E. Parente, M. Farano, J. Robinet, P. De Palma, S. Cherubini
A new mathematical framework is proposed for characterizing the coherent motion of fluctuations around a mean turbulent channel flow. We search for statistically invariant coherent solutions of the unsteady Reynolds-averaged Navier–Stokes equations written in a perturbative form with respect to the turbulent mean flow, using a suitable approximation of the Reynolds stress tensor. This is achieved by setting up a continuation procedure of known solutions of the perturbative Navier–Stokes equations, based on the continuous increase of the turbulent eddy viscosity towards its turbulent value. The recovered solutions, being sustained only in the presence of the Reynolds stress tensor, are representative of the statistically coherent motion of turbulent flows. For small friction Reynolds number and/or domain size, the statistically invariant motion is almost identical to the corresponding invariant solution of the Navier–Stokes equations. Whereas, for sufficiently large friction number and/or domain size, it considerably departs from the starting invariant solution of the Navier–Stokes equations, presenting spatial structures, main wavelengths and scaling very close to those characterizing both large- and small-scale motion of turbulent channel flows. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.
{"title":"Continuing invariant solutions towards the turbulent flow","authors":"E. Parente, M. Farano, J. Robinet, P. De Palma, S. Cherubini","doi":"10.1098/rsta.2021.0031","DOIUrl":"https://doi.org/10.1098/rsta.2021.0031","url":null,"abstract":"A new mathematical framework is proposed for characterizing the coherent motion of fluctuations around a mean turbulent channel flow. We search for statistically invariant coherent solutions of the unsteady Reynolds-averaged Navier–Stokes equations written in a perturbative form with respect to the turbulent mean flow, using a suitable approximation of the Reynolds stress tensor. This is achieved by setting up a continuation procedure of known solutions of the perturbative Navier–Stokes equations, based on the continuous increase of the turbulent eddy viscosity towards its turbulent value. The recovered solutions, being sustained only in the presence of the Reynolds stress tensor, are representative of the statistically coherent motion of turbulent flows. For small friction Reynolds number and/or domain size, the statistically invariant motion is almost identical to the corresponding invariant solution of the Navier–Stokes equations. Whereas, for sufficiently large friction number and/or domain size, it considerably departs from the starting invariant solution of the Navier–Stokes equations, presenting spatial structures, main wavelengths and scaling very close to those characterizing both large- and small-scale motion of turbulent channel flows. 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":"2022-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80845163","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 quasi-geostrophic (QG) equations play a crucial role in our understanding of atmospheric and oceanic fluid dynamics. Nevertheless, the traditional QG equations describe ‘dry’ dynamics that do not account for moisture and clouds. To move beyond the dry setting, precipitating QG (PQG) equations have been derived recently using formal asymptotics. Here, we investigate whether the moist Boussinesq equations with phase changes will converge to the PQG equations. A priori, it is possible that the nonlinearity at the phase interface (cloud edge) may complicate convergence. A numerical investigation of convergence or non-convergence is presented here. The numerical simulations consider cases of ϵ=0.1, 0.01 and 0.001, where ϵ is proportional to the Rossby and Froude numbers. In the numerical simulations, the magnitude of vertical velocity w (or other measures of imbalance and inertio-gravity waves) is seen to be approximately proportional to ϵ as ϵ decreases, which suggests convergence to PQG dynamics. These measures are quantified at a fixed time T that is O(1), and the numerical data also suggests the possibility of convergence at later times. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.
