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Some Connections Between the Classical Calogero–Moser Model and the Log-Gas

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Abstract

In this work we discuss connections between a one-dimensional system of N particles interacting with a repulsive inverse square potential and confined in a harmonic potential (Calogero–Moser model) and the log-gas model which appears in random matrix theory. Both models have the same minimum energy configuration, with the particle positions given by the zeros of the Hermite polynomial. Moreover, the Hessian describing small oscillations around equilibrium are also related for the two models. The Hessian matrix of the Calogero–Moser model is the square of that of the log-gas. We explore this connection further by studying finite temperature equilibrium properties of the two models through Monte–Carlo simulations. In particular, we study the single particle distribution and the marginal distribution of the boundary particle which, for the log-gas, are respectively given by the Wigner semi-circle and the Tracy–Widom distribution. For particles in the bulk, where typical fluctuations are Gaussian, we find that numerical results obtained from small oscillation theory are in very good agreement with the Monte–Carlo simulation results for both the models. For the log-gas, our findings agree with rigorous results from random matrix theory.

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References

  1. Forrester, P.J.: Log-Gases and Random Matrices (LMS-34). Princeton University Press, Princeton (2010)

    Book  MATH  Google Scholar 

  2. Nagao, T., Forrester, P.J.: Asymptotic correlations at the spectrum edge of random matrices. Nucl. Phys. B. 435(3), 401 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Bourgade, P.: Bulk universality for one-dimensional log-gases. In XVIIth International Congress on Mathematical Physics. World Scientific, pp. 404–416 (2014)

  4. Erdos, L.: Universality for random matrices and log-gases. arXiv:1212.0839 (2012)

  5. Ameur, Y., Hedenmalm, H., Makarov, N., et al.: Fluctuations of eigenvalues of random normal matrices. Duke Math. J. 159(1), 31 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Deift, P.: Universality for mathematical and physical systems. math-ph/0603038 (2006)

  7. Tracy, C.A., Widom, H.: Correlation functions, cluster functions, and spacing distributions for random matrices. J. Stat. Phys. 92(5–6), 809 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Tracy, C.A., Widom, H.: The distributions of random matrix theory and their applications. In New trends in mathematical physics. Springer, New York, pp. 753–765 (2009)

  9. Widom, H.: On the relation between orthogonal, symplectic and unitary matrix ensembles. J. Stat. Phys. 94(3–4), 347 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Baik, J., Arous, G.B., Péché, S., et al.: Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33(5), 1643 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Baker, T., Forrester, P.: Finite-N fluctuation formulas for random matrices. J. Stat. Phys. 88(5–6), 1371 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Nagao, T., Forrester, P.J.: Transitive ensembles of random matrices related to orthogonal polynomials. Nucl. Phys. B 530(3), 742 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Mehta, M.L.: Random Matrices, vol. 142. Elsevier, Amsterdam (2004)

    MATH  Google Scholar 

  14. Gustavsson, J.: Gaussian fluctuations of eigenvalues in the GUE. Ann. L’Inst. Henri Poincare Sect. B Probab. Stat. 41, 151 (2005). https://doi.org/10.1016/j.anihpb.2004.04.002

  15. O’Rourke, S.: Gaussian fluctuations of eigenvalues in Wigner random matrices. J. Stat. Phys. 138(6), 1045 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Zhang, D.: Gaussian fluctuations of eigenvalues in log-gas ensemble: bulk case I. Acta Math. Sin. Engl. Ser. 31(9), 1487 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bornemann, F.: On the numerical evaluation of distributions in random matrix theory: a review. arXiv:0904.1581 (2009)

  18. Calogero, F.: Exactly solvable one-dimensional many-body problems. Lett. Nuovo Cimento (1971–1985) 13(11), 411 (1975)

    Article  MathSciNet  Google Scholar 

  19. Calogero, F.: Solution of a three-body problem in one dimension. J. Math. Phys. 10(12), 2191 (1969)

    Article  ADS  MathSciNet  Google Scholar 

  20. Calogero, F.: Solution of the one-dimensional n-body problems with quadratic and/or inversely quadratic pair potentials. J. Math. Phys. 12(3), 419 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  21. Moser, J.: Three integrable Hamiltonian systems connnected with isospectral deformations. Adv. Math. 16, 197 (1975). https://doi.org/10.1016/0001-8708(75)90151-6

    Article  ADS  MATH  Google Scholar 

  22. Bogomolny, E., Giraud, O., Schmit, C.: Random matrix ensembles associated with lax matrices. Phys. Rev. Lett. 103(5), 054103 (2009)

    Article  ADS  Google Scholar 

  23. Kulkarni, M., Polychronakos, A.: Emergence of the Calogero family of models in external potentials: duality, solitons and hydrodynamics. J. Phys. A 50(45), 455202 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Polychronakos, A.P.: The physics and mathematics of Calogero particles. J. Phys. A 39(41), 12793 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Olshanetsky, M., Perelomov, A.M.: Classical integrable finite-dimensional systems related to Lie algebras. Phys. Rep. 71(5), 313 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  26. Perelomov, A.M.: Integrable Systems of Classical Mechanics and Lie Algebras. Birkhäuser, Basel (1990)

    Book  MATH  Google Scholar 

  27. Abanov, A.G., Gromov, A., Kulkarni, M.: Soliton solutions of a Calogero model in a harmonic potential. J. Phys. A 44(29), 295203 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Aniceto, I., Avan, J., Jevicki, A.: Poisson structures of Calogero-Moser and Ruijsenaars-Schneider models. J. Phys. A 43(18), 185201 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Michael Stone, I.A., Xing, L.: The classical hydrodynamics of the Calogero-Sutherland model. J. Phys. A 41, (2008)

