Anna Klünker, Kathrin Padberg-Gehle, Jean-Luc Thiffeault
We study finite-time mixing in time-periodic open flow systems. We describe the transport of densities in terms of a transfer operator, which is represented by the transition matrix of a finite-state Markov chain. The transport processes in the open system are organized by the chaotic saddle and its stable and unstable manifolds. We extract these structures directly from leading eigenvectors of the transition matrix. We use different measures to quantify the degree of mixing and show that they give consistent results in parameter studies of two model systems. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.
{"title":"Open-flow mixing and transfer operators","authors":"Anna Klünker, Kathrin Padberg-Gehle, Jean-Luc Thiffeault","doi":"10.1098/rsta.2021.0028","DOIUrl":"https://doi.org/10.1098/rsta.2021.0028","url":null,"abstract":"We study finite-time mixing in time-periodic open flow systems. We describe the transport of densities in terms of a transfer operator, which is represented by the transition matrix of a finite-state Markov chain. The transport processes in the open system are organized by the chaotic saddle and its stable and unstable manifolds. We extract these structures directly from leading eigenvectors of the transition matrix. We use different measures to quantify the degree of mixing and show that they give consistent results in parameter studies of two model systems. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79669895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study here the dynamics of opinion formation in a society where we take into account the internally held beliefs and externally expressed opinions of the individuals, which are not necessarily the same at all times. While these two components can influence one another, their difference, both in dynamics and in the steady state, poses interesting scenarios in terms of the transition to consensus in the society and characterizations of such consensus. Here we study this public and private opinion dynamics and the critical behaviour of the consensus forming transitions, using a kinetic exchange model. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.
{"title":"Opinion dynamics: public and private","authors":"Subhadeep Roy, Soumyajyoti Biswas","doi":"10.1098/rsta.2021.0169","DOIUrl":"https://doi.org/10.1098/rsta.2021.0169","url":null,"abstract":"We study here the dynamics of opinion formation in a society where we take into account the internally held beliefs and externally expressed opinions of the individuals, which are not necessarily the same at all times. While these two components can influence one another, their difference, both in dynamics and in the steady state, poses interesting scenarios in terms of the transition to consensus in the society and characterizations of such consensus. Here we study this public and private opinion dynamics and the critical behaviour of the consensus forming transitions, using a kinetic exchange model. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83516500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The discovery of accelerated expansion of the Universe opened up the possibility of new scenarios for the doom of our space–time, besides eternal expansion and a final contraction. In this paper, we review the chances that may await our universe. In particular, there are new possible singular fates (sudden singularities, big rip, etc.), but there also other evolutions that cannot be considered as singular. In addition to this, some of the singular fates are not strong enough in the sense that the space–time can be extended beyond the singularity. For deriving our results, we make use of generalized power and asymptotic expansions of the scale factor of the Universe. This article is part of the theme issue ‘The future of mathematical cosmology, Volume 1’.
{"title":"New futures for cosmological models","authors":"L. Fernández-Jambrina, R. Lazkoz","doi":"10.1098/rsta.2021.0333","DOIUrl":"https://doi.org/10.1098/rsta.2021.0333","url":null,"abstract":"The discovery of accelerated expansion of the Universe opened up the possibility of new scenarios for the doom of our space–time, besides eternal expansion and a final contraction. In this paper, we review the chances that may await our universe. In particular, there are new possible singular fates (sudden singularities, big rip, etc.), but there also other evolutions that cannot be considered as singular. In addition to this, some of the singular fates are not strong enough in the sense that the space–time can be extended beyond the singularity. For deriving our results, we make use of generalized power and asymptotic expansions of the scale factor of the Universe. This article is part of the theme issue ‘The future of mathematical cosmology, Volume 1’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77686790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A defining feature of three-dimensional hydrodynamic turbulence is that the rate of energy dissipation is bounded away from zero as viscosity is decreased (Reynolds number increased). This phenomenon—anomalous dissipation—is sometimes called the ‘zeroth law of turbulence’ as it underpins many celebrated theoretical predictions. Another robust feature observed in turbulence is that velocity structure functions Sp(ℓ):=⟨|δℓu|p⟩ exhibit persistent power-law scaling in the inertial range, namely Sp(ℓ)∼|ℓ|ζp for exponents ζp>0 over an ever increasing (with Reynolds) range of scales. This behaviour indicates that the velocity field retains some fractional differentiability uniformly in the Reynolds number. The Kolmogorov 1941 theory of turbulence predicts that ζp=p/3 for all p and Onsager’s 1949 theory establishes the requirement that ζp≤p/3 for p≥ 3 for consistency with the zeroth law. Empirically, ζ2⪆2/3 and ζ3⪅1, suggesting that turbulent Navier–Stokes solutions approximate dissipative weak solutions of the Euler equations possessing (nearly) the minimal degree of singularity required to sustain anomalous dissipation. In this note, we adopt an experimentally supported hypothesis on the anti-alignment of velocity increments with their separation vectors and demonstrate that the inertial dissipation provides a regularization mechanism via the Kolmogorov 4/5-law. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.
