entropy production (Glansdorff and Prigogine, 1963, 1964 and 1971). As a natural example of an out-of-equilibrium system, Life consists of processes that are perpetually changing in states following a precise evolution within a coherent framework of energy in line with the second law of thermodynamics. Thanks to Ludwig Boltzmann (Boltzmann, 1877) with his atomistic view of matter, the second law of thermodynamics has been derived from the statistics of the microscopic structure of macroscopic bodies. Changes in states are instead framed from a probabilistic perspective. In other words, which event or change that occurs is basically the most probable of all possibilities, hence like a maxim rule about natural processes in Charles Darwin's doctrine.
## Evolution theory

In the second half of the 19th century, Darwin published a pioneering work which was to revolutionize our understanding of the living world and the dynamic processes governing the evolution of species (Darwin, 1859). In particular, this work was to refute the immobilist thesis that inherent characteristics of species are fixed by proposing a theory of evolution through which species adapt in response to the imposed environmental constraints and disturbances. The theory of the evolution of species according to Darwin is based on three fundamental notions: (i) diversity, (ii) adaptation, and (iii) heredity. It eventually turns out that the first one, diversity, is the main notion from which the other two emerge, and this alone is sufficient to give substance to a theory of the evolution of species. In fact, Darwin, unwittingly, opened a much broader path that might concern all complex systems (Kauffman, 1993; Weber and Depew, 1996; Weber, 2011).

## What is a complex system?

A complex system can be defined as a large set of elementary entities or agents (e.g. atoms, molecules, grains of sand, animal species, or human beings) that interact efficaciously either by direct contact or remotely with each other at some disorder level. This disorder can relate to the spatial distribution of the entities or to the interactions between them. Whereas the positions of the entities can be randomly distributed with no repeated structural patterns, order can, however, be manifested at short range with a correlation length of the same magnitude as the size of the entities. Therefore, each entity has no information of the ensemble's behaviour, nor details of the structure of the whole. Despite this seemingly chaotic ensemble, there may yet arise a spectrum of more ordered structures or distinct patterns, giving way to emergent properties that could not have been anticipated from the local agent characteristics themselves. This certainly evokes the holism vs. reductionism conundrum.

## Emergent behaviour—role of scales

Such an emergent organization, unraveling a hidden order, brings out successive scales: (i) the microscopic scale, corresponding to the elementary interaction between two agents; (ii) one or several intermediate (mesoscopic) scale(s), consistent with a set of several agents exhibiting collective or direct mid-range interactions; and finally (iii) the macroscopic scale, which is that of the whole system itself, presenting long-range interactions such as exchange of information in the broad sense within the system under the constraint of the external environment. This orthographic projection of self-organization of disordered systems forces us to investigate to which extent this paradigm of distinctive scales and interactions can
address a vast set of complex systems, whatever the nature of the entities, including living
beings, plant associations, and non-living matter such as rocks or soils or many other
engineered materials. In short, the present work also searches for a possible universality rule
for complex system evolution that leads to universal consequences within a universal
framework.

Connection with granular materials

Let us thus try to illustrate the above quest for a unifying view of things by turning to granular materials or any other assembly of elementary interacting solid entities. They indeed constitute a paradigmatic example for demonstrating how the theory of evolution of species can be connected to the dynamics of complex systems, irrespective of their nature. Despite an *a priori* very simplistic composition, granular materials offer a variety of remarkable behavioural features as they consist of an evolving collection of disordered individual particles that interact through contacts. The mechanical properties of granular assemblies have attracted much interest within many scientific communities, ranging from soil mechanics to statistical physics (Jaeger *et al.*, 1996; Duran, 2000; De Gennes *et al.*, 2002; Edwards *et al.*, 2007; Parisi *et al.*, 2017; Wan *et al.*, 2017). Despite an apparent simplicity, granular materials are an prototype of complex systems where the existence of several separable scales—from the grain scale to the specimen scale—gives rise to a wide spectrum of emergent, salient properties.

In what follows, it will be attempted to demonstrate that granular systems can indeed serve as
an example of inanimate complex systems to uncover the intricate relationship between them
and the evolution of living species (Section 2). After discussing how the complexity of living
species can be described through a double time scale model (Section 3), a unified approach
will be articulated from the perspective of a global understanding of complex systems in the
context of evolution and emergent behaviour based on well-known concepts in the field of
classical mechanics.

2. GRANULAR SYSTEMS THROUGH THE LENS OF EVOLUTION THEORY

The connection between granular systems and evolution theory is illuminated by this key
question: "To what extent can non-living species, herein referred to as inanimate matter, adapt
to external changes"? For illustrative purposes, the discussion will focus on a wide class of
granular systems. Diluted assemblies, referred to as granular fluids, will be distinguished from
dense assemblies, denoted as granular solids. In the following, we will mainly focus on
athermal granular solids whose configuration changes only when mechanically or externally
driven, but not by fluctuations in temperature.

