Relationship between dynamical entropy and energy dissipation far from thermodynamic equilibrium

Maximum speed of dissipation

We derive statistical-mechanical speed limits on dissipation from the classical, chaotic dynamics of many-particle systems. In one, the rate of irreversible entropy production in the environment is the maximum speed of a deterministic system out of equilibrium, S[over ¯]_{e}/k_{B}≥1/2Δt, and its inverse is the minimum time to execute the process, Δt≥k_{B}/2S[over ¯]_{e}. Starting with deterministic fluctuation theorems, we show there is a corresponding class of speed limits for physical observables measuring dissipation rates. For example, in many-particle systems interacting with a deterministic thermostat, there is a trade-off between the time to evolve between states and the heat flux, Q[over ¯]Δt≥k_{B}T/2. These bounds constrain the relationship between dissipation and time during nonstationary processes, including transient excursions from steady states.

Stimuli-Responsive Langmuir Films Composed of Nanoparticles Decorated with Poly(N-isopropyl acrylamide) (PNIPAM) at the Air/Water Interface

The nanotechnology shift from static toward stimuli-responsive systems is gaining momentum. We study adaptive and responsive Langmuir films at the air/water interface to facilitate the creation of two-dimensional (2D) complex systems. We verify the possibility of controlling the assembly of relatively large entities, i.e., nanoparticles with diameter around 90 nm, by inducing conformational changes within an about 5 nm poly(N-isopropyl acrylamide) (PNIPAM) capping layer. The system performs reversible switching between uniform and nonuniform states. The densely packed and uniform state is observed at a higher temperature, i.e., opposite to most phase transitions, where more ordered phases appear at lower temperatures. The induced nanoparticles' conformational changes result in different properties of the interfacial monolayer, including various types of aggregation. The analysis of surface pressure at different temperatures and upon temperature changes, surface potential measurements, surface rheology experiments, Brewster angle microscopy (BAM), and scanning electron microscopy (SEM) observations are accompanied by calculations to discuss the principles of the nanoparticles' self-assembly. Those findings provide guidelines for designing other adaptive 2D systems, such as programable membranes or optical interfacial devices.

Density matrix formulation of dynamical systems

Physical systems that are dissipating, mixing, and developing turbulence also irreversibly transport statistical density. However, predicting the evolution of density from atomic and molecular scale dynamics is challenging for nonsteady, open, and driven nonequilibrium processes. Here, we establish a theory to address this challenge for classical dynamical systems that is analogous to the density matrix formulation of quantum mechanics. We show that a classical density matrix is similar to the phase-space metric and evolves in time according to generalizations of Liouville's theorem and Liouville's equation for non-Hamiltonian systems. The traditional Liouvillian forms are recovered in the absence of dissipation or driving by imposing trace preservation or by considering Hamiltonian dynamics. Local measures of dynamical instability and chaos are embedded in classical commutators and anticommutators and directly related to Poisson brackets when the dynamics are Hamiltonian. Because the classical density matrix is built from the Lyapunov vectors that underlie classical chaos, it offers an alternative computationally tractable basis for the statistical mechanics of nonequilibrium processes that applies to systems that are driven, transient, dissipative, regular, and chaotic.

Stochastic paths controlling speed and dissipation

Natural processes occur in a finite amount of time and dissipate energy, entropy, and matter. Near equilibrium, thermodynamic intuition suggests that fast irreversible processes will dissipate more energy and entropy than slow quasistatic processes connecting the same initial and final states. For small systems, recently discovered thermodynamic speed limits suggest that faster processes will dissipate more than slower processes. Here, we test the hypothesis that this relationship between speed and dissipation holds for stochastic paths far from equilibrium. To analyze stochastic paths on finite timescales, we derive an exact expression for the path probabilities of continuous-time Markov chains from the path summation solution to the master equation. We present a minimal model for a driven system in which relative energies of the initial and target states control the speed, and the nonequilibrium currents of a cycle control the dissipation. Although the hypothesis holds near equilibrium, we find that faster processes can dissipate less under far-from-equilibrium conditions because of strong currents. This model serves as a minimal prototype for designing kinetics to sculpt the nonequilibrium path space so that faster paths produce less dissipation.

