Abstract
In this chapter, we study the facet of noise arising in the dynamics of a physical system due to its interaction with its environment. We consider the system of Kuramoto oscillators evolving in presence of Gaussian, white noise. Introducing noise into the bare Kuramoto dynamics models its interaction with the environment. For this noisy Kuramoto model, we obtain exact results on autocorrelation of the order parameter in the nonequilibrium stationary state that the dynamics attains at long times. In obtaining the results, we employ a method based on an exact mapping of the stationary-state dynamics of the model in the thermodynamic limit to the noisy dynamics of a single, nonuniform oscillator. The method also allows to obtain the autocorrelation in the equilibrium stationary state of a related model of long-range interactions, the Brownian mean-field model. Both the models exhibit a phase transition between a synchronized and an incoherent phase at a critical value of the noise strength. We show that in the two phases as well as at the critical point, the autocorrelation for both the model decays as an exponential with a rate that increases continuously with the noise strength. This chapter is based on our published work, Ref. [1].
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Notes
- 1.
For detailed discussions on long-range interactions, see Chap. 3.
- 2.
- 3.
Unlike a nonequilibrium stationary state, an equilibrium stationary state does not support any net probability current [11].
- 4.
Here, we are considering a one-dimensional phase space. In one dimension, one has oscillations only if the motion is periodic [12]. The motion of a simple harmonic oscillator is two dimensional. The reason is that the second-order equation of motion of a simple harmonic oscillator reduces to two coupled first-order equations, thereby resulting in a two-dimensional phase space.
References
D. Das, S. Gupta, J. Phys. A: Math. Theor. 51(44), 445003 (2018). https://doi.org/10.1088/1751-8121/aae2c2
K. Huang, Statistical Mechanics, 2nd edn. (Wiley, New York, 1987)
R. Livi, P. Politi, Nonequilibrium Statistical Physics: A Modern Perspective (Cambridge University Press, Cambridge, 2017)
R. Kubo, Rep. Prog. Phys. 29(1), 255 (1966). https://doi.org/10.1088/0034-4885/29/1/306
H. Sakaguchi, Prog. Theor. Exp. Phys. 79(1), 39 (1988). https://doi.org/10.1143/PTP.79.39
R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford University Press, Oxford, 2001)
I.H.d. Mendoza, L.A. Pachón, J. Gómez-Gardeñes, D. Zueco, Phys. Rev. E 90(5), 052904 (2014). https://doi.org/10.1103/PhysRevE.90.052904
P.H. Chavanis, Eur. Phys. J. B 87(5), 120 (2014). https://doi.org/10.1140/epjb/e2014-40586-6
M. Antoni, S. Ruffo, Phys. Rev. E 52(3), 2361 (1995). https://doi.org/10.1103/PhysRevE.52.2361
R.K. Pathria, P.D. Beale, Statistical Mechanics, 3rd edn. (Elsevier, Amsterdam, 2011)
C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer, Berlin, 1983)
S.H. Strogatz, Nonlinear Dynamics and Chaos (CRC Press, Boca Raton, 2018). https://doi.org/10.1201/9780429492563
H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer, Berlin, 1996)
S. Gupta, A. Campa, S. Ruffo, Statistical Physics of Synchronization (Springer, Cham, 2018). https://doi.org/10.1007/978-3-319-96664-9
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Das, D., Gupta, S. (2023). Stationary Correlations in the Noisy Kuramoto Model. In: Facets of Noise. Fundamental Theories of Physics, vol 214. Springer, Cham. https://doi.org/10.1007/978-3-031-45312-0_6
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