Visualizations of simulations

	Coarsening in combinatorial algorithms. Coarsening is familiar in physical systems, where a disordered system establishes a phase by expanding volumes of uniform phase. Here, the coarsening occurs during the application of an algorithm borrowed from computer science and applied to finding ground states in disordered magnets. The two rows show the coarsening of an auxiliary variable, the height, in the push-relabel algorithm, during the deterministic search for the solution. In the cited paper, connections are made between the dynamics of the algorithms and the nature of the ground states. In particular, the algorithms, though quite rapid, slow down near critical points. [Middleton, cond-mat/0104185.] (32 KB)
	Droplets in disordered systems. These are low energy excitations above the ground state configuration - the regions indicate where the excited state differs from the ground state. The picture on the left is a sample droplet (in a two dimensional elastic medium) constrained to contain the origin of a sample, but unconstrained in scale (except by sample size.) The measurements of the indicated radii show that the droplets are "compact" in the sense that their areas grow like the square of their linear dimension. The picture on the right is of droplets excited by bulk perturbation of the bond strengths in a two dimensional spin glass. [Middleton, cond-mat/0007375.] (32 KB)
	Introduction of domain walls by the expansion of sample size in a 2-dimensional spin glass. The ground state is computed for two system sizes (L and L'), with the disorder in the common region identical. The solid lines indicate the relative domain walls introduced and the window w is where changes in the ground state are measured. [Middleton, cond-mat/9904285.] (23 KB)
	Visualization of disorder-induced defects in a two-dimensional elastic medium at zero temperature. The ends of the strings represent +/- defect pairs. The defect density depends on the strength of the disorder. The strings themselves indicate where the strain relative to the defect-free state is localized: in contrast with defects in the pure 2D XY model, where the elastic strain is spread throughout space, disorder leads to confinement of the strain associated with the defects to a line. [Middleton, cond-mat/9807374.] (23 KB)
	Configuration of heights in a 3-dimensional elastic medium, with scalar displacements, subject to quenched disorder. The black and white cubes indicate displacements up and down from the average position (transparent cubes, which are not visible!) The simulations on this model indicate a logarithmic roughness and elastic costs which support some details of the theoretical arguments (cond-mat/9610146) in favor of the existence of a Bragg glass in three dimensions [McNamara, Middleton, Zeng, cond-mat/9905058.] (50 KB)
	Defect wall in the above model (3D elastic medium) generated by twisted boundary conditions. Note that the domain wall is fractal, quite rough, and though its surface area is relatively small, it is intersected by any vertical plane. [McNamara, Middleton, Zeng, cond-mat/9905058.] (142 KB postscript)
	Not a simulation, but a figure from a proof. In the referenced paper, it is shown that determining the barriers to the motion of interfaces is generally an NP-hard problem (computationally intractable, in practice), even though finding the low-lying states can be done rapidly (polynomial time) [Middleton, PRE 59, 2571 (1999); cond-mat/9902203.] (50 KB)
	Visualization of the application of an algorithm to find ground state of lines in a random potential: shown is the exact ground state for 100 lines in a 300x300 area potential. [ Zeng, Middleton, and Shapir, Physical Review Letters 77, 3204 (1996).] (23 KB)
	Mincut algorithm applied to condensed matter physics. The application here is to find the ground state of an interface in the random bond Ising model. [Reference: A. Middleton, Phys. Rev. E 52, R3337 (1995).] (12 KB)
	Currents in a 128x128 array of mesoscopic dots, with underlying charge disorder. The top picture shows the time-averaged current just above the threshold voltage for conduction, while the lower picture shows the currents for a somewhat higher voltage. The color scale is logarithmic in the current; the red dots carry the most current, the blue dots the least. [Reference: A. Middleton, N. Wingreen, Phys. Rev. Lett. 71, 3198 (1993).] (8.4 KB)
	Avalanches generated by thermal excitation in a model of charge-density waves. [Middleton, Physical Review B 45, 9465 (1992).] (1 KB)