Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymmetric Relaxations Through the Lens of Information Geometry

Alessandro Bravetti, Miguel Ángel García Ariza, Pablo Padilla

TL;DR

This work reframes relaxation toward thermodynamic equilibrium as a gradient flow on information-geometry, using the dual affine structure of equilibrium manifolds and the divergence $D^*_q=T_q D_{ ext{KL}}(ullet||q)$. It derives a general criterion for asymmetry in endoreversible relaxations via the Amari-Chentsov tensor $C^{ijk}$, and provides concrete analyses for classical isobaric and quantum ideal-gas relaxations, including a Mpemba-like non-isobaric scenario. The results yield a thermodynamic uncertainty relation and the Horse-Carrot theorem within this geometric framework, and reveal that asymmetry directions can reverse between classical and quantum regimes. Overall, the paper offers a unified geometric mechanism for understanding when cooling or warming dominates in a wide range of thermo-physical processes, with potential implications beyond thermodynamics to fields like machine learning and evolutionary dynamics.

Abstract

We frame Newton's Law of Cooling as a gradient flow within the context of information geometry. This connects it to a thermodynamic uncertainty relation and the Horse-Carrot Theorem, and reveals novel instances of asymmetric relaxations in endoreversible processes. We present a general criterion for predicting asymmetries using the Amari-Chentsov tensor, applicable to classical and quantum thermodynamics. Examples include faster cooling of quantum ideal gases and relaxations that resemble the Mpemba effect in classical ideal gases.

Asymmetric Relaxations Through the Lens of Information Geometry

TL;DR

This work reframes relaxation toward thermodynamic equilibrium as a gradient flow on information-geometry, using the dual affine structure of equilibrium manifolds and the divergence . It derives a general criterion for asymmetry in endoreversible relaxations via the Amari-Chentsov tensor , and provides concrete analyses for classical isobaric and quantum ideal-gas relaxations, including a Mpemba-like non-isobaric scenario. The results yield a thermodynamic uncertainty relation and the Horse-Carrot theorem within this geometric framework, and reveal that asymmetry directions can reverse between classical and quantum regimes. Overall, the paper offers a unified geometric mechanism for understanding when cooling or warming dominates in a wide range of thermo-physical processes, with potential implications beyond thermodynamics to fields like machine learning and evolutionary dynamics.

Abstract

We frame Newton's Law of Cooling as a gradient flow within the context of information geometry. This connects it to a thermodynamic uncertainty relation and the Horse-Carrot Theorem, and reveals novel instances of asymmetric relaxations in endoreversible processes. We present a general criterion for predicting asymmetries using the Amari-Chentsov tensor, applicable to classical and quantum thermodynamics. Examples include faster cooling of quantum ideal gases and relaxations that resemble the Mpemba effect in classical ideal gases.
Paper Structure (11 sections, 5 theorems, 51 equations, 3 figures)

This paper contains 11 sections, 5 theorems, 51 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The information geometry of thermodynamics
  3. Newton's Law of Cooling as a gradient flow
  4. Asymmetric relaxation of a closed ideal gas
  5. The general result
  6. An example of cooling down occuring faster than warming up
  7. Non-isobaric relaxation of ideal gases: a resemblance to the Mpemba effect
  8. Concluding remarks
  9. Detailed calculations of the divergence for closed ideal gases
  10. Detailed calculations of the divergence for quantum ideal gases
  11. Detailed calculations on the nonisobaric relaxation of ideal gases

Key Result

Theorem 1

Newton's Law of Cooling is one component of the gradient flow defined by Eq. eq:grad.

Figures (3)

  • Figure 1: Left: $D^*$ as a function of $T$ for a monoatomic ideal gas ($c=3/2$), in units of $k_\text{B}=1$. We observe that $T^-$ is closer to $T_q$ than $T^+$, meaning that warming up occurs faster at constant pressure. Right: $\Delta D^*$ as a function of time in units of $k_\text{B}=1$. According to the graph, the available work is greater when the system is cooling down at every time during the relaxation. The difference in available work between the two systems is the greatest at $t=t_*$, and it gradually vanishes as the systems relax to equilibrium.
  • Figure 2: Left: $D^*$ as a function of $T$ for bosons, in units of $\kappa=1$ and $k_\text{B}=1$. Since $T^+$ is closer to $T_q$ than $T^-$, cooling down occurs faster (cf. Fig. \ref{['fig:IG']}). Right: $\Delta D^*$ for bosons as a function of time, in units of $\kappa=1$ and $k_\text{B}=1$. The shapes of $D^*$ and $\Delta D^*$ agree with the fact that $C(\dot\gamma^+,\dot\gamma^+,\dot\gamma^+)\geq C(\dot\gamma^-,\dot\gamma^-,\dot\gamma^-).$
  • Figure 3: Left: Contour plot (dark to light) of $A:=-\lambda^3C(\operatorname{grad}D_q^*,\operatorname{grad}D_q^*,\operatorname{grad}D_q^*)$ for a monoatomic ideal gas. At $t=t_*$ (red points), $A$ is greater for $\gamma^1$ (dashed) than for $\gamma^2$ (dot-dashed), whence the relaxation along the former is faster. Right: $\Delta D^*$ (orange), $\Delta(||\dot\gamma||/\lambda)^2$ (brown), and $\Delta A/\lambda^3$ (blue) along $\gamma^1$ and $\gamma^2$. The minimum of $\Delta D^*$ is attained when the relaxation speeds are equal. Observe that the two relaxations converge to $q$ at different speeds. Since $\Delta A>0$ at $t=t_*$, relaxation along $\gamma^1$ is faster, which agrees with $\Delta D^*<0$.

Theorems & Definitions (5)

  • Theorem 1
  • Corollary 1
  • Proposition 1
  • Theorem 2
  • Proposition 2