Separation of scales

It is worth emphasizing that the scale separation within granular solids is often not obvious
nor unique. For instance, in this apparently simple case of a sand pile, it is quite natural to
invoke an assembly of sand grains in mutual interactions to understand the granularity of the
system at hand. However, probing deeper into granularity, each single sand grain could be
further regarded as a complex polycrystal made up of an arrangement of atoms as shown in
Fig. 1. And this downscale cascade could certainly be extended further until reaching the
ultimate quantum mechanics frontier. In this paper, to resolve this conundrum of scale
separation, we will stop at the grain level which stands as the elementary, microscopic scale
of the considered granular assembly in the absence of a quantum scale.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Successive scales in granular materials: (a) sand pile macroscopic scale (meter); (b) intermediate scale consisting of a small set of interacting grains (cm); (c) microscopic scale of contacting grains (mm); (d) crystal structure of silicon (nanometer).</figcaption></figure>
## Disorder, diversity of configurations and adaptation

Excluding regular sand piles, granular assemblies fit within the class of disordered materials in the sense that the microstructure consists of an irregular, random arrangement of grains in contact without any repetitive patterns (Fig. 1b). This disordered packing makes it possible for the assembly to assume a diversity of topological configurations to adapt against external mechanical loading conditions. Adaptation is evolutionary in nature, operating through topological transformations that encompass relative grain motions (rotation and sliding), including grain deformation mechanisms. At the origin of granular material deformations are the diverse structural configurations under steric constraints and relative displacements between the grains that make the topology of the contact force network adapt to external evolving loads in view of achieving equilibrium balance (Tordesillas et al., 2012; Pucilowski et al., 2020; Walker et al., 2015; Radjai et al., 1998; Peters et al., 2005). By contrast, regular granular arrangements in a crystalline solid are only able to resist a limited range of external loading, the direction of which should align with the principal symmetry directions of the crystalline structure. This thought of multiplicity of configurational rearrangements in granular solids, just like in fluids, evokes the link between internal fluctuations and the external loading, which is nothing else but the seminal meaning of the fluctuation-dissipation theorem (Glansdorff and Prigogine, 1971; Onsager, 1931; Casimir, 1945; Serdyukov, 2022) as one of the pillar theorems of statistical physics.

## Stored elastic energy—memory effect

For most granular solids, local interaction can be simply described through an elastic-plastic formalism where the incremental relative displacement between contacting grains is decomposed into an elastic part and a plastic (frictional) counterpart. It should be emphasized that this elastic part contributes to the elastic energy stored within the two grains in contact. Thus, the total elastic energy of the system is simply the summation of stored elastic energies at all contacts. It turns out that a memory effect emerges since this stored elastic energy controls the part of the reversible energy that the system can restore later. This memory effect is however restricted to only low-range correlations, involving a few grains. The limitation in correlation length is due to the frictional dissipative sliding between grains, responsible for continual topological evolution that also includes subtle non-dissipative mechanisms such as contact opening and closing. In short, the characteristic length $l$ of this memory effect is small with respect to the typical size $L$ of the system.
## Critical State

As an exploration of fundamentals to further illustrate peculiarities of granular solids, consider an assembly of polydisperse grains, densely packed inside a cubical enclosure and confined by an equal stress ($\sigma_o$) from all axes. The specimen is thereafter deformed such that the lateral stresses ($\sigma_o$) are kept constant, while the vertical (axial) stress ($\sigma_1$) is being increased. This so-called triaxial compression test is widely used for granular materials and is classical in soil mechanics.

Figure 2 illustrates the evolution of the axial stress $\sigma_1$ given as a function of the axial strain $\varepsilon_1$ with a typical peak response after which an asymptotic state is reached. The latter state, at the macroscopic scale, refers to the most remarkable emergent property of granular solids through the existence of a so-called Critical State regime (Roscoe et al., 1958; Been et al., 1991; Parisi et al., 2017; Liu et al., 2020; Deng et al., 2022). A steady state regime is ultimately reached defined by an asymptotic mechanical state while shear strains are continuing to develop.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Typical stress-strain response curve of a densely packed granular specimen in a triaxial compression test</figcaption></figure>
Microstructural aspects

It is natural to explore the microscopic states of the system that are compatible with its macroscopic properties, especially the above-mentioned asymptotic state. This is done through the discrete element method (DEM) which numerically computes the collective mechanical behaviour of an externally loaded granular assembly composed of either spheres or circular disks interacting through elastic springs and a frictional slider.

Figure 3 shows the DEM simulations of a 2-D granular assembly loaded under biaxial compression (as opposed to triaxial compression in Fig. 2) for both dense and loose packings. It is found that, irrespective of packing fraction or compactness, the material approaches an identical stationary stress state where the axial stress no longer evolves while the axial strains continue to increase (Tordesillas, 2007; Zhu et al., 2016a; Kuhn, 2016a and 2016b; Deng et al., 2021; Pouragha and Wan, 2021).

