Nonequilibrium Steady-state Condition Research Articles

Steady-state probabilities for Markov jump processes in terms of powers of the transition rate matrix.

Several types of dynamics at stationarity can be described in terms of a Markov jump process among a finite number N of representative sites. Before dealing with the dynamical aspects, one basic problem consists in expressing the a priori steady-state occupation probabilities of the sites. In particular, one wishes to go beyond the mere black-box computational tools and find expressions in which the jump rate constants appear explicitly, therefore allowing for a potential design/control of the network. For strongly connected networks admitting a unique stationary state with all sites populated, here we express the occupation probabilities in terms of a formula that involves powers of the transition rate matrix up to order N - 1. We also provide an expression of the derivatives with respect to the jump rate constants, possibly useful in sensitivity analysis frameworks. Although we refer to dynamics in (bio)chemical networks at thermal equilibrium or under nonequilibrium steady-state conditions, the results are valid for any Markov jump process under the same assumptions.

Hot crystals of thermo-responsive particles with temperature dependent diameter in the presence of a temperature gradient.

Structure formation under non-equilibrium steady state conditions is poorly understood. A non-equilibrium steady state can be achieved in a system by maintaining a temperature gradient. A class of cross-linked microgel particles, such as poly-N-iso-propylacrylamide, is reported to increase in size due to the adsorption of water as the temperature decreases. Here, we study thermo-responsive particles with a temperature sensitive diameter in the presence of a temperature gradient, using molecular dynamics simulations with the Langevin thermostat. We find long-ranged structural order using bond order parameters in both cold and hot regions of the system beyond a certain diameter ratio of the cold and hot particles. This is due to an increase in packing and pressure in both regions. Our observations might be useful in understanding ordered structures under extreme conditions of a non-equilibrium steady state.

Energetics of quantum vacuum friction. II. Dipole fluctuations and field fluctuations

As a second paper in series with arXiv:2108.01539, we discuss here quantum vacuum friction on an intrinsically dissipative particle. The friction arises not only from the field fluctuations but also from the dipole fluctuations intrinsic to the particle. As a result, the dissipative particle can be out of the nonequilibrium steady state (NESS), where it loses or gains internal energy. Only if the temperature of the particle equals a special NESS temperature will the particle be in NESS. We first derive the NESS conditions which give the NESS temperature of the particle as a function of the radiation temperature and the velocity of the particle. Imposing the NESS conditions, we then obtain an expression for the quantum vacuum friction in NESS. The NESS quantum vacuum friction is shown to be always negative definite, therefore a true drag. The NESS temperature and quantum vacuum friction are calculated numerically for various models. Out of NESS, even though the quantum vacuum frictional force no longer has a definite sign in the rest frame of the radiation, we find the external force needed to keep the particle moving must be in the same direction as the motion of the particle. This then excludes the possibility of making a perpetual motion machine, which could convert the vacuum energy into useful mechanical work. In addition, we find that the deviation of the temperature of the particle from its NESS temperature causes the particle to lose or gain internal energy in such a way that the particle would return to NESS after deviating from it. This enables experimental measurements of the NESS temperature of the particle to serve as a feasible signature for these quantum vacuum frictional effects.

A long-range order in a thermally driven system with temperature-dependent interactions.

Aggregation of macro-molecules under an external force is far from being understood. An important driving situation is achieved by temperature difference. Inter-particle interactions in metallic nanoparticles with ligand capping are reported to be sensitive to temperature and the zeta potential of the particles being reduced in the cold region. Such particles form aggregates in the cold region of the system in the presence of temperature difference. Here we study the aggregation of particles in the presence of temperature difference with temperature-dependent interaction parameters using Brownian dynamics simulation. The particle interaction and particle diffusion are considered to be sensitive to the local temperature. We identify a long-range structural order in the cold region of the system using the Avrami equation for crystal growth kinetics. Our observations might be useful in designing ordered structures with macro-molecules under non-equilibrium steady-state conditions.

