doi: 10.3934/nhm.2022022
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

*Corresponding author: Gyuyoung Hwang

Received  March 2022 Early access May 2022

Fund Project: The first author is supported by NRF grant 2020R1A2C3A01003881, and the third author is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.2021R1F1A1055929)

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.

Citation: Seung-Yeal Ha, Gyuyoung Hwang, Dohyun Kim. On the complete aggregation of the Wigner-Lohe model for identical potentials. Networks and Heterogeneous Media, doi: 10.3934/nhm.2022022
References:
[1]

J. A. AcebronL. L. BonillaC. J. P. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.  doi: 10.1103/RevModPhys.77.137.

[2]

B. Andrews and C. Hopper, The Ricci Flow in Riemannian Geometry, Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-16286-2.

[3]

P. AntonelliS.-Y. HaD. Kim and P. Marcati, The Wigner-Lohe model for quantum synchronization and its emergent dynamics, Netw. Hetero. Media, 12 (2017), 403-416.  doi: 10.3934/nhm.2017018.

[4]

N. J. Balmforth and R. Sassi, A shocking display of synchrony, Phys. D, 143 (2000), 21-55.  doi: 10.1016/S0167-2789(00)00095-6.

[5]

C. S. BohunR. Illner and P. F. Zweifel, Some remarks on the Wigner transform and the Wigner-Poisson system, Le Matematiche, 46 (1991), 429-438. 

[6]

F. Brezzi and P. A. Markowich, The three-dimensional Wigner-Poisson problem: Existence, uniqueness and approximation, Math. Methods Appl. Sci., 14 (1991), 35-61.  doi: 10.1002/mma.1670140103.

[7]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0.

[8]

L.-M. DuanB. Wang and H. J. Kimble, Robust quantum gates on neutral atoms with cavity-assisted photon scattering, Phys. Rev. A, 72 (2005), 032333.  doi: 10.1103/PhysRevA.72.032333.

[9]

S.-Y. Ha and D. Kim, Collective dynamics of Lohe type aggregation models, archived as arXiv: 2108.10473.

[10]

R. Illner, Existence, uniqueness and asymptotic behavior of Wigner-Poisson and Vlasov-Poisson systems: A survey, Transport Theory Stat. Phys., 26 (1997), 195-207.  doi: 10.1080/00411459708221783.

[11]

R. IllnerP. F. Zweifel and H. Lange, Global existence, uniqueness and asymptotic behavior of solutions of the Wigner-Poisson and Schrödinger-Poisson systems, Math. Methods Appl. Sci, 17 (1994), 349-376.  doi: 10.1002/mma.1670170504.

[12]

G. B. Folland, Harmonic Analysis in Phase Space, Annals of Mathematics Studies, 122. Princeton University Press, Princeton, NJ, 1989. doi: 10.1515/9781400882427.

[13]

P. GérardP. A. MarkowichN. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379.  doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.

[14]

G. L. GiorgiF. GalveG. ManzanoP. Colet and R. Zambrini, Quantum correlations and mutual synchronization, Phys. Rev. A, 85 (2012), 052101.  doi: 10.1103/PhysRevA.85.052101.

[15]

I. GoychukJ. Casado-PascualM. MorilloJ. Lehmann and P. Hänggi, Quantum stochastic synchronization, Phys. Rev. Lett., 97 (2006), 210601.  doi: 10.1103/PhysRevLett.97.210601.

[16]

H. J. Kimble, The quantum internet, Nature, 453 (2008), 1023-1030.  doi: 10.1038/nature07127.

[17]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag. Berlin. 1984. doi: 10.1007/978-3-642-69689-3.

[18]

Y. Kuramoto, International Symposium on Mathematical Problems in Mathematical Physics, Lecture Notes in Theoretical Physics, 30, 420, 1975.

[19]

M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A, 43 (2010), 465301.  doi: 10.1088/1751-8113/43/46/465301.

[20]

M. A. Lohe, Non-Abelian Kuramoto model and synchronization, J. Phys. A, 42 (2009), 395101.  doi: 10.1088/1751-8113/42/39/395101.

