On the complete aggregation of the Wigner-Lohe model for identical potentials

Seung-Yeal Ha; Gyuyoung Hwang; Dohyun Kim; Seung-Yeal Ha; Gyuyoung Hwang; Dohyun Kim

doi:10.3934/nhm.2022022

Networks and Heterogeneous Media

2022, Volume 17, Issue 5: 665-686. doi: 10.3934/nhm.2022022

Previous Article Next Article

On the complete aggregation of the Wigner-Lohe model for identical potentials

1.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea
2.
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea
3.
School of Mathematics, Statistics and Data Science, Sungshin Women's University, Seoul 02844, Republic of Korea

Received: 01 March 2022 Published: 20 May 2022
Primary: 82C10, 82C22; Secondary: 35B40, 35Q40

We study the collective behaviors of the Wigner-Lohe (WL) model for quantum synchronization in phase space which corresponds to the phase description of the Schrödinger-Lohe (SL) model for quantum synchronization, and it can be formally derived from the SL model via the generalized Wigner transform. For this proposed model, we show that the WL model exhibits asymptotic aggregation estimates so that all the elements in the generalized Wigner distribution matrix tend to a common one. On the other hand, for the global unique solvability, we employ the fixed point argument together with the classical semigroup theory to derive the global unique solvability of mild and classical solutions depending on the regularity of initial data.
- Aggregation,
- collective behaviors,
- quantum synchronization,
- Wigner equation
Citation: Seung-Yeal Ha, Gyuyoung Hwang, Dohyun Kim. On the complete aggregation of the Wigner-Lohe model for identical potentials[J]. Networks and Heterogeneous Media, 2022, 17(5): 665-686. doi: 10.3934/nhm.2022022

Related Papers:

Abstract

We study the collective behaviors of the Wigner-Lohe (WL) model for quantum synchronization in phase space which corresponds to the phase description of the Schrödinger-Lohe (SL) model for quantum synchronization, and it can be formally derived from the SL model via the generalized Wigner transform. For this proposed model, we show that the WL model exhibits asymptotic aggregation estimates so that all the elements in the generalized Wigner distribution matrix tend to a common one. On the other hand, for the global unique solvability, we employ the fixed point argument together with the classical semigroup theory to derive the global unique solvability of mild and classical solutions depending on the regularity of initial data.

References

[1]	The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys. (2005) 77: 137-185.
[2]	B. Andrews and C. Hopper, The Ricci Flow in Riemannian Geometry, Springer Science & Business Media, 2011.
[3]	The Wigner-Lohe model for quantum synchronization and its emergent dynamics. Netw. Hetero. Media (2017) 12: 403-416.
[4]	A shocking display of synchrony. Phys. D (2000) 143: 21-55.
[5]	Some remarks on the Wigner transform and the Wigner-Poisson system. Le Matematiche (1991) 46: 429-438.
[6]	The three-dimensional Wigner-Poisson problem: Existence, uniqueness and approximation. Math. Methods Appl. Sci. (1991) 14: 35-61.
[7]	Biology of synchronous flashing of fireflies. Nature (1966) 211: 562-564.
[8]	Robust quantum gates on neutral atoms with cavity-assisted photon scattering. Phys. Rev. A (2005) 72: 032333.
[9]	S.-Y. Ha and D. Kim, Collective dynamics of Lohe type aggregation models, archived as arXiv: 2108.10473.
[10]	Existence, uniqueness and asymptotic behavior of Wigner-Poisson and Vlasov-Poisson systems: A survey. Transport Theory Stat. Phys. (1997) 26: 195-207.
[11]	Global existence, uniqueness and asymptotic behavior of solutions of the Wigner-Poisson and Schrödinger-Poisson systems. Math. Methods Appl. Sci (1994) 17: 349-376.
[12]	G. B. Folland, Harmonic Analysis in Phase Space, Annals of Mathematics Studies, 122. Princeton University Press, Princeton, NJ, 1989.
[13]	Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. (1997) 50: 323-379.
[14]	Quantum correlations and mutual synchronization. Phys. Rev. A (2012) 85: 052101.
[15]	Quantum stochastic synchronization. Phys. Rev. Lett. (2006) 97: 210601.
[16]	The quantum internet. Nature (2008) 453: 1023-1030.
[17]	Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag. Berlin. 1984.
[18]	Y. Kuramoto, International Symposium on Mathematical Problems in Mathematical Physics, Lecture Notes in Theoretical Physics, 30, 420, 1975.
[19]	Quantum synchronization over quantum networks. J. Phys. A (2010) 43: 465301.
[20]	Non-Abelian Kuramoto model and synchronization. J. Phys. A (2009) 42: 395101.
[21]	Quantum synchronization effects in intrinsic Josephson junctions. Phys. C (2008) 468: 689-694.
[22]	On the equivalence of the Schrödinger and the quantum Liouville equation. Math. Methods Appl. Sci. (1989) 11: 459-469.
[23]	An analysis of quantum Lioville equation,. Z. Angew. Math. Mech. (1989) 69: 121-127.
[24]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, Berlin, 1983.
[25]	C. S. Peskin, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York, 1975.
[26]	The one-dimensional Wigner-Poisson problem and a relation to the Schrödinger-Poisson problem. SIAM J. Math. Anal. (1991) 22: 957-972.
[27]	From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Phys. D (2000) 143: 1-20.
[28]	Superinsulator and quantum synchronization. Nature (2008) 452: 613-616.
[29]	Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. (1967) 16: 15-42.
[30]	On the quantum correction for thermodynamic equilibrium. Phys. Rev. (1932) 40: 749-759.
[31]	M. W. Wong, Weyl Transforms, Springer, New York, 1998.
[32]	P. Zhang, Wigner Measure and Semiclassical Limits of Nonlinear Schödinger Equations, Courant Lecture Notes in Mathematics, vol. 17, 2008.
[33]	Quantum synchronization and entanglement of two qubits coupled to a driven dissipative resonator. Phys. Rev. B. (2009) 80: 014519.
[34]	Quantum synchronization. Eur. Phys. J. D. (2006) 38: 375-379.
[35]	The Wigner transform and the Wigner-Poisson system. Transp. Theory Stat. Phys. (1993) 22: 459-484.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)