Onsager Relations Research Articles

Transitional steady states of exchange dynamics between finite quantum systems.

We examine energy and particle exchange between finite-sized quantum systems and find a new form of nonequilibrium state. The exchange rate undergoes stepwise evolution in time, and its magnitude and sign dramatically change according to system size differences. The origin lies in interference effects contributed by multiply scattered waves at system boundaries. Although such characteristics are utterly different from those of true steady state for infinite systems, Onsager's reciprocal relation remains universally valid.

Onsager's cross coupling effects in gas flows confined to micro-channels

In rarefied gases, mass and heat transport processes interfere with each other, leading to the mechano-caloric effect and thermo-osmotic effect, which are of interest to both theoretical study and practical applications. We employ the unified gas-kinetic scheme to investigate these cross coupling effects in gas flows in micro-channels. Our numerical simulations cover channels of planar surfaces and also channels of ratchet surfaces, with Onsager's reciprocal relation verified for both cases. For channels of planar surfaces, simulations are performed in a wide range of Knudsen number and our numerical results show good agreement with the literature results. For channels of ratchet surfaces, simulations are performed for both the slip and transition regimes and our numerical results not only confirm the theoretical prediction [Phys. Rev. Lett. 107, 164502 (2011)] for Knudsen number in the slip regime but also show that the off-diagonal kinetic coefficients for cross coupling effects are maximized at a Knudsen number in the transition regime. Finally, a preliminary optimization study is carried out for the geometry of Knudsen pump based on channels of ratchet surfaces.

Polarizability Tensor Retrieval for Subwavelength Particles of Arbitrary Shape

We present a semianalytical method to determine the full polarizability tensor of subwavelength particles with arbitrary shape and composition. This method extracts both co- and cross-coupling terms of the polarizability tensor, retrieving the full 6 × polarizability tensor with three measurements, by appropriately rotating the inclusions, when embedded in a planar periodic array, with respect to their principle axes. In order to validate this method, we present four examples, including a magnetodielectric sphere, a magnetodielectric spheroid, a conducting helix and pairs of split-ring resonators, and discuss the relation with fundamental constraints such as reciprocity and Onsager's relations. We also discuss how this retrieval method may be useful for the design and homogenization of periodic metamaterials and two-dimensional (2-D) metasurfaces formed by subwavelength inclusions.

Electron and phonon drag in thermoelectric transport through coherent molecular conductors

We study thermoelectric transport through a coherent molecular conductor connected to two electron and two phonon baths using the nonequilibrium Green's function method. We focus on the mutual drag between electron and phonon transport as a result of `momentum' transfer, which happens only when there are at least two phonon degrees of freedom. After deriving expressions for the linear drag coefficients, obeying the Onsager relation, we further investigate their effect on nonequilibrium transport. We show that the drag effect is closely related to two other phenomena: (1) adiabatic charge pumping through a coherent conductor; (2) the current-induced nonconservative and effective magnetic forces on phonons.

Mass and heat diffusion in ternary polymer solutions: A classical irreversible thermodynamics approach

Governing equations for evolution of concentration and temperature in three-component systems were derived in the framework of classical irreversible thermodynamics using Onsager’s variational principle and were presented for solvent/solvent/polymer and solvent/polymer/polymer systems. The derivation was developed from the Gibbs equation of equilibrium thermodynamics using the local equilibrium hypothesis, Onsager reciprocal relations and Prigogine’s theorem for systems in mechanical equilibrium. It was shown that the details of mass and heat diffusion phenomena in a ternary system are completely expressed by a 3 × 3 matrix whose entries are mass diffusion coefficients (4 entries), thermal diffusion coefficients (2 entries) and three entries that describe the evolution of heat in the system. The entries of the diffusion matrix are related to the elements of Onsager matrix that are bounded by some constraints to satisfy the positive definiteness of entropy production in the system. All the elements of diffusion matrix were expressed in terms of derivatives of exchange chemical potentials of the components with respect to concentration and temperature. The spinodal curves of ternary polymer solutions were derived from the governing equations and their correctness was checked by the Hessian of free energy density. Moreover, it was proved that setting cross-diffusion coefficients to zero results in a contradiction, and the governing equations without cross-diffusion coefficients do not express the actual phase behavior of the system.

From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics

This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We introduce the concept of the generalized Gibbs ensemble, and discuss its connection with ideas of prethermalization in weakly interacting systems.

Adiabatic response and quantum thermoelectrics for ac-driven quantum systems

We generalize the theory of thermoelectrics to include coherent electron systems under adiabatic ac driving, accounting for quantum pumping of charge and heat as well as the associated work exchange between electron system and driving potentials. We derive the relevant response coefficients in the adiabatic regime and show that they obey Onsager reciprocity relations. We analyze the consequences of our generalized thermoelectric framework for quantum motors, generators, heat engines, and heat pumps, characterizing them in terms of efficiencies and figures of merit.

