We consider a class of stochastic dynamical systems, called piecewise deterministic Markov processes, with states (x, sigma) is an element of Omega x Gamma, Omega being a region in R(d) or the d-dimensional torus, Gamma being a finite set. The continuous variable x follows a piecewise deterministic dynamics, the discrete variable sigma 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 sigma-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.
Non-equilibrium Thermodynamics of Piecewise Deterministic Markov Processes / Faggionato, Alessandra; D., Gabrielli; Ribezzi M., Crivellari. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - STAMPA. - 137:2(2009), pp. 259-304. [10.1007/s10955-009-9850-x]
Non-equilibrium Thermodynamics of Piecewise Deterministic Markov Processes
FAGGIONATO, ALESSANDRA;
2009
Abstract
We consider a class of stochastic dynamical systems, called piecewise deterministic Markov processes, with states (x, sigma) is an element of Omega x Gamma, Omega being a region in R(d) or the d-dimensional torus, Gamma being a finite set. The continuous variable x follows a piecewise deterministic dynamics, the discrete variable sigma 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 sigma-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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.