## 3 Results  
### Lennard-Jones Crystal  
We first tested our algorithm on an fcc Lennard-Jones crystal. Results obtained using either the isotropic or the anisotropic implementations of the SCR and using both the simpler Euler integrator or the time-reversible one are reported in Figure 1, as functions of the barostat parameters \(\tau_{P}\) and \(n_{s}\). We underline that the SimpleMD implementation tested here is the only one supporting the time-reversible scheme discussed in Section 2.3.2. Figure 1a shows that systematic errors in the volume were negligible for \(\tau_{P}\gtrapprox 0.5\), whereas statistical errors, that were computed using block bootstrap and depended on the relaxation time of the volume, grew with \(\tau_{P}\). The relative error in the average volume was extremely small in all cases. The small discrepancies between the volume distributions of the isotropic and anisotropic systems were a consequence of the finite size of the simulated system, as they appeared to shrink in the thermodynamic limit (see Table 1). A different scenario was seen in Figure 1b, where the systematic error on the volume variance was significantly dependent on the parameter \(\tau_{P}\). We noted that a \(\tau_{P}\gtrapprox 10\) should have been used to make sure fluctuations were not significantly affected. Figure 1c,d reports the dependence of the volume average and variance on the multiple-time-stepping stride \(n_{s}\). It can be seen that a too-aggressive choice of \(n_{s}\) might significantly affect the estimated variance, and that, for this specific system, any choice \(n_{s}\gtrapprox 10\) produced correct results. All the results discussed here were obtained using the formulation where momenta were scaled and, in the anisotropic case, rotations were eliminated. However, the same simulations were also carried out without scaling momenta and without eliminating rotations, producing equivalent results. Figure 2 reports the effective energy drift that could be used as a measure of detailed balance violations. We noted that the drift became negligible at \(\tau_{P}\gtrapprox 2\). Beyond this value, its estimation was difficult due to the limited simulation length and the fact that the overall drift could not be distinguished from the effective energy fluctuations; thus, resulting in highly fluctuating or even negative estimates. The drift of the isotropic version followed the same trend as in Ref. [18]; in the anisotropic case, the drift was similar but slightly larger. We then tested a Lennard-Jones system using the community code GROMACS. Here, we used physical units corresponding to the parametrization of Ar. Results were comparable to those obtained with SimpleMD, but allowed to directly compare SCR with the PR algorithm [3,4] and with the MTTK algorithm [8,9], as implemented in GROMACS (Figure 3). We noted that the MTTK algorithm was only implemented in its isotropic variant. Here, it can be seen that the averages and fluctuations for all methods were in agreement with each other when the \(\tau_{P}\) parameter was properly chosen. It is important to remark here that there is not a direct equivalence in the choice of \(\tau_{P}\) when using different methods. On the tested range of \(\tau_{P}\), we observed that the anisotropic PR algorithm had \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{N_{\text{at}}}\) & \(\langle\boldsymbol{V}\rangle_{\text{i}}\) & \(\langle\boldsymbol{V}\rangle_{\text{a}}\) & \(\Delta\langle\boldsymbol{V}\rangle\) & \(\Delta\langle V\rangle/\langle\boldsymbol{V}\rangle_{\text{a}}\) \\ \hline
256 & 238.057 & 238.077 & 0.020 & 0.0084\% \\
500 & 464.962 & 464.979 & 0.017 & 0.0037\% \\
1372 & 1275.873 & 1275.881 & 0.008 & 0.0006\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Differences between isotropic and anisotropic volume averages observed by simulating Lennard-Jones crystals of different sizes with SimpleMD.