[21]

M. MachidaT. KanoS. YamadaM. OkumuraT. Imamura and T. Koyama, Quantum synchronization effects in intrinsic Josephson junctions, Phys. C, 468 (2008), 689-694.  doi: 10.1016/j.physc.2007.11.081.

[22]

P. A. Markowich, On the equivalence of the Schrödinger and the quantum Liouville equation, Math. Methods Appl. Sci., 11 (1989), 459-469.  doi: 10.1002/mma.1670110404.

[23]

P. A. Markowich and C. A. Ringhofer, An analysis of quantum Lioville equation,, Z. Angew. Math. Mech., 69 (1989), 121-127.  doi: 10.1002/zamm.19890690303.

[24]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[25]

C. S. Peskin, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York, 1975.

[26]

H. Steinrück, The one-dimensional Wigner-Poisson problem and a relation to the Schrödinger-Poisson problem, SIAM J. Math. Anal., 22 (1991), 957-972.  doi: 10.1137/0522061.

[27]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Phys. D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4.

[28]

V. M. VinokurT. I. BaturinaM. V. FistulA. Y. MironovM. R. Baklanov and C. Strunk, Superinsulator and quantum synchronization, Nature, 452 (2008), 613-616.  doi: 10.1038/nature06837.

[29]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.  doi: 10.1016/0022-5193(67)90051-3.

[30]

E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759.  doi: 10.1103/PhysRev.40.749.

[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. doi: 10.1090/cln/017.

[33]

O. V. Zhirov and D. L. Shepelyansky, Quantum synchronization and entanglement of two qubits coupled to a driven dissipative resonator, Phys. Rev. B., 80 (2009), 014519.  doi: 10.1103/PhysRevB.80.014519.

[34]

O. V. Zhirov and D. L. Shepelyansky, Quantum synchronization, Eur. Phys. J. D., 38 (2006), 375-379.  doi: 10.1140/epjd/e2006-00011-9.

[35]

P. F. Zweifel, The Wigner transform and the Wigner-Poisson system, Transp. Theory Stat. Phys., 22 (1993), 459-484.  doi: 10.1080/00411459308203824.

show all references

References:
[1]

J. A. AcebronL. L. BonillaC. J. P. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.  doi: 10.1103/RevModPhys.77.137.

[2]

B. Andrews and C. Hopper, The Ricci Flow in Riemannian Geometry, Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-16286-2.

[3]

P. AntonelliS.-Y. HaD. Kim and P. Marcati, The Wigner-Lohe model for quantum synchronization and its emergent dynamics, Netw. Hetero. Media, 12 (2017), 403-416.  doi: 10.3934/nhm.2017018.

[4]

N. J. Balmforth and R. Sassi, A shocking display of synchrony, Phys. D, 143 (2000), 21-55.  doi: 10.1016/S0167-2789(00)00095-6.

[5]

C. S. BohunR. Illner and P. F. Zweifel, Some remarks on the Wigner transform and the Wigner-Poisson system, Le Matematiche, 46 (1991), 429-438. 

[6]

F. Brezzi and P. A. Markowich, The three-dimensional Wigner-Poisson problem: Existence, uniqueness and approximation, Math. Methods Appl. Sci., 14 (1991), 35-61.  doi: 10.1002/mma.1670140103.

[7]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0.

[8]

L.-M. DuanB. Wang and H. J. Kimble, Robust quantum gates on neutral atoms with cavity-assisted photon scattering, Phys. Rev. A, 72 (2005), 032333.  doi: 10.1103/PhysRevA.72.032333.

[9]

S.-Y. Ha and D. Kim, Collective dynamics of Lohe type aggregation models, archived as arXiv: 2108.10473.

[10]

R. Illner, Existence, uniqueness and asymptotic behavior of Wigner-Poisson and Vlasov-Poisson systems: A survey, Transport Theory Stat. Phys., 26 (1997), 195-207.  doi: 10.1080/00411459708221783.

