We discuss pattern formation in three-dimensional gels with cylindrical shapes during their shrinking such as volume phase transition. A Ginzburg–Landau theory is given for the pattern formation in shrinking gels. A characteristic feature in shrinking gels is the dense layer formed around the gel surface in the early stage of phase transition. This layer reduce considerably permeation of solvent and the shrinkage practically stops. We introduce the external osmotic pressure and the external elastic stress acting on the gel surface in order to take account of the effect of the layer. Patterns are classified according to the anisotropy and the incompressibility of gels by the linearized stability analysis of the theory. It appears that the external stress term suppresses the growth of the fluctuation with short wavelength. The results obtained by a numerical calculation for the evolution of patterns are also shown.

We examine the correlation of the limit price with the order book, when a limit order comes. We analyzed the Rebuild Order Book of Stock Exchange Electronic Trading Service, which is the centralized order book market of London Stock Exchange. As a result, the limit price is broadly distributed around the best price according to a power-law, and it isn't randomly drawn from the distribution, but has a strong correlation with the size of cumulative unexecuted limit orders on the price. It was also found that the limit price, on the coarse-grained price scale, tends to gather around the price which has a large size of cumulative unexecuted limit orders.

We give a stochastic microscopic modelling of stock markets driven by continuous double auction. If we take into account the mimetic behavior of traders, when they place limit order, our virtual markets shows the power-law tail of the distribution of returns with the exponent outside the Levy stable region, the short memory of returns and the long memory of volatilities. The Hurst exponent of our model is asymptotically 1/2. An explanation is also given for the profile of the autocorrelation function, which is responsible for the value of the Hurst exponent.

We analyze the statistics of daily price change of stock market in the framework of a statistical physics model for the collective fluctuation of stock portfolio. In this model the time series of price changes are coded into the sequences of up and down spins, and the Hamiltonian of the system is expressed by spin-spin interactions as in spin glass models of disordered magnetic systems. Through the analysis of Dow-Jones industrial portfolio consisting of 30 stock issues by this model, we find a non-equilibrium fluctuation mode on the point slightly below the boundary between ordered and disordered phases. The remaining 29 modes are still in disordered phase and well described by Gibbs distribution. The variance of the fluctuation is outlined by the theoretical curve and peculiarly large in the non-equilibrium mode compared with those in the other modes remaining in ordinary phase.The number of days in which prices of all stock issues in Dow-Jones industrial portfolio moved in the same direction is 80 within 3636 trading days in the period from 9/Jul/86 to 22/Nov/00, while its probability is 2 −29 when we assume Bernoulli trials. How can we explain the factor 10 7 in the difference between these values? The methods and the concepts, as scaling and criticality, developed in material science have been applied to the study of financial markets [1, 2, 3]. Recently a framework based on spin glass model to study the collective price changes of stock portfolios was proposed [4, 5]. The application to 1-minute price changes in D-J portfolio made clear that the concept of energy works even in financial markets as well as the above physical concepts. Here we study the properties of daily price changes in D-J portfolio based on the same model and attempt an explanation of the factor 10 7 by Gibbs factor of canonical distribution. Through this study we clarify the applicability of the spin glass picture to the price fluctuations in a wide range of time scale and find a novel feature of stock market. That is a non-equilibrium fluctuation mode on the point close to the boundary between ordered and disordered phases. D-J portfolio has been quenched into the unstable region, but does not reach equilibrium. The variance of the fluctuation, which is physically equal to susceptibility and is called (the square of) risk in financial economy, is outlined by the theoretical curve and peculiarly large in the non-equilibrium mode compared with those in the other modes remaining in ordinary phase.In this paper, we concentrate on the statistics of the sign of price change [4, 5]. The time series of price changes are coded into the sequences of up and down spins S i = ±1 (i=1 to portfolio size N) and the Hamiltonian is expressed by long-range spin-spin interactions as Sherrington-Kirkpatrick model of spin glass [6], which is given byWe consider a portfolio as a subset of the whole stock market, and the complement of the subset works as heat reservoir. Various observable quantities are obtained as the statistical averages with Gibbs weight ...