Using thermodynamic integration, we calculated the free energies of the solid and liquid phases, which we used to localize the melting transition. Both constant density and constant pressure simulations of the transition region confirmed the predictions of the free energy analysis. We initialized a series of constant pressure simulations with a configuration from a constant density simulation of the transition region.
Chapter 1: Introduction
Furthermore, the correlation function of the two points diverges at the phase boundary, which means that all particle positions in the system are correlated. Their pressure data show a van der Waals loop connecting the solid and liquid regimes of the system. The two densities they studied apparently lie on the line of constant pressure phase coexistence in the two-phase region.
Pressure
A Possible Phase Diagram of the Two-Dimensional System
Temperature
D e n sity
Chapter 2: The Monte Carlo Simulation
- Introduction
Since the system we are simulating is relatively small, the effects of boundary conditions can be significant. Thus, the numerical evaluation of the partition function of a system appears to be a method for studying a physical system. However, a more complicated situation arises in a constant pressure simulation because the simulation updates the volume of the system in addition to the positions of its particles.
Potential Energy
Particle Separation
An important property of the Lennard-Jones potential is that it is a short-range force. Proper organization of the particle coordinates in the system can greatly reduce the complexity of the Monte Carlo update algorithm. Clearly, assigning the same number of cells to each node is important because it balances the computational loads of the nodes.
In the concurrent Monte Carlo update algorithm, all nodes update particles at the same time using the same procedure. While the difference in potential energy due to a particle moving is evaluated, all other particles in the system remain fixed. Although the method works even if the clocks in all nodes are different, if you use a clock.
An interesting feature of the Monte Carlo concurrent update algorithm is that it produces non-repeatable results. Although all the nodes that make up the Mark I machine use the same clock signal, each of the commun. The total time required to execute successive components of the concurrent algorithm is t^.
In the latter case, some of the particles in the specified region may not be within the FTA.
A Typical Cell
Chapter 4: Simulations of Solid and Fluid Regions
Two important quantities we measure in such simulations are the potential energy and the pressure of the system. A measure of the system related to the particle distribution is the two-point correlation function. Since a Monte Carlo sweep of the system with fewer particles takes less time, the smaller system thermalizes faster.
The other advantage is that the same amount of computing time provides more accurate expectation values for the potential energy and pressure. The distribution of the individual measurements of a quantity such as the pressure has a standard deviation that is proportional to N 3, where N is the number of particles in the system. On a sequential computer, the competing effects balance so that a simulation achieves a statistical error that is independent of the number of particles in the system.
We now describe the sequence of constant-density simulations in the solid and liquid regimes of the Lennard-Jones system. While thermalizing the simulations of the 1024 particle system, we observed a van der Waals loop in the pressure data. Tables 4.1 to 4.4 report the measurements of the potential energy and pressure for each of the simulations.
Tables 4.S and 4.6 report the results of the simulations along the p* = .95 isochore in the same format as the other tables.
Chapter 5: Free Energy Analysis
- Thermodynamic Integration
Instead, we calculate the free energy of the system using thermodynamic integration, which is a standard method for evaluating quantities that depend on the value of the parti. The reference values of the free energy that we calculate in the next section account for the kinetic energy exactly, so we must ignore its contribution to the derivatives of the free energy. To calculate the Ff, we first write the partition function for each of the nor.
After fitting the polynomial to the set of data points, we integrate the polynomial to calculate the numerical integral of the data points. The result of the numerical integration is only accurate when the polynomial correctly describes the function that the data points represent, although the fit of a poly. We assume that each of the original data points samples a normal distribution that is centered.
The first integration that we perform calculates the Helmholtz free energy of the solid at T* = .7 and p* = .95, using the values of U* in Table 4.5. As a check on the results of the polynomial fitting method, we repeat the evaluation of (5.41) using the cubic spline method of numerical integration. Before we can use the modified cubic spline to interpolate the data in Table 4.4, we must obtain the values of the second and fourth virial coefficients for the two-dimensional Lennard-Jones system at T* = .7.
