Journal
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
Volume 49, Issue 38, Pages -Publisher
IOP Publishing Ltd
DOI: 10.1088/1751-8113/49/38/384002
Keywords
delay Langevin equation; Fokker-Planck equation for delay proceeses; generalized Langevin equation; non-Markov processes; ageing dynamics; generalized Green-Kubo formula; one and two-time probability distribution delayed processes
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Funding
- Engineering and Physical Sciences Research Council (EPSRC) UK [EP/I013717/1]
- EPSRC grant [EP/I013717/1]
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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 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.
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