We explore the effect of different spatially periodic, deterministic forces on the information geometry of stochastic processes. The three forces considered are f0=sin(πx)/π and f±=sin(πx)/π±sin(2πx)/2π , with f− chosen to be particularly flat (locally cubic) at the equilibrium point x=0 , and f+ particularly flat at the unstable fixed point x=1 . We numerically solve the Fokker–Planck equation with an initial condition consisting of a periodically repeated Gaussian peak centred at x=μ , with μ in the range [0,1] . The strength D of the stochastic noise is in the range 10−4 – 10−6 . We study the details of how these initial conditions evolve toward the final equilibrium solutions and elucidate the important consequences of the interplay between an initial PDF and a force. For initial positions close to the equilibrium point x=0 , the peaks largely maintain their shape while moving. In contrast, for initial positions sufficiently close to the unstable point x=1 , there is a tendency for the peak to slump in place and broaden considerably before reconstituting itself at the equilibrium point. A consequence of this is that the information length L∞ , the total number of statistically distinguishable states that the system evolves through, is smaller for initial positions closer to the unstable point than for more intermediate values. We find that L∞ as a function of initial position μ is qualitatively similar to the force, including the differences between f0=sin(πx)/π and f±=sin(πx)/π±sin(2πx)/2π , illustrating the value of information length as a useful diagnostic of the underlying force in the system.
CORE (COnnecting REpositories)
CORE (COnnecting REpositories)