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We show that the time-resolved dynamics of an underdamped harmonic oscillator can be used to do multifunctional computation, performing distinct computations at distinct times within a single dynamical trajectory. We consider the amplitude of an oscillator whose inputs influence its frequency. The activity of the oscillator at fixed times is a nonmonotonic function of its inputs, so it can solve problems such as XOR that are not linearly separable. The activity of the oscillator at fixed input is a nonmonotonic function of time, so it is multifunctional in a temporal sense, and able to carry out distinct nonlinear computations at distinct times within the same dynamical trajectory. We show that a single oscillator, observed at different times, can act as all of the elementary logic gates and perform binary addition, the latter usually implemented in hardware using five logic gates. We show that a set of n oscillators, observed at different times, can perform an arbitrary number of analog-to-n-bit digital conversions. We also show that oscillators can be trained by gradient descent to perform distinct classification tasks at distinct times. Computing with time-dependent functionality can be done in or out of equilibrium, and suggests a way of reducing the number of parameters or devices required to do nonlinear computations.

The Jarzynski equality allows the calculation of free-energy differences using values of work measured from nonequilibrium trajectories. The number of trajectories required to accurately estimate free-energy differences in this way grows sharply with the size of work fluctuations, motivating the search for protocols that perform desired transformations with minimum work. However, protocols of this nature can involve varying temperature, to which the Jarzynski equality does not apply. We derive a variant of the Jarzynski equality that applies to varying-temperature protocols, and show that it can have better convergence properties than the standard version of the equality. We derive this modified equality and the associated fluctuation relation within the framework of Markovian stochastic dynamics, complementing related derivations done within the framework of Hamiltonian dynamics.

Time-dependent protocols that perform irreversible logical operations, such as memory erasure, cost work and produce heat, placing bounds on the efficiency of computers. Here we use a prototypical computer model of a physical memory to show that it is possible to learn feedback-control protocols to do fast memory erasure without input of work or production of heat. These protocols, which are enacted by a neural-network "demon," do not violate the second law of thermodynamics because the demon generates more heat than the memory absorbs. The result is a form of nonlocal heat exchange in which one computation is rendered energetically favorable while a compensating one produces heat elsewhere, a tactic that could be used to rationally design the flow of energy within a computer.

Buča et al. [Phys. Rev. E 100, 020103(R) (2019)2470-004510.1103/PhysRevE.100.020103] study the dynamical large deviations of a boundary-driven cellular automaton. They take a double limit in which first time and then space is made infinite, and interpret the resulting large-deviation singularity as evidence of a first-order phase transition and the accompanying coexistence of two distinct dynamical phases. This view is characteristic of an approach to dynamical large deviations in which time is interpreted as if it were a spatial coordinate of a thermodynamic system [Jack, Eur. Phys. J. B 93, 74 (2020)1434-602810.1140/epjb/e2020-100605-3]. Here, I argue that the large-deviation function produced in this double limit is not consistent with the basic features of the model of Buča et al. I show that a modified limiting procedure results in a nonsingular large-deviation function consistent with those features, and that neither supports the idea of coexisting dynamical phases.

We show that cellular automata can classify data by inducing a form of dynamical phase coexistence. We use Monte Carlo methods to search for general two-dimensional deterministic automata that classify images on the basis of activity, the number of state changes that occur in a trajectory initiated from the image. When the number of time steps of the automaton is a trainable parameter, the search scheme identifies automata that generate a population of dynamical trajectories displaying high or low activity, depending on initial conditions. Automata of this nature behave as nonlinear activation functions with an output that is effectively binary, resembling an emergent version of a spiking neuron.

Using a model heat engine, we show that neural-network-based reinforcement learning can identify thermodynamic trajectories of maximal efficiency. We consider both gradient and gradient-free reinforcement learning. We use an evolutionary learning algorithm to evolve a population of neural networks, subject to a directive to maximize the efficiency of a trajectory composed of a set of elementary thermodynamic processes; the resulting networks learn to carry out the maximally efficient Carnot, Stirling, or Otto cycles. When given an additional irreversible process, this evolutionary scheme learns a previously unknown thermodynamic cycle. Gradient-based reinforcement learning is able to learn the Stirling cycle, whereas an evolutionary approach achieves the optimal Carnot cycle. Our results show how the reinforcement learning strategies developed for game playing can be applied to solve physical problems conditioned upon path-extensive order parameters.

