HHigh Entropy Alloys as a new metaphor insociophysics
Pawel SobkowiczJuly 7, 2020
Abstract
Most of the opinion dynamics models in sociophysics have their his-toric origin in studies two-dimensional magnetization phenomena. Thismetaphor has proven quite useful, as it allowed to use well known tech-niques, relating individual behaviours of single spins and their interac-tions, to large scale properties, such as magnetization or magnetic do-mains creation. These physical properties were then “mapped” to socialconcepts: spin orientation to a person’s views on a specific issue, mag-netization to global opinion on the issue, etc. During the past 20 years,the models were significantly expanded, using more complex individualagent characteristics and even more complex types of the interactions,but the power of the metaphor remained unchanged. In the current pa-per we propose to use a new physical system as the basis for new ideasin sociophysics. We shall argue that the concepts and tools devoted tostudies of High Entropy Alloys (HEAs) could significantly broaden therange of social concepts “addressable” by sociophysics, by focusing on awider range of global phenomena, arising from atomic properties, interac-tions and arrangements. We illustrate the new idea by calculating a fewcharacteristics of a simple HEA system and their possible “mapping” intosocial concepts.
The idea to use the concepts and rigour of physics to describe social phenomenadates back as far as 18th century. Marquis de Condorcet, based on his ex-tensive knowledge of mathematics, introduced a concept of social arithmetics,“ la science qui a pour objet l’application du calcul aux sciences politiques etmorales ” in 1794. Some decades later, August Comte proposed physique so-ciale (social physics) as a new science in 1839, in the fourth tome of his
Coursde philosophie positive , arguing from the point of view of promoting progressthrough better understanding of social phenomena, by applying to social sci-ences the samerequirements as to physics or chemistry. These ideas were sharedby Adolphe Quetelet, who, a few years earlier (1835) published
Sur l’hommeet le dveloppement de ses facults, ou Essai de physique sociale . His ambition1 a r X i v : . [ phy s i c s . s o c - ph ] J u l as to include social sciences into the mainstream composed of mathematics,physics, chemistry and biology.Have these early attempts at connecting mathematical foundations typicalfor physics with the importance of understanding our societies been successful?In certain aspects, the answer is a decisive yes. Most of social sciences today usehighly developed statistical tools and models to derive the main characteristicsof social systems and to provide mathematical strength to the reasoning or tobetter understand the data. This application of statistical methods (although inmany cases imperfect due to small sample sizes) has allowed sociologists to movebeyond the anecdotal and speculative “just so stories” stage. On the other hand,for many research questions, the complexity of the human behaviour overwhelmsthe statistical and mathematical models. The latter situation is present espe-cially when the amount of unknowns and uncertainties (typical for the complexsocial situations) is so large as to allow multiple, mutually incongruent modelsto be created and used. The appeal of the high status of physics among soci-ologists and psychologists (sometimes dubbed “physics envy”) has even led toabuses, recognized as early as 1970’s [Andreski, 1972].In the context of our work, it is important to remember that the ubiquitoususe of mathematics is not the only source of the strength of physics. Anotherobvious reason for its success is the constant interplay between theory and ex-periment, challenging and driving each other. But there is yet another aspect,often forgotten: it is the ability to pick the right level of abstraction for thestudied problems. For example, in studying the physics of gases we “forget”,for the most part, the internal complexity of atoms, their nuclei, the quark-gluoninteractions – replacing them with a simple elastic-ball approximation. Usinga two parameter Lennard-Jones potential, chosen for its mathematical simplic-ity leads to many important results and ideas, even though we know that thispotential is not “true”. Finding the right balance between the complexity ofconcepts used to build a model and the simplicity required for the model to beuseful ind insightful is still a grand challenge of social sciences.During the past 30–40 years, with the advances in statistical physics ofcomplex systems and, especially, with the appearance and ubiquity of computersand computing tools, the ideas of social physics begun to be part of mainstreamresearch. Modernized name of sociophysics denotes today a broad range of caseswhere analogies with certain physical systems are used to improve understandingof social systems and human behaviours. We are now able to use these ideas tostudy many static and dynamic processes, sometimes with quite important andilluminating results.The current work is mainly devoted to a branch of sociophysics (one of theoldest ones), namely the studies of opinion changes in societies and societalreactions to media, propaganda, rumours etc. The paper will start with a shortintroduction of the history and current status of such models, moving on to thechallenges faced by the research field (Section 2). The main part is concernedwith a proposal to use a relatively novel class of physical systems, High EntropyAlloys, as the base for sociophysical models. This could allow us to describe abroader range of social phenomena than the current approach (Section 3). A2imple example of such model is presented in Section 4, to illustrate the potentialof the approach. The attempts to describe the dynamics of social opinions using the analogiesderived from the physics of magnetic materials have a respectable history dat-ing back more than 40 years. The foundation of such approach rests on three“mappings”: a person is equated to an atom; the person’s opinion to the orien-tation of the atomic spin; and the social influences are modelled by interatomicinteractions. This simplification of human behaviour has led to coining of anotion of spinson (spin-person, Nyczka and Sznajd-Weron [2013]).There are more correspondences in such approach. The topology of socialcontacts was typically one of three variants: fully connected network, in whichany person could communicate with any other person; a regular grid (typicallytwo-dimensional square lattice, the easiest one for graphical presentation of re-sults), where the agents could interact with their neighbours (either only theclosest neighbourhood or a suitably chosen extended one); a random network ofa given density of links. Somewhat later on, the growing popularity of networkscience has led to models based on more realistic interaction connections, in-cluding the scale free topology. Volatility of individual opinions corresponds, insome models to temperature. Additionally, external influences (such as exertedby media or propaganda) were mapped to external magnetic field.The last component of the models was the interaction mechanism – describ-ing the influence of one agent on others. This could take the form of one-to-oneinteractions (one agent trying to convince the other, or two agents sharing adiscussion and trying to convince each other, for example the voter model [Coxand Griffeath, 1986, Ben-Naim et al., 1996, Galam et al., 2002, Castellano et al.,2003], the bounded confidence model [Deffuant et al., 2000, 2002, Weisbuch,2004, Weisbuch et al., 2003]) or many-to-one group influence models (wherea suitably averaged opinion of a group “surrounding” an agent influences itsopinion, for example the social impact model [Nowak et al., 1990, Nowak andLewenstein, 1996, Holyst et al., 2001, Kacperski and Holyst, 2000, 1999], theHegselmann-Krause model [Hegselmann and Krause, 2002]) or the Sznajd model[Sznajd-Weron and Sznajd, 2000, Stauffer, 2001, Stauffer and de Oliveira, 2002,Stauffer, 2002, Slanina and Lavicka, 2003, Sabatelli and Richmond, 2004, 2003,Bernardes et al., 2001]. The specific conditions required for an agent to keepor to change its opinion varied between the models, leading to different processdynamics. There was, however, an assumption which was used in most of theearly models: any interaction between two agents with differing opinions woulddiminish the difference or, at worst, leave it unchanged. Similarly, a group influ-ence would tend to bring the agent’s opinion closer to the group one. This basicindividual dynamics, in the absence of constraints, should eventually lead to aconvergence of opinions. In fact, many of the early works considered the dynam-ics of the appearance of such consensus, seemingly without taking into account3he observation that societies very seldom reach the state of global agreementof