Universal scaling function ansatz for finite-temperature jamming

TL;DR

This paper introduces a universal scaling function framework for analyzing the finite-temperature jamming transition, unifying different regimes and revealing the role of a dangerous irrelevant variable in critical behavior.

Contribution

It proposes a novel universal scaling ansatz with two branches for finite-temperature jamming, connecting zero-temperature and thermal regimes through a critical line.

Findings

Two distinct scaling regimes separated by a critical line.

The scaling function incorporates a dangerous irrelevant variable affecting exponents.

Reproduction of thermal hard sphere exponents in the high-temperature branch.

Abstract

We cast a nonzero-temperature analysis of the jamming transition into the framework of a scaling ansatz. We show that four distinct regimes for scaling exponents of thermodynamic derivatives of the free energy such as pressure, bulk and shear moduli, can be consolidated by introducing a universal scaling function with two branches. Both the original analysis and the scaling theory assume that the system always resides in a single basis in the energy landscape. The two branches are separated by a line $T^*(\Delta \phi)$ in the $T-\Delta \phi$ plane, where $\Delta \phi=\phi-\phi_c^\Lambda$ is the deviation of the packing fraction from its critical, jamming value, $\phi_c^\Lambda$, for that basin. The branch for $T<T^*(\Delta \phi)$ reduces at $T=0$ to an earlier scaling ansatz that is restricted to $T=0$, $\Delta \phi \ge 0$, while the branch for $T>T^*(\Delta \phi)$ reproduces exponents observed for thermal hard spheres. In contrast to the usual scenario for critical phenomena, the two branches are characterized by different exponents. We suggest that this unusual feature can be resolved by the existence of a dangerous irrelevant variable $u$, which can appear to modify exponents if the leading $u=0$ term is sufficiently small in the regime described by one of the two branches of the scaling function.