[11]

R. IllnerP. F. Zweifel and H. Lange, Global existence, uniqueness and asymptotic behavior of solutions of the Wigner-Poisson and Schrödinger-Poisson systems, Math. Methods Appl. Sci, 17 (1994), 349-376.  doi: 10.1002/mma.1670170504.

[12]

G. B. Folland, Harmonic Analysis in Phase Space, Annals of Mathematics Studies, 122. Princeton University Press, Princeton, NJ, 1989. doi: 10.1515/9781400882427.

[13]

P. GérardP. A. MarkowichN. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379.  doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.

[14]

G. L. GiorgiF. GalveG. ManzanoP. Colet and R. Zambrini, Quantum correlations and mutual synchronization, Phys. Rev. A, 85 (2012), 052101.  doi: 10.1103/PhysRevA.85.052101.

[15]

I. GoychukJ. Casado-PascualM. MorilloJ. Lehmann and P. Hänggi, Quantum stochastic synchronization, Phys. Rev. Lett., 97 (2006), 210601.  doi: 10.1103/PhysRevLett.97.210601.

[16]

H. J. Kimble, The quantum internet, Nature, 453 (2008), 1023-1030.  doi: 10.1038/nature07127.

[17]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag. Berlin. 1984. doi: 10.1007/978-3-642-69689-3.

[18]

Y. Kuramoto, International Symposium on Mathematical Problems in Mathematical Physics, Lecture Notes in Theoretical Physics, 30, 420, 1975.

[19]

M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A, 43 (2010), 465301.  doi: 10.1088/1751-8113/43/46/465301.

[20]

M. A. Lohe, Non-Abelian Kuramoto model and synchronization, J. Phys. A, 42 (2009), 395101.  doi: 10.1088/1751-8113/42/39/395101.

[21]

M. MachidaT. KanoS. YamadaM. OkumuraT. Imamura and T. Koyama, Quantum synchronization effects in intrinsic Josephson junctions, Phys. C, 468 (2008), 689-694.  doi: 10.1016/j.physc.2007.11.081.

[22]

P. A. Markowich, On the equivalence of the Schrödinger and the quantum Liouville equation, Math. Methods Appl. Sci., 11 (1989), 459-469.  doi: 10.1002/mma.1670110404.

[23]

P. A. Markowich and C. A. Ringhofer, An analysis of quantum Lioville equation,, Z. Angew. Math. Mech., 69 (1989), 121-127.  doi: 10.1002/zamm.19890690303.

[24]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[25]

C. S. Peskin, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York, 1975.

[26]

H. Steinrück, The one-dimensional Wigner-Poisson problem and a relation to the Schrödinger-Poisson problem, SIAM J. Math. Anal., 22 (1991), 957-972.  doi: 10.1137/0522061.

[27]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Phys. D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4.

[28]

V. M. VinokurT. I. BaturinaM. V. FistulA. Y. MironovM. R. Baklanov and C. Strunk, Superinsulator and quantum synchronization, Nature, 452 (2008), 613-616.  doi: 10.1038/nature06837.

[29]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.  doi: 10.1016/0022-5193(67)90051-3.

[30]

E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759.  doi: 10.1103/PhysRev.40.749.

[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. doi: 10.1090/cln/017.

[33]

O. V. Zhirov and D. L. Shepelyansky, Quantum synchronization and entanglement of two qubits coupled to a driven dissipative resonator, Phys. Rev. B., 80 (2009), 014519.  doi: 10.1103/PhysRevB.80.014519.

[34]

O. V. Zhirov and D. L. Shepelyansky, Quantum synchronization, Eur. Phys. J. D., 38 (2006), 375-379.  doi: 10.1140/epjd/e2006-00011-9.

[35]

P. F. Zweifel, The Wigner transform and the Wigner-Poisson system, Transp. Theory Stat. Phys., 22 (1993), 459-484.  doi: 10.1080/00411459308203824.

