# Time crystal

(Redirected from Space-time crystal)

A time crystal, space-time crystal, or four-dimensional crystal, is a periodic structure that repeats in time, as well in space. Normal three-dimensional crystals have a repeating pattern in space, but remain unchanged with respect to time; time crystals repeat themselves in time as well, leading the crystal to change from moment to moment. Time crystals extend the idea of a crystal to four dimensions. A time crystal never reaches thermal equilibrium as it is a type of non-equilibrium matter - a form of matter proposed in 2012, and first observed in 2017. This state of matter cannot be isolated from its environment - it is an open system in non-equilibrium. This allows for the crystal to be in perpetual motion.[1] In technical terms, a time crystal spontaneously breaks the symmetry of time translation and thus it is said to exhibit time translation symmetry breaking (TTSB).

The idea of a time crystal was first put forward by Nobel laureate and MIT professor Frank Wilczek in 2012. Subsequent work then developed a more precise definition for time crystals, ultimately leading to a proof that time crystals in equilibrium are not possible. However, these proofs left the door open for time crystals in non-equilibrium systems. In 2016, Norman Yao and his colleagues from the University of California, Berkeley put forward a concrete proposal that would allow time crystals to be created in a laboratory environment. Yao's blueprint was then used by two teams, a group led by Christopher Monroe at the University of Maryland and a group led by Mikhail Lukin at Harvard University, who were both able to successfully create a time crystal. Both experiments were published in the journal Nature in March 2017.

Time crystals are closely related to the concepts of zero-point energy and the dynamical Casimir effect. Time crystals are thought to exhibit topological order, an emergent phenomenon which can greatly increase the resilience of quantum computing and quantum thermodynamic tasks from the loss of quantum coherence. It is also thought that time crystals could provide deeper understanding of the theory of time.[2]

## History

The idea of a space-time crystal was first put forward by Frank Wilczek, a professor at MIT and Nobel laureate, in 2012.[a]

Xiang Zhang, a nanoengineer at University of California, Berkeley, and his team proposed creating a time crystal in the form of a constantly rotating ring of charged ions.[b]

In response to Wilczek and Zhang, Patrick Bruno, a theorist at the European Synchrotron Radiation Facility in Grenoble, France, published several papers claiming to show that space-time crystals were impossible.[c][4]

Subsequent work developed more precise definitions of time translation symmetry breaking[d] which ultimately led to a proof that quantum time crystals in equilibrium are not possible.[e]

Several realizations of time crystals, which avoid the equilibrium no-go arguments, were proposed.[f]

Krzysztof Sacha at Jagiellonian University in Krakow, Poland, has predicted the behaviour of discrete time crystals in a periodically driven many-body system.[g]

The no-go theorem leaves the door open to time translation symmetry breaking in a non-equilibrium system, and pioneering work in spin systems[h] has demonstrated that quantum systems subject to periodic driving can exhibit discrete time translation symmetry breaking.

Using Wilczek's idea, Norman Yao and his colleagues from the University of California, Berkeley, studied a different model that would allow the existence of time crystals.[i]

Yao's blueprint was then used by two teams: a group led by Mikhail Lukin at Harvard university[j] and a group led by Christopher Monroe at University of Maryland[k] both of whom were able to independently create a time crystal successfully.[9] Both experiments were published in the journal Nature in March 2017.[10]

## Time translation symmetry

### Symmetry

Symmetries are of prime importance in physics and are closely related to the hypothesis that certain physical quantities are only relative and unobservable.[11] Symmetries apply to the equations that govern the physical laws rather than the initial conditions or to themselves and state that the laws remain unchanged under a transformation.[12] If a symmetry is preserved under a transformation it is said to be invariant. Symmetries in nature lead directly to conservation laws, something which is precisely formulated by the Noether theorem.[13]

Symmetries in physics[11]
Symmetry Transformation Unobservable Conservation law
Space-translation ${\displaystyle \mathbf {r} \rightarrow \mathbf {r} +\Delta }$ absolute position in space momentum
Time-translation ${\displaystyle t\rightarrow t+\tau }$ absolute time energy
Rotation ${\displaystyle \mathbf {r} \rightarrow \mathbf {r} '}$ absolute direction in space angular momentum
Space inversion ${\displaystyle \mathbf {r} \rightarrow -\mathbf {r} }$ absolute left or right parity
Time-reversal ${\displaystyle t\rightarrow -t}$ absolute sign of time Kramers' degeneracy
Sign reversion of charge ${\displaystyle e\rightarrow -e}$ absolute sign of electric charge charge conjugation
Particle substitution distinguishability of identical particles Bose or Fermi statistics
Gauge transformation ${\displaystyle \psi \rightarrow e^{iN\theta }\psi }$ relative phase between different normal states particle number

