2007 Fiscal Year Final Research Report Summary
Extended ensemble method and its application to frustrated systems
| Project/Area Number |
17540348
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Mathematical physics/Fundamental condensed matter physics
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| Research Institution | The University of Tokyo |
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Principal Investigator |
HUKUSHIMA Koji The University of Tokyo, Graduate School of Arts and Sciences, Associate Professor (80282606)
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| Co-Investigator(Kenkyū-buntansha) |
IBA Yukito The Institute of Statistical Mathematics, Department of Statistical Modeling, Associate Professor (30213200)
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| Project Period (FY) |
2005 – 2007
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| Keywords | Condensed matter theory / Computational Physics / Algorithm / Statistical Physics |
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| Research Abstract |
(1) Rare event Monte Carlo method: An importance-sampling Monte Carlo algorithm is developed for sampling rare events that make a sense in statistical mechanical problems. This could be also useful for sampling quenched variables in randomly disordered systems. The main idea for the method is to enhance a probability for finding a set of quenched variables, which gives a rare event, in a Markov chain of Monte Carlo simulations. The applications are as follows.
(1-a)Griffiths singularity in diluted Ising models: A typical example of such rare events is a Griffiths singularity found in random spin models, that is caused by the existence of arbitrary large rare cluster. We first applied the developed MC method to a bond-diluted Ising model in two dimensions which is expected to exhibit the Griffiths singularity. Then, we found that the distribution of the inverse susceptibility has an exponential tail down to the origin which is considered the consequence of the singularity.
(1-b) Lee-Yang
… More
zero distribution in Ising spin glasses: In general, the phase transition of statistical mechanical models is characterized by a partition function zero as a function of imaginary temperature. We also applied the method for estimating the distribution of partition-function zeros in Ising spin glasses in which zeros close to the real temperature axis are rare events. We found an evidence for the existence of spin glass transition in three dimensions and the Griffiths singularity in two dimensions. This might be the first time to observe directly the singularity in spin glass systems.
(1-C) other applications: The developed method is applicable to other field such as information theory and mathematical problems with random variables. By using the method, we studied the bit-error distribution of an error-correcting code problem in an information theory and the largest-eigenvalue distribution in random matrix.
(2) We also propose an efficient method for enumerating large number of solutions, which can be mapped onto the entropy under some conditions using statistical mechanical idea. This is basis of the extended ensemble method, that enables us to estimate the free energy and the entropy with high accuracy. As a specific example, we calculate a number of possible grids in Sudoku that is a kind of constraint-satisfaction problems. A standard Sudoku consists of 9×9 grids, whose number of grids is exactly enumerated. We successfully calculate generalized Sudoku problems with 49×49 grids. Less
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