views.Another characteristic feature of the spinson and related models was the useof “high temperature state” as the initial point. In the studies of spontaneous(or induced) magnetization, preparing the sample at high temperature results ina random orientation of spins. When the temperature is lowered, magnetizationmay appear, either globally or in domains. It is tacitly assumed that the specificinitial random configuration is irrelevant for the final results. Unfortunately,social systems are very seldom in such randomized state. The evolution mightcrucially depend on even minor correlations within the initial state. For thisreason, the starting conditions for most models were largely unrealistic.The early sociophysical works were later developed to include many con-straints, inhibiting this trivial path to consensus. Among the most widely rec-ognized was the bounded confidence model, which assumed that the opinionsof the two interacting agents could get closer to each other only if the initialdifference was suitably small, below the tolerance threshold. Depending of thetolerance level, this could lead to consensus (if the tolerance was high and mostagents would be able to “talk” to each other). For small enough tolerances,a stable coexistence of two or more groups would appear. Each group havinga locally convergent opinion, separated from other groups in a way effectivelyblocking intergroup communication.It should be noted that in the standard formulation of the bounded confi-dence model, the split opinions were always within the spectrum postulated inthe initial configuration. Thus the basic bounded confidence model could notexplain growing polarization. The bounded confidence model itself has beendeveloped to cover many variants, some of them including repulsion of suitablydifferent opinions, considerations of multidimensional opinion spaces, presenceof minority groups holding fast to their opinions and many others. Eventually,some of these extensions have allowed to reproduce widening of gaps betweensocial groups.The evolution of the bounded confidence model is but one example of thechanges to the initial palette of approaches. The initial interaction terms, bor-rowed directly from spin theory of magnetism were expanded, additional classesof agents (extremists, contrarians, inflexibles, leaders) were added. New modelsincluded dynamic networks, in which topology of links evolved in parallel to theopinions. The field is active and flourishing.
One could ask why should we consider new analogy between physics and socialsciences if the current approach is so active?There are basically two answers to this question. The first is based on the ob-servation that a certain fraction of sociophysical research stopped even pretend-ing to be part of the multidisciplinary effort. Sobkowicz [2009] has called for acloser link of models with reality (not just to prove the usefulness of the models insocial contexts, but also to provide the necessary theory-experiment/observation4oop). Indeed, some authors have heeded the call, and the number of papersaimed at using sociophysics to understand real social systems is growing. Theseapproaches often steer away from the direct reference to physical counterparts,and are more accurately classified as Agent Based Models rather than socio-physical ones. The agents are in some aspects more human-like, more complexin their individual behaviours, for example incorporating emotions, biases, mo-tivated reasoning. Also the interactions between the agents become much morecomplex and non-linear.Still, there is a large number of papers which, in our opinion, use the ex-cuse of social modelling, but should be more accurately described as “physics ofnon-physical systems”. The range of properties, arrangements and interactionsbetween actual atoms in magnetic systems is limited. Once the barrier requiringthat a model should correspond to a real system is broken, one can include what-ever comes to mind: multiple “spins” and other degrees of freedom per atom,adjustable interaction networks, multiple layers of such networks etc. Thenone can apply the same methods, concepts and variables as are used for realphysical systems and – quite often – derive results not present in real physicalsystems for obvious reasons, and unusual enough to warrant a publication. If,as is often the case, there’s no corresponding social system, the work represents,unfortunately, modelling for modelling’s sake.The second reason for the potential usefulness of looking for a broadenedphysical base of social studies analogy is more fundamental. Most of the previouswork focused on the effects of putting together large numbers of agents andexamining how the interactions change their behaviours. Would the individualreactions to external stimuli become ordered/coordinated? Can a small groupof people convince larger groups to their views (and what are the conditionsnecessary for this to happen)? Can opposing views coexist – or even growapart, even when people have, potentially, access to the same information?These are important questions, no doubt. But there are others, which aremore difficult to address in using the current metaphor. They deal with be-haviours of groups and societies as complex entities themselves. Focused onsociology rather than social psychology. Can our societies survive increasedpolarisation and function effectively? What would be the effects of technolog-ical and cultural changes? is the pace of such changes an important factor,and if yes, in what way? What will be the effects of the growing strain be-tween the super-rich and low-income populations? What would result from theclash of societies in a global market and global inequalities between countriesand communities? These issues are reminiscent of looking at the macroscaleproperties of materials: their hardness, plasticity, corrosion resistance, meltingtemperatures, thermal and electrical conductance, resistance to radiation etc.,etc. Of course, ultimately, these large scale characteristics have their origin inmicro- and nanoscale composition and arrangement of the constituent atoms.Our paper comes directly from the idea of such correspondence between ideasof materials science and sociology. 5
Need for a richer metaphor
It is not the goal of the current paper present a ready solution or a fully de-veloped model, describing and predicting the behaviour of some social system.
We aim at suggesting a possibility of using a different domain fromphysical research as the source of inspiration.
The motivation is partiallydescribed in the previous sections: to allow creative and imaginative multidis-ciplinary cross-polination of ideas, and to rejuvenate sociophysics.The inspiration for the particular choice of the of the High Entropy Alloys(HEAs) as the physical system to be used as the foundation of the model wasactually the complexity of social systems. Spinson-based models (especially intheir original two-dimensional versions) are obviously too simplistic to corre-spond to most of the real social situations. We need to include much morecomplex interactions and combinations of constituents, to be able to describesystem long term as well as short term temporal behaviours and to focus on aricher set of global characteristics than magnetization alone.We should note here that the magnetic system analogy is not the only oneused in models of opinion dynamics. Other physical processes were consid-ered, for example percolation theory [Chandra, 2012, Yang and Huang, 2015] orkinetic models [Biswas, 2011, Boudin and Salvarani, 2016, Toscani, 2006, Lal-louache et al., 2010]. In this context, the breadth of physical phenomena andtools developed to understand them allows to focus on different aspects of thesocial systems for which the models are used.High Entropy Alloy studies are faced with similar challenges as social sci-ences. These include the presence of many chemical elements, each characterisedby multiple different properties and possibilities of widely varying spatial organ-isation, changeable under environmental changes. The tools and concepts thatcould provide insights into the HEA-related problems could become useful in thesocial contexts. Moreover, we already know that specific chemical compositionsand preparation processes of high entropy alloys can lead to a wide variety ofmacroscopic characteristics. Thus the link between the nano- and micro-scalesand global properties is quite fruitful in terms of designing materials capableof withstanding external forces and preserving desired functionality. At thesame time, the path from nano-scale composition to the macro-scale behaviouris challenging to discover and describe.It is our hope that an eventual “mapping” of various physical characteris-tics of HEAs to their social systems equivalents could enhance our capacity tounderstand, human societies, arguably the most complex systems we know of.