The basic idea of time-translation symmetry is that a translation in time has no effect on physical laws, that the laws of nature that apply today were the same in the past and will be the same in the future.[12] For example, if we measure the energy levels of hydrogen today, tomorrow or in ten years it makes no difference - we will always observe the same energy. Moreover, when we look at distant stars we are really looking back in time, so the fact that the energy levels of hydrogen are the same in all stars we ever looked at tells us something about the symmetry of both space and time. Invariance of time-translation implies absolute time is unobservable and a direct consequence is the conservation of energy.[11] A violation in time-translation symmetry means that under certain conditions or select cases energy is not a conserved quantity and that laws of nature themselves are variable with time.

### Broken symmetry

For a long time physicists believed that symmetries in the laws of nature were absolute, but deviations do occur. In 1957 scientists confirmed experimentally[l] a broken symmetry in space inversion (a right-left asymmetry) in weak interactions and that therefore parity was not a conserved quantity (known as a P violation). This led to the 1957 Nobel Prize in physics being awarded to Tsung-Dao Lee and Chen-Ning Yang, who had put forward the original idea in 1956. It was also established in 1957 that there were not only right-left asymmetries but also asymmetries between the positive and negative signs of electric charge (a charge conjugation or C violation). At around the same time questions of possible asymmetries under time-reversal T and CP violations (the product of C and P transformations) were also raised, though actual experimental confirmation did not come until quite a few years later.[14]

A consequence of these asymmetries in nature is that it is possible to determine an absolute left or right, absolute charge or absolute direction of time in the universe; such terms are not merely relative or a subjective naming convention.[15] If two advanced civilisations were separated on different sides of the universe with no possibility of physical contact but could somehow send signals to communicate with each other, they would be able to convey the results of experiments that will lead them to objectively agree on the definitions of what direction is left and right, whether they are made of particle or antiparticles and whether time was flowing in the same direction.[16][m]

### Broken symmetry in normal crystals

Figure 2. Normal process (N-process) and Umklapp process (U-process). While the N-process conserves total phonon momentum, the U-process changes phonon momentum.

Different states of matter can be classified by the symmetries they spontaneously break. In a magnet, for example, spins are limited to a few possible orientations along a common direction chosen spontaneously from ones of less orientation but greater freedom and symmetry; thus a ferromagnet breaks symmetry as the process of magnetisation occurs. Normal crystals exhibit broken translation symmetry. For example, a gas is said to have translational symmetry, as its atoms can move freely to occupy any point in a given area. A crystal by contrast does not have the same degree of symmetry; only certain spatial points are permitted and there is a requirement that the atoms have a particular structure or order. If a gas cools to form a crystal, the symmetry is said to be continuously broken as the crystal gains order.

Since crystals are not invariant under arbitrary translations, strictly speaking, momentum is not conserved. No strict conservation law can be applied but a discrete translation symmetry may sometimes be achieved. The crystal momentum is called quasimomentum[20] which determines the crystal's Bloch state and is the cause of Umklapp processes.[21] This technical violation of the conservation of momentum is important in establishing some of the properties of crystals; for example, thermal conductivity of crystals cannot be understood without taking into consideration Umklapp processes.[22] The violation in the conservation of momentum can be accounted for as a transfer to the vacuum state (i.e. the zero-point field).[23]

### Quasienergy

While ordinary crystals break spatial translational symmetries leading to repeated spatial patterns, time crystals spontaneously break time-translation symmetry (TTS) and have repeated patterns in time. Fields or particles in the presence of a time crystal background will appear to violate the conservation of energy, analogous to the apparent violation of the conservation of momentum in crystalline Umklapp processes. In either case the apparent non-conservation is in reality a transfer to the vacuum field (i.e. zero-point field).[23] The term quasienergy has been coined to explain some of the predicted properties of time crystals.

