1. |
Sheldon Goldstein, Takashi Hara, Hal Tasaki, Extremely quick thermalization in a macroscopic quantum system for a typical nonequilibrium subspace, *New Journal of Physics*, 10.1088/1367-2630/17/4/045002, 17, 045002-1-045002-7, 2015.04, The fact that macroscopic systems approach thermal equilibrium may seem puzzling, for example, because it may seem to conflict with the time-reversibility of the microscopic dynamics. We here prove that in a macroscopic quantum system for a typical choice of 'nonequilibrium subspace', any initial state indeed thermalizes, and in fact does so very quickly, on the order of the Boltzmann time . Therefore what needs to be explained is, not that macroscopic systems approach thermal equilibrium, but that they do so slowly.. |

2. |
Sheldon Goldstein, Takashi Hara, Hal Tasaki, Time Scales in the Approach to Equilibrium of Macroscopic Quantum Systems, *Physical Review Letters*, 111, 140401-1-140401-5, 2013.10, We prove two theorems concerning the time evolution in general isolated quantum systems. The theorems are relevant to the issue of the time scale in the approach to equilibrium. The first theorem shows that there can be pathological situations in which the relaxation takes an extraordinarily long time, while the second theorem shows that one can always choose an equilibrium subspace, the relaxation to which requires only a short time for any initial state.. |

3. |
Takashi HARA, Decay of Correlations in Nearest-Neighbor Self-Avoiding Walk, Percolation, Lattice Trees and Animals, *Annals of Probability*, vol. 36, pp.530-593, 2008.03. |

4. |
Takashi Hara, Remco van der Hofstad, Gordon Slade, Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models., *Annals of Probability*, vol. 31, pp. 349-408, 2003.01. |

5. |
Takashi Hara, Tetsuya Hattori, Hiroshi Watanabe, Triviality of hierarchical Ising model in four dimensions., *Communications in Mathematical Physics*, vol. 220, pp.13-40, 2001.01. |

6. |
Takashi Hara, Gordon Slade, The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents., *J. of Statistical Physics*, 99, 1075-1168, vol. 99, pp. 1075--1168, 2000.01. |

7. |
Takashi Hara and Gordon Slade, The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion., *Journal of Mathematical Physics*, vol. 41, pp. 1244--1293, 2000.01. |

8. |
Tatsuhiko Koike, Takashi Hara and Satoshi Adachi
Tatsuhiko Koike, Takashi Hara and Satoshi Adachi, Critical behavior in gravitational collapse of a perfect fluid., *Physical Review D*, vol D59, pp. 104008, 1999.01. |

9. |
Takashi Hara and Gordon Slade.
The incipient infinite cluster in high-dimensional percolation.
Elec. Reseach Announcements of AMS, 4 (1998) 48--55.
Takashi Hara and Gordon Slade.
The incipient infinite cluster in high-dimensional percolation.
Elec. R, The incipient infinite cluster in high-dimensional percolation., *Electronic Research Annoucement of AMS*, vol. 4, pp. 48-55, 1998.01. |