Large-scale dynamics of event-chain Monte Carlo

Event-chain Monte Carlo (ECMC) accelerates the sampling of hard-sphere systems, and has been generalized to the potentials used in classical molecular simulations. Rather than imposing detailed balance on the transition probabilities, the method enforces a weaker global-balance condition in order to guarantee convergence to equilibrium. In this paper, we generalize the factor-field variant of ECMC to higher space dimensions. In the two-dimensional fluid phase, factor-field ECMC saturates the lower bound $z=0$ for the dynamical scaling exponent for local dynamics, whereas molecular dynamics is characterized by $z=1$ and local Metropolis Monte Carlo by $z=2$. In the presence of hexatic order, factor fields are not found to speed up the convergence. We note that generalizations of factor fields could couple to orientational order.

Figure 1

Spheres in a 2D box, with a grating of lanes derived from a 2D cell system, shown as squares. Hard-sphere lifting moves correspond to collisions (for example of the active sphere $B$ with $C$). Factor-field lifting moves always concern same-lane spheres (for example $A$, for the active sphere $B$).

Figure 2

Active-sphere trajectory featuring anisotropic 2D diffusion of a single event chain with direction ${\widehat{\mathbf{e}}}_x$ for the 2D hard-sphere system at $\eta =0.67$. The optimal $h_{\text{ff}}=P{w}_{\text{lane}}$ is used ($N={192}^2$).

Figure 3

Maximum distance of the empirical cumulated distribution function of $rg\left(r\right)$ for $2\sigma <r<4\sigma$, with and without a factor field for $n_c$ recordings (i.e., $n_c$ $N$-sphere samples). The $\sim 1/\sqrt{n_c}$ scaling of this Kolmogorov–Smirnov-like statistic indicates that $rg\left(r\right)$ (shown in the inset) is independent of $h_{\text{ff}}$ ($N={64}^2$, $\eta =0.67$).

Figure 4

Selected eigenmodes $v^{\left(k\right)}$ with eigenvalues ${\lambda }^{\left(k\right)}$ of the steady-state equal-time correlation matrix of Eq. (5) [moves in ${\widehat{\mathbf{e}}}_x$ from a given initial configuration $\mathbf{x}(t=0)]$. Red and blue denote positive and negative displacements of $x_i$ around its mean. (a), (b) Simple shears coexist with (c) higher-order shear, (d) compression, and mixed modes ($N={256}^2$, $\eta =0.67$).

Figure 5

Autocorrelation of the static eigenmodes from Eq. (6) [legends refer to Fig. 4, displacements in ${\widehat{\mathbf{e}}}_x$ from a given initial configuration $\mathbf{x}(t=0)$, center-of-mass motion subtracted]. (a) Factor field $h_{\text{ff}}=0$, large chain times ${\tau }_{\text{chain}}$ (corresponding to ${\mathrm{\Delta }}_{\text{chain}}=3.1L$): Compression eigenmodes relax more slowly that shear eigenmodes and are oscillatory. (b) Factor field $h_{\text{ff}}=0$, small ${\tau }_{\text{chain}}$ (corresponding to ${\mathrm{\Delta }}_{\text{chain}}=0.54L$): Shear eigenmodes may relax more slowly than compression eigenmodes. (c) Factor fields $h_{\text{ff}}=P{w}_{\text{lane}}$: All eigenmodes decay rapidly, on similar timescales ($N={256}^2$, $\eta =0.67$).

Figure 6

Integrated ECMC autocorrelation time for the eigenmodes of Figs. 4 and 4 in the harmonic approximation of Eq. (7). [Chain direction ${\widehat{\mathbf{e}}}_x$ moves from given initial configurations $\mathbf{x}(t=0)$.] The scaling exponent for $h_{\text{ff}}=P{w}_{\text{lane}}$ appears to saturate the lower bound $z=0$ (density $\eta =0.67$).

Figure 7

Autocorrelation $R^S\left(\tau \right)$ of the structure factor $S_{\mathbf{q}}$ for full ECMC in the fluid phase (density $\eta =0.67$). With factor fields, the autocorrelations seem to decay on a timescale that is independent of $N$, indicating $z=0$.

Figure 8

Autocorrelation $R^{\psi }\left(\tau \right)$ of the squared amplitude ${\left|\psi \right|}^2$ of the global orientational order parameter in the two-phase region ($\eta =0.708$). The autocorrelations decay on a timescale that increases with $N$ both for $h_{\text{ff}}=0$ and for $h_{\text{ff}}=P{w}_{\text{lane}}$, where the simulation is slightly slowed down