Equilibrium Transition Research Articles

Inferring equilibrium transition rates from nonequilibrium protocols

We develop a theory for inferring equilibrium transition rates from trajectories driven by a time-dependent force using results from stochastic thermodynamics. Applying the Kawasaki relation to approximate the nonequilibrium distribution function in terms of the equilibrium distribution function and the excess dissipation, we formulate a nonequilibrium transition state theory to estimate the rate enhancement over the equilibrium rate due to the nonequilibrium protocol. We demonstrate the utility of our theory in examples of pulling of harmonically trapped particles in one and two dimensions, as well as a semiflexible polymer with a reactive linker in three dimensions. We expect our purely thermodynamic approach will find use in both molecular simulation and force spectroscopy experiments.

Generation of stochastic mixed-mode oscillations in a pair of VDP oscillators with direct-indirect coupling.

Environmental noise can lead to complex stochastic dynamical behavior in nonlinear systems. In this paper, we studied the phenomenon of a pair of Van der Pol (VDP) oscillators with direct-indirect coupling affected by Gaussian white noise. That is to say, a noise-induced equilibrium transition oscillation was observed in three types of different parameter regions, where the deterministic system had two kinds of stable equilibrium points. Meanwhile, with the noise intensity increasing, we found that the stochastic system will constantly switch between two stable equilibrium points. To analyze the stochastic behavior, we used the stochastic sensitivity equation and confidence ellipse method. When the confidence ellipsoid crossed the boundary of the attraction basin of the equilibrium point, the system entered into the state of stochastic mixed-mode oscillations, which was consistent with the simulation results.

Mean-field models for the chemical fueling of transient soft matter states.

The chemical fueling of transient states (CFTS) is a powerful process to control the nonequilibrium structuring and the homeostatic function of adaptive soft matter systems. Here, we introduce a simple mean-field model of CFTS based on the activation of metastable equilibrium states in a tilted 'Landau' bistable energy landscape along a coarse-grained reaction coordinate (or 'order parameter') triggered by a nonmonotonic two-step chemical fueling reaction. Evaluation of the model in the quasi-static (QS) limit-valid for fast system relaxation-allows us to extract useful analytical laws for the critical activation concentration and duration of the transient states in dependence of physical parameters, such as rate constants, fuel concentrations, and the system's distance to its equilibrium transition point. We apply our model in the QS limit explicitly to recent experiments of CFTS of collapsing responsive microgels and find a very good performance with only a few global and physically interpretable fitting parameters, which can be employed for programmable material design. Moreover, our model framework also allows a thermodynamic analysis of the energy and performed work in the system. Finally, we go beyond the QS limit, where the system's response is slow and retarded versus the chemical reaction, using an overdamped Smoluchowski approach. The latter demonstrates how internal system time scales can be used to tune the time-dependent behavior and programmed delay of the transient states in full nonequilibrium.

Ultrafast Martensitic Phase Transition Driven by Intense Terahertz Pulses

We report on an ultrafast nonequilibrium phase transition with a strikingly long-lived martensitic anomaly driven by above-threshold single-cycle terahertz pulses with a peak field of more than 1 MV/cm. A nonthermal, terahertz-induced depletion of low-frequency conductivity in Nb 3 Sn indicates increased gap splitting of high-energy Γ 12 bands by removal of their degeneracies, which induces the martensitic phase above their equilibrium transition temperature. In contrast, optical pumping leads to a Γ 12 gap thermal melting. Such light-induced nonequilibrium martensitic phase exhibits a substantially enhanced critical temperature up to ∼100 K, i.e., more than twice the equilibrium temperature, and can be stabilized beyond technologically relevant, nanosecond time scales. Together with first-principle simulations, we identify a compelling terahertz tuning mechanism of structural order via Γ 12 phonons to achieve the ultrafast phase transition to a metastable electronic state out of equilibrium at high temperatures far exceeding those for equilibrium states.

Relativistic ab initio study on the spectroscopic and radiative properties of the lowest states and modeling of the optical cycles for the LiFr molecule

The LiFr diatomic represents a promising candidate for indirect laser cooling that has not yet been investigated not theoretically or experimentally. The potential energy curves of the ground and low-lying excited states of the LiFr heteronuclear alkali metal dimer are calculated using the Fock-space relativistic coupled cluster theory for the first time. A number of properties such as the electronic term energies, equilibrium internuclear distances, transition and permanent dipole moments, sequences of vibrational energies, harmonic vibrational frequencies, Franck–Condon factors, and radiative lifetimes (including bound and free transitions) are predicted. The probabilities of the two-step schemes (optical cycles) for the transfer process of the LiFr molecules from high excited vibrational states to the ground vibronic state are also predicted. The data obtained would be useful for laser cooling and spectral experiments with LiFr molecules.

