Onsager's Regression Research Articles

Onsager's regression hypothesis adjusted to quantum systems

Onsager's regression hypothesis adjusted to quantum systems

Observation of nonlinear response and Onsager regression in a photon Bose-Einstein condensate

The quantum regression theorem states that the correlations of a system at two different times are governed by the same equations of motion as the single-time averages. This provides a powerful framework for the investigation of the intrinsic microscopic behaviour of physical systems by studying their macroscopic response to a controlled external perturbation. Here we experimentally demonstrate that the two-time particle number correlations in a photon Bose-Einstein condensate inside a dye-filled microcavity exhibit the same dynamics as the response of the condensate to a sudden perturbation of the dye molecule bath. This confirms the regression theorem for a quantum gas, and, moreover, demonstrates it in an unconventional form where the perturbation acts on the bath and only the condensate response is monitored. For strong perturbations, we observe nonlinear relaxation dynamics which our microscopic theory relates to the equilibrium fluctuations, thereby extending the regression theorem beyond the regime of linear response.

Level frequency postulate and the dice analog.

A level frequency postulate is proposed in the context of the Onsager regression hypothesis, and is utilized to demonstrate Fourier fluctuation time between levels in an analog system composed of red and white dice. This dice system is shown to be analogous to an isolated composite system of particles through derivation of the level probability distribution. Level fluctuation time is developed as an algebraic expression involving average energy and a Gaussian parameter, with quasistatic evolution demonstrated as an integral over fluctuation time.

Nonequilibrium Green-Kubo relations for hydrodynamic transport from an equilibrium-like fluctuation-response equality

Near equilibrium, Green-Kubo relations provide microscopic expressions for macroscopic transport coefficients in terms of equilibrium correlation functions. At their core, they are based on the intimate relationship between response and fluctuations as embodied by the equilibrium fluctuation-dissipation theorem, a connection generically broken far from equilibrium. In this work, we identify a class of perturbations whose response around far-from-equilibrium steady states is linked to steady-state correlation functions via an equilibrium-like fluctuation-response equality. We then utilize this prediction to substantiate linearized hydrodynamic transport equations that describe how spatial inhomogeneities in macroscopic nonequilibrium systems relax. As a consequence, we derive nonequilibrium Green-Kubo relations for the transport coefficients of two types of hydrodynamic variables: local conserved densities and broken-symmetry modes. A byproduct of this work is to provide a theoretical foundation for the validity of Onsager's regression hypothesis around nonequilibrium steady states. Our predictions are analytically and numerically corroborated for two model systems: density diffusion in a fluid of soft, spherical active Brownian particles and phase diffusion in the noisy Kuramoto model on a square lattice.

Phonon-mediated lipid raft formation in biological membranes

Short-wavelength collective molecular motions, also known as phonons, have recently attracted much interest in revealing dynamic properties of biological membranes through the use of neutron and X-ray scattering, infrared and Raman spectroscopies, and molecular dynamics simulations. Experimentally detecting unique vibrational patterns such as, shear phonon excitations, viscoelastic crossovers, transverse acoustic phonon gaps, and continuous and truncated optical phonon modes in cellular membranes, to name a few, has proven non-trivial. Here, we review recent advances in liquid thermodynamics that have resulted in the development of the phonon theory of liquids. The theory has important predictions regarding the shear vibrational spectra of fluids, namely the emergence of viscoelastic crossovers and transverse acoustic phonon gaps. Furthermore, we show that these vibrational patterns are common in soft (non-crystalline) materials, including, but not limited to liquids, colloids, liquid crystals (mesogens), block copolymers, and biological membranes. The existence of viscoelastic crossovers and acoustic phonon gaps define the self-diffusion properties of cellular membranes and provide a molecular picture of the transient nature of lipid rafts (Bolmatov et al., 2020). Importantly, the timescales (picoseconds) for the formation and dissolution of transient lipid rafts match the lifetime of the formation and breakdown of interfacial water hydrogen bonds. Apart from acoustic propagating phonon modes, biological membranes can also support more energetic non-propagating optical phonon excitations, also known as standing waves or breathing modes. Importantly, optical phonons can be truncated due to the existence of finite size nanodomains made up of strongly correlated lipid–cholesterol molecular pairs. These strongly coupled molecular pairs can serve as nucleation centers for the formation of stable rafts at larger length scales, due to correlations of spontaneous fluctuations (Onsager's regression hypothesis). Finally and importantly, molecular level viscoelastic crossovers, acoustic phonon gaps, and continuous and truncated optical phonon modes may offer insights as to how lipid–lipid and lipid–protein interactions enable biological function.

Time-reversal symmetry breaking in two-dimensional nonequilibrium viscous fluids.

