3.4 Transport Theory Approach to Phase Transition

Phase transition is an important topic in statistical mechanics. There is singularity behaviour of transport properties such as specific heat, viscosity, Reynolds number during phase transition. The study of singularities of transport properties of the region surrounding the critical point of phase transition will enable understanding of phase transition. There are two interesting problems in nonlinear acoustics: turbulence and sonoluminescence which until today are still unsolved. Turbulence is a phase transition from laminar flow to turbulence flow. There is a tremendous huge increase in the Reynolds number or the viscosity tends to zero during the phase transition from laminar flow to turbulence flow. This is an example of the singularity behaviour of the transport properties: Reynolds number and viscosity. Besides this, there is spontaneous symmetry breaking (SSB) during the phase transition. The meaning of spontaneous symmetry breaking is explained as follows. In Gan [3] proposed turbulence as a second order phase transition with spontaneous symmetry breaking (SSB). Exampes of second order phase transition are magnetization,superconductivity and superfluidity in condensed matter physics. Hence turbulence is also a field in condensed matter physics. My hypothesis has been subsequently supported by the work of Goldenfeld [4]’s group which showed that turbulence has the same behaviour as magnetization. In his paper he presented experimental evidence that turbulence flows are closely analogous to critical phenomena from a reanalysis of friction factor measurements in rough pipe. He found experimentally two aspects that confirm that turbulence is similar to second order phase transition such as magnetization in a ferromagnet. These are experimentally verified power law scaling of correlation function which

is reminiscent of the power law fluctuation on many length scales that accompany critical phenomena for example in a ferromagnet near its critical point which is second order phase transition. Another aspect is the phenomena of data collapse or Widom scaling [5]. For example, in a ferromagnet, the equation of state, nominally a function of two variables is expressible in terms of a single reduced variable which depends on a combination of external field and temperature. This has been confirmed by the experiments of Nikuradz [6] in 1932 and 1933 which showed data collapse. Goldenfeld [4]’s work proposed that the feature of the turbulence can be understood as arising from a singularity at infinite Reynolds number and zero roughness. Such singularities are known to arise as second order phase transition such as that occurs when iron is cooled down below the Curie temperature and becomes magnetic. This theory predicts that the small scale fluctuation in the fluid speed, a characteristic of turbulence are connected to the friction and can be demonstrated by plotting the data in a special way that causes all of the Nikuradze [8] curves at different roughness collapse into a single curve. According to Goldenfeld [4]’s study the formation of eddies in turbulence might be a similar phenomenon to the lining of spins in magnetization. Eddies are thus similar to the cluster of atoms. Goldenfeld et al. [6] hope that as a result of these discoveries, the approaches that solved the problem of phase transition will now find a new and unexpected application in providing a fundamental understanding of turbulence. In Gan [7] extended Landau [8]’s theory of the second order phase transition from phenomenology to a more rigorous approach by using statistical mechanics and gauge theory. We propose turbulence as a classical analog of Bose-Einstein condensation and the Gross-Pitaevskii equation is used to derive the condensate free energy. The critical value of the order parameter, the condensation wave function is determined.This is the value when turbulence occurs and SSB in the ground state of the condensation free energy takes place. Being a condensate, there is molecular pairing in turbulence. We determine the numerical value of the condensate free energy with the use of the coupled oscillation model for the pair of molecules.

Our expression for the condensation free energy also yields a power series in terms of the order parameter in agreement with the Landau phenomenology of second order phase transition. This confirms that turbulence is a condensate since Gross-Pitaevskii equation is the equation for condensate. We conclude that our understanding of turbulence is a second order phase transition and is a condensate with molecular pairing.

References

1.Gan, Woon Siong. 1969. Transport Theory in Magnetoacoustics, PhD Thesis. Imperial College London.

2.Spelman, G.M., and R.S. Langley. 2015. Statistical energy analysis of nonlinear vibrating system.

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences A373: 20140403.

3.Gan, Woon Siong. 2009. Turbulence is a second order phase transition with spontaneous symmetry breaking. In Proceedings of the ICSV16. Krakow, Poland.

4.Goldenfeld, N. 2006. Roughness-induced critical phenomena in a turbulent flow. Physical Review Letter 96: 044503.

5.Widom, B. 1965. Equation of state in the neighbourhood of critical point. The Journal of Chemical Physics 43: 3898–3905.

6.Nikuradze, J. 1933. VDI, Forschungsheft, Stromungsgesetze in rauhen Rohren (Laws of flow in rough pipes), NACA Technical Memorandum 1292, vol. 361. Berlin: Springer-Verlag.

7.Gan, Woon Siong. 2013. Towards the understanding of turbulence. Journal of Basic and Applied Physics 12 (5): 1–9.

8.Landau, L. 1937. Theory of phase transformation, I, Theory of phase transformation, I. Zh.Eksp.Teor.Fiz. 7: 19, 627.