Paul Siders Chemistry
The driven lattice gas (DLG) is a statistical mechanical model for the effects of an external field on a simple phase transition. The gas particles occupy half of the sites on a two-dimensional lattice. Particles are attracted to each other but also hop randomly between lattice sites with a temperature-dependent hopping rate. The field biases hops preferentially along one direction.
This project focused on converting the three dimensional data from the ECOPHYS program into a realistic image of the tree in simulation. We focused especially on generating images of a whole forest of young simulated trees. We used a free program called POV-Ray to generate realistic scenery and ray-trace the image of a small experimental forest.
The current overall project is a study of
the driven lattice gas on triangular and hexagonal lattices, and
especially the relationship of the field to the temperature at which the
lattice gas freezes. Two field-lattice combinations are particularly
interesting: the hexagonal lattice with the field aligned with a lattice
vector, and the triangular lattice with the field bisecting lattice
vectors. In these cases a strong field does not simply prevent freezing
but changes the nature and temperature of the transition. The two graphs
below show energy versus temperature at various field strengths. Each
data point is the result of a Monte Carlo simulation. For the
equilibrium (zero field) cases, freezing occurs at approximately
T/Tc(0)=1. Higher fields shift freezing to lower temperature and may
introduce a second transition (the triangular lattice at field strength
e=6J). Strong fields result in a higher-energy less-ordered frozen state
at low temperature.
Conclusions about phase transitions on lattices require study of the effect of lattice size. Increasing the system size on the available serial computers is difficult because typical serial simulations of the current size require one to several cpu-days. My main summer Viz Lab goal was to use the beowulf cluster to simulate larger systems in parallel.
At the Viz Lab, the first task was to learn basics of the message passing interface LAM/MPI that is installed on bwulf.d.umn.edu. Then an existing serial code was modified to run in parallel. The simplest modification proved inadequate because passing streams of pseudo-random numbers dominated the cpu time. The SPRNG scalable parallel random number generation library solved that problem, allowing each process to access an independent stream of random numbers.
The beowulf cluster quickly (i.e., in just 2700 hours total wall time) quadrupled the earlier serial simulation size for an infinitely strong field on a triangular lattice. The results at right show strong system-size dependence. Analogous calculations elsewhere showed there is almost no size dependence for the hexagonal lattice. The debugged and tested parallel code was ported to the netfinity cluster at the Minnesota Supercomputer Center where it also ran in parallel until granted cpu time was (quickly) exhausted. The UMD Viz Lab Beowulf cluster provides, comparatively, an excellent code development environment because users of the Beowulf cluster can run interactively and can select individual nodes in the cluster.
Further increasing the system size on bwulf failed for some unknown reason. The code executed and made progress but simulation times extrapolated from incomplete runs were unacceptably long, on the order of months.
Visualization of simulation results was an unexpected benefit of the Viz Lab summer grant. A C++ interface to gnuplot was installed on the Linux computer wall.d.umn.edu in the Viz Lab. Code calling that interface quickly and simply displayed configurations of the gas particles on their lattices. Two low-temperature configurations are shown below. The images show how different are the low-temperature states of the two lattices. The triangular lattice supports a single but imperfectly ordered strip oriented along the field while the hexagonal lattice breaks into narrow zig-zag patterns aligned with the field. Presumably it is because of the small spatial extent of its low-temperature configurations that the hexagonal lattice shows little dependence on system size.
The driven lattice gas is a dynamic model but the Monte Carlo simulations sought steady states only and did not show time dependence. However, a sense of time can be added to the simulation results by combining a sequence of steady states into an animated gif. The program “gifsicle” was installed on computer wall and used to combine single images into animations that crudely show the process of freezing. These simple animations suggested that during cooling, and at low-to-intermediate field strengths, the triangular lattice first orders along the field, then loses that order and converts to the equilibrium frozen state. At high field, however, the images show that the early order along the field is preserved to the lowest temperatures that could be simulated. That ordered state is so imperfect (see the triangular configuration above) that the final energy (i.e., in the zero-temperature limit) is higher than the energy in the field-free frozen state. This explanation suggested by visualized configurations is supported by calculated structure factors.
Details about parallel programming with LAM/MPI and SPRNG on the Viz Lab beowulf cluster are available at http://www.d.umn.edu/~psiders/Viz Lab. Samples of the images and animated gifs described above are also posted on that web site.
The weak size dependence observed for
hexagonal lattices suggested that much smaller lattices might suffice to
explain qualitatively the effects of temperature and field strength.
The 2x2 hexagonal lattice consists of only four particles on 8 sites.
The 70 configurations of such particles group into 13 classes based on
symmetries. The rates of transitions among those classes were placed in a
13x13 matrix from which the steady-state probabilities were calculated
analytically with the program Mathematica at the Viz Lab. The energy of
this tiny system is shown at right. Behavior is similar to that of
larger systems except that the low-temperature energy is independent of
field strength, not rising to the less-ordered value seen in the larger
lattices. Even though the low-temperature energy is independent of field
strength, the arrangement of particles is not. Two classes (class 0 and
class 12) dominate at low temperature. The mix of the two is highly
field dependent, as shown at right.
The field-dependent mixture of just a few 2x2 configurations suggested that a slightly larger lattice might allow the transition from an ordered frozen state to a disordered low-temperature state, as seen in the large-lattice Monte Carlo results. The 3x3 hexagonal lattice has 9 particles on 18 sites, 2710 classes. Nearly exact results for the probabilities of the 2710 classes have been calculated and indeed show the field and temperature cooperating to select one or a few low-energy classes at low temperature and low field strengths, with probability spreading to many higher-energy classes when the field strength is large. Even a 3x3 lattice is large enough to capture most of what happens on larger hexagonal lattices. For triangular lattices, however, no small model has been found to be applicable.
Work in the Viz Lab under the summer 2004 program led to fast parallel Monte Carlo simulations. Visualization of configurations was easier and more helpful than anticipated. Exact results for small systems were also obtained at the Viz Lab. The Viz Lab environment emphasizes creativity and interdisciplinary discussion. That pleasant and stimulating setting was great for summer work.