  30. Franchini, F., Gromov, A., Kulkarni, M., Trombettoni, A.: Universal dynamics of a soliton after an interaction quench. J. Phys. A 48(28), 28FT01 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Franchini, F., Kulkarni, M., Trombettoni, A.: Hydrodynamics of local excitations after an interaction quench in 1D cold atomic gases. N. J. Phys. 18(11), 115003 (2016)

    Article  Google Scholar 

  32. Calogero, F.: Equilibrium configuration of the one-dimensionaln-body problem with quadratic and inversely quadratic pair potentials. Lett. Nuovo Cimento. (1971–9185) 20(7), 251 (1977)

    Article  Google Scholar 

  33. Sutherland, B.: Quantum many-body problem in one dimension: ground state. J. Math. Phys. 12(2), 246 (1971)

    Article  ADS  Google Scholar 

  34. Sutherland, B.: Exact results for a quantum many-body problem in one dimension. II. Phys. Rev. A 5(3), 1372 (1972)

    Article  ADS  Google Scholar 

  35. Dhar, A., Kundu, A., Majumdar, S.N., Sabhapandit, S., Schehr, G.: Exact extremal statistics in the classical 1D Coulomb gas. Phys. Rev. Lett. 119, 060601 (2017)

    Article  ADS  Google Scholar 

  36. Forrester, P., Rogers, J.: Electrostatics and the zeros of the classical polynomials. SIAM J. Math. Anal. 17(2), 461 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  37. Calogero, F.: Matrices, differential operators, and polynomials. J. Math. Phys. 22(5), 919 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Wigner, E.P.: On the statistical distribution of the widths and spacings of nuclear resonance levels. In Mathematical Proceedings of the Cambridge Philosophical Society, vol. 47, pp. 790–798. Cambridge University Press, Cambridge (1951)

  39. Nadal, C., Majumdar, S.N.: A simple derivation of the Tracy–Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix. J. Stat. Mech. 2011(04), P04001 (2011)

    Article  MathSciNet  Google Scholar 

  40. Szeg, G.: Orthogonal Polynomials, vol. 23. American Mathematical Soc, Providence, RI (1939)

    Google Scholar 

  41. Pathria, R.: Statistical mechanics. International Series in Natural Philosophy (1986)

Download references

Acknowledgements

We thank Anirban Basak, Manjunath Krishnapur, Anupam Kundu, Arul Lakshminarayan, Joseph Samuel, Herbert Spohn, Patrik Ferrari and Satya Majumdar for useful discussions. AD would like to acknowledge support from the Project 5604-2 of the Indo-French Centre for the Promotion of Advanced Research (IFCPAR). MK would like to acknowledge support from the Project 6004-1 of the Indo-French Centre for the Promotion of Advanced Research (IFCPAR). MK gratefully acknowledges the Ramanujan Fellowship SB/S2/RJN-114/2016 from the Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India. SA would like to acknowledge support from the Long Term Visiting Students’ Programme, 2018 at International Centre for Theoretical Sciences, Tata Institute of Fundamental Research (ICTS-TIFR), Bengaluru. SA is also grateful to Aditya Vijaykumar and Junaid Bhat for helpful discussions. We would like to thank the ICTS program “Universality in random structures: Interfaces, Matrices, Sandpiles (Code: ICTS/URS2019/01)” for enabling valuable discussions with many participants.

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Authors and Affiliations

  1. Birla Institute of Technology and Science, Pilani, 333031, India

    Sanaa Agarwal

  2. International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru, 560089, India

    Sanaa Agarwal, Manas Kulkarni & Abhishek Dhar

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  1. Sanaa Agarwal
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  2. Manas Kulkarni
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  3. Abhishek Dhar
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Correspondence to Sanaa Agarwal.

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Communicated by Hal Tasaki.

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Equilibration Checks of the Monte–Carlo Dynamics

Equilibration Checks of the Monte–Carlo Dynamics

As a test of equilibration in the system with our Monte–Carlo dynamics, we checked the level of equipartition that is attained. The general form of the equipartition theorem, for a physical system with Hamiltonian H and positional degrees of freedom \(x_i\) is [41]:

$$\begin{aligned} \left\langle x_i \frac{\partial H}{\partial x_i} \right\rangle = k_B T, \end{aligned}$$
(A.1)

for \(i=1,2,\ldots ,N\). We computed the left hand side by averaging over microstates generated by the MC dynamics. The samples were collected after every 5 MC cycles. In Fig. 11 we show the results of the equipartition check for the two systems, for \(N = 100, 200\) and averaging over \(16\times 10^7\) samples. This figure demonstrates that the numerics are already accurate and give good agreement with the generalized equipartition theorem. Some general observations are: (i) Equilibration times increase with system size, (ii) equilibration is better for bulk particles and (iii) equilibration is somewhat better for the LG model than the CM model.

Fig. 11
figure 11

The results on verification of thermal equilibration from equipartition using equation (A.1) for the CM and LG systems. We present results for two system sizes and for a sample size of \(16\times 10^7\)

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Agarwal, S., Kulkarni, M. & Dhar, A. Some Connections Between the Classical Calogero–Moser Model and the Log-Gas. J Stat Phys 176, 1463–1479 (2019). https://doi.org/10.1007/s10955-019-02349-6

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