{"title":"Convergence to precipitating quasi-geostrophic equations with phase changes: asymptotics and numerical assessment","authors":"Yeyu Zhang, L. Smith, S. Stechmann","doi":"10.1098/rsta.2021.0030","DOIUrl":"https://doi.org/10.1098/rsta.2021.0030","url":null,"abstract":"The quasi-geostrophic (QG) equations play a crucial role in our understanding of atmospheric and oceanic fluid dynamics. Nevertheless, the traditional QG equations describe ‘dry’ dynamics that do not account for moisture and clouds. To move beyond the dry setting, precipitating QG (PQG) equations have been derived recently using formal asymptotics. Here, we investigate whether the moist Boussinesq equations with phase changes will converge to the PQG equations. A priori, it is possible that the nonlinearity at the phase interface (cloud edge) may complicate convergence. A numerical investigation of convergence or non-convergence is presented here. The numerical simulations consider cases of ϵ=0.1, 0.01 and 0.001, where ϵ is proportional to the Rossby and Froude numbers. In the numerical simulations, the magnitude of vertical velocity w (or other measures of imbalance and inertio-gravity waves) is seen to be approximately proportional to ϵ as ϵ decreases, which suggests convergence to PQG dynamics. These measures are quantified at a fixed time T that is O(1), and the numerical data also suggests the possibility of convergence at later times. 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":"2022-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85170199","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}
Vincent Bouillaut, Benoît Flesselles, B. Miquel, S. Aumaitre, B. Gallet
Three-dimensional convection driven by internal heat sources and sinks (CISS) leads to experimental and numerical scaling laws compatible with a mixing-length—or ‘ultimate’—scaling regime Nu∼Ra. However, asymptotic analytic solutions and idealized two-dimensional simulations have shown that laminar flow solutions can transport heat even more efficiently, with Nu∼Ra. The turbulent nature of the flow thus has a profound impact on its transport properties. In the present contribution, we give this statement a precise mathematical sense. We show that the Nusselt number maximized over all solutions is bounded from above by const.×Ra, before restricting attention to ‘fully turbulent branches of solutions’, defined as families of solutions characterized by a finite non-zero limit of the dissipation coefficient at large driving amplitude. Maximization of Nu over such branches of solutions yields the better upper-bound Nu≲Ra. We then provide three-dimensional numerical and experimental data of CISS compatible with a finite limiting value of the dissipation coefficient at large driving amplitude. It thus seems that CISS achieves the maximal heat transport scaling over fully turbulent solutions. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.
{"title":"Velocity-informed upper bounds on the convective heat transport induced by internal heat sources and sinks","authors":"Vincent Bouillaut, Benoît Flesselles, B. Miquel, S. Aumaitre, B. Gallet","doi":"10.1098/rsta.2021.0034","DOIUrl":"https://doi.org/10.1098/rsta.2021.0034","url":null,"abstract":"Three-dimensional convection driven by internal heat sources and sinks (CISS) leads to experimental and numerical scaling laws compatible with a mixing-length—or ‘ultimate’—scaling regime Nu∼Ra. However, asymptotic analytic solutions and idealized two-dimensional simulations have shown that laminar flow solutions can transport heat even more efficiently, with Nu∼Ra. The turbulent nature of the flow thus has a profound impact on its transport properties. In the present contribution, we give this statement a precise mathematical sense. We show that the Nusselt number maximized over all solutions is bounded from above by const.×Ra, before restricting attention to ‘fully turbulent branches of solutions’, defined as families of solutions characterized by a finite non-zero limit of the dissipation coefficient at large driving amplitude. Maximization of Nu over such branches of solutions yields the better upper-bound Nu≲Ra. We then provide three-dimensional numerical and experimental data of CISS compatible with a finite limiting value of the dissipation coefficient at large driving amplitude. It thus seems that CISS achieves the maximal heat transport scaling over fully turbulent solutions. 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":"2022-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76052119","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}
Nonlinear simple invariant solutions representing the ultimate scaling have been discovered to the Navier–Stokes equations for thermal convection between horizontal no-slip permeable walls with a distance H and a constant temperature difference ΔT. On the permeable walls, the vertical transpiration velocity is assumed to be proportional to the local pressure fluctuations, i.e. w=±βp/ρ (Jiménez et al. 2001 J. Fluid Mech., 442, 89–117. (doi:10.1017/S0022112001004888)). Two-dimensional steady solutions bifurcating from a conduction state have been obtained using a Newton–Krylov iteration up to the Rayleigh number Ra∼108 for the Prandtl number Pr=1, the horizontal period L/H=2 and the permeability parameter βU=0–3, U being the buoyancy-induced terminal velocity. The wall permeability has a significant impact on the onset and the scaling properties of the found steady ‘wall-bounded’ thermal convection. The ultimate scaling Nu∼Ra1/2 has been observed for βU>0 at high Ra, where Nu is the Nusselt number. In the steady ultimate state, large-scale thermal plumes fully extend from one wall to the other, inducing the strong vertical velocity comparable with the terminal velocity U as well as intense temperature variation of O(ΔT) even in the bulk region. As a consequence, the wall-to-wall heat flux scales with UΔT independent of thermal diffusivity, although the heat transfer on the walls is dominated by thermal conduction. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.
