Diverse phenomena, common themes

INSIGHT | COMMENTARY

Diverse phenomena, common themes Christopher Jarzynski Our framework for understanding non-equilibrium behaviour is yet to match the simplicity and power of equilibrium statistical physics. But recent theoretical and experimental advances reveal key principles that unify seemingly unrelated topics.

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lassical thermodynamics is built around the concept of equilibrium states. If we leave a system undisturbed, it relaxes to an apparently quiescent state in which nothing much seems to be happening. From the late nineteenth century through much of the twentieth century, a comprehensive theoretical framework — informed by ever more precise experiments — was developed to explain how the macroscopic properties of these equilibrium states arise from interactions between their microscopic constituents. A grand triumph of this effort was the success in understanding phase transitions and critical phenomena, which provided new tools, stimulated careful experiments and influenced our view of what constitutes a successful physical theory. If we shift our focus away from equilibrium states, we find a rich universe of non-equilibrium behaviour that has been the object of increasing attention in recent decades. The contributions to this Insight highlight some of the progress in this field. Although they cover a range of topics, several unifying themes provide links. These include the central role of the second law of thermodynamics, the importance of fluctuations as a key to understanding out-of-equilibrium behaviour, the thermodynamic relevance and implications of information processing and the subtleties that arise when quantum mechanics enters the picture. Biology provides a natural setting for applying and refining the tools of non-equilibrium statistical physics. If an equilibrium state is one in which nothing seems to be happening, then living organisms — which grow, move and multiply — seem to be the exact opposite. Just how does a living organism maintain itself away from equilibrium? Processes such as growth and motion arise from intricate networks of chemical

reactions driven by chemical imbalances, that is, chemical potential differences. But these same reactions simultaneously tend to balance the imbalances, driving the system toward equilibrium. As pointed out by Erwin Schrödinger in his 1944 monograph What is Life?, to avoid a quick decay to equilibrium, a living organism must continually increase the entropy of its surroundings1. By digesting food, breathing air and absorbing photons, living organisms channel, or transduce, a continual flow of energy that produces the dissipation required to sustain life away from equilibrium. The field of active matter is devoted to the non-equilibrium behaviour of systems composed of many particles, which individually produce movement by transducing energy. The particles can range from micron-sized ‘swimming’ colloids that generate entropy by catalysing the oxidation of hydrogen peroxide, to flocks of birds and schools of fish2. Despite this diversity, such systems and the fascinating collective behaviours they exhibit can be analysed quantitatively using common sets of tools, often borrowed from equilibrium statistical physics. In a Progress Article in this issue, Jacques Prost and colleagues3 summarize recent research on the cytoskeleton — a network of protein filaments crosslinked to one another, which endows a living cell with its shape. The cytoskeleton can evolve by a process known as treadmilling, in which new protein molecules attach to one end of a filament as molecules at the other end detach. Additionally, molecular motors such as myosin can exert forces on the filaments. The result is an active gel whose complex dynamics are driven by the continual transduction of chemical energy. As Prost et al.3 explain, active gels are amenable to field-theoretic analyses, in which microscopic details manifest

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themselves as effective parameters in a simplified hydrodynamic treatment, and underlying symmetries determine relevant order parameters. Although these treatments cannot be expected to capture all biologically relevant details of the cytoskeleton, they often provide agreement with and insight into experiments, and they have expanded the scope of soft condensed-matter physics. The Review Article by Jens Eisert and co-workers4 addresses an entirely different aspect of non-equilibrium behaviour. The authors ask: how does a closed quantum many-body system self-thermalize? In others words, does the Schrödinger equation enable us to understand why a thermally isolated system relaxes spontaneously to equilibrium? For a classical many-body system, like a gas of atoms modelled as a collection of Newtonian particles, selfthermalization involves concepts such as ergodicity and mixing — related to a system’s ability to self-randomize effectively — as well as typicality, which expresses the idea that most accessible microscopic states describe the same macroscopic state. The linear Schrödinger equation, however, does not give rise to the sort of nonlinear, chaotic dynamics responsible for ergodicity and mixing in classical many-body systems. This suggests that new concepts are needed to understand thermalization in isolated quantum systems. One approach invokes the eigenstate thermalization hypothesis5, which states that individual energy eigenstates of complex systems are effectively thermal, in the sense that expectation values of suitable observables are equal to their thermal average values. The tendency of a system to relax to equilibrium is then inherited from this conjectured property of the eigenstates themselves. The still-open question of selfthermalization has always been important to the foundations of statistical physics, but as Eisert et al.4 emphasize, the study of 105

COMMENTARY | INSIGHT

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Although fluctuation theorems were originally developed for systems whose microscopic evolution obeys classical equations of motion, the Perspective by Hänggi and Talkner describes recent progress related to fluctuation theorems for quantum systems7. Focusing on far-from-equilibrium relations between work and free energy, they emphasize the conceptual subtlety of defining work in the setting of quantum mechanics, stating that “work is not an observable.” In other words, there is no Hermitian operator corresponding to work, in contrast to observables such as position and momentum. For a closed quantum system — that is, in the absence of a heat bath — the work

performed on the system can be defined in terms of projective measurements of its initial and final energy. These are idealized measurements that cause the state of a quantum system to ‘collapse’ onto an eigenstate of the quantity being measured, which in this case is