{"title":"Self-regularization in turbulence from the Kolmogorov 4/5-law and alignment","authors":"T. Drivas","doi":"10.1098/rsta.2021.0033","DOIUrl":"https://doi.org/10.1098/rsta.2021.0033","url":null,"abstract":"A defining feature of three-dimensional hydrodynamic turbulence is that the rate of energy dissipation is bounded away from zero as viscosity is decreased (Reynolds number increased). This phenomenon—anomalous dissipation—is sometimes called the ‘zeroth law of turbulence’ as it underpins many celebrated theoretical predictions. Another robust feature observed in turbulence is that velocity structure functions Sp(ℓ):=⟨|δℓu|p⟩ exhibit persistent power-law scaling in the inertial range, namely Sp(ℓ)∼|ℓ|ζp for exponents ζp>0 over an ever increasing (with Reynolds) range of scales. This behaviour indicates that the velocity field retains some fractional differentiability uniformly in the Reynolds number. The Kolmogorov 1941 theory of turbulence predicts that ζp=p/3 for all p and Onsager’s 1949 theory establishes the requirement that ζp≤p/3 for p≥ 3 for consistency with the zeroth law. Empirically, ζ2⪆2/3 and ζ3⪅1, suggesting that turbulent Navier–Stokes solutions approximate dissipative weak solutions of the Euler equations possessing (nearly) the minimal degree of singularity required to sustain anomalous dissipation. In this note, we adopt an experimentally supported hypothesis on the anti-alignment of velocity increments with their separation vectors and demonstrate that the inertial dissipation provides a regularization mechanism via the Kolmogorov 4/5-law. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90595058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by the search for sharp bounds on turbulent heat transfer as well as the design of optimal heat exchangers, we consider incompressible flows that most efficiently cool an internally heated disc. Heat enters via a distributed source, is passively advected and diffused, and exits through the boundary at a fixed temperature. We seek an advecting flow to optimize this exchange. Previous work on energy-constrained cooling with a constant source has conjectured that global optimizers should resemble convection rolls; we prove one-sided bounds on energy-constrained cooling corresponding to, but not resolving, this conjecture. In the case of an enstrophy constraint, our results are more complete: we construct a family of self-similar, tree-like ‘branching flows’ whose cooling we prove is within a logarithm of globally optimal. These results hold for general space- and time-dependent source–sink distributions that add more heat than they remove. Our main technical tool is a non-local Dirichlet-like variational principle for bounding solutions of the inhomogeneous advection–diffusion equation with a divergence-free velocity. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.
{"title":"Optimal cooling of an internally heated disc","authors":"Ian Tobasco","doi":"10.1098/rsta.2021.0040","DOIUrl":"https://doi.org/10.1098/rsta.2021.0040","url":null,"abstract":"Motivated by the search for sharp bounds on turbulent heat transfer as well as the design of optimal heat exchangers, we consider incompressible flows that most efficiently cool an internally heated disc. Heat enters via a distributed source, is passively advected and diffused, and exits through the boundary at a fixed temperature. We seek an advecting flow to optimize this exchange. Previous work on energy-constrained cooling with a constant source has conjectured that global optimizers should resemble convection rolls; we prove one-sided bounds on energy-constrained cooling corresponding to, but not resolving, this conjecture. In the case of an enstrophy constraint, our results are more complete: we construct a family of self-similar, tree-like ‘branching flows’ whose cooling we prove is within a logarithm of globally optimal. These results hold for general space- and time-dependent source–sink distributions that add more heat than they remove. Our main technical tool is a non-local Dirichlet-like variational principle for bounding solutions of the inhomogeneous advection–diffusion equation with a divergence-free velocity. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87705022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Saeedian, E. Pigani, A. Maritan, S. Suweis, S. Azaele
Understanding the conditions of feasibility and stability in ecological systems is a major challenge in theoretical ecology. The seminal work of May in 1972 and recent developments based on the theory of random matrices have shown the existence of emergent universal patterns of both stability and feasibility in ecological dynamics. However, only a few studies have investigated the role of delay coupled with population dynamics in the emergence of feasible and stable states. In this work, we study the effects of delay on generalized Loka–Volterra population dynamics of several interacting species in closed ecological environments. First, we investigate the relation between feasibility and stability of the modelled ecological community in the absence of delay and find a simple analytical relation when intra-species interactions are dominant. We then show how, by increasing the time delay, there is a transition in the stability phases of the population dynamics: from an equilibrium state to a stable non-point attractor phase. We calculate analytically the critical delay of that transition and show that it is in excellent agreement with numerical simulations. Finally, following a similar approach to characterizing stability in empirical studies, we investigate the coefficient of variation, which quantifies the magnitude of population fluctuations. We show that in the oscillatory regime induced by the delay, the variability at community level decreases for increasing diversity. This article is part of the theme issue ‘Emergent phenomena in complex physical and socio-technical systems: from cells to societies’.
