We explore the relationship between symmetry and entropy, distinguishing between symmetries of state and dynamical symmetries, and in the context of quantum thermodynamics between symmetries of pure and mixed states. Ultimately, we will argue that symmetry in thermodynamics is best understood as a means of control within the control theory paradigm, and we will describe an interesting technological application of symmetry-based control in the context of a quantum coherence capacitor. Symmetry, the concept from which Noether derived the conservation laws of physics, is one of the most important guiding principles of modern physics. Moreover, symmetry is often regarded as a form of order, and entropy is sometimes regarded as a measure of disorder, so it is natural to suppose that symmetry and entropy are related in some way. In this article, we will explore the relationship between symmetry and entropy, demonstrating that this relationship is by no means a simple one: in particular, it is important to distinguish between symmetries of state and dynamical symmetries, and in the context of quantum thermodynamics to distinguish between symmetries of pure and mixed states. Ultimately, we will argue that symmetry in thermodynamics is best understood as a means of control within the control theory paradigm, and we will describe an interesting technological application of symmetry-based control in the context of a quantum coherence capacitor.

1.
H.
Callen
,
Found. Phys.
4
(
4
),
423
(
1974
).
2.
H. E.
Stanley
,
Introduction to Phase Transitions and Critical Phenomena
, International Series of Monographs on Physics (
Oxford University Press
,
Oxford
,
1971
).
3.
E. T.
Jaynes
,
Am. J. Phys.
33
(
5
),
391
398
(
1965
).
4.
D.
Wallace
, “Thermodynamics as a Control Theory”
Entropy
16
,
699
725
(
2013
); available at https://philpapers.org/rec/WALTAC-6.
5.
W. C.
Myrvold
,
Found. Phys.
50
(
10
),
1219
1251
(
2020
).
6.
E.
Chitambar
and
G.
Gour
,
Rev. Mod. Phys.
91
(
2
),
025001
(
2019
).
7.
N.
Huei
,
Y.
Ng
, and
M. P.
Woods
, “
Resource theory of quantum thermodynamics: Thermal operations and second laws
,” in
Thermodynamics in the Quantum Regime
(
Springer International Publishing
,
2018
), pp.
625
650
.
8.
M.
Lostaglio
,
Rep. Prog. Phys.
82
(
11
),
114001
(
2019
).
9.
F.
Brandão
,
M.
Horodecki
,
N.
Ng
,
J.
Oppenheim
, and
S.
Wehner
,
Proc. Natl. Acad. Sci.
112
(
11
),
3275
3279
(
2015
).
10.
G. E.
Crooks
,
Phys. Rev. Lett.
99
(
10
),
100602
(
2007
).
11.
P.
Salamon
and
R. S.
Berry
,
Phys. Rev. Lett.
51
,
1127
(
1983
).
12.
D. A.
Sivak
and
G. E.
Crooks
,
Phys. Rev. Lett.
108
,
190602
(
2012
).
13.
X.
Chen
,
Z.-C.
Gu
,
Z.-X.
Liu
, and
X.-G.
Wen
,
Phys. Rev. B
87
,
155114
(
2013
).
14.
K.
Wierschem
and
P.
Sengupta
,
Mod. Phys. Lett. B
28
(
32
),
1430017
(
2014
).
15.
J. K.
Asbóth
,
L.
Oroszlány
, and
A.
Pályi
,
A Short Course on Topological Insulators
, Lecture Notes in Physics (
Springer International Publishing
,
2016
).
16.
P.
Walker
and
J.
Carroll
,
Nucl. Phys. News
17
,
11
15
(
2007
).
17.
M.
König
,
S.
Wiedmann
,
C.
Brüne
,
A.
Roth
,
H.
Buhmann
,
L. W.
Molenkamp
,
X.-L.
Qi
, and
S.-C.
Zhang
,
Science
318
(
5851
),
766
(
2007
).
18.
A. M.
Bozkurt
,
B.
Pekerten
, and
Î.
Adagideli
,
Phys. Rev. B
97
(
24
),
245414
(
2018
).
19.