Dissipation-driven selection of states in non-equilibrium chemical networks

Life has most likely originated as a consequence of processes taking place in non-equilibrium conditions (e.g. in the proximity of deep-sea thermal vents) selecting states of matter that would have been otherwise unfavorable at equilibrium. Here we present a simple chemical network in which the selection of states is driven by the thermodynamic necessity of dissipating heat as rapidly as possible in the presence of a thermal gradient: states participating to faster reactions contribute the most to the dissipation rate, and are the most populated ones in non-equilibrium steady-state conditions. Building upon these results, we show that, as the complexity of the chemical network increases, the velocity of the reaction path leading to a given state determines its selection, giving rise to non-trivial localization phenomena in state space. A byproduct of our studies is that, in the presence of a temperature gradient, thermophoresis-like behavior inevitably appears depending on the transport properties of each individual state, thus hinting at a possible microscopic explanation of this intriguing yet still not fully understood phenomenon.

Nonequilibrium thermodynamics of quantum friction

Thermodynamic principles are often deceptively simple and yet surprisingly powerful. We show how a simple rule, such as the net flow of energy in and out of a moving atom under nonequilibrium steady state condition, can expose the shortcomings of many popular theories of quantum friction. Our thermodynamic approach provides a conceptual framework in guiding atom-optical experiments, thereby highlighting the importance of fluctuation-dissipation relations and long-time correlations between subsystems. Our results introduce consistency conditions for (numerical) models of nonequilibrium dynamics of open quantum systems.

Non-Equilibrium Thermodynamic Analysis of Flash Evaporation Process in Vacuum Ice Making

Abstract Traditional equilibrium thermodynamics can only solve a few equilibrium processes composed of continuous stable equilibrium states. However, the vacuum flash evaporation process is a typical unsteady process. The study of non-equilibrium thermodynamics of the two-phase flow model is helpful to improve our understanding of the basic law of the flash evaporation process. Based on the theory of non-equilibrium thermodynamics, the flash chamber in the vacuum flash ice making system was studied in this paper, and the possibility of non-equilibrium steady state evaporation with superheat was obtained. The chemical potential difference between liquid water and water vapor under non-equilibrium steady state conditions was determined, and the corresponding evaporation entropy was calculated. It is shown that the results obtained by equilibrium thermodynamics are only related to the temperature difference, while the results obtained by non-equilibrium thermodynamics are not only related to the temperature difference, but also the state of the evaporation process. This is because non-equilibrium thermodynamics considers the cooling of liquid water and the evaporation of water vapor as a whole, taking into account the interaction between the two processes. However, the traditional equilibrium thermodynamics theory divides the steady state evaporation process into two independent processes and ignores the influence of each other.

Relationship between Macrostep Height and Surface Velocity for a Reaction-Limited Crystal Growth Process

This work examined the effect of macrostep height on the growth velocity of a vicinal surface during reaction- (interface-) limited crystal growth under non-equilibrium steady state conditions. The Monte Carlo method was employed, based on a restricted solid-on-solid (RSOS) model with point-contact-type step-step attraction (termed the p-RSOS model). Although this is a simple lattice model, the model surface shows a variety of distinctive configurations depending on the temperature and the driving force for crystal growth. The results demonstrate that the surface velocity decreases as the height of the faceted macrostep increases. In addition, the significant variation in surface velocity recently reported by Onuma {\it et al.} in a study based on 4H-SiC was reproduced. This work also shows that the terrace slope, elementary step velocity and elementary step kinetic coefficient are all affected by the faceted macrostep height.

Extreme reductions of entropy in an electronic double dot

We experimentally study negative fluctuations of stochastic entropy production in an electronic double dot operating in nonequilibrium steady-state conditions. We record millions of random electron tunneling events at different bias points, thus collecting extensive statistics. We show that for all bias voltages the experimental average values of the minima of stochastic entropy production lie above $-k_B$, where $k_B$ is the Boltzmann constant, in agreement with recent theoretical predictions for nonequilibrium steady states. Furthermore, we also demonstrate that the experimental cumulative distribution of the entropy production minima is bounded, at all times and for all bias voltages, by a universal expression predicted by the theory. We also extend our theory by deriving a general bound for the average value of the maximum heat absorbed by a mesoscopic system from the environment and compare this result with experimental data. Finally, we show by numerical simulations that these results are not necessarily valid under non-stationary conditions.

