Near-equilibrium Regime Research Articles

The Compressible Euler and Acoustic Limits from Quantum Boltzmann Equation with Fermi–Dirac Statistics

This paper justifies the compressible Euler and acoustic limits from quantum Boltzmann equation with Fermi–Dirac statistics rigorously. By employing Hilbert expansion, in particular analyzing the nonlinear implicit transformation between the classical form of compressible Euler equations and the one obtained directly from BFD, and some new type of Grad–Caflisch type decay estimate of the linearized collision operator, we establish the compressible Euler limit from scaled BFD equation, which was formally derived by Zakrevskiy in (Kinetic models in the near-equilibrium regime. Thesis at Polytechnique, 2015) by moment method. Consequently, the acoustic limit is obtained in optimal scaling with respect to Knudsen number.

On the derivation of the Landau–Lifshitz frame in relativistic kinetic theory

The Landau–Lifshitz frame has been widely used to represent the macroscopic quantities of relativistic hydrodynamics in the presence of the dissipative process. In this paper, we derive the Landau–Lifshitz frame in the near-equilibrium regime under self-contained assumptions that do not require comparison with the Eckart frame. And then we revisit the relativistic BGK model proposed by Anderson and Witting to provide an application example of the Landau–Lifshitz frame.

Optimization and Efficiency Studies of Heat Engines: A Review

This review aims to study the various theoretical and numerical investigations in the optimization of heat engines. The main focus is to discuss the procedures to derive the efficiency of heat engines under different operating regimes (or optimization criteria) for different models of heat engines such as endreversible models, stochastic models, low-dissipation models, quantum models etc. Both maximum power and maximum efficiency operational regimes are desirable but not economical, so to meet the thermo-ecological considerations, some other compromise-based criteria have been proposed such as Ω criterion (ecological criterion) and efficient power criterion. Thus, heat engines can be optimized to work at an efficiency which may not be the maximum (Carnot) efficiency. The optimization efficiency obtained under each criterion shows a striking universal behaviour in the near-equilibrium regime. We also discussed a multi-parameter combined objective function of heat engines. The optimization efficiency derived from the multi-parameter combined objective function includes a variety of optimization efficiencies, such as the efficiency at the maximum power, efficiency at the maximum efficiency-power state, efficiency at the maximum criterion, and Carnot efficiency. Thus, a comparison of optimization of heat engines under different criteria enables to choose the suitable one for the best performance of heat engine under different conditions.

Generic theory of the dynamic magnetic response of ferrofluids.

Ferrofluids belong to an important class of highly functional soft matter, benefiting from their magnetically controllable physical properties. Therefore, it is of central importance to quantitatively predict the dynamic magnetic response of ferrofluids. Traditional dynamic theories, however, are often restricted to the near-equilibrium regime and/or only apply to nearly ideal ferrofluids that are monodisperse, dilute enough, and weakly interacting. In this paper I develop a self-consistent and nonperturbative dynamical mean field theory for typical ferrofluids which are often polydisperse, concentrated, and strongly interacting, possibly driven far from equilibrium. I obtain a general nonperturbative expression for the dynamic magnetic susceptibility, quantitatively agreeing with the spectra obtained from Brownian Dynamics simulations on both mono- and bidisperse samples. Furthermore, I derive a generic magnetization relaxation equation (MRE) for both mono- and polydisperse ferrofluids by employing the projection operator technique in nonequlibrium statistical mechanics. This MRE is in simple closed form and independent of which model is employed to approximate the equilibrium magnetization curve. Existing models can be recovered as low-order approximations of my generic and nonperturbative MRE. My theory can play a key role in studying the dynamics of ferrofluids and other polar fluids. It may also have substantial and immediate consequences to various ferrofluid applications.

Large deviations and optimal control forces for hard particles in one dimension

We analyse large deviations of the dynamical activity in one-dimensional systems of diffusing hard particles. Using an optimal-control representation of the large-deviation problem, we analyse effective interaction forces which can be added to the system, to aid sampling of biased ensembles of trajectories. We find several distinct regimes, as a function of the activity and the system size: we present approximate analytical calculations that characterise the effective interactions in several of these regimes. For high activity the system is hyperuniform and the interactions are long-ranged and repulsive. For low activity, there is a near-equilibrium regime described by macroscopic fluctuation theory, characterised by long-ranged attractive forces. There is also a far-from-equilibrium regime in which one of the interparticle gaps becomes macroscopic and the interactions depend strongly on the size of this gap. We discuss the extent to which transition path sampling of these ensembles is improved by adding suitable control forces.