We study the rheological signatures of departure from equilibrium in two-dimensional viscous fluids with and without internal spin. Under the assumption of isotropy, we provide the most general linear constitutive relations for stress and couple stress in terms of the velocity and spin fields. Invoking Onsager's regression hypothesis for fluctuations about steady states, we derive the Green-Kubo formulas relating the transport coefficients to time-correlation functions of the fluctuating stress. In doing so, we show that one of the nonequilibrium transport coefficients, the odd viscosity, requires time-reversal symmetry breaking in the case of systems without internal spin. However, the Green-Kubo relations for systems with internal spin also show that there is a possibility for nonvanishing odd viscosity even when time-reversal symmetry is preserved. Furthermore, we find that breakdown of equipartition in nonequilibrium steady states results in the decoupling of the two rotational viscosities relating the vorticity and the internal spin.

Time reversal symmetry breaking and odd viscosity in active fluids: Green–Kubo and NEMD results

Active fluids, which are driven at the microscale by non-conservative forces, are known to exhibit novel transport phenomena due to the breaking of time reversal symmetry. Recently, Epstein and Mandadapu [arXiv:1907.10041 (2019)] obtained Green-Kubo relations for the full set of viscous coefficients governing isotropic chiral active fluids, including the so-called odd viscosity, invoking Onsager's regression hypothesis for the decay of fluctuations in active non-equilibrium steady states. In this Communication, we test these Green-Kubo relations using molecular dynamics simulations of a canonical model system consisting of actively torqued dumbbells. We find the resulting odd and shear viscosity values from the Green-Kubo relations to be in good agreement with values measured independently through non-equilibrium molecular dynamics flow simulations. This provides a test of the Green-Kubo relations and lends support to the application of the Onsager regression hypothesis in relation to viscous behaviors of active matter systems.

Optical controlled non-equilibrium processes: effect of excitation wavelength

Experimental results on optically controlled non-equilibrium solvation dynamics show their dependence on excitation wavelength, However, the renowned Onsager regression hypothesis and Smoluchowski equation-based approach fail to explain such dependance. Failure of this theory prompted us to develop a novel theory by projecting the dynamics in a higher-dimensional space onto a one-dimensional energy gap space, where the effect of is included explicitly. We have then derived hierarchical equations of moments in time domain for an anharmonic potential based on the new kinetic equation derived here in the energy gap space to evaluate the non-equilibrium solvation time correlation function (NSTCF). The new methodlogy presented here also removes the difficulties in perturbative approaches to evaluate the NSTCF in a condensed phase. We have also shown that the NSTCF is independent of excitation wavelength for harmonic potential but dependent on the same for anharmonic potential, which is in accord with experimental observation. However, Onsager’s regression hypothesis-based prediction for the NSTCF is independent of excitation wavelength for all kinds of potentials. More importantly, the calculated results of NSTCF is in excellent agreement with the experimental results where the Onsager regression hypothesis-based prediction fails.

Non-equilibrium by molecular dynamics: a dynamical approach

Some time ago, Ciccotti and Jacucci [Phys. Rev. Lett. 1975;35:789-792] – on the basis of Onsager regression hypothesis as explained by Kubo – suggested and implemented an original approach to study both stationary and time-dependent situations in non-equilibrium systems. The key idea of their approach was to tackle separately the dynamical evolution from the problem of sampling the initial condition. In this review, we discuss the evolution of this approach dubbed D-NEMD, the dynamical approach to non-equilibrium molecular dynamics, to differentiate it from the stationary NEMD methods. We will go through the D-NEMD theoretical framework and illustrate how it allows to compute time-dependent macroscopic dynamical behaviours by averaging on a large sample of non-equilibrium trajectories starting from an ensemble of initial conditions associated to a suitable distribution (either equilibrium or non-equilibrium) at time zero. We discuss a few case studies where the D-NEMD method is illustrated: first for ‘historical’ applications, like the study of transport properties in the linear and non-linear regimes, including a divertissement on the calculation of time correlation functions in the Gran Canonical ensemble. We finally briefly illustrate D-NEMD applications to the study of problems arising in key hydrodynamic processes such as the formation of convective cells and the relaxation of a non-uniform density profile in a fluid, giving attention to the problem of sampling the conditional probability density ensemble associated with the initial states.

