On the Calculation of Pressure Derivatives in Mean-Field Thermal Field Theories

TL;DR

This paper introduces a symbolic Jacobian-based method for accurately calculating higher-order pressure derivatives in mean-field thermal field theories, improving stability near phase transitions.

Contribution

It presents a systematic symbolic derivation approach for pressure derivatives, reducing numerical instabilities in complex mean-field models.

Findings

Effective in the two-flavor Nambu-Jona-Lasinio model

Improves accuracy near phase transitions

Reduces numerical instability in derivative calculations

Abstract

Accurate determination of higher-order pressure derivatives with respect to temperature $T$ and chemical potential $\mu$ is essential for analyzing critical phenomena, transport properties, and phase transitions in strongly interacting matter. However, standard numerical differentiation methods often suffer from large numerical instabilities, especially in more complex mean-field thermal field theories. In this work, we present an approach that systematically derives symbolic expressions for these higher-order derivatives, bypassing the numerical instabilities commonly encountered in conventional methods. Our formalism is based on a Jacobian technique, which ensures that the dependence of internal mean-field parameters is fully incorporated into the final symbolic expressions. We illustrate the effectiveness of this method using the two-flavor Nambu-Jona-Lasinio model as an example and show that it is particularly advantageous near phase transitions and at low temperatures, where numerical differentiation becomes highly sensitive.