Abstract
We generalize the Clausius (in)equality to overdamped mesoscopic and macroscopic diffusions in the presence of nonconservative forces. In contrast to previous frameworks, we use a decomposition scheme for heat which is based on an exact variant of the Minimum Entropy Production Principle as obtained from dynamical fluctuation theory. This new extended heat theorem holds true for arbitrary driving and does not require assumptions of local or close to equilibrium. The argument remains exactly intact for diffusing fields where the fields correspond to macroscopic profiles of interacting particles under hydrodynamic fluctuations. We also show that the change of Shannon entropy is related to the antisymmetric part under a modified time-reversal of the time-integrated entropy flux.
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Notes
Studies on the heat theorem were started by Clausius since about 1855, with the final emergence of the entropy concept in [1].
References
Clausius, R.: On a modified form of the second fundamental theorem in the mechanical theory of heat. Ann. Phys. (Leipz.) 125, 353 (1865)
Carnot, S.: Réflexions sur la puissance motrice du feu et sur les machines propres à déveloper cette puissance. Paris (1824)
Maes, C.: Nonequilibrium entropies. Phys. Scr. 86, 058509 (2012)
Komatsu, T.S., Nakagawa, N., Sasa, S.-I., Tasaki, H.: Phys. Rev. Lett. 100, 230602 (2008)
Komatsu, T.S., Nakagawa, N., Sasa, S.-I., Tasaki, H.: J. Stat. Phys. 134, 401 (2009)
Komatsu, T.S., Nakagawa, N.: An expression for stationary distribution in nonequilibrium steady states. Phys. Rev. Lett. 100, 030601 (2008)
Maes, C., Netočný, K.: Minimum entropy production principle from a dynamical fluctuation law. J. Math. Phys. 48, 053306 (2007)
Maes, C., Netočný, K.: Europhys. Lett. 82, 30003 (2008)
Hatano, T., Sasa, S.-I.: Steady-state thermodynamics of Langevin systems. Phys. Rev. Lett. 86, 3463 (2001)
Oono, Y., Paniconi, M.: Steady state thermodynamics. Prog. Theor. Phys. Suppl. 130, 29 (1998)
Bertini, L., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Thermodynamic transformations of nonequilibrium states. J. Stat. Phys. 149, 773–802 (2012)
Bertini, L., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Clausius inequality and optimality of quasistatic transformations for nonequilibrium stationary states. Phys. Rev. Lett. 110, 020601 (2013)
Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991)
Boksenbojm, E., Maes, C., Netočný, K., Pešek, J.: Heat capacity in nonequilibrium steady states. Europhys. Lett. 96, 40001 (2011)
Maes, C., Salazar, A.: Linear response in the nonequilibrium zero range process. arXiv:1305.4157v1 [cond-mat.stat-mech]
Lebowitz, J.L., Spohn, H.: Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs. Adv. Chem. Phys. 39, 109–142 (1978)
Pesek, J., Boksenbojm, E., Netočný, K.: Model study on steady heat capacity in driven stochastic systems. Cent. Eur. J. Phys. 10, 692 (2012)
Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Phys. Rev. Lett. 94, 030601 (2005)
Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Macroscopic fluctuation theory for stationary non-equilibrium states. J. Stat. Phys. 110, 635 (2002)
De Groot, S., Mazur, P.O.: Non-Equilibrium Thermodynamics. Dover, New York (1962)
Chetrite, R., Gawedzki, K.: Fluctuation relations for diffusion processes. Commun. Math. Phys. 282, 469–518 (2007)
Trepagnier, E.H., Jarzynski, C., Ritort, F., Crooks, G.E., Bustamante, C.J., Liphardt, J.: Experimental test of Hatano and Sasa’s nonequilibrium steady-state equality. Proc. Natl. Acad. Sci. USA 101, 15038 (2004)
Maes, C., Netočný, K., Wynants, B.: Steady state statistics of driven diffusions. Physica A 387, 2675–2689 (2008)
Maes, C., Netočný, K., Shergelashvili, B.: Nonequilibrium relation between potential and stationary distribution for driven diffusion. Phys. Rev. E 80, 011121 (2009)
Maes, C., Netočný, K., Wynants, B.: On and beyond entropy production: the case of Markov jump processes. Markov Process. Relat. Fields 14, 445 (2008)
Acknowledgements
This work started from discussions with Keiji Saito, Hal Tasaki, Shin-Ichi Sasa and Naoko Nakagawa in January–February 2012. CM wishes to thank the hospitality of the Yukawa institute in Kyoto and of Hisao Hayakawa in particular. KN thanks Shin-Ichi Sasa for fruitful discussions and acknowledges the support from the Grant Agency of the Czech Republic, Grant no. P204/12/0897.
The results of the present paper were first reported on 6 June 2012 at the Isola del Giglio during the workshop on Non-equilibrium fluctuation-response relations, June 5–8, 2012, in parallel with the talk by G. Jona-Lasinio on the work in [11, 12]. In that same context we are also grateful for private communication with K. Gawedzki.
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In honor of Herbert Spohn on the occasion of his 65th birthday.
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Maes, C., Netočný, K. A Nonequilibrium Extension of the Clausius Heat Theorem. J Stat Phys 154, 188–203 (2014). https://doi.org/10.1007/s10955-013-0822-9
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DOI: https://doi.org/10.1007/s10955-013-0822-9