Compared to the microscopic studies magnetic phenomena in physics, whichare now over 100 years old (or even to the sociophysical spinson-based opin-ion dynamics models, in their mid-forties), the field of High Entropy Alloysis very young. Miracle and Senkov [2017] place its birth at 2004. Gao [2016]notes that the interest in alloys composed of near-equal ratios of various ele-6ents predates these first publications by more than a decade. Nevertheless,the past sixteen years were extremely active, mainly due to the combination ofinteresting science (including many unexpected results and fundamental discus-sions) and significant potential applications – materials with designed properties(functional materials) can become crucial in many applications.The literature on specific properties and applications of HEAs is alreadyvery broad. For an the introduction, in addition to the two general reviewsmentioned above, recent reviews providing good starting points are Pickeringand Jones [2016], George et al. [2019], George et al. [2020].To present the concepts related to the high entropy alloys, we chose to fo-cus on these properties that could, in our opinion, provide bridges to interestingsocial phenomena, in the hope that such mapping would indicate that the poten-tial offered by the HEA modelling toolset extends beyond that of the classical,spinson-based analogy.
As noted above, it is definitely too early to describe in detail how the specificaspects of HEAs physics can be used to describe social phenomena. The listpresented below offers suggestions of HEA characteristic features and their po-tential relevance in social modelling context. Note that there is no particularimportance order in the list. • HEAs, as their name suggests, derive much of their properties and com-plex thermodynamics from the high entropy, especially configurationalentropy. The presence of many different atomic species (elements) andmultiple ways they can be arranged locally is directly reminiscent of hu-man societies. This similarity promises that some tools and techniquesdeveloped for HEAs could be used to describe societal characteristics. • In contrast to the spinson models, where the dynamics of the wholesystem resulted from the combination of individual changes of a singleatom changeable characteristic (its spin), the many properties of the con-stituents of HEAs are “fixed” (in the sense of atomic properties) while oth-ers are flexible (e.g. spin, type of chemical bonds). The presence and roleof the fixed atomic properties suggests that the focus of research shouldshift from the individual to group dynamics. In fact, the interest in HEAsis in their global properties, for example mechanical ones (hardness, stress-strain properties, brittleness, ductility) or thermal or electrical properties(e.g. electric and thermal conduction). These describe the behaviour ofthe alloys under external influences, which would suggest similar appli-cation of the HEA based models in sociophyscis, focusing on the societalreaction to environment and external forces and influences, in particularthe resilience or lack of it. • One of the features of HEAs is the presence of so-called ‘cocktail effects’– unexpected synergies resulting from mixtures of different atoms. These7ould result in exceptional structural properties, for example increasedhardness, fatigue resistance, ductility etc. Such “cocktail effects” are ob-viously present in human societies, where various characteristics and rolesof individual people combine to create specific groups and contribute totheir characteristics. • HEAs exhibit rich processing dependence: instead of the simple influenceof temperature (characteristic for magnetic systems), the properties ofthe alloys heavily depend on the history of treatment, rate of temperaturechanges, mechanical treatment, irradiation etc., etc. Again, this showssignificant similarity to the complexity and history effects in human soci-eties. • Another, related similarity, results from the possibility that specific al-loys may be treated as systems away from thermodynamic equilibrium.For example, one often studies HEAs in “as-cast” conditions, where non-equilibrium or metastable phases are common. It leads to an interestingquestion whether sociophysics should treat societies as systems at equilib-rium (as most of the spinson models do) or as non-equilibrium ones. HEAmodelling toolset is ready to handle these issues. • Both HEAs and social systems exhibit mixtures of slow and fast phenom-ena, combining in complex ways. The physical examples may be providedby relatively slow atomic diffusion and fast effects of ionizing radiation.One could imagine corresponding social phenomena, for example globaleconomic trends and associated cultural changes and local effects of spe-cific events. The use of tools developed to understand HEAs, could be aninteresting opportunity in these contexts. • Mechanical studies of the behaviour under stress offer an interesting ex-ample of physical/social analogy. Application of slow, gradual changesin stress load may lead to abrupt and unpredictable changes in mate-rial structure: avalanches of dislocation movements and serrated (jerky)strain-stress curves. These processes often exhibit power-law behaviour.Their presence is similar to certain stress-relieving social phenomena, suchas appearance of social protests. Moreover, the power-law dependence isquite ubiquitous in our social life [Newman, 2005]. • In addition to the complexity brought by the presence of many differ-ent elements, HEAs behaviour is further complicated by the potentialco-existence of various atomic arrangements. A specific sample may crys-talline or amorphous phases, the former corresponding to different struc-tures (FCC, BCC, HCP. . . ). Depending on their preparation, the alloysmay exhibit non-random arrangements: presence of single element pre-cipitates, nanocrystals, condensates along grain boundaries, etc. Thisallows us to look for correspondences in social systems, differentiating or-ganized and dis-organized social networks as consequences of the societalhistory (“preparation”) and natural processes: assortative grouping and8ther locally non-random configurations and their effects on the overallcharacteristics of the society. • In certain circumstances, the multi-component alloys may exhibit spin-odal decomposition, leading to the appearance of local pockets composedof single elements – which could be compared to assortative matching andspontaneous segregation in social systems. In fact, as Pickering and Jones[2016] notes, the experimental evidence indicates that only very few HEAsare stable as solid solutions. Most of them, under certain conditions de-compose into multiple phases of differing chemical composition. This cor-responds to observation that majority of human societies exhibit divisioninto classes and groups, segregated by different individual characteristics. • A similar example of non-uniform distribution of atoms of different kindsis called the Contrell clouds – in which solute atoms gather around dis-locations. And as dislocations represent imperfections in the “organized”single crystal structure, this could provide an analogy to increased assor-tative gathering of certain minorities where the social structure is less rigidand controlled.As shown by these examples, the field of HEA research offers quite largenumber of potential “matchings” to social studies. Moreover, in the context ofmodelling, there are multiple ready tools which can be used in specific situations.In fact, there is a similar “grand challenge” facing HEA theoretical studies, asthe one faced by sociology. This is often referred to as the “Holy Grail of mul-tiscale modelling”: creating a continuous set of tools and methods, spanningthe range from first-principles, quantum mechanical descriptions at nanoscalelevel, through various approaches valid for the intermediate scales (hundreds,millions, billions of atoms) to the prediction of macroscopic behaviour of analloy under specific treatment and conditions. This corresponds to the chal-lenge of social systems description. We do not have a universal understandingspanning the scale starting from psychological characteristics of individual peo-ple and their histories and environments, through description of interpersonalinteractions ans small group behaviour to large scale societies.What is especially interesting are the transitions between tools used at differ-ent scales: quantum mechanical calculations, Monte Carlo simulations, Molecu-lar Dynamics, finite element methods. While a large number of highly developedtools and programs exists in each of these fields, construction of a consistentand usable “modelling factory”, with proper validation and feedback, is still achallenging task for materials sciences. This is mainly due to the difficulty ofchoice of the simplifications necessary to move from one “resolution scale” tothe next one. But finding such bridging solution might suggest a path to befollowed by social sciences which face similar problems related to complexityand disorder. 9
A simple example
To illustrate the potential of the HEA metaphor for sociophyscis we will presentbelow some results for a simple toy model of an alloy (technically not even aproper HEA, as it is composed from just 3 elements). The goal was to show howspecific properties of the alloy may depend on its preparation, and to suggestpossible social “interpretations” for these properties.
