Temperature and coordinate dependence of the perturbation energy φ

Determine whether the perturbation energy φ that represents cross-correlation between the bond-length and angle Hamiltonians in the coarse-grained SWCNT model depends on both temperature and spatial coordinates; specifically, verify whether the partial derivatives ∂φ/∂T and ∂φ/∂x are nonzero.

Background

Within the stochastic thermodynamics framework applied to the coarse-grained SWCNT, the authors introduce a perturbation energy φ to encode the influence of fast dynamics and cross-correlation between Hamiltonians. They relate φ to heat diffusion and discuss its role in entropy, work, and heat balances.

In the Thermodynamics subsection of the Discussion, the authors state a conjecture that φ varies with temperature and spatial coordinates, implying nonzero partial derivatives with respect to T and x. Establishing this dependence would clarify how thermal and spatial variations modulate the cross-correlation energy and associated diffusion processes.