Dynamic and Renormalization-Group Extensions of the Landau Theory of Critical Phenomena

Abstract

We place the Landau theory of critical phenomena into the larger context of multiscale thermodynamics. The thermodynamic potentials, with which the Landau theory begins, arise as Lyapunov like functions in the investigation of the relations among different levels of description. By seeing the renormalization-group approach to critical phenomena as inseparability of levels in the critical point, we can adopt the renormalization-group viewpoint into the Landau theory and by doing it bring its predictions closer to results of experimental observations.

Multiscale Thermodynamics

“…This multiscale viewpoint of the Landau theory is formulated and illustrated by the van der Waals theory in [ 27 , 28 ]. The mathematical results of the universality of thermodynamic potentials in critical regions that have arisen in catastrophe theory [ 25 ] may become less surprising if we allegorically compare them with the very familiar observation that our two friends, Bill and Bob, are very different in many respects, but their behavior in critical situations is very similar.…”

“…In this formulation, the microscopic Hamiltonians approach in the pattern recognition process (consisting usually of a spatial coarse graining) fixed points. In the context of multiscale thermodynamics, the renormalization group approach to critical phenomena was formulated in [ 27 , 28 ]. In this formulation, the coefficients of the Landau polynomials approach the fixed points.…”

Multiscale Thermodynamics

“…This multiscale viewpoint of the Landau theory is formulated and illustrated on the van der Waals theory in [25] [26]. The mathematical results of the universality of thermodynamic potentials in critical regions, that have arisen in the catastrophe theory [23], may become less surprising if we allegorically compare them with the very familiar observation that our two friends, Bill and Bob, are very different in many respects but their behavior in critical situations is very similar.…”

“…In this formulation the microscopic Hamiltonians approach in the pattern recognition process (consisting usually of a spatial coarse graining) to fixed points. In the context of the multiscale thermodynamics, the renormalization-group approach to critical phenomena has been formulated in [25] [26]. In this formulation the coefficients of the Landau polynomials approach the fixed points.…”

The Statistical Foundations of Entropy

“…The theory of critical phenomena is one of the essential parts of statistical physics with important overlaps to other scientific fields. The authors of the second paper [2] extend the classic Landau theory of critical phenomena in the context of multiscale dynamics. By using renormalization group theory, the authors can describe critical points through the inseparability of levels at certain points.…”