Generalized Fluctuation-dissipation Relation Research Articles

Generalized Langevin equation with fluctuating diffusivity

A generalized Langevin equation with fluctuating diffusivity (GLEFD) is proposed, and it is shown that the GLEFD satisfies a generalized fluctuation-dissipation relation. If the memory kernel is a power law, the GLEFD exhibits subdiffusion, non-Gaussianity, and stretched-exponential relaxation. The case in which the memory kernel is given by a single exponential function is also investigated as an analytically tractable example. In particular, the mean-square displacement and the self-intermediate-scattering function of this system show plateau structures. A numerical scheme to integrate the GLEFD is also presented.

Extracting causation from millennial-scale climate fluctuations in the last 800 kyr

The detection of cause-effect relationships from the analysis of paleoclimatic records is a crucial step to disentangle the main mechanisms at work in the climate system. Here, we show that the approach based on the generalized Fluctuation–Dissipation Relation, complemented by the analysis of the Transfer Entropy, allows the causal links to be identified between temperature, CO_2 concentration and astronomical forcing during the glacial cycles of the last 800 kyr based on Antarctic ice core records. When considering the whole spectrum of time scales, the results of the analysis suggest that temperature drives CO_2 concentration, or that are both driven by the common astronomical forcing. However, considering only millennial-scale fluctuations, the results reveal the presence of more complex causal links, indicating that CO_2 variations contribute to driving the changes of temperature on such time scales. The results also evidence a slow temporal variability in the strength of the millennial-scale causal links between temperature and CO_2 concentration.

Depinning and flow of a vortex line in a uniaxial random medium

We study numerically and analytically the dynamics of a single directed elastic string driven through a three-dimensional disordered medium. In the quasistatic limit the string is super-rough in the direction of the driving force, with roughness exponent ${\ensuremath{\zeta}}_{\ensuremath{\parallel}}=1.25\ifmmode\pm\else\textpm\fi{}0.01$, dynamic exponent ${z}_{\ensuremath{\parallel}}=1.43\ifmmode\pm\else\textpm\fi{}0.01$, correlation-length exponent $\ensuremath{\nu}=1.33\ifmmode\pm\else\textpm\fi{}0.02$, depinning exponent $\ensuremath{\beta}=0.24\ifmmode\pm\else\textpm\fi{}0.01$, and avalanche-size exponent ${\ensuremath{\tau}}_{\ensuremath{\parallel}}=1.09\ifmmode\pm\else\textpm\fi{}0.03$. In the transverse direction we find ${\ensuremath{\zeta}}_{\ensuremath{\perp}}=0.5\ifmmode\pm\else\textpm\fi{}0.01, {z}_{\ensuremath{\perp}}=2.27\ifmmode\pm\else\textpm\fi{}0.05$, and ${\ensuremath{\tau}}_{\ensuremath{\perp}}=1.17\ifmmode\pm\else\textpm\fi{}0.06$. Our results show that transverse fluctuations do not alter the critical exponents in the driving direction, as predicted by the planar approximation (PA) proposed by Ertas and Kardar (EK) [Phys. Rev. B 53, 3520 (1996)]. We check the PA for the measured force-force correlator, comparing to the functional renormalization-group and numerical simulations. Both random-bond (RB) and random-field (RF) disorder yield a single universality class, indistinguishable from the one of an elastic string in a two-dimensional random medium. While relations ${z}_{\ensuremath{\perp}}={z}_{\ensuremath{\parallel}}+1/\ensuremath{\nu}$ and $\ensuremath{\nu}=1/(2\ensuremath{-}{\ensuremath{\zeta}}_{\ensuremath{\parallel}})$ of EK are satisfied, the transversal movement is that of a Brownian, with a clock set locally by the forward movement. This implies ${\ensuremath{\zeta}}_{\ensuremath{\perp}}=(2\ensuremath{-}d)/2$, distinct from EK. Finally, at small driving velocities the distribution of local parallel displacements has a negative skewness, while in the transverse direction it is a Gaussian. For large scales, the system can be described by anisotropic effective temperatures defined from generalized fluctuation-dissipation relations. In the fast-flow regime the local displacement distributions become Gaussian in both directions and the effective temperatures vanish as ${T}_{\mathtt{eff}}^{\ensuremath{\perp}}\ensuremath{\sim}1/v$ and ${T}_{\mathtt{eff}}^{\ensuremath{\parallel}}\ensuremath{\sim}1/{v}^{3}$ for RB disorder and as ${T}_{\mathtt{eff}}^{\ensuremath{\perp}}\ensuremath{\approx}{T}_{\mathtt{eff}}^{\ensuremath{\parallel}}\ensuremath{\sim}1/v$ for RF disorder.

