Numerical Analysis of Statistical Manifolds Associated with Nonequilibrium Processes
Project/Area Number  07680320 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Statistical science

Research Institution  Gunma National College of Technology 
Principal Investigator 
OBATA Tsunehiro Gunma National College of Technology/Department of Electrical Engineering/assistant professor/, 電気工学科, 助教授 (50005534)

CoInvestigator(Kenkyūbuntansha) 
HARA Hiroaki Tohoku University/Graduate School of Information Science/assistant professor/, 大学院・情報科学研究科, 助教授 (60005296)

Project Fiscal Year 
1995 – 1996

Project Status 
Completed(Fiscal Year 1996)

Budget Amount *help 
¥2,100,000 (Direct Cost : ¥2,100,000)
Fiscal Year 1996 : ¥400,000 (Direct Cost : ¥400,000)
Fiscal Year 1995 : ¥1,700,000 (Direct Cost : ¥1,700,000)

Keywords  information geometry / correlated walk / random walk / stability / nonequilibrium / viscoelastic / complex system / quantum gase / 情報幾何学 / 相関歩 / ランダムウォーク / 安定性 / 非平衡過程 / 粘弾性 / 複雑系 / 量子気体 / 粘弾性物質 
Research Abstract 
Metrics and connections are introduced on parameter spaces constituted from the jump probabilities specifying correlated walks, by the method of information geometry. These geometrical objects depend on a step time in general. Relations between the time evolution of curvature tensors and physical properties such as the stability of processes are investigated. Two correlatedwalk models on a linear lattice are treated. In a model, the walkers do not stay, and the parameter space is then twodimensional. In another model, the walkers sometimes stay, and the parameter space is threedimensional in case of symmetric walking. The threespace is foliated into twodimensional spaces by a stay parameter, and the time evolution of the Riemann scalar curvature R of each leaf is investigated. It is shown that the dynamic properties of the R's for both models can be well understood in terms of the stability of processes, the correlation between successive two steps, and also the activity of steppi
… More
ng. As time goes by, the parameter spaces approach uniform spaces. The R's in the infinite time are shown to reflect the stability of systems and also the regularity of paths. The alphacurvature is also investigated. The alpha=1 curvature converges to zero for any case. This property is suggested to be an universal property in a broad class of systems including thermodynamic equilibrium systems. The mathematical structure of time evolbing parameter spaces is pointed out to be very analogous to that of the NewtonCartan theory of gravity. Moreover, new models of stochastic processes are proposed. Stochastic equations prescribing the motion of objects whose inner states change in time are obtained through considering the behavior of animals. Another model is introduced, using reponse functions of hypothetical complex viscoelastic materials made from an infinitude of socalled viscoelastic materials. Using an infinitude of complex viscoelastic materials, one can construct more complicated viscoelastic materials. The highly complicated materials are found to be subject to superslow dynamics. The geometrical properties of the parameter spaces for the new models are still unknown. Less

Report
(4results)
Research Output
(18results)