<figure><img src="image_4.png" /><figcaption>Fig. 3. Evolution of deviatoric stress *q* and volumetric strain *ε*<sub>v</sub> with respect to axial strain *ε*<sub>1</sub> along a biaxial compression: dense packing (left); loose packing (right). After Wang *et al*., 2024.</figcaption></figure>

Furthermore, for a sufficiently dense packing, a break in symmetry (phase transition) is observed after a certain level of shearing, consistent with the formation of a strain localization pattern (one or multiple shear bands develop; Desrues and Andò, 2015; Darve *et al*., 2022). Whereas the kinematic pattern remains homogeneous (with no apparent shear band pattern) in loose packing (low compactness, Fig. 4b), localization patterns develop within denser specimens (high compactness, Fig. 4a).
<figure><img src="image_5.png" /><figcaption>Fig. 4. Comparing kinematical patterning in dense and loose specimens at the critical state regime from the distribution of the incremental deviatoric strain $\delta\varepsilon_d$ (see footnote <fnref n="1" />). A shear band has clearly developed within the dense specimen after the stress peak is reached (a), whereas the distribution remains much more random and roughly homogeneous inside the loose specimen (b). The distribution has been computed for an axial strain</figcaption></figure>

increment $\delta\varepsilon_1 = 10^{-3}$; (Liu et al., 2018).

## Mesostructures and their mutations

The microstructure of the ensemble can be described by a distribution of grain loops in 2-D conditions as illustrated in Fig. 5a. The notion of loops can be extended to 3-D as clusters of particles of different orders to form mesoscale structures. The clusters evolve to mutate into new ones of different orders as the entire assembly (specimen) assumes a new configuration under external constraints. The mutation of clusters follows a certain blueprint, here specific to the granular material.

These mesostructures have been amply discussed in Tordesillas et al., 2007; Tordesillas et al., 2012; Walker et al., 2015; Zhu et al., 2016a and 2016b. Interestingly, the microstructural construct of grain loops or clusters turns out to be the same at the critical state regime within the entire loose specimen and within the localized patterns in denser specimens (Fig. 6). This quantitative result suggests that irrespective of their initial packing characteristics, granular assemblies have the same traits at the mesoscopic scale, i.e. the same distribution of loop orders, once the stationary critical state is reached. In other words, regardless of their initial

<footnote n="1">¹ In 2-D conditions, $\delta\varepsilon_d = \sqrt{\frac{1}{2}\left(\delta\varepsilon_{ij} - \frac{1}{2}(\delta\varepsilon_{11} + \delta\varepsilon_{22})I_{ij}\right)\left(\delta\varepsilon_{ij} - \frac{1}{2}(\delta\varepsilon_{11} + \delta\varepsilon_{22})I_{ij}\right)}$, where **I** is the identity matrix, and $\delta\varepsilon$ is the incremental strain tensor</footnote>
characteristics, all specimen structures converge toward the same traits when subjected to a given external loading path; see Deng et al., 2021, for a thorough analysis of this attractor effect under a class of generalized loading paths.

Furthermore, the simulations in Fig. 6 show that all individual types (i.e., loops) evolve in a similar manner. The fraction of L₃ loops, consisting of exactly three grains, decreases as loading progresses, while the fraction of L₆⁺ loops, containing at least six grains, follows an opposite trend. Under the current loading conditions, L₃ loops are less adapted and more likely to disappear, whereas L₆⁺ loops, being more deformable and directionally adapted, tend to persist and grow. This transformation of mesoscopic structures resembles a mutation-like process, where L₃ loops evolve into L₆⁺ loops, akin to genetic mutations in cellular machinery.

It is important to note that this evolutionary trend for L₃ and L₆⁺ loops is not universal but depends on the specific loading path. For instance, under contractant loading paths, L₃ loops would be more adapted and thus increase as loading progresses, unlike larger loops such as L₆⁺.

<figure><img src="image_6.png" /><figcaption>Fig. 5. (a) In 2-D granular assemblies, the topological structure is defined by a mosaic of grain loops (minimal closed chain of grains for a given number of grains). Each polygon corresponds to a specific loop, the number of sides giving the order of the loop (after Liu *et al.*, 2020). (b) Extension to 3-D where clusters of particles are instead identified as mesostructures that mutate in response to external loading of the granular assembly.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 6. Evolution of four granular specimens (from the densest one, S1, to the loosest one, S4), under a biaxial loading with constant lateral pressure obtained from numerical simulations (DEM): (a) deviatoric stress as a function of the axial strain, (b) fraction of loops $L_3$ with exactly 3 grains with axial strain, (c) fraction of loops $L_{6+}$ with at least 6 grains with axial strain, after Zhu *et al.*, 2016a and 2016b. The * symbol means that the analysis is carried out within the shear band domain, when a shear band has developed in the dense specimens. It is found that whatever the initial packing characteristics of the specimens, the ratio of each loop category evolves nearly monotonously until a stationary regime corresponding to the critical state regime. During the critical state regime, some fluctuations develop as loop transformations continue to operate in a way that the microstructure composition remains stationary on average (Nicot *et al.*, 2024).</figcaption></figure>