## Topological order

Time crystals are thought to exhibit topological order,[7] an emergent phenomenon, in which nonlocal correlations encoded in the whole wave-function of the system allow for fault tolerance against perturbations, thus allowing quantum states to stabilize against decoherence effects that usually limit their useful lifetime.

### Topological states

Albert Einstein insisted that all fundamental laws of nature could be understood in terms of geometry and symmetry.[24] Before 1980 all states of matter could be classified by the principle of broken symmetry. The quantum Hall state provided the first example of a quantum state that had no spontaneously broken symmetry. Its behaviour depends only on its topology and not its specific geometry. The quantum Hall effect earned von Klitzing the Nobel Prize in Physics for 1985 and though it was not understood at the time, the quantum Hall effect is an example of topological order.[n] Topological order violates the long-held belief that ordering requires symmetry breaking. Fundamental laws can be studied under the context of topological field theory.

Recently a new class of topological states has emerged called quantum spin Hall (QSH) states or topological insulators. Inside a topological insulator Maxwell's equations of electromagnetism are dramatically altered to include an extra topological term which gives rise to novel new physics.[25] A dipole such as an electron above the surface of a topological insulator induces an emergent quasi-particle image magnetic monopole, known as a dyon, which is a composite of electric and magnetic charges.[o] This new particle obeys neither Bose nor Fermi statistics but behaves like a so-called anyon, named as such because it is governed by "any possible" statistics. When a superconductor is close to the surface of a topological insulator, Majorana fermions occur inside vortices.[p] These particles are governed by non-abelian statistics[q] and could have radical applications in a new form of electronics called spintronics and topological quantum computers.[24] Non-local effects analogous to the Aharonov Bohm effect have been observed in topological insulators, and certain conditions are expected to give rise to the ability of Majorana fermions to teleport, a test of which has been proposed.[r]

### Floquet topological states

Floquet topological states combine ideas from photonics and condensed matter physics. A system that is driven by a periodic external field shows a discrete time-translation symmetry. In the framework of the Floquet theory the concepts of quasienergy and Floquet states[s] were introduced to account for this time periodicity: the term quasienergy reflects the formal analogy with the quasimomentum characterizing the Bloch eigenstates in a periodic solid.[t] Recently it has been shown that the topological properties can be "tuned" by applying a time-dependent electromagnetic field.[u] For example, when microwaves periodically drive a crystalline material (i.e. a combined spacial and time periodicity) it may become a Floquet topological insulator. The crystal's quasienergy spectrum causes the emergence of new forms of topological order. It is hoped that many topological properties may be transmuted into the material at will simply by using low energy electromagnetic fields, acting like a topological switch.[v][35]

### Floquet time crystals

Time crystals can extend the idea of Floquet topological insulators still further, by enabling entirely new non-equilibrium dynamical phases. These dynamical phases are characterised by properties forbidden in the thermal equilibrium, such as spontaneous time-translation symmetry breaking or dynamical topological order. The latter opens the door to a new realm of quantum topological phenomena, which has only barely begun to be explored.[citation needed]

### Fault-tolerance against decoherence

Preventing decoherence via topological order has a wide range of implications: The efficiency of some computing and information theory tasks can be greatly enhanced when using quantum correlated states; quantum correlations are an equally valuable resource in the realm of quantum thermodynamics[36] New types of quantum devices in non-equilibrium states function very differently than their classical counterparts: For example, it has been theoretically shown that non-equilibrium quantum ratchet systems function far more efficiently than that predicted by classical thermodynamics.[w] It has also been shown that quantum coherence can be used to enhance the efficiency of systems beyond the classical Carnot limit. This is because it could be possible to extract work, in the form of photons, from a single heat bath. Quantum coherence can be used in effect to play the role of Maxwell's demon[38] allowing a hypothetical bypassing of the second law of thermodynamics.[x][43]

## Thermodynamics

### Compatibility with the laws of thermodynamics

Because a time crystal is a driven (i.e. open) quantum system that is in perpetual motion, it does not violate the laws of thermodynamics:[44]

• A time crystal does not produce work as it rotates in its ground state; energy is conserved so that the first law of thermodynamics is not violated. (Otherwise such a device would be a perpetuum mobile of the first kind.)
• A time crystal does not spontaneously convert thermal energy into mechanical work so that the second law of thermodynamics is not violated. (Otherwise such a device would be a perpetuum mobile of the second kind.)
• A time crystal cannot serve as a perpetual store of work, so that the third law of thermodynamics is not violated (Otherwise the device would be a perpetuum mobile of the third kind.)