{"title":"Effect of delay on the emergent stability patterns in generalized Lotka–Volterra ecological dynamics","authors":"M. Saeedian, E. Pigani, A. Maritan, S. Suweis, S. Azaele","doi":"10.1098/rsta.2021.0245","DOIUrl":"https://doi.org/10.1098/rsta.2021.0245","url":null,"abstract":"Understanding the conditions of feasibility and stability in ecological systems is a major challenge in theoretical ecology. The seminal work of May in 1972 and recent developments based on the theory of random matrices have shown the existence of emergent universal patterns of both stability and feasibility in ecological dynamics. However, only a few studies have investigated the role of delay coupled with population dynamics in the emergence of feasible and stable states. In this work, we study the effects of delay on generalized Loka–Volterra population dynamics of several interacting species in closed ecological environments. First, we investigate the relation between feasibility and stability of the modelled ecological community in the absence of delay and find a simple analytical relation when intra-species interactions are dominant. We then show how, by increasing the time delay, there is a transition in the stability phases of the population dynamics: from an equilibrium state to a stable non-point attractor phase. We calculate analytically the critical delay of that transition and show that it is in excellent agreement with numerical simulations. Finally, following a similar approach to characterizing stability in empirical studies, we investigate the coefficient of variation, which quantifies the magnitude of population fluctuations. We show that in the oscillatory regime induced by the delay, the variability at community level decreases for increasing diversity. This article is part of the theme issue ‘Emergent phenomena in complex physical and socio-technical systems: from cells to societies’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90717695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The first part of this paper is a brief survey of the approaches to economic inequality based on ideas from statistical physics and kinetic theory. These include the Boltzmann kinetic equation, the time-reversal symmetry, the ergodicity hypothesis, entropy maximization and the Fokker–Planck equation. The origins of the exponential Boltzmann–Gibbs distribution and the Pareto power law are discussed in relation to additive and multiplicative stochastic processes. The second part of the paper analyses income distribution data in the USA for the time period 1983–2018 using a two-class decomposition. We present overwhelming evidence that the lower class (more than 90% of the population) is described by the exponential distribution, whereas the upper class (about 4% of the population in 2018) by the power law. We show that the significant growth of inequality during this time period is due to the sharp increase in the upper-class income share, whereas relative inequality within the lower class remains constant. We speculate that the expansion of the upper-class population and income shares may be due to increasing digitization and non-locality of the economy in the last 40 years. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.
{"title":"Physics-inspired analysis of the two-class income distribution in the USA in 1983–2018","authors":"D. Ludwig, V. Yakovenko","doi":"10.1098/rsta.2021.0162","DOIUrl":"https://doi.org/10.1098/rsta.2021.0162","url":null,"abstract":"The first part of this paper is a brief survey of the approaches to economic inequality based on ideas from statistical physics and kinetic theory. These include the Boltzmann kinetic equation, the time-reversal symmetry, the ergodicity hypothesis, entropy maximization and the Fokker–Planck equation. The origins of the exponential Boltzmann–Gibbs distribution and the Pareto power law are discussed in relation to additive and multiplicative stochastic processes. The second part of the paper analyses income distribution data in the USA for the time period 1983–2018 using a two-class decomposition. We present overwhelming evidence that the lower class (more than 90% of the population) is described by the exponential distribution, whereas the upper class (about 4% of the population in 2018) by the power law. We show that the significant growth of inequality during this time period is due to the sharp increase in the upper-class income share, whereas relative inequality within the lower class remains constant. We speculate that the expansion of the upper-class population and income shares may be due to increasing digitization and non-locality of the economy in the last 40 years. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87393211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gianluca Crippa, T. Elgindi, Gautam Iyer, A. Mazzucato
We consider transport of a passive scalar advected by an irregular divergence-free vector field. Given any non-constant initial data ρ¯∈Hloc1(Rd), d≥2, we construct a divergence-free advecting velocity field v (depending on ρ¯) for which the unique weak solution to the transport equation does not belong to Hloc1(Rd) for any positive time. The velocity field v is smooth, except at one point, controlled uniformly in time, and belongs to almost every Sobolev space Ws,p that does not embed into the Lipschitz class. The velocity field v is constructed by pulling back and rescaling a sequence of sine/cosine shear flows on the torus that depends on the initial data. This loss of regularity result complements that in Ann. PDE, 5(1):Paper No. 9, 19, 2019. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.