Kibble-Zurek Mechanism for Dynamical Ordering in a Driven Vortex System.

The Kibble-Zurek mechanism describes the formation of topological defects in systems crossing a continuous symmetry-breaking phase transition at a finite quench rate. While this mechanism has been extensively studied for equilibrium transitions, its applicability to nonequilibrium transitions has not yet been fully examined. Recent simulation has shown the applicability of the Kibble-Zurek mechanism to dynamical ordering transitions in particlelike assemblies, including superconducting vortices, driven over random disorder. Here, we experimentally study the configurational order of vortices in the course of dynamical ordering with various quench rates. We verify a power-law scaling of the defect density with the quench rate and an impulse-adiabatic crossover on the ordered side of the transition, which are key predictions of the Kibble-Zurek mechanism. Our results suggest the applicability of the Kibble-Zurek mechanism to other nonequilibrium phase transitions.

Learning and equilibrium transitions: Stochastic stability in discounted stochastic fictitious play

In this paper I study how adaptive learning leads to switches between multiple stable equilibria, and I develop tools to characterize the rate of transition. While the methods I develop are relatively general, I focus on a two-player model of stochastic fictitious play, where agents’ payoffs are subject to random shocks. Each player forecasts his opponent’s future play as a discounted average of past play. I analyze the behavior of agents’ beliefs as the discount rate on past information becomes small, but the payoff shock variance remains fixed. I show that agents tend to be drawn toward an equilibrium, but occasionally the stochastic shocks lead to endogenous shifts between equilibria. I then calculate the limiting transition rates and the invariant distribution of players’ beliefs, and use it to determine the most likely outcome observed in long run. I show this stochastically stable equilibrium satisfies an important continuity condition which allows for relatively direct characterization.

Theoretical Investigation of Wenzel and Cassie Wetting States on Porous Films and Fiber Meshes.

In this study, the wetting phenomenon on a thin, porous film covering a substrate was investigated on the basis of a classical equilibrium theory. The equilibrium contact angles of the Wenzel, hydrophobic Cassie, and hydrophilic Cassie states and the transition points between them were derived as functions of parameters, such as the porosity and specific surface area of the porous film. These expressions were applied to describe wetting on fiber mats/meshes. The equilibrium contact angles and transition points of the three wetting states on the fiber mesh covering the substrate are expressed as functions of parameters, such as fiber radius, roughness factor, and volume porosity of the fiber mesh. A fiber mesh can attain superhydrophobicity when it is composed of thin hydrophobic fibers with surface roughness and has an appropriate volume porosity.

Free energy landscape within the hysteresis regime for fluids confined in disordered mesoporous solids.

Adsorption/desorption and melting/freezing in structurally disordered nanoporous solids exhibit strongly non-equilibrium behavior as revealed by the formation of a hysteresis region populated by the multitude of different states. Many questions concerning the free energy spectrum of these states, including the existence of the equilibrium transition, if any, their accessibility in the experiments, and internal relaxation dynamics toward the global energy minimum, still remain poorly addressed. By using a serially connected pore model with the statistical disorder as a minimal model of the pore networks, we explore the system free energies along the solid-liquid and liquid-gas transitions in the pore systems. The rigorous results obtained with this model shed light on the occurrence and nature of the equilibrium transition line in porous solids with arbitrary pore topology. We discuss further the free energies along the experimentally measured boundary and scanning transitions and how close the equilibrium states can be approached in these experiments.

A surprisingly simple three-state generic process for reversible protein denaturation by trifluoroethanol

Despite the rich knowledge of the influence of 2,2,2-trifluoroethanol (TFE) on the structure and conformation of peptides and proteins, the mode(s) of TFE-protein interactions and the mechanism by which TFE reversibly denatures a globular protein remain elusive. This study systematically examines TFE-induced equilibrium transition curves for six paradigmatic globular proteins by using basic fluorescence and circular dichroism measurements under neutral pH conditions. The results are remarkably simple. Low TFE invariably unfolds the tertiary structure of all proteins to produce the obligate intermediate (I) which retains nearly all of native-state secondary structure, but enables the formation of extra α-helices as the level of TFE is raised higher. Inspection of the transitions at once reveals that the tertiary structure unfolding is always a distinct process, necessitating the inclusion of at least one obligate intermediate in the TFE-induced protein denaturation. It appears that the intermediate in the minimal unfolding mechanism N⇌I⇌D somehow acquires higher α-helical propensity to generate α-helices in excess of that in the native state to produce the denatured state (D), also called the TFE state. The low TFE-populated intermediate I may be called a universal intermediate by virtue of its α-helical propensity. Contrary to many earlier suggestions, this study dismisses molten globule (MG)-like attribute of I or D.