Dynamical Non-Equilibrium Molecular Dynamics

In this review, we discuss the Dynamical approach to Non-Equilibrium Molecular Dynamics (D-NEMD), which extends stationary NEMD to time-dependent situations, be they responses or relaxations. Based on the original Onsager regression hypothesis, implemented in the nineteen-seventies by Ciccotti, Jacucci and MacDonald, the approach permits one to separate the problem of dynamical evolution from the problem of sampling the initial condition. D-NEMD provides the theoretical framework to compute time-dependent macroscopic dynamical behaviors by averaging on a large sample of non-equilibrium trajectories starting from an ensemble of initial conditions generated from a suitable (equilibrium or non-equilibrium) distribution at time zero. We also discuss how to generate a large class of initial distributions. The same approach applies also to the calculation of the rate constants of activated processes. The range of problems treatable by this method is illustrated by discussing applications to a few key hydrodynamic processes (the “classical” flow under shear, the formation of convective cells and the relaxation of an interface between two immiscible liquids).

Transient Analysis of Data Traffic in Cognitive Radio Networks: A Non-Equilibrium Statistical Mechanics Approach

In contrast to traditional communication networks, the data traffic of cognitive radio network is interrupted by primary users. To evaluate the impact of primary user activities, the transient data traffic dynamics of cognitive radio networks, after being interrupted by primary users, are studied. The Onsager's regression approach, a powerful principle in non-equilibrium statistical mechanics, is applied. The autocorrelation function in the equilibrium state is used to study the time evolution of system deviation from the equilibrium state, according to the principle of Onsager's regression. Analytic approximations have been derived for the autocorrelation function for both generic and fluid approximation cases. Numerical simulations are carried out for a Jackson network model. The validity of the Onsager's regression is verified. The derived approximation of the correlation function is demonstrated to match the true value at the beginning of the relaxation. A scheduling algorithm for fast recovery of equilibrium is also proposed and demonstrated by numerical results.

Regression relation for pure quantum states and its implications for efficient computing.

We obtain a modified version of the Onsager regression relation for the expectation values of quantum-mechanical operators in pure quantum states of isolated many-body quantum systems. We use the insights gained from this relation to show that high-temperature time correlation functions in many-body quantum systems can be controllably computed without complete diagonalization of the Hamiltonians, using instead the direct integration of the Schrödinger equation for randomly sampled pure states. This method is also applicable to quantum quenches and other situations describable by time-dependent many-body Hamiltonians. The method implies exponential reduction of the computer memory requirement in comparison with the complete diagonalization. We illustrate the method by numerically computing infinite-temperature correlation functions for translationally invariant Heisenberg chains of up to 29 spins 1/2. Thereby, we also test the spin diffusion hypothesis and find it in a satisfactory agreement with the numerical results. Both the derivation of the modified regression relation and the justification of the computational method are based on the notion of quantum typicality.

Macrostatistics and Fluctuating Hydrodynamics

We extend our earlier macrostatistical treatment of hydrodynamical fluctuations about nonequilibrium steady states to viscous fluids. Since the scale dependence of the Navier-Stokes equations precludes the applicability of any infinite scale (hydrodynamical) limit, this has to be based on the generic model of a large but finite system, rather than an infinite one. On this basis, together with assumptions of Onsager's regression hypothesis and conditions of local equilibrium and chaoticity, we show that the hydrodynamical fluctuations of a reservoir driven fluid about a nonequilibrium steady state execute a Gaussian Markov process that constitutes a mathematical structure for a generalised version of Landau's fluctuating hydrodynamics and generically carries long range spatial correlations.

Relaxation in finite and isolated classical systems: An extension of Onsager's regression hypothesis

In order to derive the reciprocity relations, Onsager formulated a relation between thermal equilibrium fluctuations and relaxation widely known as regression hypothesis. It is shown in the present work how such a relation can be extended to finite and isolated classical systems. This extension is derived from the fluctuation-dissipation theorem for the microcanonical ensemble. The results are exemplified with a nonintegrable system in order to motivate possible applications to dynamical systems and statistical mechanics of finite systems.

Extended fluctuation-dissipation theorem for soft matter in stationary flow

For soft matter systems strongly driven by stationary flow, we discuss an extended fluctuation-dissipation theorem (FDT). Beyond the linear-response regime, the FDT for the stress acquires an additional contribution involving the observable that is conjugate to the strain rate with respect to the dissipation function. This extended FDT is evaluated both analytically for Rouse polymers and in numerical simulations for colloidal suspensions. More generally, our results suggest an extension of Onsager's regression principle to nonequilibrium steady states.

Dielectric Dispersion Interpretation of Single Enzyme Dynamic Disorder, Spectral Diffusion, and Radiative Fluorescence Lifetime

A formulation based on measurable dielectric dispersion of enzymes is developed to estimate fluctuations in electrostatic interaction energy on time scales as long as milliseconds to seconds at a local site in enzymes. Several single molecule experimental obsevations occur on this time scale, currently unreachable by real time computational trajectory simulations. We compare the experimental results on the autocorrelation function of the fluctuations of catalysis rate with the calculations using the dielectric dispersion formulation. We also discuss the autocorrelation functions of the fluorescence lifetime and of spectral diffusion. We use a previously derived relation between the observables and the electric field fluctuations and calculate the latter using dielectric dispersion data for the proteins and the Onsager regression hypothesis.