Theory of the nonlinear Rashba-Edelstein effect: The clean electron gas limit

It is well known that a current driven through a two-dimensional electron gas with Rashba spin-orbit coupling induces a spin polarization in the perpendicular direction (Edelstein effect). This phenomenon has been extensively studied in the linear response regime, i.e., when the average drift velocity of the electrons is a small fraction of the Fermi velocity. Here we investigate the phenomenon in the nonlinear regime, meaning that the average drift velocity is comparable to or exceeds the Fermi velocity. This regime is realized when the electric field is very large or when electron-impurity scattering is very weak. We consider the limiting case of a two-dimensional noninteracting electron gas with no impurities. In this case, the quantum kinetic equation for the density matrix is exactly and analytically solvable, reducing to a problem of spin dynamics for ``unpaired'' electrons near the Fermi surface. The crucial parameter is $\ensuremath{\gamma}=eE{L}_{s}/{E}_{F}$, where $E$ is the electric field, $e$ is the absolute value of the electron charge, ${E}_{F}$ is the Fermi energy, and ${L}_{s}=\ensuremath{\hbar}/(2m\ensuremath{\alpha})$ is the spin-precession length in the Rashba spin-orbit field with coupling strength $\ensuremath{\alpha}$. If $\ensuremath{\gamma}\ensuremath{\ll}1$, the evolution of the spin is adiabatic, resulting in a spin polarization that grows monotonically in time and eventually saturates at the maximum value $n(\ensuremath{\alpha}/{v}_{F})$, where $n$ is the electron density and ${v}_{F}$ is the Fermi velocity. If $\ensuremath{\gamma}\ensuremath{\gg}1$ the evolution of the spin becomes strongly nonadiabatic and the spin polarization is progressively reduced and eventually suppressed for $\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\infty}$. We also predict an inverse nonlinear Edelstein effect, in which an electric current is driven by a magnetic field that grows linearly in time. The ``conductivities'' for the direct and the inverse effects satisfy generalized Onsager reciprocity relations, which reduce to the standard ones in the linear response regime.

Nonequilibrium thermodynamic potentials for continuous-time Markov chains.

We connect the rare fluctuations of an equilibrium (EQ) process and the typical fluctuations of a nonequilibrium (NE) stationary process. In the framework of large deviation theory, this observation allows us to introduce NE thermodynamic potentials. For continuous-time Markov chains, we identify the relevant pairs of conjugated variables and propose two NE ensembles: one with fixed dynamics and fluctuating time-averaged variables, and another with fixed time-averaged variables, but a fluctuating dynamics. Accordingly, we show that NE processes are equivalent to conditioned EQ processes ensuring that NE potentials are Legendre dual. We find a variational principle satisfied by the NE potentials that reach their maximum in the NE stationary state and whose first derivatives produce the NE equations of state and second derivatives produce the NE Maxwell relations generalizing the Onsager reciprocity relations.

Li+ Transference Numbers in Liquid Electrolytes Obtained by Very-Low-Frequency Impedance Spectroscopy at Variable Electrode Distances

The Li+ transference numbers of three different liquid electrolytes for Li-ion batteries were measured in a symmetrical Li | electrolyte | Li cell by means of very-low-frequency impedance spectroscopy (VLF-IS). The electrolytes were: (i) The standard battery electrolyte LP30; (ii) an equimolar mixture of tetraglyme (G4) and lithium bis(trifluoromethylsulfonyl)imide (Li-TFSI); (iii) Li-TFSI dissolved in the ionic liquid 1-butyl-1-methyl-pyrrolidinium bis(trifluoromethylsulfonyl)imide (BMP-TFSI). We found that the Li+ transference numbers of the two electrolytes LP30 and G4/Li-TFSI are much smaller than the Li+ transport numbers obtained from pulsed-field gradient NMR measurements. On the other hand, in the case of BMP-TFSI/Li-TFSI, the values for and are more similar. In order to rationalize the large differences between and found for LP30 and G4/Li-TFSI, we combined the Onsager reciprocal relations with linear response theory, and we derived expressions for , which take into account all correlations between ionic movements in the electrolyte. Thereby, we show that can be considerably smaller than , if strong directional correlations exist between the movements of cations and anions. Finally, we discuss differences in Li+ transference numbers obtained by VLF-IS and by potentiostatic polarization measurements.

