4,667 research outputs found
Mean Field Games models of segregation
This paper introduces and analyses some models in the framework of Mean Field
Games describing interactions between two populations motivated by the studies
on urban settlements and residential choice by Thomas Schelling. For static
games, a large population limit is proved. For differential games with noise,
the existence of solutions is established for the systems of partial
differential equations of Mean Field Game theory, in the stationary and in the
evolutive case. Numerical methods are proposed, with several simulations. In
the examples and in the numerical results, particular emphasis is put on the
phenomenon of segregation between the populations.Comment: 35 pages, 10 figure
The non-centrosymmetric lamellar phase in blends of ABC triblock and ac diblock copolymers
The phase behaviour of blends of ABC triblock and ac diblock copolymers is
examined using self-consistent field theory. Several equilibrium lamellar
structures are observed, depending on the volume fraction of the diblocks,
phi_2, the monomer interactions, and the degrees of polymerization of the
copolymers. For segregations just above the order-disorder transition the
triblocks and diblocks mix together to form centrosymmetric lamellae. As the
segregation is increased the triblocks and diblocks spatially separate either
by macrophase-separating, or by forming a non-centrosymmetric (NCS) phase of
alternating layers of triblock and diblock (...ABCcaABCca...). The NCS phase is
stable over a narrow region near phi_2=0.4. This region is widest near the
critical point on the phase coexistence curve and narrows to terminate at a
triple point at higher segregation. Above the triple point there is two-phase
coexistence between almost pure triblock and diblock phases. The theoretical
phase diagram is consistent with experiments.Comment: 9 pages, 8 figures, submitted to Macromolecule
Structure–function relations in diF-TES-ADT blend organic field effect transistors studied by scanning probe microscopy
We develop structure–property relations for organic field effect transistors using a polymer/small-molecule blend active layer. An array of bottom gate, bottom contact devices using a polymeric dielectric and a semiconductor layer of 2,8-difluoro-5,11-bis(triethylsilylethynyl)anthradithiophene (diF-TES-ADT) is described and shown to have good device-to-device uniformity. We describe the nucleation and growth processes that lead to the formation of four structurally distinct regimes of the diF-TES-ADT semiconductor film, including evidence of layer-by-layer growth when spin-coated onto silver electrodes and an organic dielectric as part of a polymer blend. Devices exhibiting a maximum saturation mobility of 1.5 cm2 V−1 s−1 and maximum current modulation ratio (Ion/Ioff) of 1.20 × 105 are visualised by atomic force microscopy and appear to have excellent domain connectivity and aligned crystallography across the channel. In contrast, poorly performing devices tend to show a phase change in semiconductor crystallinity in the channel centre. These observations are enhanced by direct visualisation of the potential drop across the channel using Kelvin probe microscopy, which confirms the importance of large, well-aligned and well-connected semiconductor domains across the transistor channel
Quadratic Mean Field Games
Mean field games were introduced independently by J-M. Lasry and P-L. Lions,
and by M. Huang, R.P. Malham\'e and P. E. Caines, in order to bring a new
approach to optimization problems with a large number of interacting agents.
The description of such models split in two parts, one describing the evolution
of the density of players in some parameter space, the other the value of a
cost functional each player tries to minimize for himself, anticipating on the
rational behavior of the others.
Quadratic Mean Field Games form a particular class among these systems, in
which the dynamics of each player is governed by a controlled Langevin equation
with an associated cost functional quadratic in the control parameter. In such
cases, there exists a deep relationship with the non-linear Schr\"odinger
equation in imaginary time, connexion which lead to effective approximation
schemes as well as a better understanding of the behavior of Mean Field Games.
The aim of this paper is to serve as an introduction to Quadratic Mean Field
Games and their connexion with the non-linear Schr\"odinger equation, providing
to physicists a good entry point into this new and exciting field.Comment: 62 pages, 4 figure
Coarse-grained distributions and superstatistics
We show an interesting connexion between the coarse-grained distribution
function arising in the theory of violent relaxation for collisionless stellar
systems (Lynden-Bell 1967) and the notion of superstatistics introduced
recently by Beck & Cohen (2003). We also discuss the analogies and differences
between the statistical equilibrium state of a multi-components
self-gravitating system and the metaequilibrium state of a collisionless
stellar system. Finally, we stress the important distinction between mixing
entropies, generalized entropies, H-functions, generalized mixing entropies and
relative entropies
Coordinated optimization of visual cortical maps : 2. Numerical studies
In the juvenile brain, the synaptic architecture of the visual cortex remains in a state of flux for months after the natural onset of vision and the initial emergence of feature selectivity in visual cortical neurons. It is an attractive hypothesis that visual cortical architecture is shaped during this extended period of juvenile plasticity by the coordinated optimization of multiple visual cortical maps such as orientation preference (OP), ocular dominance (OD), spatial frequency, or direction preference. In part (I) of this study we introduced a class of analytically tractable coordinated optimization models and solved representative examples, in which a spatially complex organization of the OP map is induced by interactions between the maps. We found that these solutions near symmetry breaking threshold predict a highly ordered map layout. Here we examine the time course of the convergence towards attractor states and optima of these models. In particular, we determine the timescales on which map optimization takes place and how these timescales can be compared to those of visual cortical development and plasticity. We also assess whether our models exhibit biologically more realistic, spatially irregular solutions at a finite distance from threshold, when the spatial periodicities of the two maps are detuned and when considering more than 2 feature dimensions. We show that, although maps typically undergo substantial rearrangement, no other solutions than pinwheel crystals and stripes dominate in the emerging layouts. Pinwheel crystallization takes place on a rather short timescale and can also occur for detuned wavelengths of different maps. Our numerical results thus support the view that neither minimal energy states nor intermediate transient states of our coordinated optimization models successfully explain the architecture of the visual cortex. We discuss several alternative scenarios that may improve the agreement between model solutions and biological observations
Sharp interface limit of an energy modelling nanoparticle-polymer blends
We identify the -limit of a nanoparticle-polymer model as the number
of particles goes to infinity and as the size of the particles and the phase
transition thickness of the polymer phases approach zero. The limiting energy
consists of two terms: the perimeter of the interface separating the phases and
a penalization term related to the density distribution of the infinitely many
small nanoparticles. We prove that local minimizers of the limiting energy
admit regular phase boundaries and derive necessary conditions of local
minimality via the first variation. Finally we discuss possible critical and
minimizing patterns in two dimensions and how these patterns vary from global
minimizers of the purely local isoperimetric problem.Comment: Minor changes. Rephrased introduction. This version is to appear in
Interfaces and Free Boundarie
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