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Fokker–Planck description for a linear delayed Langevin equation with additive Gaussian noise

Luca Giuggioli, Thomas John McKetterick, V M Kenkre, Matthew Chase
2016 Journal of Physics A: Mathematical and Theoretical  
General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: Abstract We construct an equivalent probability description of linear multi-delay Langevin equations subject to additive Gaussian white noise. By exploiting the time-convolutionless transform and a time variable transformation we are able to write a Fokker-Planck equation (FPE) for the 1-time and for the 2-time
more » ... nd for the 2-time probability distributions valid irrespective of the regime of stability of the Langevin equations. We solve exactly the derived FPEs and analyze the aging dynamics by studying analytically the conditional probability distribution. We discuss explicitly why the initially conditioned distribution is not sufficient to describe fully out a non-Markov process as both preparation and observation times have bearing on its dynamics. As our analytic procedure can also be applied to linear Langevin equations with memory kernels, we compare the non-Markov dynamics of a one-delay system with that of a generalized Langevin equation with an exponential as well as a power law memory. Application to a generalization of the Green-Kubo formula is also presented. Introduction Markov assumptions, at the heart of the celebrated Smoluchowski's article on the equation for a diffusing particle in an external field [1], were so fruitful for the development of the modern theory of statistical physics [2] [3] [4] [5] [6] , that, as is well-known, the equation has become a workhorse for all sorts of calculations of stochastic phenomena. Over the last century, while Markov studies and their formalism grew and found applications in physics and beyond, the same cannot be said of non-Markov analysis, which has remained in its infancy. Part of the reasons for the lack of progress is the inherent difficulty in the description of non-Markov processes, but also because the limited resolution of the observations did not present a real need for it. In recent years, however, as experimental and statistical advances have brought the ability to extract history-dependent signatures in variety of systems, from the physical [7] and engineering [8] areas to financial processes [9], there is renewed interest to develop a general theory of non-Markov processes. It is thus natural on the 100th anniversary of Marian Smoluchoski's seminal contribution to look forward and report on a formalism to study some specific non-Markov systems. Among those systems for which the Markov assumption does not hold, we have selected linear delayed systems representing a classic example where the dynamics may display a remarkably different behavior compared to their Markoffian counterparts. The interplay of noise and time delays has a profound impact on many artificial and natural systems. In engineering systems, such as communication networks and sensor-actuator feedback applications, it is well known that finite transmission speeds introduce time-lags that affect process efficiency and control [10] [11] [12] . Biological systems are also affected by lagged dynamics. Delays have in fact been measured both with whole organisms, e.g. in gene regulatory networks [13, 14] and human coordination [15] , and among different organisms, e.g. in collectively moving animals [16] [17] [18] . Explaining these and other observations in economical [19] and physiological systems [20] requires modeling random processes with delayed dynamics. Langevin equations and Fokker-Planck equations (FPEs) are widely used to study the dynamics of stochastic processes. In Markov systems, there exists a well known procedure that allows one to construct an equivalent FPE description starting from a Langevin equation (see e.g. [21, 22] ). In non-Markov systems such a construction generally remains an open problem. While for Markov processes knowing the system at any (arbitrary) moment in the past allows to describe the future dynamics of the system, this is not the case for non-Markov systems that may display ageing effects. Given the history dependence of non-Markov processes, to describe the dynamics, it is necessary to have full knowledge of how the system has evolved from some time in the past. The so called bona fide FPE representation of a non-Markov process [23, 24] thus requires to construct the conditional probability distribution at any time in the past. A proper FPE description would thus need to contain information about the initial localization as well as the the initial preparation of the state of the system. Specific attempts to develop a procedure to write a FPE formalism from a non-Markov Langevin equation have already appeared in the past in the context of linear time non-local Langevin equations with additive Gaussian noise. The FPE construction consists of making J. Phys. A: Math. Theor. 49 (2016) 384002 L Giuggioli et al 2 J. Phys. A: Math. Theor. 49 (2016) 384002 L Giuggioli et al 4 x t x 1 J. Phys. A: Math. Theor. 49 (2016) 384002 L Giuggioli et al 6 J. Phys. A: Math. Theor. 49 (2016) 384002 L Giuggioli et al 7 J. Phys. A: Math. Theor. 49 (2016) 384002 L Giuggioli et al 8
doi:10.1088/1751-8113/49/38/384002 fatcat:g4aclptiwjgi5mvc5et5h3uauy