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Reversibility, irreversibility, friction and nonequilibrium ensembles in N–S equations

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Abstract

Viscosity, as a physical property of fluids, reflects an average effect over a chaotic microscopic motion described by Hamiltonian equations. It is proposed, as an example, that stationary states of an incompressible fluid subject to a constant force, can be described via several ensembles, in strict analogy with equilibrium Statistical Mechanics.

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Notes

  1. In the subatomic world time reversal is not a symmetry but another more fundamental symmetry, CPT, could replace it in the following discussion

  2. In the 3D case the equations are very similar (for instance the \(u_{\textbf{k}}\) are replaced by vectors orthogonal to \({\textbf{k}}\), ..): the main and important difference is that the expression of \(\alpha ({\textbf{u}})\) receives a contribution from the quadratic transport term which, although present also in 2D, cancels from \(\alpha \) essentially because in 2D when \(\nu =0\) the \({{\mathcal {D}}}({\textbf{u}})\) is conserved.

  3. Hence depend on finitely many Fourier’s harmonics of \({\textbf{u}}\), i.e. are “large scale” observables”, but unlike the forcing, which only has the harmonics \(|{\textbf{k}}|<k_{max}\) with \(k_{max}\) fixed, no limit is set on the size scale

  4. In many cases no “intermittency” is expected, i.e. \(n^N_\nu =m^N_D=1\).

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Acknowledgements

This is a redacted and updated version of a talk at the DinAmici meeting on 21/Dec/2018 at the Accademia dei Lincei, Roma.

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  1. Università “La Sapienza” and INFN, Rome, Italy

    Giovanni Gallavotti

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Appendix: a path through the theme

Appendix: a path through the theme

  1. (1)

    A first equivalence example: [2].

  2. (2)

    Path to the conjecture: [5, 16, 19, 20].

  3. (3)

    3D enstrophy ensemble: [16, 21].

  4. (4)

    3D energy ensemble: [7].

  5. (5)

    Shell model: [22].

  6. (7)

    Stat-Mech: [12, 13, 23,24,25].

  7. (8)

    Turbulence physics: [26,27,28,29,30].

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Gallavotti, G. Reversibility, irreversibility, friction and nonequilibrium ensembles in N–S equations. Boll Unione Mat Ital 16, 351–361 (2023). https://doi.org/10.1007/s40574-022-00343-7

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