A FESHCHENK O ANN

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equilibrium satisfy a number of strong and unexpected results, which are collectively called fluctuation theorems. Most fluctuation theorems are expressed either in terms of entropy production, ΔS (see, for instance, Jukka Pekola’s Progress Article6), or else as relations between work, W, and free energy differences, ΔF (as discussed in the Perspective by Peter Hänggi and Peter Talkner 7). In either case, they can be viewed as extensions of the second law of thermodynamics, replacing CK _ 0 TO inequalities such as ΔS > _ ΔF, familiar from or W > introductory textbooks on thermodynamics, with stronger equalities, expressed in terms of fluctuations in work or entropy production8.

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quantum many-body dynamics has been invigorated in recent years by advances in both experiment and theory. Their Review Article covers topics such as transport, localization, quantum phase transitions and periodically driven quantum systems. It provides an overview of the impressive progress that has been made by the interplay between rigorous analysis and numerical simulations, and it discusses the exciting prospect that a new class of tools is coming online: analogue quantum simulators, in which the dynamics of model quantum systems are replicated in laboratory experiments involving (for example) cold atoms in optical lattices, or chains of trapped ions. The remaining three articles in this collection focus on the fluctuations of systems away from thermal equilibrium. Matter in equilibrium is composed of constituents — atoms and molecules — that individually evolve erratically. In macroscopic systems these fluctuations effectively cancel one another due to the sheer number of particles involved (N ~ 1023), and we discern only smooth and predictable average material properties rather than the underlying molecular chaos and noise. But if the system of interest itself is microscopic then these fluctuations do not cancel quite so neatly, and randomness becomes inherent in our description of the system’s behaviour. Thus, whereas the tension of a stretched rubber band (which arises from the thermal agitations of its constituent polymer molecules) is perceived to be a smooth function of length and temperature, the tension of a molecule of RNA that is stretched using optical tweezers fluctuates unpredictably. The fluctuations of a system in thermal equilibrium are described within the durable theoretical framework pioneered by Maxwell, Boltzmann and Gibbs, and linear response theory provides a scheme for extending this framework to systems that are not too far from thermal equilibrium. Indeed, much of what we traditionally call non-equilibrium statistical physics — such as the Onsager reciprocal relations and the Green–Kubo formulas for transport coefficients — is really near-equilibrium statistical physics. However, in recent years it has become appreciated that the fluctuations of small systems far from thermal

the system’s energy. Imagine that we first measure the initial energy of the quantum system, we then allow the system to evolve under the Schrödinger equation as a time-dependent perturbation is applied, and then we measure its final energy. We can define the work performed on the system as the difference between the two values obtained from these projective measurements: W = Ef – Ei. If we accept this definition, then fluctuation theorems can be derived rather easily, and a recent experiment by Shuoming An and colleagues9, also reported in this issue, has validated this approach using a single ytterbium ion in a harmonic potential. An alternative scheme for measuring the fluctuations in the work performed on a quantum system makes clever use of interferometry to construct the Fourier transform of the probability distribution of work values, without directly measuring the amount of work performed during a given realization of the process. This approach has also recently been validated, in an experiment using a liquid-state nuclear magnetic resonance set-up10. Pekola’s Progress Article6 discusses the use of micrometre-sized electronic circuits at temperatures below 1 K to probe the validity of fluctuation theorems and related predictions. In such circuits the tunnelling of individual electrons can be observed, providing an exquisite platform for measuring microscopic fluctuations.



INSIGHT | COMMENTARY For example, a double quantum dot (DQD) set-up involves two semiconductor quantum dots, which we can imagine as a pair of artificial atoms. Each dot can accommodate a number of excess electrons (the artificial atoms can be ionized), thus the charge state of the DQD is specified by integers (nL, nR), denoting the number of excess electrons on the left and right dot. The two dots are coupled to separate electron reservoirs, maintained at a voltage difference VDQD. At sufficiently low temperatures, Coulomb repulsion ensures that only three charge states are possible: (0, 0), (1, 0) and (0, 1). A transition from (0, 0) to (1, 0) means that an electron has tunnelled from the left reservoir to the left quantum dot; a transition from (1, 0) to (0, 1) signals the tunnelling of an electron from the left to the right dot, and so on. By monitoring the charge state, the flow of current down the electronic potential can be observed, one electron at a time, and the fluctuation theorem — which makes a prediction about the fluctuations of this tiny current — can be tested. In another set-up, a single-electron box (SEB) contains two metallic islands, with a combined excess charge corresponding to one electron. This provides a paradigm of a two-state system, as the excess electron tunnels back and forth, randomly, between the two islands. (There are no electron reservoirs in this set-up.) By varying a gate voltage, Vg, with time, the energy difference between the two states is varied, and work is performed on the electron. Pekola describes how such experiments have been used to test a variety of fluctuation theorems, including ones involving measurement and feedback6. He notes that although both the DQD and the SEB make use of quantum tunnelling, they “operate essentially in the classical regime”: quantum coherence effects are absent in these experiments, as the system effectively makes instantaneous transitions between discrete classical states. However, this does not rule out the possibility that future experiments involving tiny electronic circuits may be used to probe fluctuation theorems in the genuinely quantum regime. Finally, the Review Article by Juan M. R. Parrondo and co-workers11 explores the link between thermodynamics and information, a subject whose roots trace back to one of the most famous thought experiments in physics: Maxwell’s