The system used in this work is an alloy of three metallic elements: Ag, Al andAu. This choice was motivated by several factors: atoms of the elements havesignificantly different masses, but their pure elemental crystal structures are thesame (FCC) and show very similar lattice constants (4.09˚A for silver, 4.05˚A foraluminum and 4.08˚A for gold). Moreover, despite the differences in masses, theatomic radii are also quite similar (1.447˚A for Ag, 1.432˚A for Al, 144.2˚A forAu). This has allowed to construct an alloy in which the competition betweendifferent “native” crystalline structures would ba absent, as well as the size andlattice distortion effects.The system has been constructed using the Atomsk package [Hirel, 2015]. Itconsisted of a periodically repeated box with 200˚A × × http://lammps.sandia.gov ), adopted to the use ofGraphics Processing Unit (GPU) [Brown et al., 2011, 2012, Brown and Masako,2013, Nguyen and Plimpton, 2015, Nguyen, 2017]. All subsequent simulationswere run using LAMMPS. Interatomic forces were calculated using the embed-ded atom method (EAM) alloy setfl potential files [Johnson, 1989, Zhou et al.,2004]The final atomic configurations were visualized and analysed (for example inthe context of finding the local crystalline structure) using the OVITO package[Stukowski, 2009, 2012]. The initial, artificially constructed configuration, glued together from unequi-librated chunks of pure metals, was allowed to thermodynamically equilibrateat 300K at zero pressure. This resulted in slight change of the box volume (in-creased by about 1%) and some diffusion of atoms across boundaries separatingdifferent elements. The resulting multicomponent polycrystal is visualized inFigure 1. 10igure 1: Three component polycrystalline box with approximately 200˚A × × The first test involves measuring the system response to external forces. Thetotal configuration consisted of creation of “nanowires” by periodic repetitionof the basic boxed along one of the orthogonal axes (X, Y or Z) and applyingstrain along this direction. The nanowires of (initially) square cross-sectionwere created by repeating the basic simulation cube along the strain directionand separating it from others by adding empty space in the two perpendiculardirections. The LAMMPS software allowed to simulate the system reaction tosuch strain at the level of atoms and to calculate the stress corresponding to agiven strain.Figure 4 shows the resulting stress-strain curves for the polycrystalline,14lassy and mixed crystal structures. The stress-strain approach may be con-sidered a macroscopic way of characterizing response to external forces. Forsmall values of strain (below 0.05) we observe a linear behaviour, typical forreversible, elastic regime. This could be interpreted in social terms as smalladjustments to external drivers, which do not change the social structure. Forstrain above 0.1 we observe the plastic, irreversible deformation regime, in whichmere reversal of strain does not return the sample to the original configuration.Here again an intuitive correspondence exists. For large enough external influ-ences faced by a society (economic crisis, pandemic, war, internal conflict butalso certain examples of technological change) the very structure of society maybecome changed. Specific institutions or social links may become broken and ei-ther be replaced by different ones, or vanish entirely. With increasing strain, thesample (and its social analogue) may “stretch”, exhibiting increasing internalchanges, until the point at which it would break.As expected, the differences observed for strain in different directions forthe configuration with quenched disorder were relatively small. Figures 5 and 6show atomic composition and crystalline structure cut-outs for strain in the Xand Z directions. While there are differences in specific atomic arrangements,the general behaviour is quite similar (and leads to similar strain-stress curves).The Molecular Dynamics model and results described above could providethe ground for an attempt of a “mapping” into the language of social phe-nomena. We shall present such mapping in a tabular form, stressing certainsimilarities. However, it should be borne in mind, that the “mapping” is ona very crude and qualitative level, corresponding to the stage of the spinsonmodels when spontaneous magnetic ordering below the Curie temperature wasequated to social consensus.In addition to the correspondences shown in Tables 2 and 3 one could alsothink of looking into a potential social interpretation of the microscopic con-cepts and phenomena occurring under stress: creation and mobility of disloca-tions, slip planes, twin boundaries, avalanches of local stress release and others.Other areas for model and interpretation development might cover the differ-ences brought by macroscopic types of stress: in addition to the tension (aspresented in out toy model), one could study the effects of compression, shearor torsion, leading to different types of nanoscale behaviours, and correspondingto different societal pressures and challenges.In a similar vein, not related to the simulations presented in this paper,one could exploit the analogy between the influence of minor components oftraditional alloys used to improve the characteristics of the material (such asrelatively small additions to iron used to create dedicated steels) and the roleof specific minorities in strengthening societies.