An information-theoretic approach to infer the underlying interaction domain among elements from finite length trajectories in a noisy environment

Transfer entropy in information theory was recently demonstrated [Basak et al., Phys. Rev. E 102, 012404 (2020)] to enable us to elucidate the interaction domain among interacting elements solely from an ensemble of trajectories. Therefore, only pairs of elements whose distances are shorter than some distance variable, termed cutoff distance, are taken into account in the computation of transfer entropies. The prediction performance in capturing the underlying interaction domain is subject to the noise level exerted on the elements and the sufficiency of statistics of the interaction events. In this paper, the dependence of the prediction performance is scrutinized systematically on noise level and the length of trajectories by using a modified Vicsek model. The larger the noise level and the shorter the time length of trajectories, the more the derivative of average transfer entropy fluctuates, which makes the identification of the interaction domain in terms of the position of global minimum of the derivative of average transfer entropy difficult. A measure to quantify the degree of strong convexity at the coarse-grained level is proposed. It is shown that the convexity score scheme can identify the interaction distance fairly well even while the position of the global minimum of the derivative of average transfer entropy does not. We also derive an analytical model to explain the relationship between the interaction domain and the change in transfer entropy that supports our cutoff distance technique to elucidate the underlying interaction domain from trajectories.

Typical Stochastic Paths in the Transient Assembly of Fibrous Materials

When chemically fueled, molecular self-assembly can sustain dynamic aggregates of polymeric fibers-hydrogels-with tunable properties. If the fuel supply is finite, the hydrogel is transient, as competing reactions switch molecular subunits between active and inactive states, drive fiber growth and collapse, and dissipate energy. Because the process is away from equilibrium, the structure and mechanical properties can reflect the history of preparation. As a result, the formation of these active materials is not readily susceptible to a statistical treatment in which the configuration and properties of the molecular building blocks specify the resulting material structure. Here, we illustrate a stochastic-thermodynamic and information-theoretic framework for this purpose and apply it to these self-annihilating materials. Among the possible paths, the framework variationally identifies those that are typical-loosely, the minimum number with the majority of the probability. We derive these paths from computer simulations of experimentally-informed stochastic chemical kinetics and a physical kinetics model for the growth of an active hydrogel. The model reproduces features observed by confocal microscopy, including the fiber length, lifetime, and abundance as well as the observation of fast fiber growth and stochastic fiber collapse. The typical mesoscopic paths we extract are less than 0.23% of those possible, but they accurately reproduce material properties such as mean fiber length.

Critical fluctuations and slowing down of chaos

Fluids cooled to the liquid–vapor critical point develop system-spanning fluctuations in density that transform their visual appearance. Despite a rich phenomenology, however, there is not currently an explanation of the mechanical instability in the molecular motion at this critical point. Here, we couple techniques from nonlinear dynamics and statistical physics to analyze the emergence of this singular state. Numerical simulations and analytical models show how the ordering mechanisms of critical dynamics are measurable through the hierarchy of spatiotemporal Lyapunov vectors. A subset of unstable vectors soften near the critical point, with a marked suppression in their characteristic exponents that reflects a weakened sensitivity to initial conditions. Finite-time fluctuations in these exponents exhibit sharply peaked dynamical timescales and power law signatures of the critical dynamics. Collectively, these results are symptomatic of a critical slowing down of chaos that sits at the root of our statistical understanding of the liquid–vapor critical point.

It is well known that fluids become opaque at the liquid–vapor critical point, but a description of the underlying mechanical instability is still missing. Das and Green leverage nonlinear dynamics to quantify the role of chaos in the emergence of this critical phenomenon.

Self-assembling outside equilibrium: emergence of structures mediated by dissipation

Self-assembly under non-equilibrium conditions may give rise to the formation of structures not available at equilibrium.

From dynamic self-assembly to networked chemical systems

Although dynamic self-assembly, DySA, is a relatively new area of research, the past decade has brought numerous demonstrations of how various types of components - on scales from (macro)molecular to macroscopic - can be arranged into ordered structures thriving in non-equilibrium, steady states. At the same time, none of these dynamic assemblies has so far proven practically relevant, prompting questions about the field's prospects and ultimate objectives. The main thesis of this Review is that formation of dynamic assemblies cannot be an end in itself - instead, we should think more ambitiously of using such assemblies as control elements (reconfigurable catalysts, nanomachines, etc.) of larger, networked systems directing sequences of chemical reactions or assembly tasks. Such networked systems would be inspired by biology but intended to operate in environments and conditions incompatible with living matter (e.g., in organic solvents, elevated temperatures, etc.). To realize this vision, we need to start considering not only the interactions mediating dynamic self-assembly of individual components, but also how components of different types could coexist and communicate within larger, multicomponent ensembles. Along these lines, the review starts with the discussion of the conceptual foundations of self-assembly in equilibrium and non-equilibrium regimes. It discusses key examples of interactions and phenomena that can provide the basis for various DySA modalities (e.g., those driven by light, magnetic fields, flows, etc.). It then focuses on the recent examples where organization of components in steady states is coupled to other processes taking place in the system (catalysis, formation of dynamic supramolecular materials, control of chirality, etc.). With these examples of functional DySA, we then look forward and consider conditions that must be fulfilled to allow components of multiple types to coexist, function, and communicate with one another within the networked DySA systems of the future. As the closing examples show, such systems are already appearing heralding new opportunities - and, to be sure, new challenges - for DySA research.