In any case, the statistical error in the final results of the free energy analysis will correctly explain its uncertainty.
Chapter 6: Direct Simulation of Transition Region
- Constant-Density Simulations
- Constant-Pressure Simulations
The results of both methods of direct simulation of the system in the transition region will be compared with the predictions of the free energy analysis presented in the previous chapter. After observing the pressure increase at p* = 0.84, we performed several updates of the simulations at higher densities that lie in the transition region. Simulations of the larger system showed the same type of long-term fluctuations in the pre.
In Figures 6.3 to 6.6, we show the pressure versus sweep number for each of the densities we simulated in the transition region. In order to determine the width of the melting transition in pressure using constant density simulations, we must. Thus, constant pressure simulations should allow for much more accurate measurements of the pressure and width of the melting transition.
On the other hand, if the pressure is slightly lower than the melting pressure, all the solid will melt and the system will become completely liquid. Thus, the results of constant pressure simulations further support our interpretation that the melting transition at T* = 0.7 is a first-order phase transition. The location of the melting transition is consistent with the results of the free energy analysis in Section 5 and the constant density simulations in the previous section.
The constant-density simulation of the 4096-particle system reported in the previous section obtained the pressure at p* = 0.84 with the horizontal error bar in Figure 6.13.
Residual Corrections vs Density OlOr
The horizontal lines indicate a range of one standard deviation from the values of p/ and pf predicted by the free energy analysis.
Chapter 7: Measurement of Elastic Constants
- The Effective Lame Coefficients
In our derivation of the elastic constants we will consider only the effects of homogeneous deformations, in which u,y are constant throughout the system. We first use the inversion symmetry of the triangular lattice to show that two of the elastic constants are zero. Transforming the Cm component of the elastic tensor with the rotation matrix (7.24) and applying the invariance of the elastic constants gives
We now derive expressions for the Lamd coefficients in terms of the elastic constants in (7.29) and (7.30) so that we can estimate the Kosterlitz-Thouless parameter K. However, we do not measure the elastic constants in an undeformed crystal since the solid Leonard pressure- Jones at T* = .7 is nonzero. Differentiating t-j to obtain the second-order terms in the free energy expansion gives
Using the values of the four measurements, we can construct the expected values entering (7.29) and (7.30). Other simulations of the solid phase at different densities gave similar results for the behavior of the fluctuation terms. From table 7.4 we see that both the shear modulus and the bulk modulus of the solid decrease with decreasing density.
Thus we avoid the large statistical errors in the fluctuation terms and obtain values of the isothermal bulk modulus whose errors are about an order of magnitude smaller.
Chapter 8: Topological Defects and Angular Correlations
- Topological Defects
A closer inspection of the figure reveals that nearby vacancies in the lattice are responsible for both such defects. Thus, the predictions of the Halperin-Nelson theory seem to describe the defect structure in Figure 8.3. We see that the defect structure of the fluid appears to be similar to that of the system at p* = .84, which is in the transition region, except that the concen.
We cannot determine if any of the disclinations are free in the liquid as the large number of defects obscures their interactions. Starting with the density p* = .86, we were able to fit the angular correlation function to a function of the form given in (8.3) with a non-zero value of it. In addition, we plot the function of the form given in (8.3) that best fits the data.
We observed large fluctuations in the angular correlation functions of the configurations, causing errors in the . In our measurements of the angular correlation function, we saw each of the behaviors predicted by Halperin and Nelson. We also observed exponential decay of the correlation function at densities that lie in the fluid regime.
Thus, measurements of the angular correlation function support the interpretation of the transition region as a two-phase region rather than a hexatic phase region.
The dots indicate the positions of the particles and the additional symbols identify the defects according to the key at the top of the figure. The dots indicate the positions of the particles and the additional symbols identify the defects according to the key at the top of the figure.
Dots indicate particle positions, and additional symbols indicate errors relative to the key at the top of the figure. Dots indicate particle positions, and additional symbols indicate errors relative to the key at the top of the figure.
