Stochastic $\phi^4-$Theory in the Strong Coupling Limit

TL;DR

This paper develops a statistical theory for the stochastic $^4$-theory in high dimensions, analyzing domain wall structures, deriving probability density functions, and exploring intermittency and scaling laws in the strong coupling limit.

Contribution

It provides exact expressions for PDFs and moments, and investigates intermittency and scaling laws in the strong coupling regime of the stochastic $^4$-theory.

Findings

Exact expressions for the one-point PDF and moments of the order parameter.

Derived the tail behavior of the PDF of field increments.

Established scaling laws for structure functions of field increments.

Abstract

The stochastic $\phi^4$-theory in $d-$dimensions dynamically develops domain wall structures within which the order parameter is not continuous. We develop a statistical theory for the $\phi^4$-theory driven with a random forcing which is white in time and Gaussian-correlated in space. A master equation is derived for the probability density function (PDF) of the order parameter, when the forcing correlation length is much smaller than the system size, but much larger than the typical width of the domain walls. Moreover, exact expressions for the one-point PDF and all the moments $<\phi^n>$ are given. We then investigate the intermittency issue in the strong coupling limit, and derive the tail of the PDF of the increments $\phi(x_2) - \phi(x_1)$. The scaling laws for the structure functions of the increments are obtained through numerical simulations. It is shown that the moments of field increments defined by, $C_b=< |\phi(x_2)-\phi(x_1)|^b>$, behave as $|x_1-x_2|^{\xi_b}$, where $\xi_b=b$ for $b\leq 1$, and $\xi_b=1$ for $b\geq1$