Self-Averaging Fluctuations in the Chaoticity of Simple Fluids

Bulk properties of equilibrium liquids are a manifestation of intermolecular forces. Here, we show how these forces imprint on dynamical fluctuations in the Lyapunov exponents for simple fluids with and without attractive forces. While the bulk of the spectrum is strongly self-averaging, the first Lyapunov exponent self-averages only weakly and at a rate that depends on the length scale of the intermolecular forces; short-range repulsive forces quantitatively dominate longer-range attractive forces, which act as a weak perturbation that slows the convergence to the thermodynamic limit. Regardless of intermolecular forces, the fluctuations in the Kolmogorov-Sinai entropy rate diverge, as one expects for an extensive quantity, and the spontaneous fluctuations of these dynamical observables obey fluctuation-dissipation-like relationships. Together, these results are a representation of the van der Waals picture of fluids and another lens through which we can view the liquid state.

Lagrangian descriptors of driven chemical reaction manifolds

The persistence of a transition state structure in systems driven by time-dependent environments allows the application of modern reaction rate theories to solution-phase and nonequilibrium chemical reactions. However, identifying this structure is problematic in driven systems and has been limited by theories built on series expansion about a saddle point. Recently, it has been shown that to obtain formally exact rates for reactions in thermal environments, a transition state trajectory must be constructed. Here, using optimized Lagrangian descriptors [G. T. Craven and R. Hernandez, Phys. Rev. Lett. 115, 148301 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.148301], we obtain this so-called distinguished trajectory and the associated moving reaction manifolds on model energy surfaces subject to various driving and dissipative conditions. In particular, we demonstrate that this is exact for harmonic barriers in one dimension and this verification gives impetus to the application of Lagrangian descriptor-based methods in diverse classes of chemical reactions. The development of these objects is paramount in the theory of reaction dynamics as the transition state structure and its underlying network of manifolds directly dictate reactivity and selectivity.

Light-Induced Proton Pumping with a Semiconductor: Vision for Photoproton Lateral Separation and Robust Manipulation

Energy-transfer reactions are the key for living open systems, biological chemical networking, and the development of life-inspired nanoscale machineries. It is a challenge to find simple reliable synthetic chemical networks providing a localization of the time-dependent flux of matter. In this paper, we look to photocatalytic reaction on TiO₂ from different angles, focusing on proton generation and introducing a reliable, minimal-reagent-consuming, stable inorganic light-promoted proton pump. Localized illumination was applied to a TiO₂ surface in solution for reversible spatially controlled "inorganic photoproton" isometric cycling, the lateral separation of water-splitting reactions. The proton flux is pumped during the irradiation of the surface of TiO₂ and dynamically maintained at the irradiated surface area in the absence of any membrane or predetermined material structure. Moreover, we spatially predetermine a transient acidic pH value on the TiO₂ surface in the irradiated area with the feedback-driven generation of a base as deactivator. Importantly we describe how to effectively monitor the spatial localization of the process by the in situ scanning ion-selective electrode technique (SIET) measurements for pH and the scanning vibrating electrode technique (SVET) for local photoelectrochemical studies without additional pH-sensitive dye markers. This work shows the great potential for time- and space-resolved water-splitting reactions for following the investigation of pH-stimulated processes in open systems with their flexible localization on a surface.

Extensivity and additivity of the Kolmogorov-Sinai entropy for simple fluids

According to the van der Waals picture, attractive and repulsive forces play distinct roles in the structure of simple fluids. Here, we examine their roles in dynamics; specifically, in the degree of deterministic chaos using the Kolmogorov-Sinai (KS) entropy rate and the spectra of Lyapunov exponents. With computer simulations of three-dimensional Lennard-Jones and Weeks-Chandler-Andersen fluids, we find repulsive forces dictate these dynamical properties, with attractive forces reducing the KS entropy at a given thermodynamic state. Regardless of interparticle forces, the maximal Lyapunov exponent is intensive for systems ranging from 200 to 2000 particles. Our finite-size scaling analysis also shows that the KS entropy is both extensive (a linear function of system-size) and additive. Both temperature and density control the "dynamical chemical potential," the rate of linear growth of the KS entropy with system size. At fixed system-size, both the KS entropy and the largest exponent exhibit a maximum as a function of density. We attribute the maxima to the competition between two effects: as particles are forced to be in closer proximity, there is an enhancement from the sharp curvature of the repulsive potential and a suppression from the diminishing free volume and particle mobility. The extensivity and additivity of the KS entropy and the intensivity of the largest Lyapunov exponent, however, hold over a range of temperatures and densities across the liquid and liquid-vapor coexistence regimes.