Wiedemann-Franz law for magnon transport

One of the main goals of spintronics is to improve transport of information carriers and to achieve new functionalities with ultra-low dissipation. A most promising strategy for this holy grail is to use pure magnon currents created and transported in insulating magnets, in the complete absence of any conducting metallic elements. Here we propose a realistic solution to this fundamental challenge by analyzing magnon and heat transport in insulating ferromagnetic junctions. We calculate all transport coefficients for magnon transport and establish Onsager relations between them. We theoretically discover that magnon transport in junctions has a universal behavior, i.e. is independent of material parameters, and establish a magnon analog of the celebrated Wiedemann-Franz law which governs charge transport at low temperatures. We calculate the Seebeck and Peltier coefficients which are crucial quantities for spin caloritronics and demonstrate that they assume universal values in the low temperature limit. Finally, we show that our predictions are within experimental reach with current device and measurement technologies.

Boosting thermoelectric efficiency using time-dependent control.

Thermoelectric efficiency is defined as the ratio of power delivered to the load of a device to the rate of heat flow from the source. Till date, it has been studied in presence of thermodynamic constraints set by the Onsager reciprocal relation and the second law of thermodynamics that severely bottleneck the thermoelectric efficiency. In this study, we propose a pathway to bypass these constraints using a time-dependent control and present a theoretical framework to study dynamic thermoelectric transport in the far from equilibrium regime. The presence of a control yields the sought after substantial efficiency enhancement and importantly a significant amount of power supplied by the control is utilised to convert the wasted-heat energy into useful-electric energy. Our findings are robust against nonlinear interactions and suggest that external time-dependent forcing, which can be incorporated with existing devices, provides a beneficial scheme to boost thermoelectric efficiency.

Generalized Born and Explicit Solvent Models for Free Energy Calculations in Organic Solvents: Cyclodextrin Dimerization.

Evaluation of solvation (binding) free energies with implicit solvent models in different dielectric environments for biological simulations as well as high throughput ligand screening remain challenging endeavors. In order to address how well implicit solvent models approximate explicit ones we examined four generalized Born models (GB(Still), GB(HCT), GB(OBC)I, and GB(OBC)II) for determining the dimerization free energy (ΔG(0)) of β-cyclodextrin monomers in 17 implicit solvents with dielectric constants (D) ranging from 5 to 80 and compared the results to previous free energy calculations with explicit solvents ( Zhang et al. J. Phys. Chem. B 2012 , 116 , 12684 - 12693 ). The comparison indicates that neglecting the environmental dependence of Born radii appears acceptable for such calculations involving cyclodextrin and that the GB(Still) and GB(OBC)I models yield a reasonable estimation of ΔG(0), although the details of binding are quite different from explicit solvents. Large discrepancies between implicit and explicit solvent models occur in high-dielectric media with strong hydrogen bond (HB) interruption properties. ΔG(0) with the GB models is shown to correlate strongly to 2(D-1)/(2D+1) (R(2) ∼ 0.90) in line with the Onsager reaction field ( Onsager J. Am. Chem. Soc. 1936 , 58 , 1486 - 1493 ) but to be very sensitive to D (D < 10) as well. Both high-dielectric environments where hydrogen bonds are of interest and low-dielectric media such as protein binding pockets and membrane interiors therefore need to be considered with caution in GB-based calculations. Finally, a literature analysis of Gibbs energy of solvation of small molecules in organic liquids shows that the Onsager relation does not hold for real molecules since the correlation between ΔG(0) and 2(D-1)/(2D+1) is low for most solutes. Interestingly, explicit solvent calculations of the solvation free energy ( Zhang et al. J. Chem. Inf. Model . 2015 , 55 , 1192 - 1201 ) reproduce the weak experimental correlations with 2(D-1)/(2D+1) very well.

Fluctuation theorem for a two-terminal conductor connected to a thermal probe

We analyze the full-counting statistics of the electric heat current flowing in a two-terminal quantum conductor coupled to a probe electrode. Based on the stochastic path-integral method, we derive the full-counting statistics of the two-terminal system, which obeys the fluctuation theorem in the presence of a magnetic field. We also calculate the heat-current correlation (the noise) in the linear response and the second-order nonlinear conductance, which explicitly satisfy an extension of the Onsager relation beyond the linear-response theory.

On the thermal diffusion in polymer solutions: Case study on bicomponent film drying

This paper aims to theoretically explain experimental observations regarding the thermal diffusion of polymers in polymer solutions. 3D governing equations for mass and heat diffusion in polymer solutions were derived in the framework of classical irreversible thermodynamics using local equilibrium hypothesis, Onsager reciprocal relations and Prigogine's theory in systems in mechanical equilibrium. The mutual diffusion coefficient, thermal diffusion coefficient and thermal conductivity enter the governing equations as functions of thermodynamic variables and phenomenological coefficients that are bounded to some theoretical constraints. It was shown that the derivative of the linear combination of chemical potentials of polymer and solvent with respect to temperature plays an important role in thermal diffusion in polymer solutions. Thermal diffusion coefficient derived in the model can qualitatively explain the experimental observations in the literature regarding the dependence of thermal diffusion of polymers to molecular weight, polymer chain rigidity and viscosity of the solvent. Numerical simulation of the governing equations for a 1D drying process of polymer solutions indicates that the model is able to capture the effect of thermal diffusion. This effect manifests itself as an increase in local concentration of the solvent on the warm side of a temperature gradient during a solution casting process.