We use a neural-network ansatz originally designed for the variational optimization of quantum systems to study dynamical large deviations in classical ones. We use recurrent neural networks to describe the large deviations of the dynamical activity of model glasses, kinetically constrained models in two dimensions. We present the first finite size-scaling analysis of the large-deviation functions of the two-dimensional Fredrickson-Andersen model, and explore the spatial structure of the high-activity sector of the South-or-East model. These results provide a new route to the study of dynamical large-deviation functions, and highlight the broad applicability of the neural-network state ansatz across domains in physics.

Within simulations of molecules deposited on a surface we show that neuroevolutionary learning can design particles and time-dependent protocols to promote self-assembly, without input from physical concepts such as thermal equilibrium or mechanical stability and without prior knowledge of candidate or competing structures. The learning algorithm is capable of both directed and exploratory design: it can assemble a material with a user-defined property, or search for novelty in the space of specified order parameters. In the latter mode it explores the space of what can be made, rather than the space of structures that are low in energy but not necessarily kinetically accessible.

Singularities of dynamical large-deviation functions are often interpreted as the signal of a dynamical phase transition and the coexistence of distinct dynamical phases, by analogy with the correspondence between singularities of free energies and equilibrium phase behavior. Here we study models of driven random walkers on a lattice. These models display large-deviation singularities in the limit of large lattice size, but the extent to which each model's phenomenology resembles a phase transition depends on the details of the driving. We also compare the behavior of ergodic and nonergodic models that present large-deviation singularities. We argue that dynamical large-deviation singularities indicate the divergence of a model timescale, but not necessarily one associated with cooperative behavior or the existence of distinct phases.

We show that neural networks trained by evolutionary reinforcement learning can enact efficient molecular self-assembly protocols. Presented with molecular simulation trajectories, networks learn to change temperature and chemical potential in order to promote the assembly of desired structures or choose between competing polymorphs. In the first case, networks reproduce in a qualitative sense the results of previously known protocols, but faster and with higher fidelity; in the second case they identify strategies previously unknown, from which we can extract physical insight. Networks that take as input the elapsed time of the simulation or microscopic information from the system are both effective, the latter more so. The evolutionary scheme we have used is simple to implement and can be applied to a broad range of examples of experimental self-assembly, whether or not one can monitor the experiment as it proceeds. Our results have been achieved with no human input beyond the specification of which order parameter to promote, pointing the way to the design of synthesis protocols by artificial intelligence.

We describe a simple form of importance sampling designed to bound and compute large-deviation rate functions for time-extensive dynamical observables in continuous-time Markov chains. We start with a model, defined by a set of rates, and a time-extensive dynamical observable. We construct a reference model, a variational ansatz for the behavior of the original model conditioned on atypical values of the observable. Direct simulation of the reference model provides an upper bound on the large-deviation rate function associated with the original model, an estimate of the tightness of the bound, and, if the ansatz is chosen well, the exact rate function. The exact rare behavior of the original model does not need to be known in advance. We use this method to calculate rate functions for currents and counting observables in a set of network- and lattice models taken from the literature. Straightforward ansätze yield bounds that are tighter than bounds obtained from Level 2.5 of large deviations via approximations that involve uniform scalings of rates. We show how to correct these bounds in order to recover the rate functions exactly. Our approach is complementary to more specialized methods and offers a physically transparent framework for approximating and calculating the likelihood of dynamical large deviations.

Molecular self-assembly is usually done at low supersaturation, leading to low rates of growth, in order to allow time for binding mistakes to anneal. However, such conditions can lead to prohibitively long assembly times where growth proceeds by the slow nucleation of successive layers. Here we use a lattice model of molecular self-assembly to show that growth in this regime can be sped up by impurities, which lower the free-energy cost of layer nucleation. Under certain conditions impurities behave almost as a catalyst in that they are present at high concentration at the surface of the assembling structure, but at low concentration in the bulk of the assembled structure. Extrapolation of our numerics using simple analytic arguments suggests that this mechanism can reduce growth times by orders of magnitude in parameter regimes applicable to molecular systems.