Anomalous relaxation and dielectric response

It is shown that all the known experimental (quasi)stationary dielectric response functions of glassy media can be derived from a standard generalized Langevin description of overdamped torsional dipole oscillators in trapping potentials with random orientations under some minimal assumptions. The non-Markovian theory obeys the fluctuation-dissipation theorem and the Onsager regression theorem. Moreover, it displays no aging on the time scale of the dielectric response, all in assumption of local thermal (quasi)equilibrium. Aging might come from jumping among metastable traps. It occurs on a quite different time scale which is not related to the principal dielectric response. We put the old phenomenological theory of Cole and Cole, Davidson and Cole, and others on a firm basis within a stochastic, thermodynamically consistent approach.

Calibration of trapping force and response function of optical tweezers in viscoelastic media

At present, a major obstacle to the quantitative application of optical tweezers as a force spectrometer in living cells is the lack of a method to calibrate the tweezers. Calibration with approved methods such as the power spectrum method (Berg-Sorensen and Flyvbjerg 2004 Rev. Sci. Instrum. 75 594; Berg-Sorensen et al 2006 Rev. Sci. Instrum. 77 063106) is not possible as the viscoelastic properties of the bio-active medium are a priori unknown. Here, we present an approach that neither requires explicit assumptions about the size of the trapped particle nor about the viscoelastic properties of the medium. Instead, the interaction between the medium and the trapped particle is described in a general manner, through velocity and acceleration memory. Our method is applicable to general, at least locally homogeneous, viscoelastic media. The procedure combines active and passive approaches by the application of Onsager's regression hypothesis. It allows extraction of the trapping stiffness κ of the optical tweezers and of the response function χ(ω), which is the frequency-dependent effective inverse spring constant of the system. Finally, information about the viscoelastic properties of the medium may also be found. To test the method, we have performed simulations in which the system is driven sinusoidally. These simulations serve as an example of how to deal with real experimental data. For realistic parameters, we calibrate the trap stiffness κ with ~1% stochastic error.

Theory for non-equilibrium statistical mechanics

This paper reviews a new theory for non-equilibrium statistical mechanics. This gives the non-equilibrium analogue of the Boltzmann probability distribution, and the generalization of entropy to dynamic states. It is shown that this so-called second entropy is maximized in the steady state, in contrast to the rate of production of the conventional entropy, which is not an extremum. The relationships of the new theory to Onsager's regression hypothesis, Prigogine's minimal entropy production theorem, the Langevin equation, the formula of Green and Kubo, the Kawasaki distribution, and the non-equilibrium fluctuation and work theorems, are discussed. The theory is worked through in full detail for the case of steady heat flow down an imposed temperature gradient. A Monte Carlo algorithm based upon the steady state probability density is summarized, and results for the thermal conductivity of a Lennard-Jones fluid are shown to be in agreement with known values. Also discussed is the generalization to non-equilibrium mechanical work, and to non-equilibrium quantum statistical mechanics. As examples of the new theory two general applications are briefly explored: a non-equilibrium version of the second law of thermodynamics, and the origin and evolution of life.

ERRATUM: "QUANTUM MACROSTATISTICAL THEORY OF NONEQUILIBRIUM STEADY STATES"

We provide a general macrostatistical formulation of nonequilibrium steady states of reservoir driven quantum systems. This formulation is centred on the large scale properties of the locally conserved hydrodynamical observables, and our basic assumptions comprise (a) a chaoticity hypothesis for the nonconserved currents carried by these observables, (b) an extension of Onsager's regression hypothesis to the fluctuations about nonequilibrium states, and (c) a certain mesoscopic local equilibrium hypothesis. On this basis we obtain a picture wherein the fluctuations of the hydrodynamical observables about a nonequilibrium steady state execute a Gaussian Markov process of a generalized Onsager-Machlup type, which is completely determined by the position dependent transport coefficients and the equilibrium entropy function of the system. This picture reveals that the transport coefficients satisfy a generalized form of the Onsager reciprocity relations in the nonequilibrium situation and that the spatial correlations of the hydrodynamical observables are generically of long range. This last result constitutes a model-independent generalization of that obtained for special classical stochastic systems and marks a striking difference between the steady nonequilibrium and equilibrium states, since it is only at critical points that the latter carry long range correlations.