demon. In a letter written in 1867, Maxwell imagined a “neat-fingered being” guarding a tiny opening separating two chambers filled with gas. This creature amplifies an existing temperature difference between the chambers without performing any work — in apparent violation of the second law of thermodynamics — simply by letting unusually fast particles pass through the opening in one direction, and slow ones in the opposite direction. This thought experiment highlights the statistical nature of the second law. In an analogy with betting against a casino in a card game — you may win some hands, but in the long run the odds are against you — Maxwell’s demon is a devious player who possesses information about which cards are likely to be dealt, and uses this knowledge to reverse the odds. A central element in Maxwell’s thought experiment is the demon’s ability to gather microscopic information about the trajectories of individual molecules — information that is not ordinarily accessible to the macroscopic observer. Although discussions of Maxwell’s demon have never fully gone out of style, there has recently been a burst of activity in this area, bringing together tools from information theory and non-equilibrium statistical physics. Parrondo et al.11 describes this progress, including generalized fluctuation theorems for processes involving the measurement and feedback of thermal fluctuations, as well as recent implementations of Maxwell’s demon (and related ideas) in actual laboratory experiments. Parrondo et al.11 also sketch the outlines of an emerging theoretical framework for understanding the thermodynamics of information. In this approach, a measurement brings about a change in the statistical description of the state of a system, effectively shifting the system from an equilibrium to a non-equilibrium state. As a result, the system’s free energy increases. This non-equilibrium free energy plays a central role in the thermodynamics of processes involving the manipulation of information, and is used to formulate an extension of the second law to feedback processes. In the final section of their article, Parrondo et al.11 focus on the idea that information is not just an abstract quantity, but must reside in a physical



memory storage device. This point of view leads to an illuminating thermodynamic assessment of a model feedback control process, involving a measurement step followed by a feedback step. Each step obeys a second law-like inequality that relates work, free energy and the establishment (or consumption) of correlations between the system and the memory. When combined, the two inequalities show that in a physical implementation of Maxwell’s demon, any reduction in entropy brought about by the actions of the demon is compensated by an increase in the entropy (or information) of the memory device. This is entirely consistent with the second law of thermodynamics, as argued originally by Charles Bennett 12. The striking feature of equilibrium statistical physics, and in large measure the key to its success, is the simplicity of its foundations in Boltzmann–Gibbs theory. It may be unreasonable to hope for a comparably simple, far-reaching, and as-yet-undiscovered framework for understanding non-equilibrium phenomena. But as the articles in this Insight illustrate, the current state of non-equilibrium statistical physics is one of exciting experimental and theoretical progress, characterized by the emergence of unifying principles as well as open questions. ❐ Christopher Jarzynski is in the Department of Chemistry and Biochemistry and the Institute for Physical Science and Technology at the University of Maryland, Maryland 20742 USA. e-mail: [email protected] References

1. Schrödinger, E. What is Life? (Cambridge Univ. Press, 1944). 2. Marchetti, M. C. et al. Rev. Mod. Phys. 85, 1143–1189 (2013). 3. Prost, J., Jülicher, F. & Joanny, J-F. Nature Phys. 11, 111–117 (2015). 4. Eisert, J., Friesdorf, M. & Gogolin, C. Nature Phys. 11, 124–130 (2015). 5. Srednicki, M. Phys. Rev. E 50, 888–901 (1994). 6. Pekola, J. P. Nature Phys. 11, 118–123 (2015). 7. Hänggi, P. & Talkner, P. Nature Phys. 11, 108–110 (2015). 8. Jarzynski, C. Annu. Rev. Cond. Matter Phys. 2, 329–351 (2011). 9. An, S. et al. Nature Phys. 11, 193–199 (2015). 10. Batalhao, T. B. et al. Phys. Rev. Lett. 113, 140601 (2014). 11. Parrondo, J. M. R., Horowitz, J. M. & Sagawa, T. Nature Phys. 11, 131–139 (2015). 12. Bennett, C. H. Int. J. Theor. Phys. 21, 905–940 (1982).

Acknowledgements

The author acknowledges financial support from the National Science Foundation (USA), under DMR-1206971, and the US Army Research Office under contract number W911NF-13-1-0390.

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