Cooperative adsorption of gases by porous frameworks, which permits more efficient uptake and removal than the more usual noncooperative (Langmuir-type) adsorption, usually results from a phase transition of the framework. Here we show how cooperativity emerges in the class of metal-organic frameworks mmen-M_{2}(dobpdc) in the absence of a phase transition. Our study provides a microscopic understanding of the emergent features of cooperative binding, including the position, slope, and height of the isotherm step, and indicates how to optimize gas storage and separation in these materials.

We study simple models of intermittency, involving switching between two states, within the dynamical large-deviation formalism. Singularities appear in the formalism when switching is cooperative or when its basic time scale diverges. In the first case the unbiased trajectory distribution undergoes a symmetry breaking, leading to a change in shape of the large-deviation rate function for a particular dynamical observable. In the second case the symmetry of the unbiased trajectory distribution remains unbroken. Comparison of these models suggests that singularities of the dynamical large-deviation formalism can signal the dynamical equivalent of an equilibrium phase transition but do not necessarily do so.

We describe a simple method that can be used to sample the rare fluctuations of discrete-time Markov chains. We focus on the case of Markov chains with well-defined steady-state measures, and derive expressions for the large-deviation rate functions (and upper bounds on such functions) for dynamical quantities extensive in the length of the Markov chain. We illustrate the method using a series of simple examples, and use it to study the fluctuations of a lattice-based model of active matter that can undergo motility-induced phase separation.

We compute probability distributions of trajectory observables for reversible and irreversible growth processes. These results reveal a correspondence between reversible and irreversible processes, at particular points in parameter space, in terms of their typical and atypical trajectories. Thus key features of growth processes can be insensitive to the precise form of the rate constants used to generate them, recalling the insensitivity to microscopic details of certain equilibrium behavior. We obtained these results using a sampling method, inspired by the "s-ensemble" large-deviation formalism, that amounts to umbrella sampling in trajectory space. The method is a simple variant of existing approaches, and applies to ensembles of trajectories controlled by the total number of events. It can be used to determine large-deviation rate functions for trajectory observables in or out of equilibrium.

Models of bacterial growth tend to be "irreversible," allowing for the number of bacteria in a colony to increase but not to decrease. By contrast, models of molecular self-assembly are usually "reversible," allowing for the addition and removal of particles to a structure. Such processes differ in a fundamental way because only reversible processes possess an equilibrium. Here we show at the mean-field level that dynamic trajectories of reversible and irreversible growth processes are similar in that both feel the influence of attractors, at which growth proceeds without limit but the intensive properties of the system are invariant. Attractors of both processes undergo nonequilibrium phase transitions as model parameters are varied, suggesting a unified way of describing typical properties of reversible and irreversible growth. We also establish a connection at the mean-field level between an irreversible model of growth (the magnetic Eden model) and the equilibrium Ising model, supporting the findings made by other authors using numerical simulations.

A challenge of molecular self-assembly is to understand how to design particles that self-assemble into a desired structure and not any of a potentially large number of undesired structures. Here we use simulation to show that a strategy of minimal positive design allows the self-assembly of networks equivalent to the 8 semiregular Archimedean tilings of the plane, structures not previously realized in simulation. This strategy consists of identifying the fewest distinct types of interparticle interaction that appear in the desired structure, and does not require enumeration of the many possible undesired structures. The resulting particles, which self-assemble into the desired networks, possess DNA-like selectivity of their interactions. Assembly of certain molecular networks may therefore require such selectivity.

Colloidal particles of two types, driven in opposite directions, can segregate into lanes [Vissers et al., Soft Matter 7, 2352 (2011)1744-683X10.1039/c0sm01343a]. This phenomenon can be reproduced by two-dimensional Brownian dynamics simulations of model particles [Dzubiella et al., Phys. Rev. E 65, 021402 (2002)1063-651X10.1103/PhysRevE.65.021402]. Here we use computer simulation to assess the generality of lane formation with respect to variation of particle type and dynamical protocol. We find that laning results from rectification of diffusion on the scale of a particle diameter: oppositely driven particles must, in the time taken to encounter each other in the direction of the drive, diffuse in the perpendicular direction by about one particle diameter. This geometric constraint implies that the diffusion constant of a particle, in the presence of those of the opposite type, grows approximately linearly with the Péclet number, a prediction confirmed by our numerics over a range of model parameters. Such environment-dependent diffusion is statistically similar to an effective interparticle attraction; consistent with this observation, we find that oppositely driven nonattractive colloids display features characteristic of the simplest model system possessing both interparticle attractions and persistent motion, the driven Ising lattice gas [Katz, Leibowitz, and Spohn, J. Stat. Phys. 34, 497 (1984)JSTPBS0022-471510.1007/BF01018556]. These features include long-ranged correlations in the disordered regime, a critical regime characterized by a change in slope of the particle current with the Péclet number, and fluctuations that grow with system size. By analogy, we suggest that lane formation in the driven colloid system is a phase transition in the macroscopic limit, but that macroscopic phase separation would not occur in finite time upon starting from disordered initial conditions.