{"title":"Steady thermal convection representing the ultimate scaling","authors":"Shingo Motoki, G. Kawahara, M. Shimizu","doi":"10.1098/rsta.2021.0037","DOIUrl":"https://doi.org/10.1098/rsta.2021.0037","url":null,"abstract":"Nonlinear simple invariant solutions representing the ultimate scaling have been discovered to the Navier–Stokes equations for thermal convection between horizontal no-slip permeable walls with a distance H and a constant temperature difference ΔT. On the permeable walls, the vertical transpiration velocity is assumed to be proportional to the local pressure fluctuations, i.e. w=±βp/ρ (Jiménez et al. 2001 J. Fluid Mech., 442, 89–117. (doi:10.1017/S0022112001004888)). Two-dimensional steady solutions bifurcating from a conduction state have been obtained using a Newton–Krylov iteration up to the Rayleigh number Ra∼108 for the Prandtl number Pr=1, the horizontal period L/H=2 and the permeability parameter βU=0–3, U being the buoyancy-induced terminal velocity. The wall permeability has a significant impact on the onset and the scaling properties of the found steady ‘wall-bounded’ thermal convection. The ultimate scaling Nu∼Ra1/2 has been observed for βU>0 at high Ra, where Nu is the Nusselt number. In the steady ultimate state, large-scale thermal plumes fully extend from one wall to the other, inducing the strong vertical velocity comparable with the terminal velocity U as well as intense temperature variation of O(ΔT) even in the bulk region. As a consequence, the wall-to-wall heat flux scales with UΔT independent of thermal diffusivity, although the heat transfer on the walls is dominated by thermal conduction. 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":"2022-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86022727","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}
Recent direct numerical simulations (DNS) and computations of exact steady solutions suggest that the heat transport in Rayleigh–Bénard convection (RBC) exhibits the classical 1/3 scaling as the Rayleigh number Ra→∞ with Prandtl number unity, consistent with Malkus–Howard’s marginally stable boundary layer theory. Here, we construct conditional upper and lower bounds for heat transport in two-dimensional RBC subject to a physically motivated marginal linear-stability constraint. The upper estimate is derived using the Constantin–Doering–Hopf (CDH) variational framework for RBC with stress-free boundary conditions, while the lower estimate is developed for both stress-free and no-slip boundary conditions. The resulting optimization problems are solved numerically using a time-stepping algorithm. Our results indicate that the upper heat-flux estimate follows the same 5/12 scaling as the rigorous CDH upper bound for the two-dimensional stress-free case, indicating that the linear-stability constraint fails to modify the boundary-layer thickness of the mean temperature profile. By contrast, the lower estimate successfully captures the 1/3 scaling for both the stress-free and no-slip cases. These estimates are tested using marginally-stable equilibrium solutions obtained under the quasi-linear approximation, steady roll solutions and DNS data. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.
{"title":"Heat transport in Rayleigh–Bénard convection with linear marginality","authors":"Baole Wen, Zijing Ding, G. Chini, R. Kerswell","doi":"10.1098/rsta.2021.0039","DOIUrl":"https://doi.org/10.1098/rsta.2021.0039","url":null,"abstract":"Recent direct numerical simulations (DNS) and computations of exact steady solutions suggest that the heat transport in Rayleigh–Bénard convection (RBC) exhibits the classical 1/3 scaling as the Rayleigh number Ra→∞ with Prandtl number unity, consistent with Malkus–Howard’s marginally stable boundary layer theory. Here, we construct conditional upper and lower bounds for heat transport in two-dimensional RBC subject to a physically motivated marginal linear-stability constraint. The upper estimate is derived using the Constantin–Doering–Hopf (CDH) variational framework for RBC with stress-free boundary conditions, while the lower estimate is developed for both stress-free and no-slip boundary conditions. The resulting optimization problems are solved numerically using a time-stepping algorithm. Our results indicate that the upper heat-flux estimate follows the same 5/12 scaling as the rigorous CDH upper bound for the two-dimensional stress-free case, indicating that the linear-stability constraint fails to modify the boundary-layer thickness of the mean temperature profile. By contrast, the lower estimate successfully captures the 1/3 scaling for both the stress-free and no-slip cases. These estimates are tested using marginally-stable equilibrium solutions obtained under the quasi-linear approximation, steady roll solutions and DNS data. 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":"2022-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91481790","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}
This article presents, within a multiscale framework, a search for a unified approach towards modelling the COVID-19 pandemic, from contagion to within-host dynamics. The modelling is focused on vaccination and therapeutical actions in general. The first part of our article is devoted to understanding the complex features of the system and to the design of a modelling rationale. Then, the modelling approach follows mainly focused on the competition between the proliferating virus and the immune system. Modelling considers also the action of vaccination plans related to the onset of new variants. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.
