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Body.toc
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\contentsline {chapter}{\numberline {1}Introduction}{7}{chapter.1}
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\contentsline {chapter}{\numberline {2}Velocity-driven Non-equilibrium Phase Transition in Ising Models}{11}{chapter.2}
\defcounter {refsection}{0}\relax
\contentsline {section}{\numberline {2.1}Overview of the results}{11}{section.2.1}
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\contentsline {section}{\numberline {2.2}Concrete Examples: One-dimension and Two-dimension}{13}{section.2.2}
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\contentsline {chapter}{\numberline {3}Numerical Simulations}{17}{chapter.3}
\defcounter {refsection}{0}\relax
\contentsline {section}{\numberline {3.1}Setup of the Model}{17}{section.3.1}
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\contentsline {section}{\numberline {3.2}Non-equilibrium Monte Carlo Simulation}{19}{section.3.2}
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\contentsline {subsection}{\numberline {3.2.1}Introduction of the Time Scale to Ising Models}{19}{subsection.3.2.1}
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\contentsline {subsection}{\numberline {3.2.2}Slip Plane with the Velocity $v$}{20}{subsection.3.2.2}
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\contentsline {subsection}{\numberline {3.2.3}Convergence to the Non-equilibrium Stationary State}{21}{subsection.3.2.3}
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\contentsline {subsection}{\numberline {3.2.4}Concrete examples of matrices $\mathaccentV {hat}05E{M}(\beta )$ and $\mathaccentV {hat}05E{S}$}{22}{subsection.3.2.4}
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\contentsline {section}{\numberline {3.3}Definitions of Physical Quantities}{23}{section.3.3}
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\contentsline {chapter}{\numberline {4}Results}{25}{chapter.4}
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\contentsline {section}{\numberline {4.1}Observed Quantities}{25}{section.4.1}
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\contentsline {subsection}{\numberline {4.1.1}Frictional Force Density $f(L_{z}, T)$}{25}{subsection.4.1.1}
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\contentsline {subsection}{\numberline {4.1.2}Bulk Energy Density $\epsilon (L_{z}, T)$}{26}{subsection.4.1.2}
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\contentsline {subsection}{\numberline {4.1.3}Temperature Derivatives $\partial f(L_{z}, T)/\partial T$ and $c(L_{z}, T)$}{26}{subsection.4.1.3}
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\contentsline {section}{\numberline {4.2}Checking the Convergence in the Limit $L_{x}\to \infty $}{26}{section.4.2}
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\contentsline {chapter}{\numberline {5}Summary and Discussion}{37}{chapter.5}
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\contentsline {chapter}{\numberline {A}Analysis based on Stochastic Matrices}{39}{appendix.A}
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\contentsline {section}{\numberline {A.1}A Simple Example: Stochastic Ising Model with $N$-spins}{39}{section.A.1}
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\contentsline {section}{\numberline {A.2}General Theory of Stochastic Matrices}{40}{section.A.2}
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\contentsline {section}{\numberline {A.3}Construction of the Stochastic Matrix based on the Detailed Balanced Condition}{46}{section.A.3}
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\contentsline {subsection}{\numberline {A.3.1}Metropolis Matrix for the Model of the Size $3\times 2$}{47}{subsection.A.3.1}