Appearance of Thermal Time

In this paper a viewpoint that time is an informational and thermal entity is presented. We consider a model for a simple relaxation process for which a relationship among event, time and temperature is mathematically formulated. It is then explicitly illustrated that temperature and time are statistically inferred through measurement of events. The probability distribution of the events thus provides an intrinsic correlation between temperature and time, which can relevantly be expressed in terms of the Fisher information metric. The two-dimensional differential geometry of temperature and time then leads us to a finding of a simple equation for the scalar curvature, $$R = -1$$ R = - 1 , in this case of relaxation process. This basic equation, in turn, may be regarded as characterizing a nonequilibrium dynamical process and having a solution given by the Fisher information metric. The time can then be interpreted so as to appear in a thermal way.