Another global characteristic of societies is their capacity to transmit “some-thing” between remote parts through interactions between people. This “some-thing” might be a material entity (various goods, or even carriers of infectious15igure 4: Stress-strain relationship for the three types of structures and strainapplied along nanowires in three orthogonal directions. The nanowires wereconstructed by repeating the basic box in a given direction (X, Y or Z) andseparating the boxes in the two other directions. Configurations starting fromhigh proportion of crystalline domains (polycrystal and mixed crystal) showhigher values of the yield stress point, indicating higher resistance in the elasticregime. They show, however, quite different behaviours in the plastic regime:there is much less directional dependence for both mixed crystal and glassyconfigurations, than for the original polycrystal. In the latter case, the presenceof continuous domains of Ag and Al along the Z axis results in a radically greaterductility in this direction. 16igure 5: Microscopic view of the effect of 1.6 strain along X axis (total X direc-tion length equal to 2.6 of the original) for the quenched disorder configuration.Top figure: overall view of atomic elements. Bottom view: slice of the sampleby Y=100˚A plane, showing internal crystal structure. Due to strain effects,part of the sample, previously in disordered state (grey), has transitioned tocrystal inclusions (green corresponds to FCC, red to HCP), composed of atomsof various types. 17igure 6: Microscopic view of the effect of 1.6 strain along Z axis for thequenched disorder configuration. Top figure: overall view of atomic elements.Bottom view: slice of the sample by Y=100˚A plane, showing internal crystalstructure. Due to strain effects, part of the sample, previously in disorderedstate, has transitioned to crystal inclusions (green corresponds to FCC, red toHCP), composed of atoms of various types.18igure 7: Microscopic view of the effect of 1.6 strain along X axis for the poly-crystalline configuration. Top figure: overall view of atomic elements. Bottomview: slice of the sample by Y=100˚A plane, showing internal crystal structure.At this strain in the X direction, the sample is close to the breaking point.19igure 8: Microscopic view of the effect of 1.6 strain along Y axis for the poly-crystalline configuration. Top figure: overall view of atomic elements. Bottomview: slice of the sample by X=100˚A plane, showing internal crystal structure.As in the case of the strain in X direction (Figure 7) the sample is close to thebreaking point. 20igure 9: Microscopic view of the effect of 1.6 strain along Z axis (the samevalue as for the X and Y strain directions shown in the previous figures) for thequenched disorder configuration. Top figure: overall view of atomic elements.Bottom view: slice of the sample by Y=100˚A plane, showing internal crystalstructure. Due to the presence of continuous domains of Ag and Al atoms alongthe Z direction of the sample, its resilience to tensile strain in this directionwas much greater than for the X or Y directions. The breaking point strain isalmost twice the value for the X and Y directions (Figure 4).21
D model concept Social correspondence
Concepts and characteristicsAtom chemical type and the associatedproperties driving interactions with otheratoms. Specific individual personality traits (e.g.Big Five categories: openness, consci-entiousness, extraversion, agreeableness,neuroticism, Furnham et al. [2009]);moral foundations (e.g. dichotomies be-tween care/harm, fairness/cheating, loy-alty/betrayal, authority/subversion andsanctity/degradation, [Haidt and Joseph,2007, Haidt, 2007, Graham et al., 2013]);behavioural patterns, education and cul-tural heritage – driving individual be-haviours and responses.Differences in interactions forces betweenpairs of atoms or larger groups, dependingon atomic type, distances, spatial orienta-tion etc. Differences in interactions between individ-uals of differing personality traits, moralfoundations, behavioural patterns, educa-tion and cultural heritage, further compli-cated by the nature of their relationships.Local and mid-range spatial order (crys-talline structure); partial deviation fromsuch order (defects, dislocations, grainboundaries, twinning planes . . . ); disor-dered structures (e.g. glassy state orquenched disorder). Presence of organized social structure, fix-ing “positions” and relationships betweenindividuals through institutions and pro-cesses. Examples of partial deviations fromfully institutionally organized society couldbe given by local disregard for certain lawsor processes, local absence or inactivity ofcertain social institutions or services. Thefully disordered state could correspond tosocieties in transition or anarchist societies,where little or no institutional social orderexists.Local order in chemical composition (forexample the existence of segregation ofatoms of different elements into separatevolumes), precipitation of certain atomsclose to crystal defects, nano-inclusions.On the opposite scale, the existence of afully mixed solutions and alloys. Existence of local communities of individu-als sharing most/some of their characteris-tics, either by design or due to assortativematching. Societies based on segregation.Creation of local minority enclaves. On theopposite scale: societies with high levels ofdiversity and no segregation.External stress and sample strain; existenceof elastic (reversible) deformation regimeand plastic regime; notion of maximumstrain (breakdown of the sample). Effects of external influences on society’sor societal fragment structure (externalpolitical pressures, technological changes,environmental changes. . . ). Small andreversible responses, preserving societalstructure and relative positions of individ-uals; irreversible changes in social struc-ture re-modelling it to new circumstances;breakdown of society as a unified whole.
Table 2: Mapping of concepts of the strain-stress Molecular Dynamics simula-tion onto the social systems 22
D model result Potential social interpretation
Simulation resultsLarge difference between the breakingpoint strain values observed for the poly-crystal (elemental segregation) and themixed crystal and glassy samples (chemi-cal disorder). Similarity of results for themixed crystal and glassy samples – low im-pact of crystalline order in plastic deforma-tion regime. Dominant role of diversity in response tovery large social stress and changes (whenthe local social structure changes irre-versibly).Higher Young modulus for configurationswith crystalline ordering (polycrystal andmixed crystal samples) than for the disor-dered case. Effect of local organisational and institu-tional ordering, enhancing the resilience ofthe society to external forces below thebreakdown of the institutional order (re-versible regime: when external forces dis-appear the society can revert to the statusquo ante .)Directional dependence in the plasticregime behaviour for the polycrystal sam-ple, large differences in breaking point fordifferent strain directions; small directionaldifferences for chemically disordered sam-ples. Highly segregated societies might exhibitwidely different reactions to external forces,depending on the nature of the stress (cor-responding in model case to physical direc-tions). The society might be more resilientto certain stresses thanks to the high co-hesion of the segregated groups – if thesegroups “span” the whole society. In othercases, the separation between different seg-regated groups might actually weaken theresilience of the society, causing breakdownmuch earlier. Fully diverse society wouldshow similar resilience to a broader rangeof forces.Appearance of local spatial ordering (crys-talline structures) in disordered (glassy)sample under high strain. May be interpreted as local self-organization under high levels of societalchange and challenges, even if these insti-tutions and ordering were absent in theoriginal, “relaxed” state.