{"title":"Pandemics of mutating virus and society: a multi-scale active particles approach","authors":"N. Bellomo, D. Burini, N. Outada","doi":"10.1098/rsta.2021.0161","DOIUrl":"https://doi.org/10.1098/rsta.2021.0161","url":null,"abstract":"This article presents, within a multiscale framework, a search for a unified approach towards modelling the COVID-19 pandemic, from contagion to within-host dynamics. The modelling is focused on vaccination and therapeutical actions in general. The first part of our article is devoted to understanding the complex features of the system and to the design of a modelling rationale. Then, the modelling approach follows mainly focused on the competition between the proliferating virus and the immune system. Modelling considers also the action of vaccination plans related to the onset of new variants. 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":"2022-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88083896","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}
B. Boghosian, Matthew Hudes, G. Khachatryan, Jeremy Marcq
Despite their highly idealized nature, certain agent-based models of asset exchange, proposed for the most part by physicists and mathematicians, have been shown to exhibit remarkable agreement with empirical wealth distribution data. While this agre- ement is comforting, there is widespread sentiment that further progress will require a detailed under- standing of the connection between these idealized models and the more realistic microeconomic models of exchange used by economists. In this paper, we examine that connection for a three-parameter asset exchange model, the Affine Wealth Model (AWM), that has demonstrated fraction-of-a-per cent agreement with empirical wealth data. We compare certain properties of this model with those of three great milestones of twentieth century economics, namely (i) Expected Utility Theory, (ii) General Equilibrium Theory and (iii) Prospect Theory. We find that the phenomenology exhibited by the AWM is fundamentally incompatible with Expected Utility Theory and General Equilibrium Theory, but very similar to that exhibited by Prospect Theory. Based on these observations, we argue that AWM transactions are, in a particular sense, an approximation to those described by Prospect Theory, and that Prospect Theory provides the sought-for connection between econophysics and microeconomics, at least for the topic of wealth distribution. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.
{"title":"An economically realistic asset exchange model","authors":"B. Boghosian, Matthew Hudes, G. Khachatryan, Jeremy Marcq","doi":"10.1098/rsta.2021.0167","DOIUrl":"https://doi.org/10.1098/rsta.2021.0167","url":null,"abstract":"Despite their highly idealized nature, certain agent-based models of asset exchange, proposed for the most part by physicists and mathematicians, have been shown to exhibit remarkable agreement with empirical wealth distribution data. While this agre- ement is comforting, there is widespread sentiment that further progress will require a detailed under- standing of the connection between these idealized models and the more realistic microeconomic models of exchange used by economists. In this paper, we examine that connection for a three-parameter asset exchange model, the Affine Wealth Model (AWM), that has demonstrated fraction-of-a-per cent agreement with empirical wealth data. We compare certain properties of this model with those of three great milestones of twentieth century economics, namely (i) Expected Utility Theory, (ii) General Equilibrium Theory and (iii) Prospect Theory. We find that the phenomenology exhibited by the AWM is fundamentally incompatible with Expected Utility Theory and General Equilibrium Theory, but very similar to that exhibited by Prospect Theory. Based on these observations, we argue that AWM transactions are, in a particular sense, an approximation to those described by Prospect Theory, and that Prospect Theory provides the sought-for connection between econophysics and microeconomics, at least for the topic of wealth distribution. 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":"2022-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80104429","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}
In this work, the dynamics of agents below a threshold line in some modified CCM type kinetic wealth exchange models are studied. These agents are eligible for subsidy as can be seen in any real economy. An interaction is prohibited if both of the interacting agents’ wealth fall below the threshold line. A walk for such agents can be conceived in the abstract gain–loss space (GLS) and is macroscopically compared to a lazy walk. The effect of giving subsidy once to such agents is checked over, giving repeated subsidy from the point of view of the walk in GLS. It is seen that the walk has more positive drift if the subsidy is given once. The correlations and other interesting quantities are studied. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.