Influence functional for two mirrors interacting via radiation pressure

We study the effective dynamics of two mirrors, forming an optical cavity, and interacting with the cavity field via radiation pressure. We pursue a perturbative influence functional approach to trace out the degrees-of-freedom of the field, and obtain the second order effective action for the system composed by the mirrors. We find that the interaction between the mirrors is mediated by pairs of field modes, which combine in such a way to give rise to two different interaction channels. We find that the quantum and thermal fluctuations of the cavity field result in coloured, Gaussian stochastic noises acting on the mirrors. To each of these noises is associated a dissipative effect, and the corresponding power spectra and susceptibilities are related via generalized fluctuation-dissipation relations. We finally demonstrate that the dynamics of the mirrors admits a stochastic interpretation, and give the relative quantum Langevin equations.

Data-driven learning of Mori–Zwanzig operators for isotropic turbulence

Developing reduced-order models for turbulent flows, which contain dynamics over a wide range of scales, is an extremely challenging problem. In statistical mechanics, the Mori–Zwanzig (MZ) formalism provides a mathematically exact procedure for constructing reduced-order representations of high-dimensional dynamical systems, where the effects due to the unresolved dynamics are captured in the memory kernel and orthogonal dynamics. Turbulence models based on MZ formalism have been scarce due to the limited knowledge of the MZ operators, which originates from the difficulty in deriving MZ kernels for complex nonlinear dynamical systems. In this work, we apply a recently developed data-driven learning algorithm, which is based on Koopman's description of dynamical systems and Mori's linear projection operator, on a set of fully resolved isotropic turbulence datasets to extract the Mori–Zwanzig operators. With data augmentation using known turbulence symmetries, the extracted Markov term, memory kernel, and orthogonal dynamics are statistically converged and the generalized fluctuation–dissipation relation can be verified. The properties of the memory kernel and orthogonal dynamics, and their dependence on the choices of observables are investigated to address the modeling assumptions that are commonly used in MZ-based models. A series of numerical experiments are then constructed using the extracted kernels to evaluate the memory effects on prediction. The results show that the prediction errors are strongly affected by the choice of observables and can be further reduced by including the past history of the observables in the memory kernel.

The interplay between memory and potentials of mean force: A discussion on the structure of equations of motion for coarse-grained observables

The underdamped, non-linear, generalized Langevin equation is widely used to model coarse-grained dynamics of soft and biological materials. By means of a projection operator formalism, we show under which approximations this equation can be obtained from the dynamics of the underlying microscopic system and in which cases it makes sense to introduce a potential of mean force. We discuss shortcomings of previous derivations presented in the literature and demonstrate the implications of our derivation for the structure of memory terms and for generalized fluctuation-dissipation relations. We show, in particular, that the widely used, simple structure which contains a potential of mean force, a memory term which is linear in the observable, and a fluctuating force which is related to the memory term by a fluctuation-dissipation relation, is neither exact nor can it, in general, be derived as a controlled approximation to the exact dynamics.

Orbital magnetism of an active particle in viscoelastic suspension.

We consider an active (self-propelling) particle in a viscoelastic fluid. The particle is charged and constrained to move in a two-dimensional harmonic trap. Its dynamics is coupled to a constant magnetic field applied perpendicular to its plane of motion via Lorentz force. Due to the finite activity, the generalized fluctuation-dissipation relation (GFDR) breaks down, driving the system away from equilibrium. While breaking GFDR, we have shown that the system can have finite classical orbital magnetism only when the dynamics of the system contains finite inertia. The orbital magnetic moment has been calculated exactly. Remarkably, we find that when the elastic dissipation timescale of the medium is larger (smaller) than the persistence timescale of the self-propelling particle, it is diamagnetic (paramagnetic). Therefore, for a given strength of the magnetic field, the system undergoes a transition from diamagnetic to paramagnetic state (and vice versa) simply by tuning the timescales of underlying physical processes, such as active fluctuations and viscoelastic dissipation. Interestingly, we also find that the magnetic moment, which vanishes at equilibrium, behaves nonmonotonically with respect to increasing persistence of self-propulsion, which drives the system out of equilibrium.