Table 3: Mapping of results of the strain-stress Molecular Dynamics simulationonto the social systems 23iseases) or not (rumours, ideas, innovations, emotions). Once could thus lookfor physical equivalents of such transmission processes, for example electricalconduction or heat transport in material samples. The analogy is limited bythe fact that the range of the individual interactions between society membersis not necessarily local. In this aspect, the use of Molecular Dynamics to “map”social phenomena where long range or one-to-many communication is dominant(e.g. information spreading) might not be suitable. Quantum ab initio or Kohn-Sham calculations of electronic band structure could provide a better tool forsuch comprehensive calculations. On the other hand, we could look for exampleswhere the social transmission model is largely local in nature. In such cases theproposed approach might provide important contribution. Examples may beprovided by the already mentioned epidemic transmission in local communities,spread of rumours via word-of-mouth or panic effects in crowds with visibilitylimited to near neighbours.Our choice of the toy-model alloy, while providing such advantages as thesimilarity of lattice constants, atomic sizes and crystal structures of the com-posing elements, has one disadvantage with respect to thermal conductivity. Allcomponent metals have heat transmission which is dominated by the contribu-tion of electrons. At 300K only about 1% of the thermal conductivity κ is due tolattice vibrations [Jain and McGaughey, 2016, Wang et al., 2016]. In addition,it is quite a challenge to separate the two processes, as the electron-phonon scat-tering may also play a role for specific materials. As our model disregards theelectronic contribution, the absolute values of the thermal conductivity wouldbe incomparable with the measured values. Instead, we’d use, as a reference,the relative phonon (lattice) components, normalized to the value for gold. Jainand McGaughey [2016] provide values of κ p of 2.6 for Au, 5.2 for Ag and 5.8for Al, while Wang et al. [2016] estimates the values as 2 for Au, 4 for Ag and6 for Al. The ratios (normalized to the gold value) are then about 2 for silverand between 2.2 and 3 for aluminum.Before we move to the discussion of the lattice based thermal conductivity,let us note that the electronic contribution to the heat conduction may alsofind some analogy in social systems. Because the conduction band electrons inmetals are delocalized, their contribution might be compared to long range (oreven global) communication patterns (such as media). The varying relationsbetween lattice and electronic contributions for various materials could allowmodelling of various social circumstances.The adjustment of the lattice component of the heat conduction in advancedmaterials is an active research subject in physics. It is important for selectingmaterials with low conductivity, leading to higher resistance to radiation damage[Caro et al., 2015] or for semiconductor materials, such as Si-Ge compounds anddevices [Abs da Cruz et al., 2013, Hahn et al., 2014, Baker and Norris, 2015, Leeet al., 2016]. These studies have looked at the combination of effects of chemicalorder/disorder as well as positional ordering (crystalline, large and nano-grainsand amorphous phases, but also artificially constructed arrangements, such assuperlattices [Mizuno et al., 2015, Wang et al., 2015]). At the theoretical level,many of these works have used the same toolsets (e.g. LAMMPS codes) as our24 aterial Configuration Thermal con-ductivity κ p ,relative to goldAu Mixed crystal with all atomsreplaced by Au 1 (by definition)Ag Mixed crystal with all atomsreplaced by Ag 1.54Al Mixed crystal with all atomsreplaced by Al 2.18AgAuAl alloy Polycrystal, X direction 1.20AgAuAl alloy Polycrystal, Y direction 1.20AgAuAl alloy Polycrystal, Z direction 1.44AgAuAl alloy Mixed crystal 0.68AgAuAl alloy Glassy (quenched disorder) 0.64Table 4: Thermal conductivity of various configurations.current model.Our question in this part of the paper was quite similar to the actual : whichcharacteristic feature of the three configuration influences thermal conductionthe most? Is it the elemental separation (polycrystal) versus chemical disorder(mixed crystal and glassy state), or is it the presence of a local ordering in formof lattice structure (polycrystal and mixed crystal) vs. the quenched disorderconfiguration?In contrast to Caro et al. [2015], we have used the same interatomic potentialsfrom the EAM model, rather than the simplified Lennard-Jones ones. Thesimulations of the heat transfer were based on the Langevin model included inthe LAMMPS system.In addition to the three alloy configurations used in the strain-stress studies,we have also used the mixed crystal geometry to provide the baseline single-element heat conductivity. To do so, we have replaced in the mixed crystalconfiguration all atoms with single species: Au, Ag or Al. This has allowedus to calculate thermal conductivity of pure materials, for comparison with thepreviously reported results.In the case of pure elements we observe the expected order of the thermalconductivity values. The Al/Au and Ag/Au ratios are, however, smaller thanthose estimated by Jain and McGaughey [2016] and Wang et al. [2016]. One ofthe reasons might be that even if our mixed crystal sample is made chemicallyuniform, it still contains differently oriented grains of a size of a few tens toa hundred Angstrom. Such structure has been already indicated as a sourceof decrease of phonon mean free path by Hahn et al. [2014]. This geometricdisorder is the same for all pure element configurations, which would add ascattering mechanism to all of them in a similar way.As expected, the glassy and mixed crystal have much smaller heat conduc-25ivity than pure materials or the polycrystalline configuration. The reason isthat in the two configurations the dominant chemical disorder (and resultingdisorder of the masses of atoms) is present. The mass distribution disorder hasbeen noted as the primary source of phonon scattering in SiGe [Hahn et al.,2014, Baker and Norris, 2015, Lee et al., 2016] and in NiFe and NiCo alloys[Caro et al., 2015]. The κ p values for the glassy and mixed crystal systemswere independent on the direction of the heat flow. A small, but significantdecrease of κ p in the quenched disorder case comes from additional contributionof phonon scattering due to lack of crystalline order.The polycrystal configuration, where the atoms of various types are largelyseparated and the mean free path of phonons is longer, exhibits much highervalues of thermal conductivity. Moreover, the conductivity in the Z directionis greater than that in X and Y directions. As in the case of greater resilience,this can be explained by the asymmetry of the initial domains distribution, andthe presence of continuous domains of Ag and Al atoms in Z direction.Assignment of social meaning to the above results is less intuitive than in thecase of resistance to destructive forces. Still, if we take into account that peoplesharing the same characteristics (cultural heritage, political views, nationality,language) communicate more often and with greater ease and depth we couldattempt to create such mapping. Phonons may be compared to messages, heattransfer to information flow; all scattering mechanisms (mass disorder, differ-ences in forces between atoms, and positional disorder) decreasing the meanfree path of the phonon to limits of local person-to-person communication, themessage range; the ratios of various communication inhibitions (who speaks towhom and how often) would correspond to the hierarchy of the scattering mech-anisms. A similar reasoning could be applied to communicative diseases, wherea similarity of lifestyles and more frequent contacts among people sharing thesame characteristics could drive higher disease communication rates. Table 5presents suggestions for the mapping between physical and social phenomena inthis context. The current paper is focused on a simple idea of broadening the physical baseof sociophysics applications. Whether the idea will lead to successful, creativeapplications remains to be seen. There are certain advantages which it offers:large number of interesting phenomena present in High Entropy Alloys offerspotentially a wider range of social mappings. The fact that HEA research fieldis growing very fast provides a source of ideas and comparisons. Lastly, thereare several well developed modelling techniques and associated software systemswhich can be used immediately. In addition to Molecular Dynamics (exempli-fied by LAMMPS used in this work to provide an example), one could point outthat the similarity of challenges faced by social studies and the physics of com-plex condensed matter results provides interesting opportunities. The studyof human societies spans a range from individual behaviours (including their26
D model Social interpretation
Model conceptsVibrational (phonon-based) heat conduc-tivity Transmission of information vial local andsmall group interactions.Model resultsThe heat conductivity for pure elements ishigher than for alloys. Societies composed of agents sharing simi-lar personality traits, moral foundations orcultural heritage have higher rates of shar-ing and transmission and acceptance of in-formation, due to higher trust, confirma-tion bias and other psychological effects.Polycrystal conductivity significantlyhigher than for the chemically disor-dered cases (mixed crystal and quencheddisorder). The presence of large socially uniform do-mains (large scale segregation) enhancesthe transmission within these domains.The transmission is slowed-down only atthe boundaries between different socialgroups, due to mistrust, echo chamber andselective attention effects.Little difference between conductivity insample with crystalline order but chemicaldisorder and fully random glassy state. The presence of organized structures maybe less important then diversity of indi-vidual characteristics for local informationtransmission. (Institutional ordering mightbecome dominant for other forms of infor-mation transmission.)