The intriguing feature of the critical state regime that occurs in granular solids under shearing has been a major question for the scientific community over many decades (Roscoe *et al.*, 1958; Been *et al.*, 1991; Parisi *et al.*, 2017; Liu *et al.*, 2020; Deng *et al.*, 2022). Interestingly, this has a clear linkage with out-of-equilibrium systems as it was recently shown that the break in symmetry observed in dense granular solids via shear banding is consistent with the extremal entropy production theorems (Nicot *et al.*, 2023), which was key to the development of out-of-equilibrium thermodynamics (Glansdorff and Prigogine, 1954, 1963 and 1964; Prigogine and Lefever, 1968; Ziegler, 1983; Bažant, 1988; Dewar, 2005; Veveakis and Regenauer-Lieb, 2015; Janečka and Pavelka, 2018; Benfenati, and Beretta, 2018; Porporato *et al.*, 2020; Lucarini *et al.*, 2020).
## Out-of-equilibrium, dissipation and entropy

Out-of-equilibrium systems have internal thermodynamic forces that are responsible for the development of corresponding fluxes that drive systems back to equilibrium. In fact, thermodynamic forces and fluxes are conjugate to each other (as dual quantities) with respect to the internal entropy production and are linked by a tensorial relation which is no longer linear far from equilibrium. This absence of linearity favors the occurrence of self-organized patterns through a phase transition, leading to an optimal dissipative structure<fnref n="2" /> by way of a shear band pattern. It is through the shear band that most part of the external energy put into the system is dissipated, and potentially allowing the system to sustain external loading without collapsing (Nicot *et al.*, 2003; Wang *et al.*, 2024).

The local dissipation processes within the assembly are consistent with configurational and stress fluctuations occurring continuously (Deng *et al.*, 2021; Clerc *et al.*, 2021). Furthermore, it has been pointed out that the elastic energy stored within the material before the onset of bifurcation plays a fundamental role. In particular, the transition of the system from a homogenous regime towards a heterogeneous one is strongly governed by the maximal capacity of the system to store elastic energy (Wang *et al.*, 2024). The system is no longer able to store more elastic energy during the post-bifurcation critical state regime. Simultaneously, the entropy production reaches an extremum upon entering the critical state regime (Nicot *et al.*, 2023).

After peak stress, the entropy production decreases to a constant value as the critical state is sustained (Fig. 7). Thus, the bifurcation of the system from a homogeneous state towards a heterogeneous one into a shear band corresponds to a structuration of the material that ultimately extremizes the entropy production. In essence, for the system to dissipate energy in an optimal manner, this process must occur through a shear band where frictional dissipation is the most intense. This result suggests a new interpretation of the intriguing shear banding phenomenon as the emergence of spontaneous dissipative structures fuelled by the exhaustion of available energy, namely elastic energy, no matter how small it is. While the domain located outside the shear band pattern remains densely packed with nearly no dissipation

<footnote n="2">² The term "dissipative structure" was coined in 1969 by Ilya Prigogine for systems crossed by flows of matter and energy, where spontaneous structuring and organization processes can occur to dissipate the external energy. When they are far from equilibrium, extremal entropy theorems show that these structures conform so that the dissipation is optimal, hence the name optimal dissipative structures.</footnote>
processes occurring, the inside of the shear band undergoes a marked dilatancy associated
with an intense frictional dissipation.

<figure><img src="image_8.png" /><figcaption>Fig. 7. Left-hand side: evolution of the deviatoric stress (q) and of the volumetric strain (ε<sub>v</sub>) with axial strain along a biaxial compression under constant lateral pressure; the dashed line B corresponds to the deviatoric stress peak, whereas the dashed line C corresponds to the beginning of the Critical State regime. Right-hand side: evolution of the entropy production (times the thermodynamic temperature considered constant); while the entropy production is maximal during the softening regime (BC), it decreases to a plateau in the Critical State regime. The kinematic pattern (incremental deviatoric strain) within the assembly is given in the two insets for states B and C, where a transition toward a localized pattern is visible (after Nicot *et al.*, 2023).</figcaption></figure>

Optimized dissipation via structuration, an exemplar of the living and non-living

Structuration of a material through a phase transition that optimizes the dissipative capacity of the system reveals to be a very general emergent mechanism. It can be encountered in various fields such as atmospheric heat transfers (Rayleigh-Bénard cells; Bénard, 1901 and Rayleigh, 1916), chemical engineering (chemical clocks; Hudson and Mankin, 1981), or many biological processes based on cellular self-organizations. All these transitions might suggest that the second principle of thermodynamics is being violated, as they give rise to a gain in order. This apparent paradox is in fact resolved by noting that this gain in order induced by the structured pattern is mitigated by the increased dissipation capacity of these dissipative structures. In other words, the system can adapt to external disturbances by transforming its internal structure: a volume of fluid can accommodate the heat transferred from the outside because a structured pattern (Rayleigh-Bénard cells) enables an optimized internal dissipation; a granular assembly can sustain shearing under external loading because a structured pattern through shear-banding emerges and optimizes the plastic dissipation. In these adaptive processes, the memory effects (stored elastic energy) play a governing role (Nicot *et al.*, 2023) by controlling the information propagation within the system.
Returning to microstructural arguments along the lines of statistics of the microscopic composition of macroscopic bodies, the diversity of potential topological arrangements of particles allows the granular system to “select” only adapted configurations that can sustain the external conditions to achieve a balance. In this context, “Selection” should be understood as a statistical process operating within a granular assembly where multiple configurations coexist. A granular system evolves through configurational changes at the mesoscopic scale, with only the most resilient structures persisting, aligned with the prevailing loading conditions. This is analogous to natural selection in living systems, where only the best-suited organisms survive. Configurations that are less suited to environmental conditions, here mechanical loading, are more likely to collapse. In contrast, more adapted configurations will better withstand external forces and remain stable as long as the loading conditions remain unchanged. This process closely mirrors Darwinian natural selection, particularly in the broader classification of living species.