A time crystal has been said to be a perpetuum mobile of the fourth kind: it does not produce work and it cannot serve as a perpetual energy storage. But it rotates perpetually.[45]

### Zero-point energy

Zero-point radiation continually imparts random impulses on an electron, so that it never comes to a complete stop. Zero-point radiation gives the oscillator an average energy equal to the frequency of oscillation multiplied by one-half of Planck's constant ${\displaystyle {\tfrac {1}{2}}\hbar \omega }$

Time crystals are closely related to concepts of zero-point energy and the dynamical Casimir effect.[y] As temperature is reduced to absolute zero, it might be thought that all motion ceases and particles come completely to rest. In fact, however, kinetic energy is retained by particles even at the lowest possible temperature. The random fluctuation corresponding to this zero-point energy never vanishes as a consequence of the uncertainty principle of quantum mechanics.[47] According to modern physics (i.e. quantum field theory) the universe is made up of matter fields whose quanta are fermions (i.e. leptons and quarks) and force fields, whose quanta are bosons (e.g. photons and gluons). All these fields have zero-point energy.[48] The predicted zero-point energy contained in the vacuum is very large: physicists John Wheeler and Richard Feynman calculated that there is enough energy in the vacuum inside a single light bulb to boil all the world's oceans.[49]

Zero-point energy has many observed physical consequences such as spontaneous emission, Casimir force, Lamb shift and the magnetic moment of the electron.[50][ad] According to the fluctuation-dissipation theorem, fluctuations and dissipation go hand in hand; we cannot have one without the other, and the vacuum therefore dissipates energy. It is relatively easy to show that zero-point motion of a particle is in fact sustained by the driving zero-point field, or vacuum state.[ae][57] Zero-point energy may even be the cause of dark energy and the current acceleration of the universe, though this idea has been disputed.[af]

The zero-point field is reminiscent of the discredited aether theory prevalent before the advent of Einstein's relativity, all broken symmetries can be attributed to the influence of this all pervading vacuum state.[60] If we imagine the entire universe was immersed in a vast magnet, the presence of this magnet might cause the background to be (In the words of Wolfgang Pauli) "weakly left handed" i.e. it might cause a preference of left over right and account for the broken symmetry we observe; the idea of a complex vacuum state that has a rich structure to account for all broken symmetries in nature is in the same spirit of this idea.[61]

### No-go theorem in equilibrium

There is a proof that quantum time crystals in thermal equilibrium are not possible.[ag]

### Non-equilibrium systems

Non-equilibrium quantum fluctuations have been studied for some time[ah] and the past few years have seen a surge of interest in this topic. A comprehensive definition of energy and work in these contexts is yet to be formulated and is an open area of research.[ai]

## Experiments

### University of Maryland

In October 2016, researchers at the University of Maryland, College Park, claimed to have created the world's first discrete time crystal. Using the idea from the March proposal, they trapped a chain of 171Yb+ (ytterbium) ions in a Paul trap, confined by radio frequency electromagnetic fields. One of the two spin states was selected by a pair of laser beams. The lasers were pulsed, with the shape of the pulse controlled by an acousto-optic modulator using the Tukey window to avoid too much energy at the wrong optical frequency. The hyperfine electron states are called 2S1/2 |F=0, mF = 0⟩ and |F = 1, mF = 0⟩. The different energy levels of these are very close, separated by 12.642831 GHz. Ten Doppler cooled ions were used in a line 0.025 mm long. The ions were coupled together. The researchers observed a subharmonic oscillation of the drive. The experiment also showed "rigidity" of the time crystal, where the oscillation frequency remained unchanged even when the time crystal was perturbed. However, if the perturbation drive was too great, the time crystal "melted" and lost its oscillation.[k]

### Harvard University

Mikhail Lukin led a Harvard University team who also reported the creation of a driven time crystal. The group used a diamond crystal doped with a high concentration of Nitrogen-vacancy center which have strong dipole-dipole coupling and relatively long-lived spin coherence. By driving this strongly-interacting dipolar spin system using microwave fields and reading out the ensemble spin state using an optical (laser) field, it was observed that the spin polarization evolves at half the frequency of the microwave drive. The oscillations persist for over 100 cycles.[j] This sub-harmonic response to the drive frequency is a key signature of time-crystalline order.

