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Phys Rev E. 2016 Jun;93(6):062127. doi: 10.1103/PhysRevE.93.062127. Epub 2016 Jun 20.

Geometric structure and geodesic in a solvable model of nonequilibrium process.

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School of Mathematics and Statistics, University of Sheffield, Sheffield, S3 7RH, United Kingdom.
Department of Ecology and Evolution, University of Chicago, Chicago, Illinois 60637, USA.
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom.


We investigate the geometric structure of a nonequilibrium process and its geodesic solutions. By employing an exactly solvable model of a driven dissipative system (generalized nonautonomous Ornstein-Uhlenbeck process), we compute the time-dependent probability density functions (PDFs) and investigate the evolution of this system in a statistical metric space where the distance between two points (the so-called information length) quantifies the change in information along a trajectory of the PDFs. In this metric space, we find a geodesic for which the information propagates at constant speed, and demonstrate its utility as an optimal path to reduce the total time and total dissipated energy. In particular, through examples of physical realizations of such geodesic solutions satisfying boundary conditions, we present a resonance phenomenon in the geodesic solution and the discretization into cyclic geodesic solutions. Implications for controlling population growth are further discussed in a stochastic logistic model, where a periodic modulation of the diffusion coefficient and the deterministic force by a small amount is shown to have a significant controlling effect.

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