Direct approach for the fluctuation-dissipation theorem under nonequilibrium steady-state conditions

The test mass suspensions of cryogenic gravitational-wave detectors such as the KAGRA project are tasked with extracting the heat deposited on the optics. Thus these suspensions have a non-uniform temperature, requiring the calculation of thermal noise in non-equilibrium conditions. While it is not possible to describe the whole suspension system with one temperature, the local temperature anywhere in the system is still well defined. We therefore generalize the application of the fluctuation-dissipation theorem to mechanical systems, pioneered by Saulson and Levin, to non-equilibrium conditions in which a temperature can only be defined locally. The result is intuitive in the sense that the temperature-averaging relevant for the thermal noise in the observed degree of freedom is given by averaging the temperature field, weighted by the dissipation density associated with that particular degree of freedom. After proving this theorem we apply the result to examples of increasing complexity: a simple spring, the bending of a pendulum suspension fiber, as well as a model of the KAGRA cryogenic suspension. We conclude by outlining the application to non-equilibrium thermo-elastic noise.

Acceleration and sensitivity analysis of lattice kinetic Monte Carlo simulations using parallel processing and rate constant rescaling.

Kinetic Monte Carlo (KMC) simulation provides insights into catalytic reactions unobtainable with either experiments or mean-field microkinetic models. Sensitivity analysis of KMC models assesses the robustness of the predictions to parametric perturbations and identifies rate determining steps in a chemical reaction network. Stiffness in the chemical reaction network, a ubiquitous feature, demands lengthy run times for KMC models and renders efficient sensitivity analysis based on the likelihood ratio method unusable. We address the challenge of efficiently conducting KMC simulations and performing accurate sensitivity analysis in systems with unknown time scales by employing two acceleration techniques: rate constant rescaling and parallel processing. We develop statistical criteria that ensure sufficient sampling of non-equilibrium steady state conditions. Our approach provides the twofold benefit of accelerating the simulation itself and enabling likelihood ratio sensitivity analysis, which provides further speedup relative to finite difference sensitivity analysis. As a result, the likelihood ratio method can be applied to real chemistry. We apply our methodology to the water-gas shift reaction on Pt(111).

Quantum entanglement at high temperatures? Bosonic systems in nonequilibrium steady state

This is the second of a series of three papers examining how viable it is for entanglement to be sustained at high temperatures for quantum systems in thermal equilibrium (Case A), in nonequilibrium (Case B) and in nonequilibrium steady state conditions (Case C). The system we analyze here consists of two coupled quantum harmonic oscillators each interacting with its own bath described by a scalar field, set at temperatures $T_1 > T_2$. For \textit{constant bilinear inter-oscillator coupling} studied here (Case C1) owing to the Gaussian nature, the problem can be solved exactly at arbitrary temperatures even for strong coupling. We find that the valid entanglement criterion in general is not a function of the bath temperature difference, in contrast to thermal transport in the same NESS setting [1]. Thus lowering the temperature of one of the thermal baths does not necessarily help to safeguard the entanglement between the oscillators. Indeed, quantum entanglement will disappear if any one of the thermal baths has a temperature higher than the critical temperature $T_c$. With the Langevin equations derived we give a full display of how entanglement dynamics in this system depends on $T_{1}$, $T_{2}$ , the inter-oscillator coupling and the system-bath coupling strengths. For weak oscillator-bath coupling the critical temperature $T_c$ is about the order of the inverse oscillator frequency, but for strong oscillator-bath coupling it will depend on the bath cutoff frequency. We conclude that in most realistic circumstances, for bosonic systems in NESS with constant bilinear coupling, `hot entanglement' is largely a fiction. In Paper III we will examine the case (C2) of \textit{time-dependent driven coupling } which contains the parametric pumping type described in [2] wherein entanglement was first shown to sustain at high temperatures.

Quantum heat transport of a two-qubit system: Interplay between system-bath coherence and qubit-qubit coherence.

We consider a system consisting of two interacting qubits that are individually coupled to separate heat baths at different temperatures. The quantum effects in heat transport are investigated in a numerically rigorous manner with a hierarchial equations of motion (HEOM) approach for non-perturbative and non-Markovian system-bath coupling cases under non-equilibrium steady-state conditions. For a weak interqubit interaction, the total system is regarded as two individually thermostatted systems, whereas for a strong interqubit interaction, the two-qubit system is regarded as a single system coupled to two baths. The roles of quantum coherence (or entanglement) between the two qubits (q-q coherence) and between the qubit and bath (q-b coherence) are studied through the heat current calculated for various strengths of the system-bath coupling and interqubit coupling for high and low temperatures. The same current is also studied using the time convolutionless (TCL) Redfield equation and using an expression derived from the Fermi golden rule (FGR). We find that the HEOM results exhibit turnover behavior of the heat current as a function of the system-bath coupling strength for all values of the interqubit coupling strength, while the results obtained with the TCL and FGR approaches do not exhibit such behavior, because they do not possess the capability of treating the q-b and q-q coherences. The maximum current is obtained in the case that the q-q coherence and q-b coherence are balanced in such a manner that coherence of the entire heat transport process is realized. We also find that the heat current does not follow Fourier's law when the temperature difference is very large, due to the non-perturbative system-bath interactions.

