Abstract
We continue our study of the linear response of a nonequilibrium system. This Part II concentrates on models of open and driven inertial dynamics but the structure and the interpretation of the result remain unchanged: the response can be expressed as a sum of two temporal correlations in the unperturbed system, one entropic, the other frenetic. The decomposition arises from the (anti)symmetry under time-reversal on the level of the nonequilibrium action. The response formula involves a statistical averaging over explicitly known observables but, in contrast with the equilibrium situation, they depend on the model dynamics in terms of an excess in dynamical activity. As an example, the Einstein relation between mobility and diffusion constant is modified by a correlation term between the position and the momentum of the particle.
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Baiesi, M., Maes, C., Wynants, B.: Fluctuations and response of nonequilibrium states. Phys. Rev. Lett. 103, 010602 (2009)
Baiesi, M., Maes, C., Wynants, B.: Nonequilibrium linear response for Markov dynamics, I: Jump processes and overdamped diffusions. J. Stat. Phys. 137, 1094–1116 (2009).
Balakrishnan, V.: Elements of Nonequilibrium Statistical Mechanics. Ane Books India (2008)
Blickle, W., Speck, T., Lutz, C., Seifert, U., Bechinger, C.: Einstein relation generalized to nonequilibrium. Phys. Rev. Lett. 98, 210601 (2007)
Callen, H.B., Welton, T.A.: Irreversibility and generalized noise. Phys. Rev. 83, 34 (1951)
Chetrite, R., Falkovich, G., Gawędzki, K.: Fluctuation relations in simple examples of non-equilibrium steady states. J. Stat. Mech. P08005 (2008)
Diezemann, G.: Fluctuation-dissipation relations for Markov processes. Phys. Rev. E 72, 011104 (2005)
Gomez-Solano, J.R., Petrosyan, A., Ciliberto, S., Chetrite, R., Gawędzki, K.: Experimental verification of a modified fluctuation-dissipation relation for a micron-sized particle in a non-equilibrium steady state. Phys. Rev. Lett. 103, 040601 (2009)
Harada, T., Sasa, S.-Y.: Equality connecting energy dissipation with violation of fluctuation-response relation. Phys. Rev. Lett. 95, 130602 (2005)
Katz, S., Lebowitz, J.L., Spohn, H.: Phase transitions in stationary non-equilibrium states of model lattice systems. Phys. Rev. B 28, 1655–1658 (1983)
Krüger, M., Fuchs, M.: Phys. Rev. Lett. 102, 135701 (2009)
Krüger, M., Fuchs, M.: Non-Equilibrium relation between mobility and diffusivity of interacting Brownian particles under shear. Prog. Theor. Phys. Suppl. arXiv:0911.1632v1 [cond-mat.soft]
Kubo, R.: The fluctuation-dissipation theorem. Rep. Prog. Phys. 29, 255–284 (1966)
Lebowitz, J.L.: Stationary nonequilibrium Gibbsian ensembles. Phys. Rev. 114, 1192–1202 (1959)
Lebowitz, J.L., Spohn, H.: A Gallavotti–Cohen type symmetry in large deviation functional for stochastic dynamics. J. Stat. Phys. 95, 333–365 (1999)
Lippiello, E., Corberi, F., Zannetti, M.: Fluctuation dissipation relations far from equilibrium. J. Stat. Mech. P07002 (2007)
Lippiello, E., Corberi, F., Sarracino, A., Zannetti, M.: Nonlinear susceptibilities and the measurement of a cooperative length. Phys. Rev. B 77, 212201 (2008)
Maes, C.: Fluctuation theorem as a Gibbs property. J. Stat. Phys. 95, 367–392 (1999)
Maes, C.: On the origin and the use of fluctuation relations for the entropy. In: Dalibard, J., Duplantier, B., Rivasseau, V. (eds.) Séminaire Poincaré, vol. 2, pp. 11–29. Birkhäuser, Basel (2003)
Maes, C., Netočný, K.: Time-reversal and entropy. J. Stat. Phys. 110, 269–310 (2003)
Maes, C., Wynants, B.: On a response function and its interpretation. Markov Proc. Rel. Fields (2009)
Maes, C., Netočný, K., Wynants, B.: Steady state statistics of driven diffusions. Physica A 387, 2675–2689 (2008)
Maes, C., Netočný, K., Wynants, B.: On and beyond entropy production; the case of Markov jump processes. Markov Proc. Rel. Fields 14, 445–464 (2008)
Martin, P.A.: Physique statistique des processus irreversibles. In: Lecture Notes of the DEA de Physique Théorique, notes by F. Coppex, ENS Lyon, Fall 2001–Spring 2004
Puglisi, A., Villamaina, D.: Irreversible effects of memory. Europhys. Lett. 88, 30004 (2009)
Puglisi, A., Baldassarri, A., Vulpiani, A.: Violation of the Einstein relation in Granular Fluids: the role of correlations. J. Stat. Mech. P08016 (2007)
Ruelle, D.: A review of linear response theory for general differentiable dynamical systems. Nonlinearity 22, 855–870 (2009)
Seifert, U., Speck, T.: The fluctuation-dissipation theorem for nonequilibrium steady states: role of stochastic entropy and a classification of variants. Europhys. Lett. 89, 10007 (2010)
Speck, T., Seifert, U.: Restoring a fluctuation-dissipation theorem in a nonequilibrium steady state. Europhys. Lett. 74, 391–396 (2006)
Tasaki, H.: Two theorems that relate discrete stochastic processes to microscopic mechanics. arXiv:0706.1032 [cond-mat.stat-mech]
Villamaina, D., Baldassarri, A., Puglisi, A., Vulpiani, A.: The fluctuation-dissipation relation: how does one compare correlation functions and responses? J. Stat. Mech. P07024 (2009)
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Baiesi, M., Boksenbojm, E., Maes, C. et al. Nonequilibrium Linear Response for Markov Dynamics, II: Inertial Dynamics. J Stat Phys 139, 492–505 (2010). https://doi.org/10.1007/s10955-010-9951-6
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DOI: https://doi.org/10.1007/s10955-010-9951-6