Event-chain Monte Carlo with factor fields

We study the dynamics of one-dimensional (1D) interacting particles simulated with the event-chain Monte Carlo algorithm (ECMC). We argue that previous versions of the algorithm suffer from a mismatch in the factor potential between different particle pairs (factors) and show that in 1D models, this mismatch is overcome by factor fields. ECMC with factor fields is motivated, in 1D, for the harmonic model, and validated for the Lennard-Jones model as well as for hard spheres. In 1D particle systems with short-range interactions, autocorrelation times generally scale with the second power of the system size for reversible Monte Carlo dynamics, and with its first power for regular ECMC and for molecular dynamics. We show, using simulations, that the autocorrelation time grows only with the square root of the system size for ECMC with factor fields. Mixing times, which bound the time to reach equilibrium from an arbitrary initial configuration, grow with the first power of the system size.

Figure 1

ECMC dynamics in the harmonic model. Particle $i$ is active. Blue solid and red dashed curves indicate the factor potentials $U_{\text{harm}}(x_{i+1}-x_i)$ and $U_{\text{harm}}(x_i-x_{i-1})$, respectively. The proposed moves are indicated by solid arrows, the temperature $T$ by vertical dashed lines. (a) For $b=0$, the factor potential with $i-1$ leads to a much smaller displacement than the potential with $i+1$ so particle $i-1$ is more likely to be triggered as the next active particle. (b) For $b=L/\left(2N\right)$, the move proposed by $i-1$ is larger than in (a), but it is still smaller than that of $i+1$. (c) For $b=L/N$, the interactions with particles $i-1$ and $i+1$ overlap, and the moves proposed by $i-1$ and by $i+1$ are comparable in amplitude.

Figure 2

Equilibrium autocorrelation times $\tau$ vs system size $N$, for the 1D periodic Lennard-Jones model. Reversible local Markov-chain dynamics, ECMC with restarts and MD. (a) $T/\epsilon =10$, combined factors, ${\mathcal{M}}_{\text{LJ}}$. (b) $T/\epsilon =0.1$, separate factors, ${\mathcal{M}}_{6+12}$. Scalings $\tau \sim N$ and $\tau \sim N^2$ are indicated with dotted lines.

Figure 3

Equilibrium autocorrelation timess $\tau$ vs system size $N$, for the 1D periodic Lennard-Jones system with factor fields. (a) Reversible local Metropolis dynamics and ECMC with and without restarts, at high $T,T/\epsilon =10$ and low $T,T/\epsilon =0.01$. Scalings $\tau \sim N^{1/2},\tau \sim N$, and $\tau \sim N^2$ are indicated with dotted lines. (b) Pressure $P\left(T\right)$ and its $T\to 0$ limit [see Eq. (12)].

Figure 4

ECMC dynamics with factor fields (indicated by inclined straight lines) for 1D hard spheres. The moves of the active sphere $i$ proposed by the factors with $i+1$ and $i-1$ are indicated by horizontal arrows and dashed sphere positions. In the optimal dynamics, the slopes of the factor fields equal $\pm P$ in Eq. (16).

Figure 5

Autocorrelation time $\tau$ (in sweeps) for 1D hard spheres. (a) Reversible local Metropolis MCMC, ECMC (with restarts) and without factor field and ECMC (without restarts) with optimal factor field $h^{\text{fact}}$. Scalings $\tau \sim N^{1/2},\tau \sim N$, and $\tau \sim N^2$ are indicated with dotted lines. (b) Evolution of autocorrelation time for nonoptimal factor fields. Use of a factor field $h^{\text{fact}}\ne P$ still gives acceleration. Data for $0.5P<h^{\text{fact}}<1.5P$ (ECMC, without restarts).

Figure 6

Variance of the half-system distance $w$ [see Eq. (17)]vs time for various $N$ [hard-sphere model with factor fields (without restarts)]. The observable relaxes to its equilibrium value at the mixing time $(1.000\pm 0.005)\times N$ sweeps for the hard-sphere model with factor fields (without restarts).

Figure 7

Position $x_i$ of the active sphere $i$ vs time $t[N=4096,\sigma =L/(2N\left)\right]$. At $t=0$, the interval $[-L/2,0]$ is close packed, the interval $[0,L/2]$ is empty as illustrated below the $x$ axis, where the rightmost sphere is active. The explored space expands through oscillations, growing as $\sqrt{t}$, and reaches $[x_1,x_N]\simeq L$ at $t\simeq N$. The inset illustrates the position of $x_i$ (without periodic wrapping) on a larger time interval, showing that $x_i$ moves on larger spatial scales in an irregular manner after $t\simeq N$.

Figure 8

Equilibrium autocorrelation of event steps $u\in \{-1,1\}$ with event time $s$ for ECMC with factor field (no restarts). (a) 1D Lennard-Jones model shows monotonic decay. (b) 1D hard spheres display oscillatory behavior with a power-law envelope. The scaling $〈u\left(0\right)u\left(s\right)〉\sim s^{-2/3}$ is indicated with dotted lines [see Eq. (18)].

Figure 9

ECMC with factor fields for a 1D Lennard-Jones system $\left(N=8192,T/\epsilon =1\right)$. Probability $p(n,s)$ to return $n$ times to the original active particle during $s$ events (for $s=2500$ and $10000)$. Mean and standard deviation of $p(n,s)$ both grow as $s^{\gamma /2}\sim s^{1/3}$ [see Eq. (25)]. Inset: data collapse using scaling variables $p(n,s)s^{1/3}$ vs $n/s^{1/3}$.