Thus, just like living species do, granular matter as an illustrative example of an inanimate complex system can evolve and potentially adapt to external conditions. This irresistible evolution of all substances towards a thermodynamic equilibrium resonates with the adage that the world is the best of all possible worlds. Behind this adage, it must be recognized that the cause as the driving force in natural processes is nothing but elementary principles that govern the statistical sorting between adapted and non-adapted configurations, which is the consequence. However, this also prompts the question of whether the evolution of living or non-living complex systems could be envisioned within a unified framework?

### 3. MULTISCALE COMPLEXITY IN LIVING SPECIES

#### Diversity, adaptation and natural selection

Living species such as animals adapt inexorably through processes that develop over time under given environmental constraints. They exhibit a multiscale complexity from the cellular scale to large groups composed of many agents in interaction (Fig. 8). The genetic transmission from parents to offspring dictates heredity principles, ensuring that most of the traits that constitute the phenotype<fnref n="3" /> of an individual can persist over successive generations.

<footnote n="3">³ In genetics, the phenotype is the set of all observable characteristics or traits of a given organism</footnote>
This complex process however excludes any short-term individual behavioural adaptation that represents a reversible mechanism such as, for example, neuromuscular adaptations following recurrent training exercises.

In living species, it is commonly admitted that complexity has its roots in the cellular scale (Ma'ayan, 2017). Without discussing the whole cellular machinery—from the diversity of the nucleotide triplet combinations to the remarkable cellular organization leading to a functional ability—we focus on the genetic fluctuations during the replication of cells. Such fluctuations occur during the nucleotide triplet combinations to assemble amino acids, the basic constituents of any functional proteins. It turns out that infinitesimal genetic (allelic) variations can occur, contributing eventually to slight changes in the traits, and leading to a progressive, random and long-term evolution of the individuals. For the sake of conciseness, we exclude thereafter the genetic drift mechanism and the extended heredity approach as well (Bonduriansky, 2018). Thus, some very slight cellular modifications can occur, possibly giving rise to phenotypic variations (Alberch, 1991).

The reproduction between species over several generations can promote specific traits, particularly if these traits are well adapted against current environmental conditions (Fisher, 1930), as those individuals are more likely to survive and reproduce. These fluctuations strongly participate in the diversity of species, some of them (namely, adapted) likely able to resist against long-standing environmental conditions whereas the others will not. Thus, a selection operates between the well-adapted individuals and those which are not, dictating which are the fittest individuals likely to survive. This long-term process requires fundamentally the size of the population to be large enough in order that the diversity of traits modifications stemming from random fluctuations can generate phenotypes that comply with the changing environmental conditions. Otherwise, size effects can occur, such as the genetic drift (see for example Star and Spencer, 2013).

It is worth mentioning that this key notion of selection, highlighted by Darwin in the evolution theory, is in fact passive selection. Many researchers have queried what could be the driving force responsible for this selection mechanism, and, as a result, for the whole evolution process. By shedding light on dualities between living and non-living species as complex systems, we put forward the hypothesis that the passive selection is nothing else but a statistical process taking place within large, diverse and disordered enough systems.
In parallel, short-term adaptive processes can take place within any community of living species. When an external warning signal is recognized within a large community (signaling for example an imminent danger), everyone will react, giving way in some circumstances to collective structured patterns. Collective behavioural organization has been extensively studied such as, for example, to better predict the population dynamics (Langston et al., 2006). In this perspective, clogging effects can occur during urgent premise evacuations, together with structuration effects along the corridors: the central part flows onward, while the lateral sides exhibit much more chaotic motions, reminiscent of a viscous fluid flowing through a pipe. When it comes to dealing with even larger communities, such as the whole of humanity, much more complex behavioural patterns can emerge, mainly when an external stress is applied. An illustrative example is given by global climate change, directing severe constraints in terms of available agricultural spaces, water supply, potentially higher epidemiological risks, or geopolitical balance (Bošnjaković, 2016). As a reaction, humanity tends to divide into a few macro-groups, as a homogeneous state can probably no longer exist to comply with the external conditions. This large-scale partitioning stems undoubtedly from very complex individual and social cultural heritages. However, it can be conjectured that the larger the size of the group (for example, the whole of humanity), the lesser the collective behavior depends on the individual characteristics.