The growth of multicomponent structures in simulations and experiments often results in kinetically trapped, nonequilibrium objects. In such cases we have no general theoretical framework for predicting the outcome of the growth process. Here we use computer simulations to study the growth of two-component structures within a simple lattice model. We show that kinetic trapping happens for many choices of growth rate and intercomponent interaction energies, and that qualitatively distinct kinds of kinetic trapping are found in different regions of parameter space. In a region in which the low-energy structure is an "antiferromagnet" or "checkerboard," we show that the grown nonequilibrium structure displays a component-type stoichiometry that is different from the equilibrium one but is insensitive to growth rate and solution conditions. This robust nonequilibrium stoichiometry can be predicted via a mapping to the jammed random tiling of dimers studied by Flory, a finding that suggests a way of making defined nonequilibrium structures in experiment.

We show that model molecules with particular rotational symmetries can self-assemble into network structures equivalent to rhombus tilings. This assembly happens in an emergent way, in the sense that molecules spontaneously select irregular fourfold local coordination from a larger set of possible local binding geometries. The existence of such networks can be rationalized by simple geometrical arguments, but the same arguments do not guarantee a network's spontaneous self-assembly. This class of structures must in certain regimes of parameter space be able to reconfigure into networks equivalent to triangular tilings.

We use analytic theory and computer simulation to study patterns formed during the growth of two-component assemblies in two and three dimensions. We show that these patterns undergo a nonequilibrium phase transition, at a particular growth rate, between mixed and demixed arrangements of component types. This finding suggests that principles of nonequilibrium statistical mechanics can be used to predict the outcome of multicomponent self-assembly, and suggests an experimental route to the self-assembly of multicomponent structures of a qualitatively defined nature.

Controlling the self-assembly of surface-adsorbed molecules into nanostructures requires understanding physical mechanisms that act across multiple length and time scales. By combining scanning tunneling microscopy with hierarchical ab initio and statistical mechanical modeling of 1,4-substituted benzenediamine (BDA) molecules adsorbed on a gold (111) surface, we demonstrate that apparently simple nanostructures are selected by a subtle competition of thermodynamics and dynamics. Of the collection of possible BDA nanostructures mechanically stabilized by hydrogen bonding, the interplay of intermolecular forces, surface modulation, and assembly dynamics select at low temperature a particular subset: low free energy oriented linear chains of monomers and high free energy branched chains.

Theories of phase change and self-assembly often invoke the idea of a "quasiequilibrium," a regime in which the nonequilibrium association of building blocks results nonetheless in a structure whose properties are determined solely by an underlying free energy landscape. Here we study a prototypical example of multicomponent self-assembly, a one-dimensional fiber grown from red and blue blocks. We find that if the equilibrium structure possesses compositional correlations different from those characteristic of random mixing, then it cannot be generated without error at any finite growth rate: there is no quasiequilibrium regime. However, by exploiting dynamic scaling, structures characteristic of equilibrium at one point in phase space can be generated, without error, arbitrarily far from equilibrium. Our results, supported by mean-field theory in higher dimensions, thus suggest a "nonperturbative" strategy for multicomponent self-assembly in which the target structure is, by design, not the equilibrium one.