Relation between photon thermal Hall effect and persistent heat current in nonreciprocal radiative heat transfer

We study the photon thermal Hall effect and the persistent heat current in radiative heat transfer. We show that the photon thermal Hall effect is not a uniquely nonreciprocal effect; it can arise in some reciprocal systems with broken mirror symmetry. This is in contrast with the persistent heat current, which is a uniquely non-reciprocal effect that can not exist in any reciprocal system. Nevertheless, for a specific class of systems with $C_4$ rotational symmetry, we note that the photon thermal Hall effect is uniquely nonreciprocal, and moreover there is a direct connection between the persistent heat current and the photon thermal Hall effect. In the near-equilibrium regime, the magnitude of the photon thermal Hall effect is proportional to the temperature derivative of the persistent heat current in such systems. Therefore, the persistent heat current as predicted for the equilibrium situation can be probed by the photon thermal Hall effect away from equilibrium.

Direct Measurement of π Coupling at the Single-Molecule Level using a Carbon Nanotube Force Sensor.

We report a carbon nanotube (CNT) force sensor that combines a suspended CNT transistor with dual-trap optical tweezers to explore the interactions between two individual molecules in the near-equilibrium regime with sub-piconewton resolution. The directly measured equilibrium force (1.2 ± 0.5 pN) is likely related to the binding force between a CNT and a single DNA base, where two aromatic rings spontaneously attract to each other due to the noncovalent forces between them. On the basis of our force measurements, the binding free energy per base is calculated (∼0.34 eV), which is in good agreement with theoretical simulations. Moreover, three-dimensional scanning photocurrent microscopy enables us to simultaneously monitor the morphology changes of the CNT, leading to a comprehensive reconstruction of the CNT-DNA binding dynamics. These experimental results shed light on the fundamental understanding of the mechanical coupling between CNTs and DNA molecules and, more importantly, provide a new platform for direct observation of intermolecular interfaces at the single-molecule level.

Optimal response to non-equilibrium disturbances under truncated Burgers–Hopf dynamics

We model and compute the average response of truncated Burgers–Hopf dynamics to finite perturbations away from the Gibbs equipartition energy spectrum using a dynamical optimization framework recently conceptualized in a series of papers. Non-equilibrium averages are there approximated in terms of geodesic paths in probability space that ‘best-fit’ the Liouvillean dynamics over a family of quasi-equilibrium trial densities. By recasting the geodesic principle as an optimal control problem, we solve numerically for the non-equilibrium responses using an augmented Lagrangian, non-linear conjugate gradient descent method. For moderate perturbations, we find an excellent agreement between the optimal predictions and the direct numerical simulations of the truncated Burgers–Hopf dynamics. In this near-equilibrium regime, we argue that the optimal response theory provides an approximate yet predictive counterpart to fluctuation-dissipation identities.

Near-equilibrium universality and bounds on efficiency in quasi-static regime with finite source and sink

We show the validity of some results of finite-time thermodynamics, also within the quasi-static framework of classical thermodynamics. First, we consider the efficiency at maximum work from finite source and sink modelled as identical thermodynamic systems. The near-equilibrium regime is characterized by expanding the internal energy up to second order (i.e. up to linear response) in the difference of initial entropies of the source and the sink. It is shown that the efficiency is given by a universal expression , where is the Carnot efficiency. Then, different sizes of source and sink are treated, by combining different numbers of copies of the same thermodynamic system. The efficiency of this process is found to be , where the parameter γ depends only on the relative size of the source and the sink. This implies that within the linear response theory, is bounded as , where the upper (lower) bound is obtained with a sink much larger (smaller) in size than the source. We also remark on the behavior of the efficiency beyond linear response.

On the magnetorotational instability and elastic buckling.

This paper demonstrates an equivalence between rotating magnetized shear flows and a stressed elastic beam. This results from finding the same form of dynamical equations after an asymptotic reduction of the axis-symmetric magnetorotational instability (MRI) under the assumption of almost-critical driving. The analysis considers the MRI dynamics in a non-dissipative near-equilibrium regime. Both the magnetic and elastic systems reduce to a simple one-dimensional wave equation with a non-local nonlinear feedback. Under transformation, the equation comprises a large number of mean-field interacting Duffing oscillators. This system was the first proven example of a strange attractor in a partial differential equation. Finding the same reduced equation in two natural applications suggests the model might result from other applications and could fall into a universal class based on symmetry.