Table 5: Mapping of concepts and results of the heat conductivity MolecularDynamics simulation onto the social systemsbiological and evolutionary roots, psychology, education); influences of imme-diate environments (family, workplace) which can be studied using small scaleexperimental psychology; up to sociology of large, diverse and complex socialgroups and whole societies. Not only the scope and type of relationships mayvary significantly, but also the timescales of changes may span the range from“immediate” reactions (such as decisions based on heuristics or stereotyping,often on the scale below 1 second), through scale of hours or days for cogni-tive and communication functions, to lifelong evolution of both individual andsocial profile of a person, ultimately to historical perspective of the changes insocieties. Thus, even within a single topic, one may be forced to combine amultidisciplinary portfolio of methods and tools, and to combine them into acoherent whole.The same situation exists in studies of HEAs (in general, in materials stud-ies). The description has to span the range from picoseconds and single atomsto years and “samples” (such as construction elements) measuring many me-ters. Just as it is not possible to “reduce” sociology to psychology effectively,it is practically impossible to use the fundamental, ab initio calculations forthe whole scope of materials science. In materials science we use a “ladder”of somewhat overlapping techniques, starting from quantum mechanical calcu-27ations on the fundamental level, through density functional theory, MolecularDynamics, kinetic Monte Carlo, continuum mechanics, constitutive equations,fracture mechanics to finite element analysis to name some of the available ele-ments of the tool-chain. Each of these methods introduces specific concepts andsimplifications, which allow it to be applied to larger structures and to look atphenomena at longer times.As in the psychology-sociology spectrum, the continuity of transitions be-tween different levels of description and methods mentioned above still requiresa lot of work. The choice of the simplifications leading from one “scale” to thenext one needs to be validated. In our opinion, the fact that both disciplinesundergo significant progress and face active challenges is not a weakness but astrength of the proposal presented in this work. Answers to the questions link-ing tools used for different system and time scales of physical systems – or eventhe acknowledgement that such links are needed – might provide interesting in-sights for sociophysical applications, and, perhaps, add value to the traditionalsocial studies as well.Lastly, the proposed expansion of the physical basis for social studies toHEAs would provide an additional, technical advantage. The use of well de-veloped, actively maintained open source programming, modelling and analysistools would enhance the capacity to cross-check the model results, using stan-dardized packages and input formats. To provide an example: the LAMMPSpackage used here has been recently used to study selected human behaviourrelated phenomena, for example urban interactions [Park and Schmidt, 2019] orcrowd movements using the social force model [Cornes et al., 2019, Sticco et al.,2020]. The key element of the idea presented in our paper is, however, notthe programming toolset, but the potential similarities between highly complexmaterial systems, such as HEAs and certain aspects of human societies.
References
C. Abs da Cruz, N. A. Katcho, N. Mingo, and R. G. Veiga. Thermal conductivityof nanocrystalline SiGe alloys using molecular dynamics simulations.
Journalof Applied Physics , 114(16):164310, 2013.S. Andreski.
Social Sciences as Sorcery . Andr´e Deutsch, 1972.C. H. Baker and P. M. Norris. Effect of long-and short-range order on SiGealloy thermal conductivity: molecular dynamics simulation.
Physical ReviewB , 91(18):180302, 2015.E. Ben-Naim, L. Frachebourg, and P. L. Krapivsky. Coarsening and persistencein the voter model.
Physical Review E , 53(4):3078–3087, 1996.A. T. Bernardes, U. M. S. Costa, A. D. Araujo, and D. Stauffer. Damagespreading, coarsening dynamics and distribution of political votes in Sznajdmodel on square lattice.
International Journal of Modern Physics C , 12(2):159–168, 2001. 28. Biswas. Mean-field solutions of kinetic-exchange opinion models.
PhysicalReview E , 84(5):056106, 2011.L. Boudin and F. Salvarani. Opinion dynamics: kinetic modelling with massmedia, application to the scottish independence referendum.
Physica A: Sta-tistical Mechanics and its Applications , 444:448–457, 2016.W. M. Brown and Y. Masako. Implementing molecular dynamics on hybridhigh performance computers – three-body potentials.
Comp. Phys. Comm. ,184:2785–2793, 2013.W. M. Brown, P. Wang, S. J. Plimpton, and A. N. Tharrington. Implementingmolecular dynamics on hybrid high performance computers – short rangeforces.
Comp. Phys. Comm. , 182:898–911, 2011.W. M. Brown, A. Kohlmeyer, S. J. Plimpton, and A. N. Tharrington. Im-plementing molecular dynamics on hybrid high performance computers –particle-particle particle-mesh.
Comp. Phys. Comm. , 183:449–459, 2012.M. Caro, L. K. B´eland, G. D. Samolyuk, R. E. Stoller, and A. Caro. Lat-tice thermal conductivity of multi-component alloys.
Journal of Alloys andCompounds , 648:408–413, 2015.C. Castellano, D. Vilone, and A. Vespignani. Incomplete ordering of the votermodel on small-world networks.
EPL (Europhysics Letters) , 63:153, 2003.A. K. Chandra. Percolation in a kinetic opinion exchange model.
PhysicalReview E , 85(2):021149, 2012.F. Cornes, G. Frank, and C. Dorso. Fear propagation and the evacuation dy-namics.
Simulation Modelling Practice and Theory , 95:112–133, 2019.J. Cox and D. Griffeath. Diffusive clustering in the two dimensional voter model.
The Annals of Probability , 14(2):347–370, 1986.G. Deffuant, D. Neau, F. Amblard, and G. Weisbuch. Mixing beliefs amonginteracting agents.
Advances in Complex Systems , 3:87–98, 2000.G. Deffuant, F. Amblard, G. Weisbuch, and T. Faure. How can extremismprevail? A study based on the relative agreement interaction model.
Journalof Artificial Societies and Social Simulation , 5(4), 2002.A. Furnham, A. Eracleous, and T. Chamorro-Premuzic. Personality, motivationand job satisfaction: Hertzberg meets the big five.
Journal of managerialpsychology , 24(8):765–779, 2009.S. Galam, B. Chopard, and M. Droz. Killer geometries in competing speciesdynamics.
Physica A: Statistical Mechanics and its Applications , 314(1):256–263, 2002. 29. C. Gao.
High-Entropy Alloys Fundamentals and Applications . SpringerInternational, 2016. doi: 10.1007/978-3-319-27013-5.E. George, W. Curtin, and C. Tasan. High entropy alloys: A focused reviewof mechanical properties and deformation mechanisms.
Acta Materialia , 188:435–474, 2020.E. P. George, D. Raabe, and R. O. Ritchie. High-entropy alloys.
Nature ReviewsMaterials , 4(8):515–534, 2019.J. Graham, J. Haidt, S. Koleva, M. Motyl, R. Iyer, S. Wojcik, and P. H. Ditto.Moral foundations theory: The pragmatic validity of moral pluralism.
Ad-vances in experimental social psychology , 47:55–130, 2013.K. Hahn, C. Melis, and L. Colombo. Effect of structural features on the thermalconductivity of SiGe-based materials.
