zbMATH — the first resource for mathematics

Some connections between the classical Calogero-Moser model and the log-gas. (English) Zbl 1426.82037
Summary: 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.

MSC:
 82C22 Interacting particle systems in time-dependent statistical mechanics 60B20 Random matrices (probabilistic aspects) 15B52 Random matrices (algebraic aspects) 82M31 Monte Carlo methods applied to problems in statistical mechanics 65C05 Monte Carlo methods
Full Text:
References:
 [1] Forrester, P.J.: Log-Gases and Random Matrices (LMS-34). Princeton University Press, Princeton (2010) · Zbl 1217.82003 [2] Nagao, T.; Forrester, PJ, Asymptotic correlations at the spectrum edge of random matrices, Nucl. Phys. B., 435, 401, (1995) · Zbl 1020.82588 [3] Bourgade, P.: Bulk universality for one-dimensional log-gases. In XVIIth International Congress on Mathematical Physics. World Scientific, pp. 404-416 (2014) · Zbl 1304.82018 [4] Erdos, L.: Universality for random matrices and log-gases. arXiv:1212.0839 (2012) · Zbl 1291.15086 [5] Ameur, Y.; Hedenmalm, H.; Makarov, N.; etal., Fluctuations of eigenvalues of random normal matrices, Duke Math. J., 159, 31, (2011) · Zbl 1225.15030 [6] Deift, P.: Universality for mathematical and physical systems. math-ph/0603038 (2006) [7] Tracy, CA; Widom, H., Correlation functions, cluster functions, and spacing distributions for random matrices, J. Stat. Phys., 92, 809, (1998) · Zbl 0942.60099 [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) · Zbl 1176.15046 [9] Widom, H., On the relation between orthogonal, symplectic and unitary matrix ensembles, J. Stat. Phys., 94, 347, (1999) · Zbl 0935.60090 [10] Baik, J.; Arous, GB; Péché, S.; etal., Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices, Ann. Probab., 33, 1643, (2005) · Zbl 1086.15022 [11] Baker, T.; Forrester, P., Finite-N fluctuation formulas for random matrices, J. Stat. Phys., 88, 1371, (1997) · Zbl 0939.82020 [12] Nagao, T.; Forrester, PJ, Transitive ensembles of random matrices related to orthogonal polynomials, Nucl. Phys. B, 530, 742, (1998) · Zbl 1047.82519 [13] Mehta, M.L.: Random Matrices, vol. 142. Elsevier, Amsterdam (2004) · Zbl 1107.15019 [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, 1045, (2010) · Zbl 1196.15036 [16] Zhang, D., Gaussian fluctuations of eigenvalues in log-gas ensemble: bulk case I, Acta Math. Sin. Engl. Ser., 31, 1487, (2015) · Zbl 1328.15053 [17] Bornemann, F.: On the numerical evaluation of distributions in random matrix theory: a review. arXiv:0904.1581 (2009) · Zbl 1222.60013 [18] Calogero, F., Exactly solvable one-dimensional many-body problems, Lett. Nuovo Cimento (1971-1985), 13, 411, (1975) [19] Calogero, F., Solution of a three-body problem in one dimension, J. Math. Phys., 10, 2191, (1969) [20] Calogero, F., Solution of the one-dimensional n-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys., 12, 419, (1971) [21] Moser, J., Three integrable Hamiltonian systems connnected with isospectral deformations, Adv. Math., 16, 197, (1975) · Zbl 0303.34019 [22] Bogomolny, E.; Giraud, O.; Schmit, C., Random matrix ensembles associated with lax matrices, Phys. Rev. Lett., 103, 054103, (2009) [23] Kulkarni, M.; Polychronakos, A., Emergence of the Calogero family of models in external potentials: duality, solitons and hydrodynamics, J. Phys. A, 50, 455202, (2017) · Zbl 1386.76045 [24] Polychronakos, AP, The physics and mathematics of Calogero particles, J. Phys. A, 39, 12793, (2006) · Zbl 1109.81086 [25] Olshanetsky, M.; Perelomov, AM, Classical integrable finite-dimensional systems related to Lie algebras, Phys. Rep., 71, 313, (1981) [26] Perelomov, A.M.: Integrable Systems of Classical Mechanics and Lie Algebras. Birkhäuser, Basel (1990) · Zbl 0717.70003 [27] Abanov, AG; Gromov, A.; Kulkarni, M., Soliton solutions of a Calogero model in a harmonic potential, J. Phys. A, 44, 295203, (2011) · Zbl 1223.37080 [28] Aniceto, I.; Avan, J.; Jevicki, A., Poisson structures of Calogero-Moser and Ruijsenaars-Schneider models, J. Phys. A, 43, 185201, (2010) · Zbl 1194.37099 [29] Michael Stone, I.A., Xing, L.: The classical hydrodynamics of the Calogero-Sutherland model. J. Phys. A 41, (2008) · Zbl 1146.76009 [30] Franchini, F.; Gromov, A.; Kulkarni, M.; Trombettoni, A., Universal dynamics of a soliton after an interaction quench, J. Phys. A, 48, 28ft01, (2015) · Zbl 1327.35333 [31] Franchini, F.; Kulkarni, M.; Trombettoni, A., Hydrodynamics of local excitations after an interaction quench in 1D cold atomic gases, N. J. Phys., 18, 115003, (2016) [32] Calogero, F., Equilibrium configuration of the one-dimensionaln-body problem with quadratic and inversely quadratic pair potentials, Lett. Nuovo Cimento. (1971-9185), 20, 251, (1977) [33] Sutherland, B., Quantum many-body problem in one dimension: ground state, J. Math. Phys., 12, 246, (1971) [34] Sutherland, B., Exact results for a quantum many-body problem in one dimension. II, Phys. Rev. A, 5, 1372, (1972) [35] Dhar, A.; Kundu, A.; Majumdar, SN; Sabhapandit, S.; Schehr, G., Exact extremal statistics in the classical 1D Coulomb gas, Phys. Rev. Lett., 119, 060601, (2017) [36] Forrester, P.; Rogers, J., Electrostatics and the zeros of the classical polynomials, SIAM J. Math. Anal., 17, 461, (1986) · Zbl 0613.33009 [37] Calogero, F., Matrices, differential operators, and polynomials, J. Math. Phys., 22, 919, (1981) · Zbl 0473.33009 [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) · Zbl 0044.44203 [39] Nadal, C.; Majumdar, SN, A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix, J. Stat. Mech., 2011, p04001, (2011) [40] Szeg, G.: Orthogonal Polynomials, vol. 23. American Mathematical Soc, Providence, RI (1939) [41] Pathria, R.: Statistical mechanics. International Series in Natural Philosophy (1986) · Zbl 0862.00007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.