Manabu Hasegawa

Iron Vacancy Ordered γ-Fe2O3(001) Epitaxial Films: The Crystal Structure and Electrical Resistivity

We report the structural characterization and transport properties of highly-ordered epitaxial γ-Fe 2 O 3 thin films grown by a pure ozone-assisted molecular beam epitaxy method. X ray diffraction ...

Unidirectional Search Dynamics in Optimization by Metropolis Algorithm

Motivated by computer experiments on physical systems at low temperatures, simulated annealing (SA) was proposed as a heuristic solution search method for combinatorial optimization problems. By replacing the microscopic state and its energy with the feasible solution and its cost, respectively, a Monte Carlo simulation using the Metropolis algorithm is run with a slow decrease in effective temperature. After the success of this method, various notions and methods in the analysis of complex physical systems have been used for the understanding and the solution of optimization problems. In the present short note, we report a new application of physical methods to a clarification of local search optimization dynamics. First, we mention a numerical finding on the dynamics of a complex physical system and consider its meaning in light of the local search optimization process. About 15 years ago, Shinjo performed a moleculardynamics experiment on the formation of a glassy solid and the temporal behavior of the system at low temperatures was investigated using a mapping-onto-minima method. In this analysis, the configuration space is divided by the basin of the metastable state and the dynamics are observed as a transition process from basin to basin. Each basin is characterized by the potential energy of the inside metastable state and the history of this energy is traced as hidden dynamics. The result shows that, after the system cools to the target temperature, a unidirectional transition to a lower level of energy appears in some temperature regime. At that temperature, moreover, the energy level at the end of the given observational time is lower than that in another regime. These observations are notable for the following two points if we see the dynamics as an optimization process by a local search method. First, this Hamiltonian dynamics has a good intensification characteristic in a low temperature regime. Intensification (exploitation) is one of the functions that are necessary for local search algorithms. Since near-optimum solutions often have similar structures, the algorithms are designed such that the search process can be intensified near promising solutions in the search space to find much better ones. The conventional Monte Carlo dynamics using the Metropolis algorithm is similar to the Hamiltonian dynamics in the aspect that a neighborhood move is repeated in the configuration space. A similar intensification effect, therefore, is expected from this algorithm within a suitable observational time. Second, this temperature regime is related to the formation of a glassy solid. With optimization by SA, it is generally believed that careful cooling is necessary to finding a better solution; thus, the appearance of glasstransition-like phenomena is considered to be unfavorable to the effective use of this method. On the other hand, recent research reveals that SA with temperatures fixed at low values can be superior to algorithms based on cooling for some moderately sized problems. It is suggested, therefore, that the intensification effect plays a significant role and can be more useful than annealing in a finite-time optimization process. On the basis of these considerations, we performed a computer experiment on the hidden search dynamics in optimization using the Metropolis algorithm. Our concerns are to see whether the above intensification effect appears in the optimization process and, if it appears, how it works there. We evaluate the cost of the basin by that of the inside locally optimal solution and see whether a unidirectional transition to a better basin appears or not. We call these unidirectional search dynamics the potentially local search dynamics and regard the appearance of this process as the intensification effect. The mapping-onto-minima process to find the locally optimal solution is realized by a simple local search and is performed independently from the original search dynamics whenever the trial solution is accepted in the Metropolis algorithm. We briefly describe the cost of the locally optimal solution as the local minimum cost. For comparison purposes, the experiment was performed on the same problem instances as used in our previous study; that is, the instances of the random Euclidean traveling salesman problem. The goal in this problem is to find the shortest tour that passes through each of the given cities once and returns to the starting city: the cost function is given by the tour length. The location of the N cities are sampled uniformly in a square region and the intercity distance is computed under the Euclidean metric. The size of the square is adjusted such that the cost of the best solution obtained in 1000 preliminary runs using a standard SA, which was introduced as a baseline implementation in ref. 11, is equal to N. A randomly constructed tour is taken as an initial solution and the algorithm is employed with a fixed temperature T . We use a 2-opt neighborhood, which is defined as a set of tours constructed by any change of two intercity paths: the trial solution is randomly selected from the neighborhood. The simple local search used in the mapping-onto-minima process is executed by means of the best admissible search strategy; that is, the move to the best solution in the neighborhood is repeated. In Fig. 1, we show the results of the hidden search dynamics at some temperatures for N 1⁄4 10: the number of steps is measured by that of cost evaluations and the results are the average of 50 independent runs on a single instance. After the initial 10 search steps, a unidirectional transition to a better basin is observed as the average behavior for T < 1:0, as was expected. When T 0:2, the cost stops decreasing in the course of time and it remains at a lower level. As the temperature decreases, the lifetime of the unidirectional state gets longer and the cost level of the stagnant state goes down. When T < 0:2, the unidirectional E-mail: hasegawa@esys.tsukuba.ac.jp Journal of the Physical Society of Japan Vol. 74, No. 10, October, 2005, pp. 2872–2873 #2005 The Physical Society of Japan

Broken ergodicmmotion of two hard particles in a one-dimensional box

Abstract The dynamical behavior of one-dimensional two hard particle systems is examined in the light of broken ergodicity. Strange motions are observed in a rearranged state space of particle velocities. The origins of the two basic motions breaking ergodicity are discussed.

Convection Flip in the Rayleigh-Bénard Problem in a Small System

Molecular dynamics study of the Rayleigh-Benard system was explored about ten years ago by Mareschal et al. 1), 3) and by Rapaport. 2) The former researchers demonstrated that fully developed convective rolls could be observed even in a small system made up of a few thousands of molecules. In the present study, we focus on an unsteady behavior of convective roll observed near the conventional critical point. Our system consists of 1600 hard disks confined to a square box at a number density of 0.2. The temperatures of the upper and the lower walls are T0(1 − ∆T/2) [K] and T0(1 +∆T/2) [K], respectively, and a gravitational field is introduced so that a disk moving between these two walls experiences a zero net energy change. Initially, the disks are uniformly spaced and their velocities are sampled from the Maxwellian distributions with linearly interpolated temperatures between the two thermal walls. Disks are assumed to be reflected specularly on the walls; however, thermal effect is taken into account to the normal component of the velocity departing from the upper and the lower walls. A series of experiments was performed by changing the value of ∆T from 0.2 to 1.8. The square box is devided into small cells and the variations of local velocity in each of these cells are successively measured by taking an average in an appropriate time scale. General observations are summarized as follows. Spontaneous single-roll structure clearly appears when ∆T > 1; at first, both clockwise and counterclockwise convections are observed irregularly. As ∆T increases, the average lifetime of a convective roll pattern, τroll, is prolonged and finally no flip motion occurs on an observational time scale τobs. We realize that there is a short range of ∆T , instead of a distinct critical point, between no-roll and fully developed single-roll states. To characterize the state in this transitional phase, we introduce two parameters. Let M∗ be the sum of nondimensional angular momentum of all disks around the center of the box. This is, so to speak, an order parameter which describes the transformation from structureless no-roll (disordered) state to fully developed singleroll (ordered) state. Using this, we introduce new parameters M1 and M2 such that