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Equilibrium

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Nonequilibrium and Irreversibility

Part of the book series: Theoretical and Mathematical Physics ((TMP))

Abstract

Mechanical systems in interaction with thermostats will be modeled by evolution equations describing the time evolution of the point \(x=( X,\dot{X})=x_1,\ldots ,x_N,\dot{x}_1,\ldots ,\dot{x}_N)\in R^{6N}\) representing positions and velocities of all particles in the ambient space \(R^3\).

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Notes

  1. 1.

    Sometimes the observations can be triggered by a clock arm indicating a chosen position on the dial: in this case the phase space will be \(R^{6N+1}\) and the space \(\varXi \) will coincide with \(R^{6N}\). But in what follows we shall consider measurements triggered by some observable taking a prefixed value, unless otherwise stated.

  2. 2.

    In the case of systems described in continuous time the data show always a \(0\)-Lyapunov exponent and this remains true it the observations are made at fixed time intervals

  3. 3.

    The meaning of the word was explained by Clausius himself [11, p. 390]: “I propose to name the quantity \(S\) the entropy of the system, after the Greek word “the transformation” [12] , [in German Verwandlung]. I have deliberately chosen the word entropy to be as similar as possible to the word energy: the two quantities to be named by these words are so closely related in physical significance that a certain similarity in their names appears to be appropriate.” More precisely the German word really employed by Clausius [11, p. 390], is Verwandlungsinhalt or “transformation content”.

  4. 4.

    The value of \(D\) depends sensitively on the assumption that the atomic interaction potential is proportional to \(r^{-4}\) (hence at constant pressure \(D\) varies as \(T^2\)). The agreement with the few experimental data available (1866 and 1873) induced Maxwell to believe that the atomic interaction would be proportional to \(r^{-4}\) (hard core interaction would lead to \(D\) varying as \(T^{\frac{3}{2}}\) as in his earlier work [15]).

  5. 5.

    For a precise formulation see p. 18.

  6. 6.

    Boltzmann [18], see also Sect. 6.1.

  7. 7.

    From [19, p. 227] Differential equations require, just as atomism does, an initial idea of a large finite number of numerical values and points ...... Only afterwards it is maintained that the picture never represents phenomena exactly but merely approximates them more and more the greater the number of these points and the smaller the distance between them. Yet here again it seems to me that so far we cannot exclude the possibility that for a certain very large number of points the picture will best represent phenomena and that for greater numbers it will become again less accurate, so that atoms do exist in large but finite number. For other relevant quotations see Sect. 1.1 and 5.2 in [16].

  8. 8.

    Today it seems unwelcome because we have adjusted, under social pressure, to think that chaotic motions are non periodic and ubiquitous, and their images fill both scientific and popular magazines. It is interesting however that the ideas and methods developed by the mentioned Authors have been the basis of the chaotic conception of motion and of the possibility of reaching some understating of it. See also Sects. 3.6, 3.7 below.

  9. 9.

    The recurrence time.

  10. 10.

    For Clausius’ view see p. 8 and for Maxwell’s view see footnote p. viii in the Introduction.

  11. 11.

    The second fundamental theorem is not the second law but a logical consequence of it, see Sect. 6.1.

  12. 12.

    From Eq. (1.4.3): \(-\delta (\overline{K} +\overline{V}) +2\delta \overline{K}+\delta \overline{{\widetilde{V}}}=-2\overline{K}\delta \log i\); i.e. \(-\delta Q=-2\delta \overline{K} -2\overline{K}\log i\), hence \(\frac{\delta Q}{\overline{K}}=2\delta \log (\overline{K} i)\).

  13. 13.

    This is an important point: the condition Eq. (1.4.3) does not give to the periodic orbits describing the state of the system any variational property (of minimum or maximum): the consequence is that it does not imply \(\int \frac{\delta Q}{T}\le 0\) in the general case of a cycle but only \(\int \frac{\delta Q}{T}=0\) in the (considered) reversible cases of cycles. This comment also applies to Clausius’ derivation. The inequality seems to be derivable only by applying the second law in Clausius formulation. It proves existence of entropy, however, see comment at p. 137.

  14. 14.

    Assume here for simplicity the gas to be monoatomic.

  15. 15.

    For there will always exist configurations for which \(H(f)\) or any other extension of it decreases, although this can possibly happen only for a very short time (of “human size”) to start again increasing forever approaching a constant (until a time \(T_\infty \) is elapsed and in the unlikely event that the system is still enclosed in its container where it has remained undisturbed and if there is still anyone around to care).

  16. 16.

    “The entropy of the universe is always increasing” is not a very good statement of the second law [47, Sect. 44.12] The second law in Kelvin-Planck’s version “A process whose only net result is to take heat from a reservoir and convert it to work is impossible”; and entropy is defined as a function \(S\) such that if heat \(\Delta Q\) is added reversibly to a system at temperature \(T\), the increase in entropy of the system is \(\Delta S=\frac{\Delta Q}{T}\) [47, 48]. The Clausius’ formulation of the second law is “It is impossible to construct a device that, operating in a cycle will produce no effect other than the transfer of heat from a cooler to a hotter body” [48, p. 148]. In both cases the existence of entropy follows as a theorem, Clausius’ “fundamental theorem of the theory of heat”, here called “heat theorem”.

  17. 17.

    Like temperature differences imposed on the boundaries.

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  1. Department of Physics, University of Rome “La Sapienza”, Rome, Italy

    Giovanni Gallavotti

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Gallavotti, G. (2014). Equilibrium. In: Nonequilibrium and Irreversibility. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-06758-2_1

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