Modeling the effects of stretch-dependent surfactant secretion on lung recruitment during variable ventilation

Variable ventilation (VV) is a novel strategy of ventilatory support that utilizes random variations in the delivered tidal volume (VT) to improve lung function. Since the stretch pattern during VV has been shown to increase surfactant release both in animals and cell culture, we hypothesized that there were combinations of PEEP and VT during VV that led to improved alveolar recruitment compared to conventional mechanical ventilation (CV). To test this hypothesis, we developed a computational model of stretch-induced surfactant release combined with abnormal alveolar mechanics of the injured lung under mechanical ventilation. We modeled the lung as a set of distinct acini with independent surfactant secretion and thus pressure-volume relationships. The rate of surfactant secretion was modulated by the stretch magnitude that an alveolus experienced per breath. Mechanical ventilation was simulated by delivering a prescribed VT at each breath. The fractional VT that each acinus received depended on its local compliance relative to the total system compliance. Regional variability in VT thus developed through feedback between stretch and surfactant release and coupling of regional VT to ventilator settings. The model allowed us to simulate patient-ventilator interactions over a wide range of PEEPs and VTs during CV and VV. Full recruitment was achieved through VV at a lower PEEP than required for CV. During VV, the acini were maintained under non-equilibrium steady-state conditions with breath-by-breath fluctuations of regional VT. In CV, alveolar injury was prevented with high-PEEP-low-VT or low-PEEP-high-VT combinations. In contrast, one contiguous region of PEEP-VT combinations allowed for full recruitment without overdistention during VV. We found that maintaining epithelial cell stretch above a critical threshold with either PEEP or VT may help stabilize the injured lung. These results demonstrate the significance of patient-ventilator coupling through the influence of cellular stretch-induced surfactant release on the whole lung stability.

An effective rate equation approach to reaction kinetics in small volumes: Theory and application to biochemical reactions in nonequilibrium steady-state conditions

Chemical master equations provide a mathematical description of stochastic reaction kinetics in well-mixed conditions. They are a valid description over length scales that are larger than the reactive mean free path and thus describe kinetics in compartments of mesoscopic and macroscopic dimensions. The trajectories of the stochastic chemical processes described by the master equation can be ensemble-averaged to obtain the average number density of chemical species, i.e., the true concentration, at any spatial scale of interest. For macroscopic volumes, the true concentration is very well approximated by the solution of the corresponding deterministic and macroscopic rate equations, i.e., the macroscopic concentration. However, this equivalence breaks down for mesoscopic volumes. These deviations are particularly significant for open systems and cannot be calculated via the Fokker-Planck or linear-noise approximations of the master equation. We utilize the system-size expansion including terms of the order of Omega(-1/2) to derive a set of differential equations whose solution approximates the true concentration as given by the master equation. These equations are valid in any open or closed chemical reaction network and at both the mesoscopic and macroscopic scales. In the limit of large volumes, the effective mesoscopic rate equations become precisely equal to the conventional macroscopic rate equations. We compare the three formalisms of effective mesoscopic rate equations, conventional rate equations, and chemical master equations by applying them to several biochemical reaction systems (homodimeric and heterodimeric protein-protein interactions, series of sequential enzyme reactions, and positive feedback loops) in nonequilibrium steady-state conditions. In all cases, we find that the effective mesoscopic rate equations can predict very well the true concentration of a chemical species. This provides a useful method by which one can quickly determine the regions of parameter space in which there are maximum differences between the solutions of the master equation and the corresponding rate equations. We show that these differences depend sensitively on the Fano factors and on the inherent structure and topology of the chemical network. The theory of effective mesoscopic rate equations generalizes the conventional rate equations of physical chemistry to describe kinetics in systems of mesoscopic size such as biological cells.