The European Physical Journal B , 87:150, 2014.J. Haidt. The new synthesis in moral psychology. science , 316(5827):998–1002,2007.J. Haidt and C. Joseph. The moral mind: How five sets of innate intuitionsguide the development of many culture-specific virtues, and perhaps evenmodules. In P. Carruthers, S. Laurence, and S. Stich, editors,
The innatemind , volume 3, pages 367–391. Oxford University Press New York, NY,2007.R. Hegselmann and U. Krause. Opinion dynamics and bounded confidencemodels, analysis, and simulation.
Journal of Artificial Societies and SocialSimulation (JASSS) vol , 5(3), 2002.P. Hirel. Atomsk: a tool for manipulating and converting atomic data files.
Computer Physics Communications , 197:212–219, 2015.J. Holyst, K. Kacperski, and F. Schweitzer. Social impact models of opiniondynamics.
Annual Review of Comput. Phys. , IX:253–273, 2001. doi: 10.1142/9789812811578.A. Jain and A. J. McGaughey. Thermal transport by phonons and electronsin aluminum, silver, and gold from first principles.
Physical Review B , 93(8):081206, 2016.R. Johnson. Alloy models with the embedded-atom method.
Physical ReviewB , 39(17):12554, 1989.K. Kacperski and J. Holyst. Opinion formation model with strong leader andexternal impact: a mean field approach.
Physica A , 269:511–526, 1999.K. Kacperski and J. Holyst. Phase transitions as a persistent feature of groupswith leaders in models of opinion formation.
Physica A , 287:631–643, 2000.30. Lallouache, A. Chakrabarti, A. Chakraborti, and B. Chakrabarti. Opin-ion formation in kinetic exchange models: Spontaneous symmetry-breakingtransition.
Physical Review E , 82(5):056112, 2010.Y. Lee, A. J. Pak, and G. S. Hwang. What is the thermal conductivity limitof silicon germanium alloys?
Physical Chemistry Chemical Physics , 18(29):19544–19548, 2016.D. B. Miracle and O. N. Senkov. A critical review of high entropy alloys andrelated concepts.
Acta Materialia , 122:448–511, 2017.H. Mizuno, S. Mossa, and J.-L. Barrat. Beating the amorphous limit in thermalconductivity by superlattices design.
Scientific reports , 5:14116, 2015.M. E. J. Newman. Power laws, Pareto distributions and Zipf’s law.
Contempo-rary Physics , 46:323–351, 2005.T. D. Nguyen. Gpu-accelerated Tersoff potentials for massively parallel molec-ular dynamics simulations.
Comp. Phys. Comm. , 212:113–122, 2017.T. D. Nguyen and S. J. Plimpton. Accelerating dissipative particle dynamicssimulations for soft matter systems.
Comput. Mater. Sci. , 100:173–180, 2015.A. Nowak and M. Lewenstein. Modeling social change with cellular automata.In R. Hegselmann, U. Mueller, and K. G. Troitzsch, editors,
Modelling andSimulation in the Social Sciences From A Philosophy of Science Point ofView , pages 249–285. Kluver, Dordrecht, 1996. URLA. Nowak, J. Szamrej, and B. Latan´e. From private attitude to public opinion:A dynamic theory of social impact.
Psychological Review , 97(3):362–376,1990.P. Nyczka and K. Sznajd-Weron. Anticonformity or independence? – insightsfrom statistical physics.
Journal of Statistical Physics , 151:174–202, 2013.S. Y. Park and A. Schmidt. Nonequilibrium dynamics of multiple urban inter-actions coupled to a thermal environment. In
American Geophysical Union,Fall Meeting 2019 , volume 2019, pages H13J–1829, 2019.E. Pickering and N. Jones. High-entropy alloys: a critical assessment of theirfounding principles and future prospects.
International Materials Reviews , 61(3):183–202, 2016.L. Sabatelli and P. Richmond. Phase transitions, memory and frustration ina Sznajd-like model with synchronous updating.
International Journal ofModern Physics C , 14:1223–1229, 2003.L. Sabatelli and P. Richmond. Non-monotonic spontaneous magnetization in aSznajd-like consensus model.
Physica A: Statistical Mechanics and its Appli-cations , 334(1):274–280, 2004. 31. Slanina and H. Lavicka. Analytical results for the Sznajd model of opinionformation.
European Physical Journal B - Condensed Matter , 35(2):279 –288, 2003.P. Sobkowicz. Modelling opinion formation with physics tools: call for closerlink with reality.
Journal of Artificial Societies and Social Simulation , 12(1):11, 2009.D. Stauffer. Monte Carlo simulations of Sznajd models.
Journal of ArtificialSocieties and Social Simulation , 5(1), 2001.D. Stauffer. Sociophysics: the Sznajd model and its applications.
Computerphysics communications , 146(1):93–98, 2002.D. Stauffer and P. M. C. de Oliveira. Persistence of opinion in the Sznajdconsensus model: computer simulation.
The European Physical Journal B-Condensed Matter , 30(4):587–592, 2002.I. Sticco, G. Frank, F. Cornes, and C. Dorso. A re-examination of the role offriction in the original social force model.
Safety science , 121:42–53, 2020.A. Stukowski. Visualization and analysis of atomistic simulation data withovito–the open visualization tool.
Modelling and Simulation in MaterialsScience and Engineering , 18(1):015012, 2009.A. Stukowski. Structure identification methods for atomistic simulations ofcrystalline materials.
Modelling and Simulation in Materials Science andEngineering , 20(4):045021, 2012.K. Sznajd-Weron and J. Sznajd. Opinion evolution in closed community.
Int.J. Mod. Phys. C , 11:1157–1166, 2000.G. Toscani. Kinetic models of opinion formation.
Commun. Math. Sci , 4(3):481–496, 2006.Y. Wang, C. Gu, and X. Ruan. Optimization of the random multilayer structureto break the random-alloy limit of thermal conductivity.
Applied PhysicsLetters , 106(7):073104, 2015.Y. Wang, Z. Lu, and X. Ruan. First principles calculation of lattice thermalconductivity of metals considering phonon-phonon and phonon-electron scat-tering.
Journal of applied Physics , 119(22):225109, 2016.G. Weisbuch. Bounded confidence and social networks.
The European PhysicalJournal B-Condensed Matter and Complex Systems , 38(2):339–343, 2004.G. Weisbuch, G. Deffuant, F. Amblard, and J.-P. Nadal. Interacting agentsand continuous opinions dynamics. In R. Cowan and N. Jonard, editors,
Heterogenous Agents, Interactions and Economic Performance , volume 521of
Lecture Notes in Economics and Mathematical Systems , pages 225–242.Springer Berlin Heidelberg, 2003. ISBN 978-3-642-55651-7.32.-X. Yang and L. Huang. Opinion percolation in structured population.
Com-puter Physics Communications , 2015.X. W. Zhou, R. A. Johnson, and H. N. G. Wadley. Misfit-energy-increasingdislocations in vapor-deposited CoFe/NiFe multilayers.