Renormalization Group: Applications in Statistical Physics

TL;DR

This paper provides a comprehensive overview of the renormalization group (RG) methods and their applications in statistical physics, covering critical phenomena, dynamic systems, and non-equilibrium phase transitions.

Contribution

It introduces Wilson's momentum shell RG, field-theoretic formulations, and applies RG techniques to both equilibrium and non-equilibrium systems, including stochastic dynamics and interacting particle models.

Findings

Critical exponents for scalar Phi^4 model determined to first order in epsilon expansion.

RG equations used to compute critical exponents for O(n)-symmetric Landau-Ginzburg-Wilson theory.

Application of RG to stochastic dynamical systems and non-equilibrium phase transitions.

Abstract

These notes provide a concise introduction to important applications of the renormalization group (RG) in statistical physics. After reviewing the scaling approach and Ginzburg-Landau theory for critical phenomena, Wilson's momentum shell RG method is presented, and the critical exponents for the scalar Phi^4 model are determined to first order in an eps expansion about d_c = 4. Subsequently, the technically more versatile field-theoretic formulation of the perturbational RG for static critical phenomena is described. It is explained how the emergence of scale invariance connects UV divergences to IR singularities, and the RG equation is employed to compute the critical exponents for the O(n)-symmetric Landau-Ginzburg-Wilson theory. The second part is devoted to field theory representations of non-linear stochastic dynamical systems, and the application of RG tools to critical dynamics. Dynamic critical phenomena in systems near equilibrium are efficiently captured through Langevin equations, and their mapping onto the Janssen-De Dominicis response functional, exemplified by the purely relaxational models with non-conserved (model A) / conserved order parameter (model B). The Langevin description and scaling exponents for isotropic ferromagnets (model J) and for driven diffusive non-equilibrium systems are also discussed. Finally, an outlook is presented to scale-invariant phenomena and non-equilibrium phase transitions in interacting particle systems. It is shown how the stochastic master equation associated with chemical reactions or population dynamics models can be mapped onto imaginary-time, non-Hermitian `quantum' mechanics. In the continuum limit, this Doi-Peliti Hamiltonian is represented through a coherent-state path integral, which allows an RG analysis of diffusion-limited annihilation processes and phase transitions from active to inactive, absorbing states.