Geometric structure and geodesic in a solvable model of nonequilibrium process

Abstract

We investigate the geometric structure of a non-equilibriump r o c e s sa n di t sg e o d e s i cs o l utions. 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 realisations of such geodesic solutions satisfying boundary conditions, we present a novel 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.

Geometric structure and information change in phase transitions

“…From Fig. 6(a), we see that the initial movement of the peaks toward the origin immediately yields a very strongly Ddependent E, since narrower peaks yield correspondingly more distinguishable states along the way (e.g., [27,28]). This is responsible for the nonexistence of a geodesic solution with constant E. As indicated in Fig.…”

“…We calculate time-dependent PDFs and the total number of different states that a system undergoes during the transition. The latter is quantified by the information length [25][26][27][28][29], which is dimensionless, defined as (see Appendix A)…”

“…[30]) and is a generalization of statistical distance [31], where the distance is set by the number of distinguishable states between two PDFs. While the latter was heavily used in equilibrium or near equilibrium of classical and quantum systems [32][33][34][35][36][37][38][39][40], our recent work [25][26][27][28][29] adapted this concept to a nonequilibrium system to elucidate geometric structure of nonequilibrium processes. Specifically, Ref.…”

“…[29] mapped out the attractor structure L ∞ vs x 0 for the linear and cubic process and showed (i) that a linear damping preserves a linear geometry L ∞ ∝ x 0 and (ii) that a nonlinear damping gives rise to a power-law scaling L ∞ ∝ x n 0 (n ∼ 1.5-1.9) of the attractor structure. Reference [28] found interesting geodesic solutions in a nonautonomous OrnsteinUhlenbeck (OU) process by modulating model parameters and by including a time-dependent external deterministic killing term. Notably, the modulation of the model parameters and the killing term were periodic or oscillatory.…”

Time-dependent probability density functions and information geometry in stochastic logistic and Gompertz models

“…Furthermore, for simplicity, we assume that ξ i (i = 1, 2, 3) are Gaussian noises with short correlation time and the following correlation PDFs were shown in our previous work for a linear model (i.e. ǫ = ξ 2 = ξ 3 = 0) [26] and for a nonlinear model with ξ 1 = ξ 3 = 0 [26][27][28].…”