Studying and exploring potential energy surfaces of compressed molecules: A fresh theory from the extreme pressure polarizable continuum model.

We present a new theory for studying and exploring the potential energy surface of compressed molecular systems as described within the extreme pressure polarizable continuum model framework. The effective potential energy surface is defined as the sum of the electronic energy of the compressed system and the pressure-volume work that is necessary in order to create the compression cavity at the given condition of pressure. We show that the resulting total energy Gt is related to the electronic energy by a Legendre transform in which the pressure and volume of the compression cavity are the conjugate variables. We present an analytical expression for the evaluation of the gradient of the total energy ∇Gt to be used for the geometry optimization of equilibrium geometries and transition states of compressed molecular systems. We also show that, as a result of the Legendre transform property, the potential energy surface can be studied explicitly as a function of the pressure, leading to an explicit connection with the well-known Hammond postulate. As a proof of concept, we present the application of the theory to studying and determining the optimized geometry of compressed methane and the transition states of the electrocyclic ring-closure of hexatriene and of H-transfer between two methyl radicals.

Relations among Unidirectional Fluxes at Equilibrium, Committors, and First Passage and Transition Path Times.

For multidimensional diffusive dynamics, we algebraically derive remarkable analytical expressions that relate the mean first passage and transition path times between two dividing surfaces with the number of unidirectional transitions per unit time (fluxes) at equilibrium between the two surfaces and the committor (the probability of reaching one surface before the other). In one dimension, such relationships can be easily derived because analytical expressions for all the above-mentioned quantities can be found. This is not possible in higher dimensions, and at first sight, the problem seems much harder. We circumvent the difficulty that the equations determining the mean first passage and transition path times cannot be solved analytically by multiplying these equations by the committor, integrating both sides and finally using the divergence theorem. A byproduct of our derivation is an analytical expression for the starting point distribution over which individual first passage and transition path times must be averaged. It turns out that this distribution is not the Boltzmann one, but it has a simple physical interpretation.

Unique extension of the maximum entropy principle to phase coexistence in heat conduction

The maximum entropy principle determines the values of thermodynamic variables in thermally isolated equilibrium systems. This paper extends the principle to a variational principle that applies to liquid--gas coexistence in heat conduction. We show the uniqueness of the extension under the assumption that the variational principle and the fundamental thermodynamic relation are simultaneously extended in the linear response regime with the total energy fixed. Using the extended variational principle, we calculate the thermodynamic quantities in this steady state and find that the temperature of the liquid--gas interface deviates from the equilibrium transition temperature, which should be verified in experiments.

Reconstructing the dynamical quantum phase transitions via dimensional expansion in a generalized Su-Schrieffer-Heeger model.

We know that a one-dimensional (1D) modulated system can simulate 2D topological states by expanding the dimension. This scenario provides a justifiable avenue to test the dilatation of the dynamical quantum phase transition (DQPT). Through a generalized Su-Schrieffer-Heeger model, we have shown how the Loschmidt echo, Fisher zero, and Dynamical topological order parameter (DTOP) transit from one to two dimensions. Owing to the introduced pseudomomentum, the derivative of the return rate does not always capture the DQPT well, but the Fisher zero and the DTOP can be treated as faithful indicators. A topology-independent parameter will also affect the occurrence of the DQPTs for quenches inside a given phase. Moreover, a comparison with the Haldane model owning the same phase diagram implies that a pair of fixed points will lead to different critical momentum distributions, thus different robustness, further reminding us that the correspondences between the equilibrium and dynamical phases transitions are multifarious.

Simple Approximation for the Ideal Reference State of Gases Adsorbed on Solid-State Surfaces

Reference states are useful as models for facilitating calculations of equilibrium constants, and they may also serve as standard states that are convenient for organizing and tabulating thermodynamic data; however, standard state conventions and appropriate reference states for adsorbed species have received less attention than those for pure substances and solutes. Here, we compare seven choices of reference states for calculations of equilibrium constants and transition state theory rate constants for flat surfaces, in particular (1) an ideal 2D harmonic oscillator, (2) an ideal rigid-molecule harmonic oscillator, (3) an ideal 2D harmonic oscillator with separable surface modes, (4) a 2D ideal gas, (5) an ideal 2D hindered translator, (6) an ideal 2D hindered translator with lowest-order barriers, and (7) a simple ideal 2D hindered translator proposed in this work. The advantage of models 5-7 is that they can treat both mobile and localized adsorbates in a consistent way, whereas models 1-3 are only appropriate for localized adsorbates, and model 4 is only appropriate for a freely translating adsorbate. Furthermore, models 6 and 7 reduce the computational cost without the user having to calculate barrier heights for diffusion. An advantage of the simple ideal 2D hindered translator is that it has a physical high-temperature limit. We also propose a reference state for nonflat surfaces. The user is encouraged to choose a reference state based on the appropriateness of the model and the practicality of the calculations.

