Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                

Geometric structure and geodesic in a solvable model of nonequilibrium process

Eun-jin Kim, UnJin Lee, James Heseltine, and Rainer Hollerbach
Phys. Rev. E 93, 062127 – Published 20 June 2016
PDFHTMLExport Citation
Abstract
Authors
Article Text
  • INTRODUCTION
  • INFORMATION CHANGE AND FLOW
  • A SOLVABLE MODEL
  • GEODESIC MOTION
  • GEODESIC EXAMPLES AND SIGNIFICANCE
  • PHYSICAL REALIZABILITY OF A GEODESIC…
  • APPLICATION TO POPULATION GROWTH
  • CONCLUSION
  • APPENDICES
    • References

    Abstract

    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.

    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    2 More
    • Received 19 December 2015
    • Revised 25 April 2016

    DOI:https://doi.org/10.1103/PhysRevE.93.062127

    ©2016 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Fluctuations & noiseNonequilibrium & irreversible thermodynamics
    Statistical Physics & Thermodynamics

    Authors & Affiliations

    Eun-jin Kim1, UnJin Lee2, James Heseltine1, and Rainer Hollerbach3

    • 1School of Mathematics and Statistics, University of Sheffield, Sheffield, S3 7RH, United Kingdom
    • 2Department of Ecology and Evolution, University of Chicago, Chicago, Illinois 60637, USA
    • 3Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom

    Article Text (Subscription Required)

    Click to Expand

    References (Subscription Required)

    Click to Expand
    Issue

    Vol. 93, Iss. 6 — June 2016

    Reuse & Permissions
    Access Options
    Author publication services for translation and copyediting assistance advertisement

    Authorization Required

    Log In
    Other Options
    ×

    Images

    • Figure 1
      Figure 1

      (a) A sketch illustrating the problem of moving a PDF from a larger (y0) to smaller mean position (yF<y0), where the temperature (width) of the PDF is the same at the initial and final times t=0 and t=tF. (b) A natural path with the same temperature β=β0 for all time between t=0 and t=tF. The units of x are arbitrary.

      Reuse & Permissions
    • Figure 2
      Figure 2

      (a) Total time; (b) information length; (c) Wξ against β0 in the nongeodesic case (in solid black) and geodesic I (in dashed blue) and II (in dash-dotted red); y(t=0)=y0=5/6 and y(t=tF)=yF=1/30. Geodesic II (dash-dotted red) is shown for the value of β0 where the diffusion (DII) is non-negative. A distinct minimum in the total time is observed in geodesic I caused by the resonance (the matching of Δ=Δm).

      Reuse & Permissions
    • Figure 3
      Figure 3

      y and β−1/2 against time for β0=0.3 and 3 in (a) and (c), respectively; the corresponding geodesic circular segments in the (y,β−1/2) upper half plane in (b) and (d), respectively. In both cases, y0=5/6 and yF=1/30.

      Reuse & Permissions
    • Figure 4
      Figure 4

      Time evolution of PDFs against x for β0=0.3,3,30, and 300 in (a)–(d), respectively. y0=〈x(t=0)〉=5/6 and yF=〈x(t=tF)〉=1/30. The initial and final PDFs are shown by thick red lines on the right and blue lines on the left, respectively.

      Reuse & Permissions
    • Figure 5
      Figure 5

      y and β−1/2 against time in (a) and (c); The circular geodesics in the upper half plane y and β−1/2 in (b) and (d) for β0=3 and 30, respectively. In both cases, y0=5/6 and yF=1/30.

      Reuse & Permissions
    • Figure 6
      Figure 6

      (a) The blue upper curve shows DI(t), and the red lower curve shows f(t). (b) The blue lower curve shows DII(t), and the red upper curve shows γ(t). For both panels β0=30, y(t=0)=y0=0.08, and y(t=tF)=yF=0.05.

      Reuse & Permissions
    • Figure 7
      Figure 7

      Time evolution of PDFs of x over the first cycle for β0=30 and 300 in (a) and (b), respectively. (c)–(d) are the evolution of PDFs over the first and the last cycles [tenth and hundredth cycle for (c) and (d), respectively] shown at the same time. y0=5/6 and yF=1/30.

      Reuse & Permissions
    • Figure 8
      Figure 8

      Time evolution of PDFs of the population u for β0=30 and 300, corresponding to Fig. 7. 〈u(t=0)〉=6xs〈u(t=tF)〉=1.0345, which is close to the carrying capacity u∞=1 (γ=1, ε=1). (a)–(b) show the PDF during the first cycle as in Figs. 7–7, where the PDF at t=0 can be identified with the peak at u=6. (c)–(d) are the evolution of PDFs over the first and the last cycles [tenth and hundredth cycle for (c) and (d), respectively] shown at the same time.

      Reuse & Permissions
    • Figure 9
      Figure 9

      (a) The red lower curve shows DI(t), and the blue upper curve shows g(t). (b) The red lower curve shows y (the mean value of x), and the blue upper curve shows u. For both panels β0=30, the carrying capacity γ/ε=u∞=1, y(t=0)=y0=〈x(t=0)〉=5/6, and y(t=tF)=yF=〈x(t=tF)〉=5/6.

      Reuse & Permissions
    ×

    Sign up to receive regular email alerts from Physical Review E

    Log In

    Cancel
    ×

    Search

    Article Lookup

    Paste a citation or DOI
    Enter a citation
    ×