Relaxation to the equilibrium in the hard disk dynamics

TL;DR

This paper investigates the relaxation process to equilibrium in hard disk dynamics, focusing on the exponential distribution of displacements in Event-Chain Monte Carlo simulations.

Contribution

It provides insights into the relaxation criteria and confirms the exponential displacement distribution in the context of hard disk dynamics.

Findings

Displacement distributions follow an exponential law.

Relaxation criteria for hard disk systems are examined.

Event-Chain Monte Carlo effectively models relaxation behavior.

Abstract

We examine the question of the criteria of the relaxation to the equilibrium in the hard disk dynamics. In the Event-Chain Monte Carlo, we check the displacement distributions which follows to the exponential law.

Figures

Tables

Table 1. Table 1: Approximation of the distributions of the displacements Δ ​ r x Δ subscript 𝑟 𝑥 \Delta r_{x} pictured in the Figure  2 . Density η 𝜂 \eta defines the disk radius σ 𝜎 \sigma and the free path value λ 𝜆 \lambda . The λ f ​ i ​ t subscript 𝜆 𝑓 𝑖 𝑡 \lambda_{fit} is the value of the fit to expression ( 1 ) and χ 2 superscript 𝜒 2 \chi^{2} is the valued of the c ​ h ​ i 2 𝑐 ℎ superscript 𝑖 2 chi^{2} .

$\eta$	$\sigma$	$\lambda$	${\lambda }_{fit}$	${\chi }^2$
0.68	0.116311	0.02474	0.02382(4)	0.037
0.71	0.118849	0.02220	0.01980(7)	0.138
0.73	0.120511	0.02056	0.01628(6)	0.226

Equations

$$P(\Delta r)=\frac{1}{\lambda }\exp \left(-\frac{\Delta r}{\lambda }\right),$$

Full text

††preprint: APS/123-QED

We restrict our current analysis to the case of the hard disks. In molecular dynamics simulations, it is important to know the relaxation time from the initial state to equilibrium. Which quantities of the equilibrium can give information that system reach equilibrium? In conventional Event Driven Molecular Dynamics it is the relaxation time to the Maxwell distribution which can be used as the estimate. In the case of the Event-Chain Monte Carlo EC09 the velocities are not defined. Instead, we can use distribution of the displacements. It is known from the computer simulations Alder65 that displacements $\delta r$ follow to the exponential law in the equilibrium

[TABLE]

and theoretical arguments based on the kinetic theory support the findings Turnbull70 .

We report the preliminary results for the system of a small number of disks $N=16$ in the box of the linear size $L=1.0$.

Definition of displacements is given in the Figure 1. The real shifts $\Delta r_{x}$ are performed along the $x$-axes, and the formal shifts $\Delta r_{y}$ are defined in the perpendicular direction, as the difference of the projections of the centrum of colliding disks on the $y$-axes. We also define the total displacement $\Delta r$ as shown in the Figure 1.

We plot distribution of the displacements $\Delta r_{x}$ in the direction of the moves for the ECMC in the Figure 2 for three values of densities, $\eta=068,0.71,0.73$. The corresponding values of the mean free path are given in the Table 1. Value of $\lambda$ is defined with the density $\eta$ as $\lambda=\sqrt{L^{2}/\pi N}-\sigma$, the free volume for the particle. Estimation of the value $\lambda_{fit}$ from the fit is close to the free-path value $\lambda$ for the low density, as can be expected from the definition of $\lambda$, and diverges for the large densities $\eta$.

In the ECMC, the direction of the moves alternates from $x$-axes to the $y$-axes, therefore all real shifts of the disks follows to the expected distribution (1) and provides the equilibrium state of the system. In a way, this is an additional argument for the validity of the Event-Chain Monte Carlo approach for simulation of hard disks (and spheres).

The work is supported by the grant 17-07-01537 from Russian Foundation for Basic Research.