Highly Ordered Supramolecular Assembled Networks Tailored by Bioinspired H-Bonding Confinement for Recyclable Ion-Transport Materials

Highly Ordered Supramolecular Assembled Networks Tailored by Bioinspired H-Bonding Confinement for Recyclable Ion-Transport Materials

Positional information as a universal predictor of freezing

Variation of positional information, measured by the two-body excess entropy , is studied across the liquid-solid equilibrium transition in a simple two-dimensional system. Analysis reveals a master relation between and the freezing temperature T 1, from which a scaling law is extracted, . Theoretical and practical implications of the observed universality are discussed.

Minimal dissipation protocols of an instantaneous equilibrium Brownian particle under time-dependent temperature and potential variations

We consider the instantaneous equilibrium (ieq) transition of an underdamped Brownian particle under arbitrary time-dependent temperature and potential variations and derive analytic results for the temperature and potential protocols for minimal dissipative work. Explicit results for the time-dependent minimal dissipation protocols and the associated energetics are obtained for the cases of pure temperature variation, pure potential parameter (stiffness) variation, and isentropic ieq processes. The minimal dissipation condition enforces the forward and backward protocols to be time-reversal related, with the same minimal dissipative work. Remarkably, it is shown that the minimal dissipation path is also the isentropic ieq transition. The energetics in the mdieq transitions are analyzed in detail with emphasis on the conditions for maximizing the mean work and power that can be extracted. Furthermore, exact results for the overdamped limit are derived, indicating that there is great freedom to choose the minimal dissipation protocols, thus allowing the realization of mdieq transitions in Brownian colloidal systems with relative ease.

Cooperativity and Folding Kinetics in a Multidomain Protein with Interwoven Chain Topology.

Although a large percentage of eukaryotic proteomes consist of proteins with multiple domains, not much is known about their assembly mechanism, especially those with intricate native state architectures. Some have a complex topology in which the structural elements along the sequence are interwoven in such a manner that the domains cannot be separated by cutting at any location along the sequence. Such proteins are multiply connected multidomain proteins (MMPs) with the three-domain (NMP, LID, and CORE) phosphotransferase enzyme adenylate kinase (ADK) being an example. We devised a coarse-grained model to simulate ADK folding initiated by changing either the temperature or guanidinium chloride (GdmCl) concentration. The simulations reproduce the experimentally measured melting temperatures (associated with two equilibrium transitions), FRET efficiency as a function of GdmCl concentration, and the folding times quantitatively. Although the NMP domain orders independently, cooperative interactions between the LID and the CORE domains are required for complete assembly of the enzyme. Kinetic simulations show that, on the collapse time scale, multiple interconnected metastable states are populated, attesting to the folding heterogeneity. The network of kinetically connected states reveals that the CORE domain folds only after the NMP and LID domains, reflecting the interwoven nature of the chain topology.

Generalized Fresnel-Floquet equations for driven quantum materials

Optical drives at terahertz and midinfrared frequencies in quantum materials are increasingly used to reveal the nonlinear dynamics of collective modes in correlated many-body systems and their interplay with electromagnetic waves. Recent experiments demonstrated several surprising optical properties of transient states induced by driving, including the appearance of photo-induced edges in the reflectivity in cuprate superconductors (SCs), observed both below and above the equilibrium transition temperature. Furthermore, in other driven materials, reflection coefficients larger than unity have been observed. In this paper we demonstrate that unusual optical properties of photoexcited systems can be understood from the perspective of a Floquet system, a system with periodically modulated parameters originating from pump-induced oscillations of a collective mode. These oscillations lead to an effective Floquet system with periodically modulated parameters. We present a general phenomenological model of reflectivity from Floquet materials, which takes into account parametric generation of excitation pairs. We find a universal phase diagram of drive-induced features in reflectivity which evidence a competition between driving and dissipation. To illustrate our general analysis, we apply our formalism to two concrete examples motivated by recent experiments: A single plasmon band, which describes Josephson plasmons (JPs) in layered SCs, and a phonon-polariton system, which describes upper and lower polaritons in materials such as insulating SiC. Finally, we demonstrate that our model can be used to provide an accurate fit to results of phonon-pump--terahertz-probe experiments in the high-temperature SC ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6.5}$. Our model explains the appearance of a pump-induced edge, which is higher in energy than the equilibrium JP edge, even if the interlayer Josephson coupling is suppressed by the pump pulse.