# Geometry and quantum thermodynamics

This is a Perspective on "Thermodynamic length in open quantum systems" by Matteo Scandi and Martí Perarnau-Llobet, published in Quantum 3, 197 (2019).

By John Goold (Department of Physics, Trinity College Dublin, Dublin 2, Ireland).

It has been known for quite some time that equilibrium thermodynamics can be recast in a general geometric framework by associating an intrinsic metric structure [1,2]. If fluctuations are included then the metric can be shown to be Riemannnian and the concept of a thermodynamic length emerges [3]. On this manifold the length of a quasistatic transformation is defined as the number of fluctuations along the path in state space, which is in stark contrast with the state functions of textbook thermodynamics. Paths of minimum distance define the geodesics and can be seen as the paths which minimize the dissipation for finite-time transformations between states.

The last two decades have witnessed a surge of activity in the thermodynamics of small systems and the development of stochastic thermodynamics [4]. Influenced by this activity, in 2007, Gavin Crooks generalised the concept of thermodynamic length to this domain [5]. In doing so he made a deep connection with statistical estimation theory, showing that the metric tensor of thermodynamics can in fact be seen as the Fisher information matrix. This allowed for the direct interpretation of thermodynamic length in terms of information theoretic quantities.

In recent years there has been a concerted effort to analyse thermodynamics deep in the quantum regime in an effort which has become known as quantum thermodynamics [6]. As shown by Sebastian Deffner and Eric Lutz, the concept of thermodynamic length generalises naturally to the situation of non equilibrium transformations on quantum systems [7]. In a recent article, Matteo Scandi and Martí Perarnau-Llobet take this a further step by generalising thermodynamic length and the associated metric to open quantum systems. They work with slowly driven systems describable by the Lindblad master equation [8]. Other than this contribution being an important technical achievement it has distinct practical application. As pointed out by the authors, the geometrical framework allows them to construct minimally dissipative trajectories which may be used in future works to optimise quantum thermal machines. The authors also point out that their technique maybe extendable beyond the Lindblad master equation to the domain of non-equilibrium steady states where recent work on thermodynamics of precision [9] show that Thermodynamic Uncertainty relations, which constrain fluctuations [10], can emerge from the underlying geometric structure.

### ► References

[1] F. Weinhold Metric geometry of equilibrium thermodynamics'' The Journal of Chemical Physics 63, 2479-2483 (1975).
https:/​/​doi.org/​10.1063/​1.431689

[2] G. Ruppeiner Thermodynamics: A Riemannian geometric model'' Physical Review A 20, 1608 (1979).
https:/​/​doi.org/​10.1103/​PhysRevA.20.1608

[3] P. Salamonand R. S. Berry Thermodynamic length and dissipated availability'' Physical Review Letters 51, 1127 (1983).
https:/​/​doi.org/​10.1103/​PhysRevLett.51.1127

[4] U. Seifert Stochastic thermodynamics, fluctuation theorems and molecular machines'' Reports on progress in physics 75, 126001 (2012).
https:/​/​doi.org/​10.1088/​0034-4885/​75/​12/​126001

[5] G. E. Crooks Measuring thermodynamic length'' Physical Review Letters 99, 100602 (2007).
https:/​/​doi.org/​10.1103/​PhysRevLett.99.100602

[6] F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso, Thermodynamics in the Quantum Regime'' Fundamental Theories of Physics (Springer, 2018) (2019).

[7] S. Deffnerand E. Lutz Thermodynamic length for far-from-equilibrium quantum systems'' Phys. Rev. E 87, 022143 (2013).
https:/​/​doi.org/​10.1103/​PhysRevE.87.022143

[8] M. Scandiand M. Perarnau-Llobet Thermodynamic length in open quantum systems'' Quantum 3, 197 (2019).
https:/​/​doi.org/​10.22331/​q-2019-10-24-197

[9] G. Guarnieri, G. T. Landi, S. R. Clark, and J. Goold, Thermodynamics of precision in quantum nonequilibrium steady states'' Phys. Rev. Research 1, 033021 (2019).
https:/​/​doi.org/​10.1103/​PhysRevResearch.1.033021

[10] J. M. Horowitzand T. R. Gingrich Thermodynamic uncertainty relations constrain non-equilibrium fluctuations'' Nature Physics (2019).
https:/​/​doi.org/​10.1038/​s41567-019-0702-6

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