We study the covering of the plane by nonoverlapping rhombus tiles, a problem well studied only in the limiting case of dimer coverings of regular lattices. We go beyond this limit by allowing tiles to take any position and orientation on the plane, to be of irregular shape, and to possess different types of attractive interactions. Using extensive numerical simulations, we show that at large tile densities there is a phase transition from a fluid of rhombus tiles to a solid packing with broken rotational symmetry. We observe self-assembly of broken-symmetry phases, even at low densities, in the presence of attractive tile-tile interactions. Depending on the tile shape and interactions, the solid phase can be random, possessing critical orientational fluctuations, or crystalline. Our results suggest strategies for controlling tiling order in experiments involving "molecular rhombi.", (© 2012 American Physical Society)

We use computer simulation to study crystal-forming model proteins equipped with interactions that are both orientationally specific and nonspecific. Distinct dynamical pathways of crystal formation can be selected by tuning the strengths of these interactions. When the nonspecific interaction is strong, liquidlike clustering can precede crystallization; when it is weak, growth can proceed via ordered nuclei. Crystal yields are in certain parameter regimes enhanced by the nonspecific interaction, even though it promotes association without local crystalline order. Our results suggest that equipping nanoscale components with weak nonspecific interactions (such as depletion attractions) can alter both their dynamical pathway of assembly and optimize the yield of the resulting material.

Recent experiments [J. van Mameren, Proc. Natl. Acad. Sci. U.S.A. 106, 18231 (2009)] provide a detailed spatial picture of overstretched DNA, showing that under certain conditions the two strands of the double helix separate at about 65 pN. It was proposed that this observation rules out the existence of an elongated, hybridized form of DNA (S-DNA). Here, we argue that the S-DNA picture is consistent with the observation of unpeeling during overstretching. We demonstrate that assuming the existence of S-DNA does not imply DNA overstretching to consist of the complete or near-complete conversion of the molecule from B to S form. Instead, this assumption implies in general a more complex dynamic coexistence of hybridized and unhybridized forms of DNA. We argue that such coexistence can rationalize several recent experimental observations.

Actin networks in certain single-celled organisms exhibit a complex pattern-forming dynamics that starts with the appearance of static spots of actin on the cell cortex. Spots soon become mobile, executing persistent random walks, and eventually give rise to traveling waves of actin. Here we describe a possible physical mechanism for this distinctive set of dynamic transformations, by equipping an excitable reaction-diffusion model with a field describing the spatial orientation of its chief constituent (which we consider to be actin). The interplay of anisotropic actin growth and spatial inhibition drives a transformation at fixed parameter values from static spots to moving spots to waves.

We study an attractive version of the East model using the real-space renormalization group (RG) introduced by Stella et al. The former is a kinetically constrained model with an Ising-like interaction between excitations and shows striking agreement with the phenomonology of attractive colloidal systems. We find that the RG predicts two nonuniversal dynamic exponents, which suggests that in the out-of-equilibrium regime the model coarsens via a two-stage mechanism. We explain this mechanism physically and verify this prediction numerically. In addition, we predict that the characteristic relaxation time of the model is a nonmonotonic function of attraction strength, again in agreement with numerical results.

We derive a dynamic field theory for a kinetically constrained model, based on the Fredrickson-Andersen model, which we expect to describe the properties of an Arrhenius (strong) supercooled liquid at the coarse-grained level. We study this field theory using the renormalization group. For mesoscopic length and time scales, and for space dimension d>/=2 , the behavior of the model is governed by a zero-temperature dynamical critical point in the directed percolation universality class. We argue that in d=1 its behavior is that of compact directed percolation. We perform detailed numerical simulations of the corresponding Fredrickson-Andersen model on the lattice in various dimensions, and find reasonable quantitative agreement with the field theory predictions.

We use a real-space renormalization group (RSRG) to study the low-temperature dynamics of kinetically constrained Ising chains (KCICs). We consider the cases of the Fredrickson-Andersen (FA) model, the East model, and the partially asymmetric KCIC. We show that the RSRG allows one to obtain in a unified manner the dynamical properties of these models near their zero-temperature critical points. These properties include the dynamic exponent, the growth of dynamical length scales, and the behavior of the excitation density near criticality. For the partially asymmetric chain, the RG predicts a crossover, on sufficiently large length and time scales, from East-like to FA-like behavior. Our results agree with the known results for KCICs obtained by other methods.

We propose that the dynamics of supercooled liquids and the formation of glasses can be understood from the existence of a zero-temperature dynamical critical point. To support our proposal, we derive a dynamic field theory for a generic kinetically constrained model, which we expect to describe the dynamics of a supercooled liquid. We study this field theory using the renormalization group (RG). Its long time behavior is dominated by a zero-temperature critical point, which for d>2 belongs to the directed percolation universality class. Molecular dynamics simulations seem to confirm the existence of dynamic scaling behavior consistent with the RG predictions.