Two-dimensional lattice liquid models

A family of models of liquid on a two-dimensional lattice (2D lattice liquid models) have been proposed as primitive models of soft-material membrane. As a first step, we have formulated them as single-component, single-layered, classical particle systems on a two-dimensional surface with no explicit viscosity. Among the family of the models, we have shown and constructed two stochastic models, a vicious walk model and a flow model, on an isotropic regular lattice and on some honeycomb lattices of various sizes. In both cases, the dynamics is governed by the nature of the frustration of the particle movements. By simulations, we have found the approximate functional form of the frustration probability and peculiar anomalous diffusions in their time-averaged mean-square displacements in the flow model. The relations to other existing statistical models and possible extensions of the models are also discussed.

Figure 1

One of the “world lines” illustrating a particle traveling on a lattice.

Figure 2

(a) An example of the frustration of the velocities on a triangular lattice. Both red and blue sequences of vectors show the frustration. (b) An example of the possible movements on a triangular lattice. Red and blue sequences of vectors show the circulations.

Figure 3

The illustration of the rectangular honeycomb lattice ${\Lambda }_{h_N}$.

Figure 4

The probability of the no-loop configuration for various sizes in the log scale in the $y$ direction. The $x$ axis stands for the area of the lattice counted by the number of hexagonal faces in $\Lambda$. There is an obvious scalability. The error bars are given at the $95\%$ confidence level.

Figure 5

Distribution of the total length of the loops generated in one time step. The $x$ axis is the length in the unit of the lattice spacing $a$. The $y$ shows the number of events in a million trials. Periodic oscillations seem to suggest multiple entries of the same type of loops, e.g., two length-six loops at once contributes to the length-twelve event.

Figure 6

An example of the two-point correlation function of $\langle |{\mathbf{v}}_i\left|\right|{\mathbf{v}}_j|\rangle$, one of which is fixed at the center of the lattice ${\Lambda }_{h_{21}}$. The $z$ axis is the number of events in a million trials in the log scale.

Figure 7

An illustration of the lattice liquid on a honeycomb lattice. The sequences of filled balls show the movements of the particles. The lattice is periodic to the horizontal direction.

Figure 8

The average probability of the fluctuations in the million steps. The probability suddenly drops down between $N=31$ and 41.

Figure 9

The distribution of the size of the individual loops. There is an almost uniform scaling region in the middle.

Figure 10

The time-averaged mean-square displacements (TAMSDs) of the stochastic flow model (II) on ${\Lambda }_{h_N}$ for various sizes in the log-log format. The $x$ axis is the time lapse $t$, while the $y$ axis is the square distance in the unit of the lattice spacing squared $a^2$. The black dashed line represents the slope of the Brownian motion in the mean-square displacements, while the colored dashed lines are all exactly calculated asymptotic values for various $N$.

Figure 11

A comparison of TAMSD between different models with flows on ${\Lambda }_{h_{21}}$. The line for the nonreversal random walk is given with the trapping time consistent with the original model, and the asymptotic value for ${\Lambda }_{h_{21}}$ is shown. The thick dashed line stands for the slope of the Brownian motion.