On exact time averages of a massive Poisson particle

Abstract

In this work we study, under the Stratonovich definition, the problem of the damped oscillatory massive particle subject to a heterogeneous Poisson noise characterized by a rate of events, λ(t), and a magnitude, Φ, following an exponential distribution. We tackle the problem by performing exact time averages over the noise in a similar way to previous works analysing the problem of the Brownian particle. From this procedure we obtain the long-term equilibrium distributions of position and velocity as well as analytical asymptotic expressions for the injection and dissipation of energy terms. Considerations on the emergence of stochastic resonance in this type of system are also set forth.

Exact cumulant Kramers–Moyal-like expansion

“…These same moments can be obtained directly from the dynamics [20]. It verifies the exactness of the CKME.…”

“…Via the CKME we obtain the correct exact value given above, which can also be derived using the techniques of Ref. [20].…”

“…(20) A master equation is the evolution equation for the Markovian transition probability p(x 2 , t 1 | x 1 , t 1 ) [23,28,29]. It is usually presented as a time-evolution equation for the PDF itself since we can assume that we know exactly the initial PDF…”

“…[20], we can calculate the jump moments exactly. They are obtained in Appendix E. The existence and origin of non equilibrium and athermal reservoirs, such as the one described by Eq.…”

“…Our approach is generic in principle, but we chose to illustrate the procedures by the use of linear Markovian mechanical models for Gaussian, and non-Gaussian noise. We use a formalism based on the Laplace transform that has been used to obtain exact results for Langevin equations in the context of Markovian and non-Markovian noise [15,16], thermal conductance for linear [17] and non-linear systems [18], fluctuation relations for the work [19], and non Gaussian noises [20,21,18,22]. This method is equivalent to taking into account all noise cumulants, which can be of finite or infinite in number.…”