A Hydrodynamical Model for Carriers and Phonons With Generation-Recombination, Including Auger Effect

The asymptotic procedure proposed allows to derive closed hydrodynamical equations from the kinetic equations of carriers and phonons (treated as a partecipating species) in a photon background. The direct generation-recombination processes are accounted for. The fluid-dynamical equations constructed for the chemical potentials of carriers, temperature, and drift velocity, are related to the extended thermodynamical (ET) ones for the chemical potentials of carriers, temperature, and drift velocity. In the drift-diffusion approximation the constitutive laws are derived and the Onsager relation recovered.

Phenomenological description of the nonlocal magnetization relaxation in magnonics, spintronics, and domain-wall dynamics

A phenomenological equation called the Landau-Lifshitz-Baryakhtar (LLBar) [Zh. Eksp. Teor. Fiz 87, 1501 (1984) [Sov. Phys. JETP 60, 863 (1984)]] equation, which could be viewed as the combination of the Landau-Lifshitz (LL) equation and an extra ``exchange-damping'' term, was derived by Baryakhtar using Onsager's relations. We interpret the origin of this exchange damping as nonlocal damping by linking it to the spin current pumping. The LLBar equation is investigated numerically and analytically for the spin-wave decay and domain-wall motion. Our results show that the lifetime and propagation length of short-wavelength magnons in the presence of nonlocal damping could be much smaller than those given by the LL equation. Furthermore, we find that both the domain-wall mobility and the Walker breakdown field are strongly influenced by the nonlocal damping.

Magnetophoretic potential at the movement of cluster products of electrochemical reactions in an inhomogeneous magnetic field

An electric field arises from the influence of a nonuniform static magnetic field on charged colloid particles with magnetic susceptibility different from that of the surrounding liquid. It arises, for example, under the influence of a nonuniform static magnetic field in clusters of electrochemical reaction products created during metal etching, deposition, and corrosion processes without an external electric current passing through an electrolyte near a magnetized electrode surface. The corresponding potential consists of a Nernst potential of inhomogeneous distribution of concentration of colloid particles and a magnetophoretic potential (MPP). This potential has been calculated using a thermodynamic approach based on the equations of thermodynamics of nonequilibrium systems and the Onsager relations for a mass flow of correlated magnetic clusters under a gradient magnetic force in the electrolyte. The conditions under which the MPP contribution to the total electric potential may be significant are discussed with a reference to the example of a corroding spherical ferromagnetic steel electrode.

Onsager’s reciprocal relations in electrolyte solutions. I. Sedimentation and electroacoustics

In the framework of irreversible thermodynamics, we show that the sedimentation current in electrolyte solutions is mathematically equivalent to the low frequency limit of the ionic vibration current, appearing in the presence of an acoustic wave. This non-trivial result is obtained thanks to a careful choice of the reference frame used to express the mass fluxes in the context of electroacoustics. Coupled transport phenomena in electrolyte solutions can also be investigated in a mechanical framework, with a set of Newtonian equations for the dynamics of charged solutes. Both in the context of sedimentation and of electroacoustics, we show that the results obtained in the mechanical framework, in the ideal case (i.e., without interactions between ions), do satisfy the Onsager's reciprocal relations. We also derive the general relation between corrective forces accounting for ionic interactions which must be fulfilled so that the Onsager's reciprocal relations are verified. Finally, we show that no additional diffusion term needs to be taken into account in the flux of solutes (far from the walls), even if local concentration gradients exist, contrarily to what was done previously in the literature.

Onsager's reciprocal relations in electrolyte solutions. II. Effect of ionic interactions on electroacoustics.

In electrolyte solutions, an electric potential difference, called the Ionic Vibration Potential (IVP), related to the ionic vibration intensity, is generated by the application of an acoustic wave. Several theories based on a mechanical framework have been proposed over the years to predict the IVP for high ionic strengths, in the case where interactions between ions have to be accounted for. In this paper, it is demonstrated that most of these theories are not consistent with Onsager's reciprocal relations. A new expression for the IVP will be presented that does fulfill the Onsager's reciprocal relations. We obtained this expression by deriving general expressions of the corrective forces describing non-ideal effects in electrolyte solutions.