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Thermodynamic Geometry Through Second Order Phase Transitions

Omer M. Basri, Oren Raz

TL;DR

The work studies thermodynamic geometry for slowly driven systems near second-order phase transitions, showing that the dissipation metric can diverge even when the thermodynamic length remains finite, depending on universality class and dynamic exponents. By combining Widom scaling with dynamical scaling, it identifies regimes where a finite dissipation path crosses the transition, and it demonstrates this with a mean-field antiferromagnetic Ising model, where the optimal path may traverse the disordered phase or even pass through infinite temperature. The paper also provides a numerical approach to compute minimal-dissipation paths and reveals four distinct classes of optimal trajectories, including cases where crossing the transition is advantageous. These insights have potential applications in optimizing protocols for heat engines and other driven processes operating near criticality.

Abstract

A common approach to quantify excess dissipation in slowly driven thermodynamic processes is through the use of a Riemannian metric on the space of control parameters, where optimal driving protocols follow geodesics. Near phase transitions, this geometric picture breaks down as the metric diverges and geodesics may cease to exist. Using Widom scaling, we analyze this framework for several universality classes and show that in some cases the thermodynamic length across the phase transition remains finite. We then demonstrate a numerical approach for computing minimal paths in such systems. We show that, in some regimes, the shortest path crosses the phase transition - even when alternative paths confined to a single phase exist.

Thermodynamic Geometry Through Second Order Phase Transitions

TL;DR

The work studies thermodynamic geometry for slowly driven systems near second-order phase transitions, showing that the dissipation metric can diverge even when the thermodynamic length remains finite, depending on universality class and dynamic exponents. By combining Widom scaling with dynamical scaling, it identifies regimes where a finite dissipation path crosses the transition, and it demonstrates this with a mean-field antiferromagnetic Ising model, where the optimal path may traverse the disordered phase or even pass through infinite temperature. The paper also provides a numerical approach to compute minimal-dissipation paths and reveals four distinct classes of optimal trajectories, including cases where crossing the transition is advantageous. These insights have potential applications in optimizing protocols for heat engines and other driven processes operating near criticality.

Abstract

A common approach to quantify excess dissipation in slowly driven thermodynamic processes is through the use of a Riemannian metric on the space of control parameters, where optimal driving protocols follow geodesics. Near phase transitions, this geometric picture breaks down as the metric diverges and geodesics may cease to exist. Using Widom scaling, we analyze this framework for several universality classes and show that in some cases the thermodynamic length across the phase transition remains finite. We then demonstrate a numerical approach for computing minimal paths in such systems. We show that, in some regimes, the shortest path crosses the phase transition - even when alternative paths confined to a single phase exist.

Paper Structure

This paper contains 13 sections, 40 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Framework
  3. Results
  4. Linear Response Applicability:
  5. An example:
  6. Conclusions
  7. Acknowledgments
  8. Calculating the dissipation metric
  9. Minimal Dissipation in Disordered Phase
  10. Eigenstructure and the zero-length direction.
  11. Scaling of the metric prefactor.
  12. Scaling of the displacement along the zero mode.
  13. Minimum at $\beta=0$.

Figures (1)

  • Figure 1: Illustration of the three types of minimal paths in the mean-field antiferromagnet. The dashed black line is the phase transition between the disordered phase in the left and the antiferromagnetic in the right. The colored lines are example for optimal trajectories.