Long-Time Behaviour and Phase Transitions for the Mckean–Vlasov Equation on the Torus

Carrillo, José A.; Gvalani, Rishabh S.; Pavliotis, Grigorios A.; Schlichting, André

doi:10.1007/s00205-019-01430-4

Cited by 96 publications

(142 citation statements)

References 67 publications

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“…As a final point, we mention that our results will also be of interest to the pure mathematics community. DDFT-3 is also referred to as the McKean-Vlasov equation and in this context there are a number of recent interesting rigorous results [64,65]. Our results for DDFT-5, showing that for a finite value of µ there is a singularity with the density profile going to zero, may well be of interest to those who study the mathematics of solutions to partial differential equations with compact supportsee for example Ref.…”

Section: Discussionmentioning

confidence: 62%

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Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

et al. 2019

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Phase field crystal (PFC) theory is extensively used for modelling the phase behaviour, structure, thermodynamics and other related properties of solids. PFC theory can be derived from dynamical density functional theory (DDFT) via a sequence of approximations. Here, we carefully identify all of these approximations and explain the consequences of each. One approximation that is made in standard derivations is to neglect a term of form ∇ · [n∇Ln], where n is the scaled density profile and L is a linear operator. We show that this term makes a significant contribution to the stability of the crystal, and that dropping this term from the theory forces another approximation, that of replacing the logarithmic term from the ideal gas contribution to the free energy with its truncated Taylor expansion, to yield a polynomial in n. However, the consequences of doing this are: (i) the presence of an additional spinodal in the phase diagram, so the liquid is predicted first to freeze and then to melt again as the density is increased; and (ii) other periodic structures, such as stripes, are erroneously predicted to be thermodynamic equilibrium structures. In general, L consists of a nonlocal convolution involving the pair direct correlation function. A second approximation sometimes made in deriving PFC theory is to replace L by a gradient expansion involving derivatives. We show that this leads to the possibility of the density going to zero, with its logarithm going to −∞ whilst being balanced by the fourth derivative of the density going to +∞. This subtle singularity leads to solutions failing to exist above a certain value of the average density. We illustrate all of these conclusions with results for a particularly simple model two-dimensional fluid, the generalised exponential model of index 4 (GEM-4), chosen because a DDFT is known to be accurate for this model. The consequences of the subsequent PFC approximations can then be examined. These include the phase diagram being both qualitatively incorrect, in that it has a stripe phase, and quantitatively incorrect (by orders of magnitude) regarding the properties of the crystal phase. Thus, although PFC models are very successful as phenomenological models of crystallisation, we find it impossible to derive the PFC as a theory for the (scaled) density distribution when starting from an accurate DDFT, without introducing spurious artefacts. However, we find that making a simple one-mode approximation for the logarithm of the density distribution log ρ(x) (rather than for ρ(x)), is surprisingly accurate. This approach gives a tantalising hint that accurate PFC-type theories may instead be derived as theories for the field log ρ(x), rather than for the density profile itself.

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Section: Discussionmentioning

confidence: 62%

“…Appendix A: Linear theory for In this appendix, we discuss how we compute the linear theory for the GEM-4 potential in (64). To be specific, in a two-dimensional periodic domain, the eigenvalue σ(k) is defined by Le ik·x = σ(k)e ik·x and (65), which can be written as…”

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confidence: 99%

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

et al. 2019

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“…Stationary states for this system in bounded domains can be fairly more complicated. Let us start by mentioning that even for the simpler aggregation-diffusion equation of the form u t = ν∆u + ∇ · (u∇(W * u)) , with W being an interaction potential subject to periodic boundary conditions, this is a case that can demonstrate a phase transition depending on the strength of the noise ν, see [20] and the references therein. These phenomena were also analyzed in [24] in the case of quadratic diffusion showing sufficient conditions on the potential W for phase transitions to happen.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Phase Transitions and Bump Solutions of the Keller--Segel Model with Volume Exclusion

Carrillo¹,

Chen²,

Wang³

et al. 2020

SIAM J. Appl. Math.

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We show that the Keller-Segel model in one dimension with Neumann boundary conditions and quadratic cellular diffusion has an intricate phase transition diagram depending on the chemosensitivity strength. Explicit computations allow us to find a myriad of symmetric and asymmetric stationary states whose stability properties are mostly studied via free energy decreasing numerical schemes. The metastability behavior and staircased free energy decay are also illustrated via these numerical simulations.

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“…Positivity. Using the same approach as in [CGPS18], we consider the following version of the (6) in the unknown function ρ withρ being the non-negative weak solution…”

Section: Sketch Of Proof Of Theorem 1 Consider the Following Sequencmentioning

confidence: 99%

Long-term Behavior of Mean-field Noisy Bounded Confidence Models with Distributed Radicals

Kolarijani

Proskurnikov

Esfahani

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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In this article, we study the nonlinear Fokker-Planck (FP) equation that arises as a mean-field (macroscopic, Eulerian) approximation of bounded confidence opinion dynamics, where opinions are influenced by environmental noises and opinions of radicals (stubborn individuals). The distribution of radical opinions serves as an infinite-dimensional control input to the FP equation, visibly influencing the steady opinion profile. We establish mathematical properties of the FP equation.In particular, we show (i) the well-posedness (existence, uniqueness, and regularity) of the dynamic equation in a certain class of initial conditions and provide (ii) existence result accompanied by a global estimate for the corresponding stationary equation. Also provided is (iii) a lower bound on the noise level that guarantees exponential convergence of the dynamics to stationary state. Combining the last two results readily yields input-output stability of the system for sufficiently large noises. Next, using Fourier analysis, the structure of opinion clusters under the uniform initial distribution is examined. Specifically, two numerical schemes are provided for identification of the order-disorder transition and characterization of the initial clustering behavior. The results of analysis are validated through several numerical simulations of the continuum-agent model (partial differential equation) and the corresponding discrete-agent model (interactive stochastic differential equations) for a particular distribution of radicals.

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Long-Time Behaviour and Phase Transitions for the Mckean–Vlasov Equation on the Torus

Cited by 96 publications

References 67 publications

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

Phase Transitions and Bump Solutions of the Keller--Segel Model with Volume Exclusion

Long-term Behavior of Mean-field Noisy Bounded Confidence Models with Distributed Radicals

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