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\BOOKMARK [0][-]{chapter.1}{Introduction}{}% 1
\BOOKMARK [0][-]{chapter.2}{Velocity-driven Non-equilibrium Phase Transition in Ising Models}{}% 2
\BOOKMARK [1][-]{section.2.1}{Overview of the results}{chapter.2}% 3
\BOOKMARK [1][-]{section.2.2}{Concrete Examples: One-dimension and Two-dimension}{chapter.2}% 4
\BOOKMARK [0][-]{chapter.3}{Numerical Simulations}{}% 5
\BOOKMARK [1][-]{section.3.1}{Setup of the Model}{chapter.3}% 6
\BOOKMARK [1][-]{section.3.2}{Non-equilibrium Monte Carlo Simulation}{chapter.3}% 7
\BOOKMARK [2][-]{subsection.3.2.1}{Introduction of the Time Scale to Ising Models}{section.3.2}% 8
\BOOKMARK [2][-]{subsection.3.2.2}{Slip Plane with the Velocity v}{section.3.2}% 9
\BOOKMARK [2][-]{subsection.3.2.3}{Convergence to the Non-equilibrium Stationary State}{section.3.2}% 10
\BOOKMARK [2][-]{subsection.3.2.4}{Concrete examples of matrices \(\) and }{section.3.2}% 11
\BOOKMARK [1][-]{section.3.3}{Definitions of Physical Quantities}{chapter.3}% 12
\BOOKMARK [0][-]{chapter.4}{Results}{}% 13
\BOOKMARK [1][-]{section.4.1}{Observed Quantities}{chapter.4}% 14
\BOOKMARK [2][-]{subsection.4.1.1}{Frictional Force Density f\(Lz, T\)}{section.4.1}% 15
\BOOKMARK [2][-]{subsection.4.1.2}{Bulk Energy Density \(Lz, T\)}{section.4.1}% 16
\BOOKMARK [2][-]{subsection.4.1.3}{Temperature Derivatives f\(Lz, T\)/T and c\(Lz, T\)}{section.4.1}% 17
\BOOKMARK [1][-]{section.4.2}{Checking the Convergence in the Limit Lx}{chapter.4}% 18
\BOOKMARK [0][-]{chapter.5}{Summary and Discussion}{}% 19
\BOOKMARK [0][-]{appendix.A}{Analysis based on Stochastic Matrices}{}% 20
\BOOKMARK [1][-]{section.A.1}{A Simple Example: Stochastic Ising Model with N-spins}{appendix.A}% 21
\BOOKMARK [1][-]{section.A.2}{General Theory of Stochastic Matrices}{appendix.A}% 22
\BOOKMARK [1][-]{section.A.3}{Construction of the Stochastic Matrix based on the Detailed Balanced Condition}{appendix.A}% 23
\BOOKMARK [2][-]{subsection.A.3.1}{Metropolis Matrix for the Model of the Size 32}{section.A.3}% 24