## Memory effects and adaptation in living and the non-living

Memory effects are responsible for the emergence of correlation lengths that control how the information diffusion is structured within a given group. Roughly speaking, memory for each individual acts on his or her behavioral anticipative capacity. This individual behavioral anticipation accounts for the estimated behavioural response of the other individuals.

Turning back to granular solids, the above complex relationship is reminiscent of the local (or non-local) character of the constitutive behaviour of materials (Pijaudier-Cabot and Bažant, 1987). Under a given external loading, the mechanical response of a given grain of the assembly depends on the behaviour of a collection of grains within a certain kernel critical domain. The influence of the other grains located outside this domain is nearly negligible. The larger the critical domain with respect to the size of the assembly, the more complex the overall behaviour. Conversely, as seen previously, the overall behaviour for very large systems is much less affected by the response of each individual grain. Similarly in living species communities, when the size of the population is very large, it can therefore be
suggested as a first approximation that the behavioural specificity of everyone becomes less
and less prominent in the emergence of large-scale collective processes.

The partitioning of the large scale initially homogeneous population into a distribution of distinct subgroups can be seen as a bifurcation process. In short, living and non-living communities, as a collection of interacting agents sharing information over finite space ranges, could share unexpected common traits under certain environmental conditions, leading to the emergence of a wide spectrum of behavioral bifurcation modes, the hallmark of complexity.

Finally, living species exhibit multiscale adaptive responses under environmental stresses (Fig. 8). While for long-term environmental conditions, both genetic fluctuations and genomic diversity will give rise to a statistical selection of the most adapted species, short-term stresses can make collective behavioural patterns emerge from a cascade of bifurcation mechanisms, as can be observed in any complex systems.

Fig. 9 illustrates a heuristic example involving a bimodal population: a part of the population holds the underlying trait “triangle”, while the other part holds the trait “disc”. Under short-term constraints, the population could get partitioned, irrespective of these two traits. Thus, the population splits into two parts located in separate camps, displaying a social structuration. When long-term constraints are applied, a natural selection can operate over several generations, promoting the individuals with the most adapted trait (“triangle” in this example). After several generations, the “triangle” population has increased, while the “disc” population vanishes progressively.
<figure><img src="image_9.png" /><figcaption>Fig. 8. Multiscale complexity in living species: (a) social interactions within a human group; (b) from the functional organs to the constitutive cells; (c) cellular organization.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9. Multiple time scale adaptation model: (a) under short-term constraints, the collection becomes structured; (b) under long-term constraints, natural selection occurs and filters the less adapted species (discs).</figcaption></figure>
4. A BUILDING BLOCK TOWARD A UNIFIED APPROACH

The previous sections reveal that granular assemblies and communities of living species share intriguing common traits as complex systems. Over their respective time scales, both can self-organize and adapt to external conditions. A formal correspondence between living and granular systems can be tentatively established, as depicted in Table A1 (Appendix A).

Building on this correspondence, this section aims to develop a unified approach by exploring key concepts such as disorder, information transmission, fluctuations, diversity, complexity, and emergence.

Disorder and information transmission

Considering a complex system under given constraints, no matter the specific intricacies of each agent, the systemic evolution stems first and foremost from the collective diversity whereby the individual features are no longer prominent. In other words, it can be thought that a huge collection of diverse and disordered, non-complex individuals might be much more prone to transform and develop an adaptive complexity than any ordered collection of complex individuals can. The nature of individuals determines the type of interactions that can take place within the population and how the information propagates, involving various space and time scales. Agents interact with each other thanks to information transmission. As depicted in Fig. 10, three modes of transmission can be identified: (i) by contact (information conduction); (ii) by the motion of individuals (information convection); and (iii) by long-range information transmission such as postal or internet mailing (information radiation).
<figure><img src="image_11.png" /><figcaption>Fig. 10. Schematic diagram summarizing information transmission modes in granular systems and their analogs in living systems.</figcaption></figure>

## Information and memory

In granular solids, the contact interaction gives rise to contact forces, resulting from both the intergranular sliding between adjoining particles and the deformation of particles themselves in the vicinity of the contact zone. While particles can flow over long distance in diluted assemblies where the order of magnitude of the mean free path is of several grain diameters, the particles are much more confined in densely packed granular solids, basically preventing information convection from occurring. However, long-range information transmission can occur in granular solids when the correlation length increases which is consistent with a long-range order. Such situations can be observed in elastic materials, allowing information to be transmitted (quasi) instantaneously over distances that can scale up with the size of the specimen. Excluding any intra-particle damage process like grain fracturing or particle crushing, the underpinning mechanisms, as exposed in Section 2, embed both elastic processes and frictional dissipation. While the frictional dissipation is responsible for the entropy increase within the specimen, elastic processes induce memory effects making the material able to store energy<fnref n="4" />. Energy storage within granular assemblies has important consequences. On the one hand, the material can build a contact force network and sustain an external loading (Radjai *et al.*, 1998; Peters *et al.*, 2005) as seen far back in Fig. 5. On the other hand, the stored elastic energy is strongly involved in the occurrence of phase transitions leading to possible abrupt changes (bifurcations) within the systems (Nicot *et al.*, 2023).

