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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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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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