N. H.
Lindner
,
G.
Refael
, and
V.
Galitski
,
Nat. Phys.
7
(
6
),
490
(
2011
).
20.
S.
Goldstein
,
J. L.
Lebowitz
,
R.
Tumulka
, and
N.
Zanghì
, “
Gibbs and Boltzmann entropy in classical and quantum mechanics
,” in
Statistical Mechanics and Scientific Explanation: Determinism, Indeterminism and Laws of Nature
(
World Scientific
,
2020
), Chap. 14, pp.
519
581
.
21.
S.
De Haro
and
J.
Butterfield
, “
On symmetry and duality
,” arXiv:1905.05966 (
2019
).
22.
B.
Dóra
,
J.
Cayssol
,
F.
Simon
, and
R.
Moessner
,
Phys. Rev. Lett.
108
(
5
),
056602
(
2012
).
23.
E.
Castellani
and
J.
Ismael
,
Philos. Sci.
83
(
5
),
1002
(
2016
).
24.
B. W.
Roberts
, “
The simple failure of Curie's principle: How to get out what hasn't gone in
,”
LSE Philosophy Blog
(
2014
); available at https://www.lse.ac.uk/philosophy/blog/2014/12/22/the-simple-failure-of-curies-principle-how-to-get-out-what-hasnt-gone-in/.
25.
J.
Rosen
,
Entropy
7
(
4
),
308
(
2005
).
26.
D.
Wallace
,
The Necessity of Gibbsian Statistical Mechanics
(
American Journal of Science
,
1878
); available at URL: http://philsci-archive.pitt.edu/15290/.
27.
H. R.
Brown
and
P.
Holland
,
Mol. Phys.
102
(
11–12
),
1133
(
2004
).
28.
T. L.
Duncan
and
J. S.
Semura
,
Found. Phys.
37
(
12
),
1767
(
2007
).
29.
E.
Jaynes
, “
The Gibbs paradox
,” in
Maximum Entropy and Bayesian Methods
(
Springer
,
Dordrecht
,
1992
).
30.
P. M.
Ainsworth
,
Philos. Sci.
79
(
4
),
542
(
2012
).
31.
J. W.
Gibbs
,
On the Equilibrium of Heterogeneous Substances
(
Transactions of the Connecticut Academy of Arts and Sciences
,
Connecticut
,
1874
).
32.
X. G.
Wen
,
Int. J. Mod. Phys. B
4
(
2
),
239
(
1990
).
33.
É.
Roldán
,
I. A.
Martínez
,
J. M. R.
Parrondo
, and
D.
Petrov
,
Nat. Phys.
10
(
6
),
457
(
2014
).
34.
G.
Benenti
,
K.
Saito
, and
G.
Casati
,
Phys. Rev. Lett.
106
(
23
),
230602
(
2011
).
35.
R.
Balian
,
Stud. Hist. Philos. Mod. Phys.
36
(
2
),
323
(
2005
).
36.
M. O.
Scully
,
M. S.
Zubairy
,
G. S.
Agarwal
, and
H.
Walther
,
Science
299
(
5608
),
862
(
2003
).
37.
R.
Landauer
,
IBM J. Res. Dev.
5
(
3
),
183
(
1961
).
38.
A.
Tuncer
and
Ö. E.
Müstecaplioğlu
, “Quantum thermodynamics and quantum coherence engines,”
Turk. J. Phys.
44
,
404
(
2020
); available at https://journals.tubitak.gov.tr/physics/vol44/iss5/1/.
39.
Of course, Curie's principle may also be applied to symmetries which are not associated with groups of automorphisms on microstates in this way, and in such cases, we require a different sort of explanation, such as Chalmers' derivation from the invariance properties of deterministic physical laws, Ref. 26.
40.
Though as noted in Ref. 28, there certainly exist some dynamical symmetry transformations which are not so linked, either because the transformation employed by the dynamical symmetry fails to leave the action invariant, as for rescaling transformations, or because the dynamical system is not amenable to a Lagrangian description at all, and therefore, it has no variational symmetries.
You do not currently have access to this content.