Time reversal in nonequilibrium thermodynamics.

The general equation of nonequilibrium reversible-irreversible coupling (GENERIC) is studied in light of time-reversal transformation. It is shown that Onsager-Casimir reciprocal relations are implied by GENERIC in the near-equilibrium regime. A general structure which gives the reciprocal relations but which is valid also far from equilibrium is identified, and Onsager-Casimir reciprocal relations are generalized to far-from-equilibrium regime in this sense. Moreover, reversibility and irreversibility are carefully discussed and the results are illustrated in Hamiltonian dynamics, classical hydrodynamics, classical irreversible thermodynamics, the quantum master equation, and the Boltzmann equation.

Solid-state phase transformation kinetics in the near-equilibrium regime

Solid-state phase transformation kinetics in the near-equilibrium regime behaves differently from that in extremely non-equilibrium regime since their thermodynamic states are different. Incorporating temperature- and transformed fraction-dependent thermodynamic terms, a thermo-kinetic model is derived to describe the transformation kinetics in the near-equilibrium regime. The model predicts a sluggish stage in isothermally conducted transformation and a temperature-dependent stage in non-isothermally conducted transformation. Then, the kinetics of γ/α transformations in two binary substitutional Fe-based alloys (i.e. Fe-3.28 at.% Mn and Fe-1.73 at.% Co), measured by isothermal and non-isothermal dilatometry, is investigated by the newly proposed model. The model quantitatively describes the retarded isothermal kinetics in Fe-3.28 at.% Mn alloy and the abnormal non-isothermal kinetics in Fe-1.73 at.% Co alloy.

Quantitative bond energetics in atomic-scale junctions.

A direct measurement of the potential energy surface that characterizes individual chemical bonds in complex materials has fundamental significance for many disciplines. Here, we demonstrate that the energy profile for metallic single-atom contacts and single-molecule junctions can be mapped by fitting ambient atomic force microscope measurements carried out in the near-equilibrium regime to a physical, but simple, functional form. We extract bond energies for junctions formed through metallic bonds as well as metal-molecule link bonds from atomic force microscope data and find that our results are in excellent quantitative agreement with density functional theory based calculations for exemplary junction structures. Furthermore, measurements from a large number of junctions can be collapsed to a single, universal force-extension curve, thus revealing a surprising degree of similarity in the overall shape of the potential surface that governs these chemical bonds. Compared to previous studies under ambient conditions where analysis was confined to trends in rupture force, our approach significantly expands the quantitative information extracted from these measurements, particularly allowing analysis of the trends in bond energy directly.

An Optimization Principle for Deriving Nonequilibrium Statistical Models of Hamiltonian Dynamics

A general method for deriving closed reduced models of Hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation. As in standard projection operator methods, a set of resolved variables is selected to capture the slow, macroscopic behavior of the system, and the family of quasi-equilibrium probability densities on phase space corresponding to these resolved variables is employed as a statistical model. The macroscopic dynamics of the mean resolved variables is determined by optimizing over paths of these probability densities. Specifically, a cost function is introduced that quantifies the lack-of-fit of such paths to the underlying microscopic dynamics; it is an ensemble-averaged, squared-norm of the residual that results from submitting a path of trial densities to the Liouville equation. The evolution of the macrostate is estimated by minimizing the time integral of the cost function. The value function for this optimization satisfies the associated Hamilton-Jacobi equation, and it determines the optimal relation between the statistical parameters and the irreversible fluxes of the resolved variables, thereby closing the reduced dynamics. The resulting equations for the macroscopic variables have the generic form of governing equations for nonequilibrium thermodynamics, and they furnish a rational extension of the classical equations of linear irreversible thermodynamics beyond the near-equilibrium regime. In particular, the value function is a thermodynamic potential that extends the classical dissipation function and supplies the nonlinear relation between thermodynamics forces and fluxes.

