To the editors:
In his essay, Michael Engel reviews a series of experiments on universality in self-assembly processes and discusses their implications for nonequilibrium statistical mechanics.1 The original work under review was published by Ghaith Makey et al. in Nature Physics in April 2020.2 I would like to add my perspective on two statements: one made in the original paper and one in Engel’s review.
The central message of the article by Makey et al. is that the authors developed a general method of self-assembly under strongly out-of-equilibrium conditions. Here the term general means that the method produces the same growth behavior of the assembled aggregates irrespective of the size and chemical composition of their constituents.
This observation will be welcomed by theoreticians. If every system and every set of external conditions called for a new theory, scientific research would be a hopeless task. We pursue our work because we can empirically find large classes of systems that share general aspects of their behavior. This property of nature, which allows theoreticians to produce meaningful effective models, is caused by the separation of timescales, energy-scales, and length-scales on which physical processes take place.
Still, the observation by Makey et al. is intriguing. To rephrase it in terms of theoretical physics, irrespective of the nature of the forces between the aggregating units and the number and nature of their internal degrees of freedom, the evolution of certain macroscopic observables—such as the size of aggregates and the magnitude of surface fluctuations—is governed by the same equation of motion. This is far from obvious.
A simple thought experiment can help to illustrate why this observation is not obvious. Imagine we start out from the full, microscopic description of each experimental realization presented in Makey et al. That is, we write down the Schrödinger equation for a set of quantum dots suspended in water subject to external driving by a laser, as well as the Schrödinger equation for all atoms contained in a few dozen M. luteus cells in Luria broth medium. Clearly, these problems are much too complex to be solved in practice. Nevertheless, imagine we systematically integrate out degrees of freedom until we obtain the exact equation of motion that governs the aggregate size in each of the experimental systems. Would we, in general, not expect these equations to differ? There is something about the choice of self-assembly protocol in Makey et al. that removes the differences between these equations of motion.
In other settings, the theoretical basis for such a coincidence of coarse-grained models is well established. For systems without external driving, Pep Español analyzed by means of a projection operator formalism the equations of motion obtained under iterative coarse-graining.3 He showed that if the equation of motion has reached the form of a Fokker–Planck equation and if there is timescale separation at each successive step, one recovers a Fokker–Planck equation on each level of coarse-graining. For theoreticians, the interesting question brought up by experiments such as those presented by Makey et al. is how to generalize this kind of result to systems far from equilibrium.
In his review, Engel briefly refers to the theoretical challenges encountered when deriving coarse-grained theories for systems out of equilibrium. In this context he writes that “it appears there is no general extremum principle that governs the evolution of systems far from equilibrium.” He gives as a reference a source from 2008. The field of nonequilibrium statistical physics has seen some interesting developments since then, and I respectfully disagree with this statement.
Power functional theory provides such a general extremum principle.4 The basic aim of power functional theory is to generalize the extremum principle of density functional theory, which determines the equilibrium structure of a thermal system, to nonequilibrium systems. The nonequilibrium analogue to the thermodynamic potential is the power functional, and the analogue to the structure is the nonequilibrium particle current. As the method is rather new, it has so far only been applied to simple model problems.
Presumably, timescale separation between different degrees of freedom is at the origin of the universal behavior observed by Makey et al. It would be interesting to see exact nonequilibrium coarse-graining methods, such as power functional theory or time-dependent projection operator formalisms,5 applied to driven self-assembly processes in order to clarify this point.
