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In this old paper on Monte Carlo simulations of Lennard-Jones solid, specific heat behaviour (both $c_p$ and $c_v$) have been reported. As you can see from the picture below, both $c_p$ and $c_v$ goes to $3$ at $0K$. [The units in the plot are dimensionless reduced units defined based on the Lennard-Jones potential parameters] enter image description here

Now, the heat capacity goes to zero, if the temperature approaches $0K$. Why this is not reflected here? Could this be due to the lack of quantum effects in the simulation?

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You are correct that this is due to not including quantum effects. Ref 1 in your figure is the paper cited below. In this paper, they explicitly mention that $C_v$ calculated using the cell-cluster method is in good agreement only for sufficiently high reduced-temperatures. From section IV of this paper:

...the calculation is in acceptable agreement with experiment, except for low reduced temperatures, where quantum- mechanical effects become significant.

As further illustration, this paper has a similar plot of the cell-cluster results, but instead of comparing against Monte Carlo, they compare with experimental results for xenon.

              Heat capacities: cell-cluster vs experiment

As you can see, the experimental results approach zero as they should. The cell-cluster results do at least satisfy Mayer's Relation, which says $C_P-C_V=0$ at $T=\pu{0K}$.

  1. K. Westera and E. R. Cowley Phys. Rev. B 11, 4008 (1975) DOI: 10.1103/PhysRevB.11.4008
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    $\begingroup$ pardon my ignorance, but do you know why it goes to 3, the specific heat using cell-cluster method? Because I have seen some MD simulation paper on the same topic (c_p/c_v of LJ solid), and specific heat always goes to 3 reduced unit at low (Zero) temperature. $\endgroup$ Dec 31, 2020 at 2:07
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    $\begingroup$ @Magic_Number I'm not exactly certain. I know 3 (Nk) shows up in a lot of models of heat capacity (the Dulong-Petit Law, Einstein solid), but I haven't seen any model where the zero temperature limit specifically converges to 3. $\endgroup$
    – Tyberius
    Dec 31, 2020 at 2:40

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