Probability and Mathematical Physics Seminar

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Renormalization Group (RG) is a powerful concept -- one calculates partition functions by successively integrating out short distance degrees of freedom, until one has an effective Hamiltonian describing only large scale modes. The most famous application is Wilson's use of RG to calculate the critical scaling exponents at the water-vapor phase transition. 

We apply RG concepts to a vastly different class of states, which are far from equilibrium. It is known that a large class of weakly interacting nonlinear systems have states that are spatially homogeneous, time-independent, and scale invariant. Such states, in which mode k has occupation number $n_k = k^{-\gamma}$, go under the name of Kolmogorov-Zakharov states in wave turbulence. Canonical examples are waves on the surface of the ocean, or waves in the nonlinear Schrodinger equation. We compute one loop beta functions in such states, which encode how the effective coupling changes with scale. The beta functions tells us if the spectrum of occupation numbers is steeper or less steep than Kolmogorov-Zakharov scaling.  Depending on the sign of the beta function, nonlinear effects may either cause a minor shift of the state in the IR, or completely change the nature of the state. 

Focusing on nearly marginal interactions (ones in which the strength of the nonlinearity is weakly scale dependent), we construct an analog of Wilson's epsilon expansion and IR fixed points, with epsilon now set by the scaling of the interaction rather than the spacetime dimension. In the language of RG flow, critical balance scaling -- having applications in fields as varied as astrophysics and ocean waves -- corresponds to the state dynamically adjusting itself along the RG flow until the interaction becomes marginal.