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Exact Equalities and Thermodynamic Relations for Nonequilibrium Steady States

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Abstract

We study thermodynamic operations which bring a nonequilibrium steady state (NESS) to another NESS in physical systems under nonequilibrium conditions. We model the system by a suitable Markov jump process, and treat thermodynamic operations as protocols according to which the external agent varies parameters of the Markov process. Then we prove, among other relations, a NESS version of the Jarzynski equality and the extended Clausius relation. The latter can be a starting point of thermodynamics for NESS. We also find that the corresponding nonequilibrium entropy has a microscopic representation in terms of symmetrized Shannon entropy in systems where the microscopic description of states involves “momenta”. All the results in the present paper are mathematically rigorous.

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Notes

  1. This part can be read as a summary of the whole paper.

  2. To be precise, this is true when there always is a nonvanishing temperature difference. When \(\beta _1(t)=\beta _2(t)\), the currents are very small and we rather have \(J^\dagger _k(t)\simeq -J_k(-t)\).

  3. The standard generator of a stochastic process is given by the transpose of \(\mathsf {R}\).

  4. Throughout the present paper, the entropy production always means the entropy production in the heat baths.

  5. The approach to equilibrium is indeed a nonequilibrium phenomenon that can be studied in the framework of equilibrium dynamics.

  6. We assume here (and in what follows) that each transition is associated with only a single bath. See also Sect. 2.4.

  7. In a formulations based on a stochastic process (as in the present work), the relation (2.17) is nothing more than an interpretation. In more microscopic formulations based on mechanics, one may justify such relations. See, e.g., [43].

  8. Such measurements are indeed possible in modern calorimetry.

  9. One strategy is to measure the back action from the system to the generator (such as a coil) of the field. In a colloidal system it may be possible to determine the work done by the field by precisely measuring the positions of charged particles.

  10. In most of realistic situations for heat conduction, only some small portions of the system is in touch with the heat baths. To model such a situation by using a system of particles on a lattice, we assume that the energy of the system changes only when a particle hops within one of the portions which are in touch with the baths. In other words, if an allowed transition \({x\rightarrow y}\) is such that a particle hops outside the portions, then one must have \(H^\nu _x=H^\nu _y\). We further use the transition rule (2.28) so as to make the corresponding transition rate (which is indeed 1) independent of any inverse temperatures.

  11. We define the force in this setting as \(f_{{x\rightarrow y}}=f d_{x\rightarrow y}\). The force \(f_{x\rightarrow y}\) is said to be conservative if one can write \(f_{{x\rightarrow y}}=U_x-U_y\) for any \(x,y\in \mathcal{S}\) with a suitable function (i.e., potential) \(U_x\).

  12. A convenient derivation is to start from (3.5) with \(\kappa =1\), and apply the cumulant expansion as above.

  13. In a system with parameters (2.34) with a nonequilibrium protocol \((\beta ,\nu (t),f(t))\), for example, we choose the equilibrium protocol as \((\beta ,\nu (t),0)\).

  14. For simplicity we only consider protocols in which the reference inverse temperature \(\beta \) is fixed. It is not difficult to treat protocols where \(\beta \) varies; one simply redefines \(\beta \) properly when one decomposes the whole protocol into a sum of step protocols.

  15. To be consistent with the notation introduced in Sect. 2.1, we are here taking \(\tau _0=\{(N-1)/(N+1)\}\tau \). In this case \(\tau _0\) also diverges when we let \(\tau \uparrow \infty \).

  16. Such a technique is common, for example, in the large deviation theory [42]. Similar technique was used for steady state thermodynamics in [7].

  17. The difference between \(\bigl \langle \Psi ^{(\alpha )}\bigr \rangle ^{\tau ,(\alpha _\mathrm {eq})}_{x\rightarrow }\) and \(\bigl \langle \Psi ^{(\alpha )}\bigr \rangle ^{\tau ,(\alpha )}_{x\rightarrow }\) is proportional to \(\epsilon ^2\), but is also roughly proportional to \(\tau \). Thus the difference diverges as \(\tau \uparrow \infty \).

  18. It may be also reasonable to consider a model in which \(\lambda ^\alpha _x\ne \lambda ^\alpha _{x^*}\). In such a model, one should include the contribution from \(\lambda ^\alpha _x/\lambda ^\alpha _{x^*}\) into the definition of \(\Theta ^{{\hat{\alpha }}}[\hat{x}]\) so as to keep the symmetry (10.3) valid (see [45]). Then all the results in the present section remain valid.

  19. One can also compute the expectation value of the entropy production rate explicitly as \(\sum _x\sigma _x^{(\beta _1',\beta _2')}\,p_x(t) =\sigma _\mathrm {st}^{(\beta _1',\beta _2')} +[\beta _1'\{u(\beta _1)-u(\beta _1')\} +\beta _2'\{u(\beta _2)-u(\beta _2')\}] (2L)^{-1}\sum _{j=1}^L\lambda ^jt^{j-1}e^{-\lambda t}\) for \(t\ge 0\). It decays exponentially to the steady value.

  20. The term \(O(\epsilon ^2\delta )\) is nonvanishing.

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Acknowledgments

It is a pleasure to thank Hisao Hayakawa, Masato Itami, Nobuyasu Ito, Chris Jarzynski, Gianni Jona-Lasinio, Joel Lebowitz, Christian Maes, Karel Netocny, Yoshi Oono, Glenn Paquette, Takahiro Sagawa, Keiji Saito, Herebert Spohn, and Akira Shimizu for valuable discussions. The present study was supported by KAKENHI Nos. 22340109, 23540435, and 25103002, by the JSPS Core-to-Core program “Non-equilibrium dynamics of soft-matter and information”, and partially by JSPS and Leading Research Organizations, namely NSERC, ANR, DFG, RFBR, RCUK and NSF as Partner Organizations under the G8 Research Councils Initiative for Multilateral Research Funding.

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Authors and Affiliations

  1. Laboratory for Computational Molecular Design, RIKEN QBiC, Kobe, 650-0047, Japan

    Teruhisa S. Komatsu

  2. College of Science, Ibaraki University, Mito, Ibaraki, 310-8512, Japan

    Naoko Nakagawa

  3. Department of Physics, Kyoto University, Kyoto, 606-8502, Japan

    Shin-ichi Sasa

  4. Department of Physics, Gakushuin University, Mejiro, Toshima-ku, Tokyo, 171-8588, Japan

    Hal Tasaki

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  1. Teruhisa S. Komatsu
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  2. Naoko Nakagawa
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  4. Hal Tasaki
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Correspondence to Hal Tasaki.

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Komatsu, T.S., Nakagawa, N., Sasa, Si. et al. Exact Equalities and Thermodynamic Relations for Nonequilibrium Steady States. J Stat Phys 159, 1237–1285 (2015). https://doi.org/10.1007/s10955-015-1221-1

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