Diverging Fluctuations of the Lyapunov Exponents

We show that in generic one-dimensional Hamiltonian lattices the diffusion coefficient of the maximum Lyapunov exponent diverges in the thermodynamic limit. We trace this back to the long-range correlations associated with the evolution of the hydrodynamic modes. In the case of normal heat transport, the divergence is even stronger, leading to the breakdown of the usual single-function Family-Vicsek scaling ansatz. A similar scenario is expected to arise in the evolution of rough interfaces in the presence of suitably correlated background noise.

Reactive symbol sequences for a model of hydrogen combustion

Transient, macroscopic states of chemical disequilibrium are born out of the microscopic dynamics of molecules. As a reaction mixture evolves, the temporal patterns of chemical species encodes some of this dynamical information, while their statistics are a manifestation of the bulk kinetics. Here, we define a chemically-informed symbolic dynamics as a coarse-grained representation of classical molecular dynamics, and analyze the sequences of chemical species for a model of hydrogen combustion. We use reactive molecular dynamics simulations to generate the sequences and derive probability distributions for sequence observables: the reaction time scales and the chain length - the total number of reactions between initiation of a reactant and termination at products. The time scales and likelihood of the sequences depend strongly on the chain length, temperature, and density. Temperature suppresses the uncertainty in chain length for hydrogen sequences, but enhances the uncertainty in oxygen sequence chain lengths. This method of analyzing a surrogate chemical symbolic dynamics reduces the complexity of the chemistry from the atomistic to the molecular level and has the potential for extension to more complicated reaction systems.

Dynamics of dissipative self-assembly of particles interacting through oscillatory forces

Dissipative self-assembly is the formation of ordered structures far from equilibrium, which continuously uptake energy and dissipate it into the environment. Due to its dynamical nature, dissipative self-assembly can lead to new phenomena and possibilities of self-organization that are unavailable to equilibrium systems. Understanding the dynamics of dissipative self-assembly is required in order to direct the assembly to structures of interest. In the present work, Brownian dynamics simulations and analytical theory were used to study the dynamics of self-assembly of a mixture of particles coated with weak acids and bases under continuous oscillations of the pH. The pH of the system modulates the charge of the particles and, therefore, the interparticle forces oscillate in time. This system produces a variety of self-assembled structures, including colloidal molecules, fibers and different types of crystalline lattices. The most important conclusions of our study are: (i) in the limit of fast oscillations, the whole dynamics (and not only those at the non-equilibrium steady state) of a system of particles interacting through time-oscillating interparticle forces can be described by an effective potential that is the time average of the time-dependent potential over one oscillation period; (ii) the oscillation period is critical to determine the order of the system. In some cases the order is favored by very fast oscillations while in others small oscillation frequencies increase the order. In the latter case, it is shown that slow oscillations remove kinetic traps and, thus, allow the system to evolve towards the most stable non-equilibrium steady state.

Dissipative self-assembly of particles interacting through time-oscillatory potentials

Dissipative self-assembly is the emergence of order within a system due to the continuous input of energy. This form of nonequilibrium self-organization allows the creation of structures that are inaccessible in equilibrium self-assembly. However, design strategies for dissipative self-assembly are limited by a lack of fundamental understanding of the process. This work proposes a novel route for dissipative self-assembly via the oscillation of interparticle potentials. It is demonstrated that in the limit of fast potential oscillations the structure of the system is exactly described by an effective potential that is the time average of the oscillatory potential. This effective potential depends on the shape of the oscillations and can lead to effective interactions that are physically inaccessible in equilibrium. As a proof of concept, Brownian dynamics simulations were performed on a binary mixture of particles coated by weak acids and weak bases under externally controlled oscillations of pH. Dissipative steady-state structures were formed when the period of the pH oscillations was smaller than the diffusional timescale of the particles, whereas disordered oscillating structures were observed for longer oscillation periods. Some of the dissipative structures (dimers, fibers, and honeycombs) cannot be obtained in equilibrium (fixed pH) simulations for the same system of particles. The transition from dissipative self-assembled structures for fast oscillations to disordered oscillating structures for slow oscillations is characterized by a maximum in the energy dissipated per oscillation cycle. The generality of the concept is demonstrated in a second system with oscillating particle sizes.

Dissipative assembly of a membrane transport system

A membrane transport system functions only when activated by a chemical fuel.