Role of the Amari–Chentsov tensor beyond gradient-flow relaxations

Ascertain the role of the Amari–Chentsov tensor in characterizing or predicting asymmetric relaxation for thermodynamic processes that are not governed by the gradient-flow dynamics \dot{\gamma} = -\lambda(t) \operatorname{grad} D^*_{q(t)} (i.e., beyond the regimes where Newton’s Law of Cooling and the gradient-flow framework apply).

Background

The paper shows that Newton’s Law of Cooling can be framed as a gradient flow on the thermodynamic state manifold and derives a general criterion for asymmetric relaxation based on the Amari–Chentsov tensor, demonstrating its effectiveness for systems whose dynamics follow \dot{\gamma} = -\lambda \operatorname{grad} D^*_{q}.

Within this framework, the Amari–Chentsov tensor provides a predictive condition for asymmetry (Theorem 3), with examples in classical and quantum gases. The authors explicitly note that while this establishes the tensor’s utility where the gradient-flow model applies, its role outside these settings has not been determined.