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Mean field theory. --- Many-body problem --- Statistical mechanics

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A mathematical theory is introduced in this book to unify a large class of nonlinear partial differential equation (PDE) models for better understanding and analysis of the physical and biological phenomena they represent. The so-called mean field approximation approach is adopted to describe the macroscopic phenomena from certain microscopic principles for this unified mathematical formulation. Two key ingredients for this approach are the notions of “duality” according to the PDE weak solutions and “hierarchy” for revealing the details of the otherwise hidden secrets, such as physical mystery hidden between particle density and field concentration, quantized blow up biological mechanism sealed in chemotaxis systems, as well as multi-scale mathematical explanations of the Smoluchowski–Poisson model in non-equilibrium thermodynamics, two-dimensional turbulence theory, self-dual gauge theory, and so forth. This book shows how and why many different nonlinear problems are inter-connected in terms of the properties of duality and scaling, and the way to analyze them mathematically.

Differential equations, Partial. --- Mathematics. --- Mean field theory. --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Differential equations. --- Partial differential equations. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Partial differential equations --- 517.91 Differential equations --- Differential equations --- Math --- Science --- Differential Equations. --- Differential equations, partial.

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Mean field games and Mean field type control introduce new problems in Control Theory. The terminology “games” may be confusing. In fact they are control problems, in the sense that one is interested in a single decision maker, whom we can call the representative agent. However, these problems are not standard, since both the evolution of the state and the objective functional is influenced but terms which are not directly related to the state or the control of the decision maker. They are however, indirectly related to him, in the sense that they model a very large community of agents similar to the representative agent. All the agents behave similarly and impact the representative agent. However, because of the large number an aggregation effect takes place. The interesting consequence is that the impact of the community can be modeled by a mean field term, but when this is done, the problem is reduced to a control problem. .

Mathematics. --- Nonlinear control theory. --- Stochastic control theory. --- System analysis. --- Mean field theory --- Control theory --- Game theory --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Control theory. --- Differential equations, Partial. --- Distribution (Probability theory) --- System theory. --- Systems, Theory of --- Systems science --- Distribution functions --- Frequency distribution --- Partial differential equations --- Partial differential equations. --- Probabilities. --- Systems Theory, Control. --- Probability Theory and Stochastic Processes. --- Partial Differential Equations. --- Science --- Characteristic functions --- Probabilities --- Dynamics --- Machine theory --- Philosophy --- Systems theory. --- Distribution (Probability theory. --- Differential equations, partial. --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Mean field theory. --- Game theory.

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Beginning with a concise introduction to the theory of mean-field games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for well-posedness in the context of mean-field problems, including stationary and time-dependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of mean-field couplings. It also explores stationary and time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a range of analytical techniques, and builds on previous results to explain classical solutions. The final chapter discusses the potential applications, models and natural extensions of MFGs. As MFGs connect common problems in pure mathematics, engineering, economics and data management, this book is a valuable resource for researchers and graduate students in these fields.

Mathematics. --- Game theory. --- System theory. --- Economic theory. --- Game Theory, Economics, Social and Behav. Sciences. --- Economic Theory/Quantitative Economics/Mathematical Methods. --- Systems Theory, Control. --- Mean field theory. --- Games, Theory of --- Theory of games --- Mathematical models --- Mathematics --- Many-body problem --- Statistical mechanics --- Systems theory. --- Economic theory --- Political economy --- Social sciences --- Economic man --- Math --- Science --- Systems, Theory of --- Systems science --- Philosophy

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The work described in this book originates from a major effort to develop a fundamental theory of the glass and the jamming transitions. The first chapters guide the reader through the phenomenology of supercooled liquids and structural glasses and provide the tools to analyze the most frequently used models able to predict the complex behavior of such systems. A fundamental outcome is a detailed theoretical derivation of an effective thermodynamic potential, along with the study of anomalous vibrational properties of sphere systems. The interested reader can find in these pages a clear and deep analysis of mean-field models as well as the description of advanced beyond-mean-field perturbative expansions. To investigate important second-order phase transitions in lattice models, the last part of the book proposes an innovative theoretical approach, based on a multi-layer construction. The different methods developed in this thesis shed new light on important connections among constraint satisfaction problems, jamming and critical phenomena in complex systems, and lay part of the groundwork for a complete theory of amorphous solids.

Phase transformations (Statistical physics) --- Mean field theory. --- Phase Transitions and Multiphase Systems. --- Ceramics, Glass, Composites, Natural Materials. --- Low Temperature Physics. --- Phase transitions (Statistical physics). --- Ceramics. --- Glass. --- Composites (Materials). --- Composite materials. --- Low temperature physics. --- Low temperatures. --- Cryogenics --- Low temperature physics --- Temperatures, Low --- Temperature --- Cold --- Composites (Materials) --- Multiphase materials --- Reinforced solids --- Solids, Reinforced --- Two phase materials --- Materials --- Amorphous substances --- Ceramics --- Glazing --- Ceramic technology --- Industrial ceramics --- Keramics --- Building materials --- Chemistry, Technical --- Clay --- Phase changes (Statistical physics) --- Phase transitions (Statistical physics) --- Phase rule and equilibrium --- Statistical physics