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Phase transition of the kinetic Justh-Krishnaprasad type model for nematic alignment

Seung-Yeal Ha, Hui Yu, Baige Zhou

TL;DR

This work derives a kinetic VMFP equation from the stochastic Justh-Krishnaprasad particle system to study nematic alignment. It shows a phase transition controlled by the ratio $\sigma/\kappa$, with a critical value $1/4$ separating uniform from nonuniform von-Mises equilibria in the constant-noise case, and proves exponential convergence to equilibria in the respective regimes using free-energy and relative-entropy methods. A nonconstant-noise analysis reveals non-symmetric two-peak equilibria on the torus and a degenerate diffusion structure, with relative entropy nonincreasing under a fixed order parameter. Overall, the results connect microscopic communications to macroscopic nematic patterns and provide a rigorous link between microscale interactions and macroscopic steady states with potential applications to collective dynamics.

Abstract

We present a stochastic Justh-Krishnaprasad flocking model and study the phase transition of the Vlasov-McKean-Fokker-Planck (VMFP) equation, which can be obtained in the mean-field limit. To describe the alignment, we use order parameters in terms of the distribution function of the kinetic model. For the constant noise case, we study the well-posedness of the VMFP equation on the torus. Based on regularity, we show that the phenomenon of phase transition is only related to the ratio between the strengths of noise and coupling. In particular, for the low-noise case, we derive an exponential convergence to the von-Mises type equilibrium, which shows a strong evidence for the nematic alignment. The multiplicative noise is also studied to obtain a non-symmetric equilibrium with two different peaks on the torus.

Phase transition of the kinetic Justh-Krishnaprasad type model for nematic alignment

TL;DR

This work derives a kinetic VMFP equation from the stochastic Justh-Krishnaprasad particle system to study nematic alignment. It shows a phase transition controlled by the ratio , with a critical value separating uniform from nonuniform von-Mises equilibria in the constant-noise case, and proves exponential convergence to equilibria in the respective regimes using free-energy and relative-entropy methods. A nonconstant-noise analysis reveals non-symmetric two-peak equilibria on the torus and a degenerate diffusion structure, with relative entropy nonincreasing under a fixed order parameter. Overall, the results connect microscopic communications to macroscopic nematic patterns and provide a rigorous link between microscale interactions and macroscopic steady states with potential applications to collective dynamics.

Abstract

We present a stochastic Justh-Krishnaprasad flocking model and study the phase transition of the Vlasov-McKean-Fokker-Planck (VMFP) equation, which can be obtained in the mean-field limit. To describe the alignment, we use order parameters in terms of the distribution function of the kinetic model. For the constant noise case, we study the well-posedness of the VMFP equation on the torus. Based on regularity, we show that the phenomenon of phase transition is only related to the ratio between the strengths of noise and coupling. In particular, for the low-noise case, we derive an exponential convergence to the von-Mises type equilibrium, which shows a strong evidence for the nematic alignment. The multiplicative noise is also studied to obtain a non-symmetric equilibrium with two different peaks on the torus.

Paper Structure

This paper contains 11 sections, 20 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Mean-field limit of the Justh-Krishnaprasad model
  3. Mean-field limit in the kinetic form
  4. The spatially homogeneous J-K model with constant noise
  5. Well-posedness and regularity
  6. Phase Transition for constant noise
  7. Equilibrium and free energy
  8. Convergence in the subcritical regime
  9. Convergence in the supercritical regime
  10. Phase transition for nonconstant noise
  11. Conclusion

Key Result

Theorem 2.2

Let $f_0\in H_{\mathrm{per}}^m$, $m>\frac{5}{2}$ be the initial probability measure. Then there exists a unique strong solution for IBVP_hom. Moreover, this solution is global-in-time, nonnegative and smooth.

Theorems & Definitions (48)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5: Continuity with respect to initial data
  • proof
  • Remark 2.6
  • Proposition 2.7: Classical solutions and nonnegativity
  • ...and 38 more