## Related concepts

### Choreographic crystals

A similar idea called a choreographic crystal has been proposed.[aj]

### Dynamical Casimir effect

Time crystals are closely related to the dynamical Casimir effect or Unruh effect. These effects are basically an instability of the quantum vacuum, which leads to an exponential growth of emitted boson pairs (known as superradiance in the form of photons or phonons) when the oscillating frequency of the medium is equal to twice the boson frequency.

## Notes

1. ^
2. ^ See Li et al. (2012a, 2012b)[2]
3. ^
4. ^
5. ^
6. ^
7. ^
8. ^
9. ^
10. ^ a b
11. ^ a b
12. ^
13. ^ While violations of C, P, T, CP, PT, TC exist it is thought that the product of these transformations CPT is always invariant.[17] The CPT theorem says that CPT symmetry holds for all physical phenomena. Thus an experiment that measures a violation of CP might infer a corresponding violation under time-reversal T in order to maintain CPT invariance. Only recently was a violation of time-reversal T symmetry directly observed (Lees et al. 2012).[18] Even with time-reversal asymmetry, CPT invariance still remain valid however.[19]
14. ^
15. ^
16. ^
17. ^
18. ^
19. ^
20. ^ See Grifoni & Hänggi (1998)[33] for a review of Floquet theory
21. ^
22. ^
23. ^ See for example Yukawa et al. (1997), Reimann et al. (1997), Tatara et al. (1998)[37]
24. ^
25. ^ See Chernodub (2012, 2013a, 2013b),[46] and Mendonça & Dodonov (2014)[46]
26. ^ See for example Enz (1974)
27. ^ In QED it is argued the entire universe is completed bathed in the zero-point electromagnetic field, and as such it can only add some constant amount to measurement values. Physical measurements will therefore reveal only deviations from this constant vacuum state. It is argued that the zero-point energy is a c-number (i.e. constant) and therefore has no physical effect.[51] It is declared by fiat that the ground state has zero energy. The zero-point energy can be dropped from the Hamiltonian by redefining the zero of energy and by stating that it has no effect on the Heisenberg equations of motion:[52]
${\displaystyle :H_{F}-\langle 0|H_{F}|0\rangle ={\frac {1}{2}}\hbar \omega {\big (}aa^{\dagger }+a^{\dagger }a{\big )}-{\frac {1}{2}}\hbar \omega =\hbar \omega aa^{\dagger }}$
The new Hamiltonian is said to be normally ordered (or Wick ordered) and is denoted by a double-dot symbol. The normally ordered Hamiltonian is denoted :${\displaystyle H_{F}}$:, i.e.:
${\displaystyle :H_{F}:\equiv \hbar \omega {\big (}aa^{\dagger }+a^{\dagger }a{\big )}:\equiv \hbar \omega aa^{\dagger }}$
However, when we do this and solve the Heisenberg equation for a field operator, we must include the vacuum field, which is the homogeneous part of the solution for the field operator. It can be shown that the vacuum field is essential for the preservation of the commutators and the formal consistency of the theory (Milonni 1994). When the field energy is calculated a contribution from the vacuum field is always present (i.e. the zero-point field energy). In other words, the zero-point field energy "reappears" even though it may have deleted it from the Hamiltonian via Wick ordering.[53]
28. ^ See Schwinger (1998a, 1998b, 1998c)
29. ^ Such a derivation was first given by Schwinger (1975) for a scalar field, and then generalised to the electromagnetic case by Schwinger et al. (1978) in which they state "the vacuum is regarded as truly a state with all physical properties equal to zero". More recently Jaffe (2005) has highlighted a similar approach in deriving the Casimir effect stating "the concept of zero-point fluctuations is a heuristic and calculational aid in the description of the Casimir effect, but not a necessity in QED."
30. ^ There is a long debate[z] over the question of whether zero-point fluctuations of quantized vacuum fields are “real” i.e. do they have physical effects that cannot be interpreted by an equally valid alternative theory? In quantum electrodynamics (QED) zero-point energy is frequently assumed to be a constant (c-number) of no physical significance as the field is in equilibrium.[aa] Julian Schwinger, in particular, attempted to formulate QED without reference to zero-point fluctuations via his "source theory".[ab] From such an approach it is possible to derive the Casimir effect without reference to a fluctuating zero-point field.[ac] While the efforts of Schwinger and others have identified the difficultly in judging the physical reality of infinite zero-point energies that are inherent in field theories, no one has shown that source theory or another S-matrix based approach can provide a complete description of QED to all orders[54] and the zero-point field can be shown to be an essential requirement to preserve the formal consistency of QED.[55] zero-point energies and all they entail would seem to be a necessity for any attempt at any grand unified theory of physics: They give an explanation as to how spontaneous symmetry breaking occurs at all levels of the standard model[54] and modern physics does not know any way to construct gauge-invariant, renormalizable theories without zero-point energy.[56]
31. ^
32. ^ See for example Beck & Mackey (2005)[58] arguing for and Jetzer & Straumann (2006)[59] arguing against.
33. ^
34. ^ See for example Yukawa (2000)[62] and Mukamel (2003)[63]
35. ^ See Esposito et al. (2009) and Campisi et al. (2011) for academic review articles on non-equilibrium quantum fluctuations[64]
36. ^

