Abstract
We consider a class of stochastic dynamical systems, called piecewise deterministic Markov processes, with states (x,σ)∈Ω×Γ, Ω being a region in ℝd or the d-dimensional torus, Γ being a finite set. The continuous variable x follows a piecewise deterministic dynamics, the discrete variable σ evolves by a stochastic jump dynamics and the two resulting evolutions are fully-coupled. We study stationarity, reversibility and time-reversal symmetries of the process. Increasing the frequency of the σ-jumps, the system behaves asymptotically as deterministic and we investigate the structure of its fluctuations (i.e. deviations from the asymptotic behavior), recovering in a non Markovian frame results obtained by Bertini et al. (Phys. Rev. Lett. 87(4):040601, 2001; J. Stat. Phys. 107(3–4):635–675, 2002; J. Stat. Mech. P07014, 2007; Preprint available online at http://www.arxiv.org/abs/0807.4457, 2008), in the context of Markovian stochastic interacting particle systems. Finally, we discuss a Gallavotti–Cohen-type symmetry relation with involution map different from time-reversal.
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Faggionato, A., Gabrielli, D. & Ribezzi Crivellari, M. Non-equilibrium Thermodynamics of Piecewise Deterministic Markov Processes. J Stat Phys 137, 259 (2009). https://doi.org/10.1007/s10955-009-9850-x
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DOI: https://doi.org/10.1007/s10955-009-9850-x