<footnote n="4">⁴ While the frictional dissipation between granules gives way to the entropy production within a given system, the elastic energy stored at contacts corresponds to the Helmholtz free energy.</footnote>
Basically, this memory effect is related to a certain persistence of contacts between particles, and therefore to the configurational information, the basic tenet of adaptivity. A remarkable connection between living and non-living species can therefore be established at two different scales.

## Fluctuations and diversity

On the individual scale, heredity processes taking place within the cellular nucleic machinery should also be regarded as a long-term memory effect allowing recurrent traits to be transmitted over successive generations. Both cellular nuclei and granular assemblies are complex systems, where fluctuations occur. In granular assemblies, these fluctuations favor the most adapted topological configurations to given loading conditions, i.e., the most adapted distribution of loop orders in 2-D conditions (Fig. 5a). Similarly, genetic transmission may suffer from slight, random errors (fluctuations) that expand the phenotypic diversity of the population. The phenotypic traits that are less adapted vis-à-vis the current environmental constraints will make some groups of individuals less able to subsist. As such, phenotypic modifications creep over a long time spanning several generations. Thus, under given external conditions, the genotype of the individuals converges towards the one corresponding to the most adapted phenotype. Once this genotype is collectively obtained, a stationary regime is reached until new external conditions are imposed.

## Diversity and stationarity

The construction of this phenotypic stationary regime is the consequence of genomic diversity. Of course, this diversity can be insufficient to allow individuals over several generations to adapt to any environmental constraints, making species disappear progressively. Insofar as granular solids are concerned, the diversity of the topological network allows the assembly to adapt to external loading conditions. The Critical State observed when deviatoric, homogeneous loading conditions are applied (Fig. 3) corresponds to a stationary regime marked by recurrent local topological fluctuations in the form of grain clusters (loop orders in 2-D conditions), whose distribution remains stationary (Zhu *et al.*, 2016a and 2016b; Deng *et al.*, 2021; Nicot *et al.*, 2024). On the other hand, when a loosely packed granular assembly is subjected to an isochoric compression, a phase transition from bulk solid to fluid-like behavior is observed, corresponding to the progressive, ultimate vanishing of the inter-granular stress (Darve and Roguiez, 1998; Wan *et al.*, 2017). When the topological microstructure diversity is not sufficient to construct a configuration able to
sustain the loading conditions, the granular system collapses in the same way that living species disappear when they cannot survive severe environmental constraints.

## Complexity favors emergent bifurcations

In the realm of living systems, the interactions between agents are regulated by complex cultural, educational and emotional processes that encapsulate memory effects. At the population scale, the collective behaviour of groups (even culturally homogeneous) facilitates sociologically and behaviourally structured patterns to emerge just like non-living complex systems. As discussed in Section 3, when the size of the groups increases, the specificity of each individual does not matter any longer, and larger scales patterns develop. Just like elastic strain energy (the material memory) plays a fundamental role in the emergence of a bifurcation regime in granular assemblies, the memory of living species is also determinant in the emergence of bifurcation processes.

## Living and non-living species: two sides of a same coin?

The above analysis supports that living species, and granular solids can be regarded as complex systems at two different levels. This multiscale complexity analysis can be illustrated in Fig. 11 where the processes occurring within granular solids and living species communities are compared within their own time scale. First, in the collective response of large living species groups, social clustering shares outstanding similarities with phase transitions operating within dense granular solids under deviatoric loading programs, before bifurcating towards a structured material through one or several shear bands (Fig. 11a). Second, living species long-term adaptation to environmental conditions reveals evident common traits with the response of granular solids under external loading where microstructural transformations operate thanks to configurational fluctuations. These make it possible for the living or non-living systems to evolve and adapt to external conditions (Fig. 11b).
<figure><img src="image_12.png" /><figcaption>Fig. 11. Double-scale complexity model: (a) self-organization process between living species groups and granular assemblies; (b) selection of specific individuals process and compared analysis between living species and granular assemblies.</figcaption></figure>
5. CONCLUDING DISCUSSION

This paper has laid the foundations of a possible unified approach whereby living and non-
living species are described as complex systems through a cascade of scales. Complex
systems irrespective of their very own nature are inherently disordered and constantly
subjected to structural fluctuations. Extremal entropy production theorems state that such
systems, when brought out of equilibrium, can exhibit various bifurcation modes (Prigogine
and Lefever, 1968; Ziegler, 1983; Veveakis and Regenauer-Lieb, 2015). It was recently
established (Nicot *et al.*, 2023) that the minimal entropy production theorem outlined by
Prigogine is perfectly consistent with the second-order derivative of kinetic energy. Both deal
with the occurrence of bifurcation modes by considering the second-order variation of a
proper quantity: the internal entropy of the system (Prigogine's theorem) and the kinetic
energy of the system (second-order work theory; Wan *et al.*, 2017). The second-order work
theory is well suited to mechanical systems, while the entropy production theorems are
broader and can thus apply to any systems with diverse bifurcation modes. The relation
established between the second-order work theory and the entropy production theorems
enriches the description of the dynamics of complex systems within a unified framework.