## References

1. ^
2. ^ a b
3. ^ a b Else et al. 2016, p. 1; Watanabe & Oshikawa 2015, p. 1; Wilczek 2013a; Sacha 2015, p. 1.
4. ^ a b
5. Yao et al. 2017, p. 1.
6. ^ a b Watanabe & Oshikawa 2015, p. 1.
7. ^ a b
8. ^ a b
9. ^
10. ^
11. ^ a b c Feng & Jin 2005, p. 18.
12. ^ a b Wilczek 2015, chpt. 3.
13. ^ Cao 2004, p. 151.
14. ^ Lee 1981, pp. 183–184.
15. ^ Aitchison 1981, p. 540.
16. ^ Lee 1981, pp. 184–187.
17. ^ Lee 1981, pp. 188.
18. ^
19. ^
20. ^ Sólyom 2007, p. 191.
21. ^ Sólyom 2007, p. 193.
22. ^ Sólyom 2007, p. 194.
23. ^ a b Wilczek 2012, p. 4.
24. ^ a b Qi & Zhang 2010, p. 38.
25. ^ a b Qi & Zhang 2010, p. 33.
26. ^
27. ^
28. ^
29. ^
30. ^ a b Hasan & Kane 2010, p. 19.
31. ^ Grifoni & Hänggi 1998, pp. 233–234, 241.
32. ^ Grifoni & Hänggi 1998, p. 237.
33. ^ Guo et al. 2013, p. 1.
34. ^
35. ^ a b
36. ^ Dillenschneider & Lutz 2009, p. 6.
37. ^ Yukawa 2000, p. 1.
38. ^ a b Maruyama et al. 2009, p. 20.
39. ^ Horodecki et al. 2009, p. 80; Maruyama et al. 2009, p. 20.
40. ^ Modi et al. 2012, p. 43.
41. ^
42. ^
43. ^ Dillenschneider & Lutz 2009, pp. 5–6.
44. ^ Chernodub 2013b, p. 2, 13.
45. ^ Chernodub 2013a, p. 10.
46. ^ a b Sacha 2015, p. 1.
47. ^ Milonni 1994, pp. 36–38.
48. ^ Milonni 1994, p. 35.
49. ^
50. ^ Milonni 1994, p. 111.
51. ^ Milonni 1994, pp. 42–43.
52. ^ Itzykson & Zuber 1980, p. 111.
53. ^ Milonni 1994, p. 73.
54. ^ a b Jaffe 2005, p. 7.
55. ^ Milonni 1994, p. 48.
56. ^ Greiner et al. 2012, p. 20.
57. ^ a b Milonni 1994, p. 54.
58. ^ Ball 2004; Copeland et al. 2006, p. 20.
59. ^ Copeland et al. 2006, p. 20.
60. ^ Lee 1981, pp. 378–381.
61. ^ Aitchison 1981, p. 541.
62. ^ Esposito et al. 2009, p. 2; Jarzynski 2011, p. 348; Campisi et al. 2011, p. 8.
63. ^ Esposito et al. 2009, p. 2, 8; Jarzynski 2011, p. 348; Campisi et al. 2011, p. 13.
64. ^ Seifert 2012, p. 9.
65. ^

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