Single-molecule nonequilibrium periodic Mg 2+ -concentration jump experiments reveal details of the early folding pathways of a large RNA

The evolution of RNA conformation with Mg(2+) concentration ([Mg(2+)]) is typically determined from equilibrium titration measurements or nonequilibrium single [Mg(2+)]-jump measurements. We study the folding of single RNA molecules in response to a series of periodic [Mg(2+)] jumps. The 260-residue catalytic domain of RNase P RNA from Bacillus stearothermophilus is immobilized in a microfluidic flow chamber, and the RNA conformational changes are probed by fluorescence resonance energy transfer (FRET). The kinetics of population redistribution after a [Mg(2+)] jump and the observed connectivity of FRET states reveal details of the folding pathway that complement and transcend information from equilibrium or single-jump measurements. FRET trajectories for jumps from [Mg(2+)] = 0.01 to 0.1 mM exhibit two-state behavior whereas jumps from 0.01 mM to 0.4 mM exhibit two-state unfolding but multistate folding behavior. RNA molecules in the low and high FRET states before the [Mg(2+)] increase are observed to undergo dynamics in two distinct regions of the free energy landscape separated by a high barrier. We describe the RNA structural changes involved in crossing this barrier as a "hidden" degree of freedom because the changes do not alter the detected FRET value but do alter the observed dynamics. The associated memory prevents the populations from achieving their equilibrium values at the end of the 5- to 10-sec [Mg(2+)] interval, thereby creating a nonequilibrium steady-state condition. The capability of interrogating nonequilibrium steady-state RNA conformations and the adjustable period of [Mg(2+)]-jump cycles makes it possible to probe regions of the free energy landscape that are infrequently sampled in equilibrium or single-jump measurements.

Design of chemically propelled nanodimer motors

The self-propelled motion of nanodimers fueled by a chemical reaction taking place under nonequilibrium steady state conditions is investigated. The nanodimer consists of a pair of catalytic and chemically inactive spheres, in general with different sizes, with a fixed internuclear separation. The solvent in which the dimer moves is treated at a particle-based mesoscopic level using multiparticle collision dynamics. The directed motion of the dimer can be controlled by adjusting the interaction potentials between the solvent molecules and the dimer spheres, the internuclear separation, and sphere sizes. Dimers can be designed so that the directed motion along the internuclear axis occurs in either direction and is much larger than the thermal velocity fluctuations, a condition needed for such nanodimers to perform tasks involving targeted dynamics.

Nonequilibrium Steady State of a Nanometric Biochemical System: Determining the Thermodynamic Driving Force from Single Enzyme Turnover Time Traces

A single enzyme molecule in a living cell is a nanometric system that catalyzes biochemical reactions in a nonequilibrium steady-state condition. The chemical driving force, Deltamu, is an important thermodynamic quantity that determines the extent to which the reaction system is away from equilibrium. Here we show that Deltamu for an enzymatic reaction in situ can be determined from the nonequilibrium time traces for enzymatic turnovers of individual enzyme molecules, which can now be recorded experimentally by single-molecule techniques. Three different Deltamu estimators are presented from principles of nonequilibrium statistical mechanics: fluctuation theorem, Kawasaki identity, and fluctuation dissipation theorem, respectively. In particular, a maximum likelihood estimation method of Deltamu has been derived based on fluctuation theorem. The statistical precisions of these three Deltamu estimators are analyzed and compared for experimental time traces with finite lengths.

Effects of differential mobility on biased diffusion of two species

Using simulations and a simple mean-field theory, we investigate jamming transitions in a two-species lattice gas under non-equilibrium steady-state conditions. The two types of particles diffuse with different mobilities on a square lattice, subject to an excluded volume constraint and biased in opposite directions. Varying filling fraction, differential mobility, and drive, we map out the phase diagram, identifying first order and continuous transitions between a free-flowing disordered and a spatially inhomogeneous jammed phase. Ordered structures are observed to drift, with a characteristic velocity, in the direction of the more mobile species.

Demagnetization of spin systems at low temperature

We report on analytical and numerical results for the time evolution of a ~lattice! model of ferromagnetic particles after field inversion. Relaxation from the metastable phase occurs by superposition of two independent spin-flip processes: with probability 1 2 p, the flip obeys the Metropolis rule at finite temperature, T, while the flip is performed at random, as for T!‘, with probability p. For small p, the latter process mimics quantum tunneling or any other sort of impurity that would induce both additional randomness and a nonequilibrium steady-state condition asymptotically. Critical avalanches and constant magnetic viscosity at low T ensue as two key features of the relaxation for any pfi0. This has some implications for experiments on demagnetization and mesoscopic quantum coherence. @S0163-1829~97!08437-3#