{"title":"A poor agent and subsidy: an investigation through CCM model","authors":"Sanchari Goswami","doi":"10.1098/rsta.2021.0166","DOIUrl":"https://doi.org/10.1098/rsta.2021.0166","url":null,"abstract":"In this work, the dynamics of agents below a threshold line in some modified CCM type kinetic wealth exchange models are studied. These agents are eligible for subsidy as can be seen in any real economy. An interaction is prohibited if both of the interacting agents’ wealth fall below the threshold line. A walk for such agents can be conceived in the abstract gain–loss space (GLS) and is macroscopically compared to a lazy walk. The effect of giving subsidy once to such agents is checked over, giving repeated subsidy from the point of view of the walk in GLS. It is seen that the walk has more positive drift if the subsidy is given once. The correlations and other interesting quantities are studied. 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":"2022-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78285743","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}
Julian Neñer, B. Cardoso, M. F. Laguna, S. Gonçalves, J. R. Iglesias
Genetic machine learning (ML) algorithms to train agents in the Yard–Sale model proved very useful for finding optimal strategies that maximize their wealth. However, the main result indicates that the more significant the fraction of rational agents, the greater the inequality at the collective level. From social and economic viewpoints, this is an undesirable result since high inequality diminishes liquidity and trade. Besides, with very few exceptions, most agents end up with zero wealth, despite the inclusion of rational behaviour. To deal with this situation, here we include a taxation–redistribution mechanism in the ML algorithm. Previous results show that simple regulations can considerably reduce inequality if agents do not change their behaviour. However, when considering rational agents, different types of redistribution favour risk-averse agents, to some extent. Even so, we find that rational agents looking for optimal wealth can always arrive to an optimal risk, compatible with a particular choice of parameters, but increasing inequality. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.
{"title":"Study of taxes, regulations and inequality using machine learning algorithms","authors":"Julian Neñer, B. Cardoso, M. F. Laguna, S. Gonçalves, J. R. Iglesias","doi":"10.1098/rsta.2021.0165","DOIUrl":"https://doi.org/10.1098/rsta.2021.0165","url":null,"abstract":"Genetic machine learning (ML) algorithms to train agents in the Yard–Sale model proved very useful for finding optimal strategies that maximize their wealth. However, the main result indicates that the more significant the fraction of rational agents, the greater the inequality at the collective level. From social and economic viewpoints, this is an undesirable result since high inequality diminishes liquidity and trade. Besides, with very few exceptions, most agents end up with zero wealth, despite the inclusion of rational behaviour. To deal with this situation, here we include a taxation–redistribution mechanism in the ML algorithm. Previous results show that simple regulations can considerably reduce inequality if agents do not change their behaviour. However, when considering rational agents, different types of redistribution favour risk-averse agents, to some extent. Even so, we find that rational agents looking for optimal wealth can always arrive to an optimal risk, compatible with a particular choice of parameters, but increasing inequality. 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":"2022-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90825961","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 propose a kinetic model to describe trade among different populations, living in different countries. The interaction rules are assumed depending on the trading propensity of each population and also on non-deterministic (random) effects. Moreover, the possible transfers of individuals from one country to another are also taken into account, by means of suitable Boltzmann-type operators. Consistent macroscopic equations for number density and mean wealth of each country are derived from the kinetic equations, and the effects of transfers on their equilibrium values are commented on. Finally, a suitable continuous trading limit is considered, leading to a simpler system of Fokker–Planck-type kinetic equations, with specific contributions accounting for transfers. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.
{"title":"Kinetic model for international trade allowing transfer of individuals","authors":"M. Bisi","doi":"10.1098/rsta.2021.0156","DOIUrl":"https://doi.org/10.1098/rsta.2021.0156","url":null,"abstract":"We propose a kinetic model to describe trade among different populations, living in different countries. The interaction rules are assumed depending on the trading propensity of each population and also on non-deterministic (random) effects. Moreover, the possible transfers of individuals from one country to another are also taken into account, by means of suitable Boltzmann-type operators. Consistent macroscopic equations for number density and mean wealth of each country are derived from the kinetic equations, and the effects of transfers on their equilibrium values are commented on. Finally, a suitable continuous trading limit is considered, leading to a simpler system of Fokker–Planck-type kinetic equations, with specific contributions accounting for transfers. 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":"2022-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72788792","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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