Gunduz Caginalp

An analysis of a phase field model of a free boundary

A mathematical analysis of a new approach to solidification problems is presented. A free boundary arising from a phase transition is assumed to have finite thickness. The physics leads to a system of nonlinear parabolic differential equations. Existence and regularity of solutions are proved. Invariant regions of the solution space lead to physical interpretations of the interface. A rigorous asymptotic analysis leads to the Gibbs-Thompson condition which relates the temperature at the interface to the surface tension and curvature.

Dynamics of Layered Interfaces Arising from Phase Boundaries

The dynamics of a material in two phases is studied in the context of phase-field models based on a Landau–Ginzburg free energy functional. They consist of a system of two nonlinear diffusion equat...

Convergence of the phase field model to its sharp interface limits

We consider the distinguished limits of the phase eld equations and prove that the corresponding free boundary problem is attained in each case. These include the classical Stefan model, the surface tension model (with or without kinetics), the surface tension model with zero specic heat, the two phase Hele{Shaw, or quasi-static, model. The Hele{Shaw model is also a limit of the Cahn{Hilliard equation, which is itself a limit of the phase eld equations. Also included in the distinguished limits is the motion by mean curvature model that is a limit of the Allen{Cahn equation, which can in turn be attained from the phase eld equations.

Financial Bubbles: Excess Cash, Momentum, and Incomplete Information

We report on a large number of laboratory market experiments demonstrating that a market bubble can be reduced under the following conditions: 1) a low initial liquidity level, i.e., less total cash than value of total shares, 2) deferred dividends, and 3) a bid-ask book that is open to traders. Conversely, a large bubble arises when the opposite conditions exist. The first part of the article is comprised of twenty-five experiments with varying levels of total cash endowment per share (liquidity level), payment or deferral of dividends and an open or closed bid-ask book. We find that the liquidity level has a very strong influence on the mean and maximum prices during an experiment (P < 1/10,000). These results suggest that within the framework of the classical bubble experiments (dividends distributed after each period and closed book), each dollar per share of additional cash results in a maximum price that is

The Predictive Power of Price Patterns

1 per share higher. There is also limited statistical support for the theory that deferred dividends (which also lower the cash per share during much of the experiment) and an open book lead to a reduced bubble. The three factors taken together show a striking difference in the median magnitude of the bubble (

Momentum and Overreaction in Experimental Asset Markets

7.30 versus

Phase Field Equations in the Singular Limit of Sharp Interface Problems

0.22 for the maximum deviation from fundamental value). Another set of twelve experiments features a single dividend at the end of fifteen trading periods and establishes a 0.8 correlation between price and liquidity during the early periods of the experiments. As a result, calibration of prices and evolution toward equilibrium price as a function of liquidity are possible.

Asset Flow and Momentum: Deterministic and Stochastic Equations

Using two sets of data, including daily prices (open, close, high and low) of all S&P 500 stocks between 1992 and 1996, we perform a satistical test of the predictive capability of candlestick patterns. Out-of-sample tests indicate statistical significance at the level of 36 standard deviations from the null hypothesis, and indicate a profit of almost 1% during a two-day holding period. An essentially non-parametric test utilizes standard definitions of three-day candlestick patterns and removes conditions on magnitudes. The results provide evidence that traders are influenced by price behaviour. To the best of our knowledge, this is the first scientific test to provide strong evidence in favour of any trading rule or pattern on a large unrestricted scale.

The role of microscopic anisotropy in the macroscopic behavior of a phase boundary

Price volatility and investor overreactions are commonplace in experimental asset markets. Understanding the price dynamics in these markets is crucial for designing successful new trading institutions. We report on a series of experiments to test the predictions of a new momentum model using a dynamical systems approach. This model is then pitted against several standard models to predict prices, as well as against expert human forecasters. The comparative results suggest that each model has its advantages and regions of best performance. Overall, the best predictive methods are the momentum model and expert human forecasters.

Computation of sharp phase boundaries by spreading: the planar and spherically symmetric cases

In one of the singular limits as interface thickness approaches zero, solutions to the phase field equations formally approach those of a sharp interface model which incorporates surface tension. Here, we use a modification of the original phase field equations and prove this convergence rigorously in the one-dimensional and radially symmetric cases. Convergence to motion by mean curvature in another distinguished limit is also proved.