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One-Dimensional Heat Conduction and Entropy Production

Part of the Advanced Structured Materials book series (STRUCTMAT,volume 87)

Abstract

This review paper analyzes the entropy concept and the second law of thermodynamics in the context of one-dimensional media. For simplicity, only thermal processes are taken into account and mechanical motions are neglected. The relation between entropy and temperature and the constraints on the direction of the heat flux are discussed. A comparison with the approach of P. A. Zhilin and the approach based on statistical mechanics is presented. The obtained conclusions are applied to three models: classical, hyperbolic and ballistic heat conduction. It is shown that the concept according to which heat flows from hot to cold is consistent only with the classical model. The peculiarities of the entropy definition and the second law of thermodynamics formulation for non-classical systems are discussed.

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Notes

  1. 1.

    Literally it is said in [8]: “the rate of change of internal entropy...”. However, the word “internal” is used only as an antithesis to entropy coming from outside. Therefore, this notation of “internal entropy” is equivalent to the notation of “entropy” used in the current work. Moreover, one cannot divide the entropy into internal and external. Entropy supply is different—it can be associated with a transfer from outside or with an internal processes. After entropy has entered the system, it “mixes,” and it is impossible to divide it into internal and external.

  2. 2.

    Due to external heat supply (12) any value for quantity \(\dot{S}\) can be realized.

  3. 3.

    Since there are no mechanical motions, the volume remains unchanged.

  4. 4.

    This inequality is sometimes called the Planck inequality or the Clausius-Planck inequality, however, in monograph [11] these terms are not used.

  5. 5.

    In monograph [11] the term “Fourier’s inequality” is not mentioned, instead the term “zeroth law of thermodynamics” is used.

  6. 6.

    \(\omega _e=\sqrt{C/m}\): the frequency of a particle with the mass m on a spring with the stiffness C, which is \(C=\varPi ''(a)\), where \(\varPi \) is the potential of the atomic interaction, a is the lattice step.

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Acknowledgements

The authors of this work would like to thank E. A. Ivanova and E. N. Vilchevskaya for the useful discussions.

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Correspondence to A. A. Sokolov .

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Krivtsov, A.M., Sokolov, A.A., Müller, W.H., Freidin, A.B. (2018). One-Dimensional Heat Conduction and Entropy Production. In: dell'Isola, F., Eremeyev, V., Porubov, A. (eds) Advances in Mechanics of Microstructured Media and Structures. Advanced Structured Materials, vol 87. Springer, Cham. https://doi.org/10.1007/978-3-319-73694-5_12

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  • DOI: https://doi.org/10.1007/978-3-319-73694-5_12

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