Structural diversity of complex systems, tightly related to the notion of disorder, calls for an
appropriate approach to describe the underpinning processes. Previous investigations of the
concept of configurational mechanics (Edwards and Oakeshott, 1989; Baule et al., 2018;
Nicot et al., 2024) offer a suitable framework to define entropy of complex systems. Under
certain circumstances, when brought out of equilibrium, the system can gain order to optimize
its dissipation capacity. This could be counter-intuitive according to the second law of
thermodynamics. The structured patterns, resulting from a self-organization process, can
optimize the dissipative capacity of the system, ensuring the global entropy to increase as
expected. This is the reason why they have been referred to as dissipative structures by
Prigogine (Glansdorff and Prigogine, 1971; Nicolis and Prigogine, 1977; Prigogine and
Lefever, 1968 and 1975). It is believed that memory effects through information storage and
dissipation are closely involved in bringing the system out of equilibrium. These different
notions are conceptually outlined in Fig. 12.
<figure><img src="image_13.png" /><figcaption>Fig. 12. General self-organization scheme in out-of-equilibrium complex systems through information storage and dissipation.</figcaption></figure>

In the light of this discussion, and following the ideas developed in (Kauffman, 1991 and 1993; Weber, 2011; Fábregas-Tejeda and Vergara-Silva, 2018), it transpires that Darwin's natural selection theory is a key building block for conceptualizing the dynamics of complex systems under given environmental conditions where adaptive self-organizational processes result from structural diversity. In essence, complex systems, such as living species or granular solids, can be regarded as multiscale adaptive systems. This is perfectly in agreement with the quest for an extended evolutionary theory beyond biology. Quoting Weber (1996), "natural selection is an emergent phenomenon of physical and chemical selection". The following two major findings of this paper support this assertion:

* Structural transformations are essential for adapting to environmental conditions. Phenotype traits evolve due to fluctuations within the cellular nuclear material, making a given type of species more prone to resist long-lasting environmental conditions. Similarly, microstructural transformations operate within granular solids thanks to configurational fluctuations to withstand the loading conditions. These processes make it possible for living or non-living systems to adapt to evolving external conditions. Comprehending the dualities between living and non-living species as complex systems reinforces our hypothesis that the passive selection is nothing else but a statistical process taking place within large, diverse and disordered enough systems.

* Social clustering within large groups of living species shares outstanding similarities with phase transitions operating within granular solids. Such bifurcations lead to self-
organized patterns and ultimately to stationary regimes compatible with long-lasting
environmental stresses. In short, the living and non-living can be understood as
communities of agents sharing finite spatial information that have unexpected
common features under certain environmental conditions. This comprehension
clarifies that the characteristic of complexity is the result of an emergence of a broad
spectrum of behavioural bifurcation modes.

Our primary objective has been to lay the groundwork, demonstrating that such a unification
is both accessible and essential as a cross-disciplinary approach to gaining deeper insights into
the dynamics of complex systems, regardless of their nature. We anticipate that elaborating a
unified approach for capturing the dynamics of complex systems could play a key role to
better understand the intertwined mechanisms governing the consequences of the global
change (related to the non-stationarity of the environmental conditions), and how humanity
can adapt in the future through interactions with the surrounding living and non-living
environmental systems.
ACKNOWLEDGMENTS

We gratefully acknowledge the CNRS International Research Network GeoMech for having offered the opportunity to make this project possible through long-standing collaboration of all the authors ([http://gdrmege.univ-lr.fr/](http://gdrmege.univ-lr.fr/)).

APPENDIX A

<table><caption>Table. A1. Correspondence between living and granular systems: general settings, diversity and adaptation.</caption><thead><tr><th>Living species</th><th>Material (granular) systems</th></tr></thead><tbody><tr><td colspan="2">General correspondence</td></tr><tr><td>Large population scale</td><td>Individual scale</td></tr><tr><td>Social group</td><td>Individual</td></tr><tr><td>Social patterns</td><td>Characters of individuals (traits)</td></tr><tr><td>Social interaction rules</td><td>Genetic machinery</td></tr><tr><td>Environmental conditions</td><td>Environmental stress</td></tr><tr><td colspan="2">Diversity</td></tr><tr><td>Social diversity</td><td>Gene diversity</td></tr><tr><td>Short- and long-range social interactions</td><td>Genes can mutate (e.g. alleles)</td></tr><tr><td>Environmental conditions</td><td>Genes define traits</td></tr><tr><td>Environmental conditions</td><td>All the traits define the phenotype</td></tr><tr><td colspan="2">Adaptation</td></tr><tr><td>Social clustering</td><td>Traits evolve</td></tr><tr><td>Some clustering patterns are more</td><td>Some traits provide greater adaptibility</td></tr></tbody></table>
<table><thead><tr><th>adapted to the general environmental conditions</th><th>to certain environmental conditions</th><th></th></tr></thead><tbody><tr><td>Those clustering patterns are more likely to persist</td><td>Those traits have more chance to appear in the offspring</td><td>Those loops have a longer lifespan</td></tr></tbody></table>

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