The Mean Field Theories of Magnetism and Turbulence

Abstract

Abstract:In the last few decades a series of experiments have revealed that turbulence is a cooperative and critical phenomenon showing a continuous phase change with the critical Reynolds number at its onset. However, the applications of phase transition models, such as the Mean Field Theory (MFT), the Heisenberg model, the XY model, etc. to turbulence, have not been realized so far. Now, in this article, a successful analogy to magnetism is reported, and it is shown that a Mean Field Theory of Turbulence (MFTT) can be built that reveals new results. In analogy to compressibility in fluids and susceptibility in magnetic materials, the vorticibility (the authors of this article propose this new name in analogy to response functions, derived and given names in other fields) of a turbulent flowing fluid is revealed, which is identical to the relative turbulence intensity. By analogy to magnetism, in a natural manner, the Curie Law of Turbulence was discovered. It is clear that the MFTT is a theory describing equilibrium flow systems, whereas for a long time it is known that turbulence is a highly non-equilibrium phenomenon. Nonetheless, as a starting point for the development of thermodynamic models of turbulence, the presented MFTT is very useful to gain physical insight, just as Kraichnan's turbulent energy spectra of 2-D and 3-D turbulence are, which were developed with equilibrium Boltzmann-Gibbs thermodynamics and only recently have been generalized and adapted to non-equilibrium and intermittent turbulent flow fields.

1 Phenomenological thermodynamics of irreversible processes

“…Therefore, the aims and goals of this carefully written paper are in its explicit comparison between the computed and measured output data for well-described input conditions. P. W. Egolf & K. Hutter [40] in "The Mean Field Theories of Magnetism and Turbulence" studied the turbulence as a cooperative and critical phenomenon with a critical Re-number at its onset. Critical or cooperative phenomena are distinctive features in physics and their analogies demonstrate near or perfect universal behavior.…”

Phenomenological Thermodynamics of Irreversible Processes

“…P. W. Egolf & K. Hutter [ 40 ] in “ The Mean Field Theories of Magnetism and Turbulence ” studied the turbulence as a cooperative and critical phenomenon with a critical Re-number at its onset. Critical or cooperative phenomena are distinctive features in physics and their analogies demonstrate near or perfect universal behavior.…”

“…On the other hand, it is surprising to observe that the entropy functional and the partition function define “ universality classes ”, which show identical “ critical exponents ”. Otherwise stated, a system close to its criticality exhibits a kind of universal behavior that is not determined by the specific microscopic interactions at the sub grid dimension (see the article by Egolf & Hutter [ 40 ] on the mean field theory of magnetism and turbulence).…”

Tsallis Extended Thermodynamics Applied to 2-d Turbulence: Lévy Statistics and q-Fractional Generalized Kraichnanian Energy and Enstrophy Spectra

“…Today it is known that the entropy functional and the partition function, respectively, define ‘universality classes’ showing identical ‘critical exponents’ (see [ 31 ]). This means that a system close to its criticality exhibits a kind of universal behavior that is not determined by the specific microscopic interactions of neighboring particles.…”

“…With our conviction that turbulence is a critical phenomenon from today’s point-of-view, this is like closing a circle. Egolf and Hutter discovered a strong analogy between magnetism and turbulence and thereby could develop the mean field theory of turbulence [ 31 ]. With this finding they could transform the Curie and the Curie-Weiss laws to the related laws describing turbulence.…”