DOI QR코드

DOI QR Code

REMARKS ON THE GRADIENT FLOW OF α ENERGY POTENTIAL ON THE LINE

  • Hyojun An (Department of Mathematics Chung-Ang University) ;
  • Hyungjin Huh (Department of Mathematics Chung-Ang University)
  • Received : 2022.12.19
  • Accepted : 2023.03.08
  • Published : 2023.10.31

Abstract

We are interested in the gradient flow of α energy potential. We provide basic estimates and study asymptotic behaviors for the case N = 2, . . . , 5.

Keywords

Acknowledgement

H. Huh was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (2020R1F1A1A01072197). We would like to thank the anonymous referee for the useful comments on the manuscript.

References

  1. S. Agarwal, M. Kulkarni, and A. Dhar, Some connections between the classical Calogero-Moser model and the log-gas, J. Stat. Phys. 176 (2019), no. 6, 1463-1479. https://doi.org/10.1007/s10955-019-02349-6
  2. S. Andraus, M. Katori, and S. Miyashita, Interacting particles on the line and Dunkl intertwining operator of type A: application to the freezing regime, J. Phys. A 45 (2012), no. 39, 395201, 26 pp. https://doi.org/10.1088/1751-8113/45/39/395201
  3. S. Andraus and M. Voit, Limit theorems for multivariate Bessel processes in the freezing regime, Stochastic Process. Appl. 129 (2019), no. 11, 4771-4790. https://doi.org/10.1016/j.spa.2018.12.011
  4. F. Calogero, Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials, J. Mathematical Phys. 12 (1971), 419-436. https://doi.org/10.1063/1.1665604
  5. F. Calogero and F. Leyvraz, Many-body problem with quadratic and/or inversely-quadratic potentials in one- and more-dimensional spaces: some retrospective remarks, J. Stat. Phys. 155 (2014), no. 4, 658-665. https://doi.org/10.1007/s10955-014-0973-3
  6. F. Calogero and C. Marchioro, Exact solution of a one-dimensional three-body scattering problem with two-body and/or three-body inverse-square potentials, J. Mathematical Phys. 15 (1974), 1425-1430. https://doi.org/10.1063/1.1666827
  7. P. J. Forrester and J. B. Rogers, Electrostatics and the zeros of the classical polynomials, SIAM J. Math. Anal. 17 (1986), no. 2, 461-468. https://doi.org/10.1137/0517035
  8. R. A. Horn and C. R. Johnson, Matrix Analysis, second edition, Cambridge Univ. Press, Cambridge, 2013.
  9. F. Marcellan Espanol, A. Martinez-Finkelshtein, and P. Martinez-Gonzalez, Electrostatic models for zeros of polynomials: old, new, and some open problems, J. Comput. Appl. Math. 207 (2007), no. 2, 258-272. https://doi.org/10.1016/j.cam.2006.10.020
  10. J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, in Surveys in applied mathematics (Proc. First Los Alamos Sympos. Math. in Natural Sci., Los Alamos, N.M., 1974), 235-258, Academic Press, New York, 1976.
  11. J. F. Peter, Log-Gases and Random Matrices, Princeton University Press, 2010.
  12. M. Voit and J. H. C. Woerner, The differential equations associated with Calogero-Moser-Sutherland particle models in the freezing regime, Hokkaido Math. J. 51 (2022), no. 1, 153-174. https://doi.org/10.14492/hokmj/2020-307