{"title":"Growth of Sobolev norms and loss of regularity in transport equations","authors":"Gianluca Crippa, T. Elgindi, Gautam Iyer, A. Mazzucato","doi":"10.1098/rsta.2021.0024","DOIUrl":"https://doi.org/10.1098/rsta.2021.0024","url":null,"abstract":"We consider transport of a passive scalar advected by an irregular divergence-free vector field. Given any non-constant initial data ρ¯∈Hloc1(Rd), d≥2, we construct a divergence-free advecting velocity field v (depending on ρ¯) for which the unique weak solution to the transport equation does not belong to Hloc1(Rd) for any positive time. The velocity field v is smooth, except at one point, controlled uniformly in time, and belongs to almost every Sobolev space Ws,p that does not embed into the Lipschitz class. The velocity field v is constructed by pulling back and rescaling a sequence of sine/cosine shear flows on the torus that depends on the initial data. This loss of regularity result complements that in Ann. PDE, 5(1):Paper No. 9, 19, 2019. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78191517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The rise of social networks as the primary means of communication in almost every country in the world has simultaneously triggered an increase in the amount of fake news circulating online. The urgent need for models that can describe the growing infodemic of fake news has been highlighted by the current pandemic. The resulting slowdown in vaccination campaigns due to misinformation and generally the inability of individuals to discern the reliability of information is posing enormous risks to the governments of many countries. In this research using the tools of kinetic theory, we describe the interaction between fake news spreading and competence of individuals through multi-population models in which fake news spreads analogously to an infectious disease with different impact depending on the level of competence of individuals. The level of competence, in particular, is subject to evolutionary dynamics due to both social interactions between agents and external learning dynamics. The results show how the model is able to correctly describe the dynamics of diffusion of fake news and the important role of competence in their containment. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.
{"title":"Spreading of fake news, competence and learning: kinetic modelling and numerical approximation","authors":"Jonathan Franceschi, L. Pareschi","doi":"10.1098/rsta.2021.0159","DOIUrl":"https://doi.org/10.1098/rsta.2021.0159","url":null,"abstract":"The rise of social networks as the primary means of communication in almost every country in the world has simultaneously triggered an increase in the amount of fake news circulating online. The urgent need for models that can describe the growing infodemic of fake news has been highlighted by the current pandemic. The resulting slowdown in vaccination campaigns due to misinformation and generally the inability of individuals to discern the reliability of information is posing enormous risks to the governments of many countries. In this research using the tools of kinetic theory, we describe the interaction between fake news spreading and competence of individuals through multi-population models in which fake news spreads analogously to an infectious disease with different impact depending on the level of competence of individuals. The level of competence, in particular, is subject to evolutionary dynamics due to both social interactions between agents and external learning dynamics. The results show how the model is able to correctly describe the dynamics of diffusion of fake news and the important role of competence in their containment. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90198503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bertram Düring, Michael Fischer, Marie-Therese Wolfram
The Elo rating system, which was originally proposed by Arpad Elo for chess, has become one of the most important rating systems in sports, economics and gaming. Its original formulation is based on two-player zero-sum games, but it has been adapted for team sports and other settings. In 2015, Junca and Jabin proposed a kinetic version of the Elo model, and showed that under certain assumptions the ratings do converge towards the players’ strength. In this paper, we generalize their model to account for variable performance of individual players or teams. We discuss the underlying modelling assumptions, derive the respective formal mean-field model and illustrate the dynamics with computational results. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.
{"title":"An Elo-type rating model for players and teams of variable strength","authors":"Bertram Düring, Michael Fischer, Marie-Therese Wolfram","doi":"10.1098/rsta.2021.0155","DOIUrl":"https://doi.org/10.1098/rsta.2021.0155","url":null,"abstract":"The Elo rating system, which was originally proposed by Arpad Elo for chess, has become one of the most important rating systems in sports, economics and gaming. Its original formulation is based on two-player zero-sum games, but it has been adapted for team sports and other settings. In 2015, Junca and Jabin proposed a kinetic version of the Elo model, and showed that under certain assumptions the ratings do converge towards the players’ strength. In this paper, we generalize their model to account for variable performance of individual players or teams. We discuss the underlying modelling assumptions, derive the respective formal mean-field model and illustrate the dynamics with computational results. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85153281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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