[1]

Paolo Antonelli, Seung-Yeal Ha, Dohyun Kim, Pierangelo Marcati. The Wigner-Lohe model for quantum synchronization and its emergent dynamics. Networks and Heterogeneous Media, 2017, 12 (3) : 403-416. doi: 10.3934/nhm.2017018

[2]

Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208

[3]

Seung-Yeal Ha, Myeongju Kang, Bora Moon. Collective behaviors of a Winfree ensemble on an infinite cylinder. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2749-2779. doi: 10.3934/dcdsb.2020204

[4]

Seung-Yeal Ha, Myeongju Kang, Hansol Park. Collective behaviors of the Lohe Hermitian sphere model with inertia. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2613-2641. doi: 10.3934/cpaa.2021046

[5]

Hyungjun Choi, Seung-Yeal Ha, Hansol Park. Emergent behaviors of discrete Lohe aggregation flows. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021308

[6]

Wolfgang Wagner. A random cloud model for the Wigner equation. Kinetic and Related Models, 2016, 9 (1) : 217-235. doi: 10.3934/krm.2016.9.217

[7]

Andrea Tosin, Paolo Frasca. Existence and approximation of probability measure solutions to models of collective behaviors. Networks and Heterogeneous Media, 2011, 6 (3) : 561-596. doi: 10.3934/nhm.2011.6.561

[8]

Thomas Chen, Ryan Denlinger, Nataša Pavlović. Moments and regularity for a Boltzmann equation via Wigner transform. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 4979-5015. doi: 10.3934/dcds.2019204

[9]

Dongsheng Yin, Min Tang, Shi Jin. The Gaussian beam method for the wigner equation with discontinuous potentials. Inverse Problems and Imaging, 2013, 7 (3) : 1051-1074. doi: 10.3934/ipi.2013.7.1051

[10]

Yanghong Huang, Andrea Bertozzi. Asymptotics of blowup solutions for the aggregation equation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1309-1331. doi: 10.3934/dcdsb.2012.17.1309

[11]

Orazio Muscato, Wolfgang Wagner. A stochastic algorithm without time discretization error for the Wigner equation. Kinetic and Related Models, 2019, 12 (1) : 59-77. doi: 10.3934/krm.2019003

[12]

Tomasz Komorowski, Lenya Ryzhik. Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 871-914. doi: 10.3934/dcdsb.2012.17.871

[13]

Andrea L. Bertozzi, Dejan Slepcev. Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1617-1637. doi: 10.3934/cpaa.2010.9.1617

[14]

Qi Wang. On some touchdown behaviors of the generalized MEMS device equation. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2447-2456. doi: 10.3934/cpaa.2016043

[15]

Xueke Pu, Min Li. Asymptotic behaviors for the full compressible quantum Navier-Stokes-Maxwell equations with general initial data. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 5149-5181. doi: 10.3934/dcdsb.2019055

[16]

Dongfen Bian, Huimin Liu, Xueke Pu. Modulation approximation for the quantum Euler-Poisson equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4375-4405. doi: 10.3934/dcdsb.2020292

[17]

Jitendra Kumar, Gurmeet Kaur, Evangelos Tsotsas. An accurate and efficient discrete formulation of aggregation population balance equation. Kinetic and Related Models, 2016, 9 (2) : 373-391. doi: 10.3934/krm.2016.9.373

[18]

Faustino Sánchez-Garduño, Philip K. Maini, Judith Pérez-Velázquez. A non-linear degenerate equation for direct aggregation and traveling wave dynamics. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 455-487. doi: 10.3934/dcdsb.2010.13.455

[19]

José A. Carrillo, Yanghong Huang. Explicit equilibrium solutions for the aggregation equation with power-law potentials. Kinetic and Related Models, 2017, 10 (1) : 171-192. doi: 10.3934/krm.2017007

[20]

Houda Hani, Moez Khenissi. Asymptotic behaviors of solutions for finite difference analogue of the Chipot-Weissler equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1421-1445. doi: 10.3934/dcdss.2016057

2021 Impact Factor: 1.41

Metrics

  • PDF downloads (60)
  • HTML views (36)
  • Cited by (0)

Other articles
by authors

[Back to Top]