^{1,2}, John D. McCoy

^{1,a)}and Brian Borchers

^{2}

### Abstract

A recently developed methodology for the calculation of the dynamic heat capacity from simulation is applied to the east Ising model. Results show stretched exponential relaxation with the stretching exponent, , decreasing with decreasing temperature. For low temperatures, the logarithm of the relaxation time is approximately proportional to the inverse of the temperature squared, which is the theoretical limiting behavior predicted by theories of facilitated dynamics. In addition, an analytical approach is employed where the overall relaxation is a composite of relaxation processes of subdomains, each with their own characteristic time. Using a Markov chain method, these times are computed both numerically and in closed form. The Markov chain results are seen to match the simulations at low temperatures and high frequencies. The dynamics of the east model are tracked very well by this analytic procedure, and it is possible to associate features of the spectrum of the dynamic heat capacity with specific domain relaxation events.

The authors thank Lorie Liebrock and Brian Hughes for time on the Exemplar minisupercomputer at New Mexico Tech.

I. INTRODUCTION

II. GENERAL PROPERTIES OF THE EAST MODEL

III. METHODOLOGY

IV. RESULTS

V. DISCUSSION

### Key Topics

- Relaxation times
- 29.0
- Heat capacity
- 26.0
- Ising model
- 10.0
- Markov processes
- 6.0
- Temperature inversion
- 5.0

## Figures

Real and imaginary components of the dynamic heat capacity plotted against frequency on a log-log scale for a range of temperatures. The squares and circles are the real and imaginary parts of the simulation data, respectively, with every other point dropped for clarity. The dashed line is the Markov chain Debye series computation, truncated by the number of terms that could be computed numerically. The solid line is the extended Markov chain Debye series fit. The parameters for the four plots are (a) a temperature of , domain length at truncation , and best fit maximum effective domain length ; (b) , , and ; (c) , , and no possible fit for ; (d) , , and no possible fit for .

Real and imaginary components of the dynamic heat capacity plotted against frequency on a log-log scale for a range of temperatures. The squares and circles are the real and imaginary parts of the simulation data, respectively, with every other point dropped for clarity. The dashed line is the Markov chain Debye series computation, truncated by the number of terms that could be computed numerically. The solid line is the extended Markov chain Debye series fit. The parameters for the four plots are (a) a temperature of , domain length at truncation , and best fit maximum effective domain length ; (b) , , and ; (c) , , and no possible fit for ; (d) , , and no possible fit for .

The phase lag plotted against frequency on a semilogarithmic scale for the same range of temperatures as Fig. 1. The circles are the simulation data and the lines have the same meaning as in Fig. 1.

The phase lag plotted against frequency on a semilogarithmic scale for the same range of temperatures as Fig. 1. The circles are the simulation data and the lines have the same meaning as in Fig. 1.

Cole–Cole plots for the same range of temperatures as Fig. 1. The circles are the simulation data and the lines have the same meaning as in Fig. 1. The range of the real axis in plots (a)–(c) is between zero and the equilibrium heat capacity. The scale on the imaginary axis is twice the scale on the real axis for clarity.

Cole–Cole plots for the same range of temperatures as Fig. 1. The circles are the simulation data and the lines have the same meaning as in Fig. 1. The range of the real axis in plots (a)–(c) is between zero and the equilibrium heat capacity. The scale on the imaginary axis is twice the scale on the real axis for clarity.

(a) The imaginary part of the dynamic heat capacity normalized by peak height plotted against frequency normalized by peak frequency on a log-log scale. Each curve is from simulation data for temperatures ranging from to (as labeled), and each successive curve differs from the next by a constant change in temperature of 0.1. (b) Equilibrium heat capacity (solid line) and imaginary peak height from simulation (squares) plotted against temperature. The vertical dashed line goes through the peak of the equilibrium heat capacity. (c) Fitted maximum isolated domain length plotted against the mean domain length . The solid line has a slope of one, and the dashed line is a linear fit to the data. (d) Relaxation time for the KWW (squares) and HN (inverted triangles) fits plotted against the relaxation time of the largest nonisolated domain on a log-log scale. The solid line has a slope of one.

(a) The imaginary part of the dynamic heat capacity normalized by peak height plotted against frequency normalized by peak frequency on a log-log scale. Each curve is from simulation data for temperatures ranging from to (as labeled), and each successive curve differs from the next by a constant change in temperature of 0.1. (b) Equilibrium heat capacity (solid line) and imaginary peak height from simulation (squares) plotted against temperature. The vertical dashed line goes through the peak of the equilibrium heat capacity. (c) Fitted maximum isolated domain length plotted against the mean domain length . The solid line has a slope of one, and the dashed line is a linear fit to the data. (d) Relaxation time for the KWW (squares) and HN (inverted triangles) fits plotted against the relaxation time of the largest nonisolated domain on a log-log scale. The solid line has a slope of one.

Plots of the fitting parameters. (a) Relaxation time plotted against inverse temperature on a semilogarithmic scale. The squares are from KWW fits to simulation data. The center curve is a Vogel–Fulcher fit, the upper curve fit is to with , the lower curve fit is to with . Upper and lower curves are shifted vertically for clarity. (b) Relaxation time plotted against inverse temperature squared on a semilogarithmic scale. The squares are from KWW fits to the simulation data. The dashed line is an EITS fit to the last six relaxation times. The solid lines have slopes of 1/ln 2 and 1/2 ln 2. (c) Shape parameters fit from simulation data, (squares) (inverted triangles), and (diamonds), plotted against temperature. (d) Shape parameters plotted against relaxation time on a semilogarithmic scale.

Plots of the fitting parameters. (a) Relaxation time plotted against inverse temperature on a semilogarithmic scale. The squares are from KWW fits to simulation data. The center curve is a Vogel–Fulcher fit, the upper curve fit is to with , the lower curve fit is to with . Upper and lower curves are shifted vertically for clarity. (b) Relaxation time plotted against inverse temperature squared on a semilogarithmic scale. The squares are from KWW fits to the simulation data. The dashed line is an EITS fit to the last six relaxation times. The solid lines have slopes of 1/ln 2 and 1/2 ln 2. (c) Shape parameters fit from simulation data, (squares) (inverted triangles), and (diamonds), plotted against temperature. (d) Shape parameters plotted against relaxation time on a semilogarithmic scale.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content