Interpreting the widespread nonlinear force spectra of intermolecular bonds

Single molecule force spectroscopy probes the strength, lifetime, and energetic details of intermolecular interactions in a simple experiment. A growing number of these studies have reported distinctly nonlinear trends in rupture force with loading rate that are typically explained in conventional models by invoking complex escape pathways. Recent analyses suggested that these trends should be expected even for simple barriers based on the basic assumptions of bond rupture dynamics and thus may represent the norm rather than the exception. Here we explore how these nonlinear trends reflect the two fundamental regimes of bond rupture: (i) a near-equilibrium regime, produced either by bond reforming in the case of a single bond or by asynchronized rupture of multiple individual bonds, and (ii) a kinetic regime produced by fast, non-equilibrium bond rupture. We analyze both single- and multi-bonded cases, describe the full evolution of the system as it transitions between near- and far-from-equilibrium loading regimes, and show that both interpretations produce essentially identical force spectra. Data from 10 different molecular systems show that this model provides a comprehensive description of force spectra for a diverse suite of bonds over experimentally relevant loading rates, removes the inconsistencies of previous interpretations of transition state distances, and gives ready access to both kinetic and thermodynamic information about the interaction. These results imply that single-molecule binding free energies for a vast number of bonds have already been measured.

Dynamical diagram and scaling in polymer driven translocation

By analyzing the real space non-equilibrium dynamics of polymers, we elucidate the physics of driven translocation and propose its dynamical scaling scenario analogous to that in the surface growth phenomena. We provide a detailed account of the previously proposed tension-propagation formulation and extend it to cover the broader parameter space relevant to real experiments. In addition to a near-equilibrium regime, we identify three distinct non-equilibrium regimes reflecting the steady-state property of a dragged polymer with finite extensibility. Finite-size effects are also pointed out. These elements are shown to be crucial for the appropriate comparison with experiments and simulations.

Apparent horizon in fluid-gravity duality

This article develops a computational framework for determining the location of boundary-covariant apparent horizons in the geometry of conformal fluid-gravity duality in arbitrary dimensions. In particular, it is shown up to second order and conjectured to hold to all orders in the gradient expansion that there is a unique apparent horizon which is covariantly expressible in terms of fluid velocity, temperature, and boundary metric. This leads to the first explicit example of an entropy current defined by an apparent horizon and opens the possibility that in the near-equilibrium regime there is preferred foliation of apparent horizons for black holes in asymptotically anti-de Sitter spacetimes.

Experimental investigation of the spreading of viscoplastic fluids on inclined planes

We report experimental results related to the dam-break problem for viscoplastic fluids. Using image processing techniques, we were able to accurately reconstruct the free-surface evolution of fixed volumes of fluid suddenly released a plane. We used Carbopol Ultrez 10 as a viscoplastic material; its rheological behavior was closely approximated by a Herschel–Bulkley model for a fairly wide range of shear-rates. Varying the Carbopol concentration allowed us to change the yield stress and bulk viscosity. The yield stress ranged from 78 to 109 Pa, producing Bingham numbers in the 0.07–0.35 range. We investigated the behavior of a 43-kg mass released on a plane, whose inclination ranged from 0° to 18°. For each run, we observed that the behavior was nearly the same: at short times, the mass accelerated vigorously on gate opening and very quickly reached a nearly constant velocity. At time t = 1 s, independently of plane inclination and yield stress, the mass reached a near-equilibrium regime, where the front position varied as a power function of time over several decades. We did not observe any run-out phase, during which the mass would have gradually come to a halt. The similarity in the flow behavior made it possible to derive an empirical scaling for the front position in the form x f = t 0.275 ( sin ⁡ α ) 1 / 3 ( sin ⁡ α ) 5 / 4 , where α and t denote plane inclination and time, respectively, and which holds for sloping beds ( α > 0).

Dynamic strength of molecularly bonded surfaces

This study reports a theoretical analysis of the forced separation of two adhesive surfaces linked via a large number of parallel noncovalent bonds. To describe the bond kinetics, we implement a three-state reaction model with kinetic rates obtained from a simple integral expression of the mean first passage time for diffusive barrier crossing in a pulled-distance-dependent potential. We then compute the rupture force for the separation of adhesive surfaces at a constant rate. The results correspond well with a Brownian dynamics simulation of the same system. The separation rate relative to the intrinsic relaxation time of the bonds defines three loading regimes and the general dependence of the adhesion on kinetic or thermodynamic parameters of the bonds. In the equilibrium regime, the rupture force asymptotically approaches the equilibrium rupture force, which increases linearly with the equilibrium bond energy. In the near-equilibrium regime, the rupture force increases with the separation rate and increasingly correlates with the bond rupture barrier. In the far-from-equilibrium regime where rebinding is irrelevant, the rupture force varies linearly with the rupture barrier.

Ilya Prigogine

Ilya Prigogine