Generalized fluctuation–dissipation relations holding in non-equilibrium dynamics

We derive generalized fluctuation–dissipation relations (FDR) holding for a general stochastic dynamics that includes as subcases both equilibrium models for passive colloids and non-equilibrium models used to describe active particles. The relations reported here differ from previous formulations of the FDR because of their simplicity: they require only the microscopic knowledge of the dynamics instead of the whole expression of the steady-state probability distribution function that, except for linear interactions, is unknown for systems displaying non-vanishing currents. From the response function, we can extrapolate generalized versions of the mesoscopic virial equation and the equipartition theorem, which still holds far from equilibrium. Our results are tested in the case of equilibrium colloids described by underdamped or overdamped Langevin equations and for models describing the non-equilibrium behavior of active particles. Both the active Brownian particle and the active Ornstein–Uhlenbeck particle models are compared in the case of a single particle confined in an external potential.

Zeroth law in quantum thermodynamics at strong coupling: In equilibrium, not at equal temperature

The zeroth law of thermodynamics involves a transitivity relation (pairwise between three objects) expressed either in terms of ``equal temperature'' (ET), or ``in equilibrium'' (EQ) conditions. In conventional thermodynamics conditional on vanishingly weak system-bath coupling these two conditions are commonly regarded as equivalent. In this work we show that for thermodynamics at strong coupling they are inequivalent: namely, two systems can be in equilibrium and yet have different effective temperatures. A recent result [J.-T. Hsiang and B. L. Hu, Phys. Rev. D 103, 065001 (2021) for Gaussian quantum systems shows that an effective temperature ${T}^{*}$ can be defined at all times during a system's nonequilibrium evolution, but because of the inclusion of interaction energy, after equilibration the system's ${T}^{*}$ is slightly higher than the bath temperature ${T}_{\mathrm{B}}$, with the deviation depending on the coupling. A second object coupled with a different strength with an identical bath at temperature ${T}_{\mathrm{B}}$ will not have the same equilibrated temperature as the first object. Thus $ET\ensuremath{\ne}EQ$ for strong coupling thermodynamics. We then investigate the conditions for dynamical equilibration for two objects 1 and 2 strongly coupled with a common bath $B$, each with a different equilibrated effective temperature. We show this is possible, and prove the existence of a generalized fluctuation-dissipation relation under this configuration. This affirms that in equilibrium is a valid and perhaps more fundamental notion which the zeroth law for quantum thermodynamics at strong coupling should be based on. Only when the system-bath coupling becomes vanishingly weak that ``temperature'' appearing in thermodynamic relations becomes universally defined and makes better physical sense.

Dissipative dynamics of a particle coupled to a field via internal degrees of freedom

We study the non-equilibrium dissipative dynamics of the center of mass of a particle coupled to a field via its internal degrees of freedom. We model the internal and external degrees of freedom of the particle as quantum harmonic oscillators in 1+1 D, with the internal oscillator coupled to a scalar quantum field at the center of mass position. Such a coupling results in a nonlinear interaction between the three pertinent degrees of freedom -- the center of mass, internal degree of freedom, and the field. It is typically assumed that the internal dynamics is decoupled from that of the center of mass owing to their disparate characteristic time scales. Here we use an influence functional approach that allows one to account for the self-consistent backaction of the different degrees of freedom on each other, including the coupled non-equilibrium dynamics of the internal degree of freedom and the field, and their influence on the dissipation and noise of the center of mass. Considering a weak nonlinear interaction term, we employ a perturbative generating functional approach to derive a second order effective action and a corresponding quantum Langevin equation describing the non-equilibrium dynamics of the center of mass. We analyze the resulting dissipation and noise arising from the field and the internal degree of freedom as a composite environment. Furthermore, we establish a generalized fluctuation-dissipation relation for the late-time dissipation and noise kernels. Our results are pertinent to open quantum systems that possess intermediary degrees of freedom between system and environment, such as in the case of optomechanical interactions.

Field-Dependent Ionic Conductivities from Generalized Fluctuation-Dissipation Relations.

We derive a relationship for the electric field dependent ionic conductivity in terms of fluctuations of time integrated microscopic variables. We demonstrate this formalism with molecular dynamics simulations of solutions of differing ionic strength with implicit solvent conditions and molten salts. These calculations are aided by a novel nonequilibrium statistical reweighting scheme that allows for the conductivity to be computed as a continuous function of the applied field. In strong electrolytes, we find the fluctuations of the ionic current are Gaussian, and subsequently, the conductivity is constant with applied field. In weaker electrolytes and molten salts, we find the fluctuations of the ionic current are strongly non-Gaussian, and the conductivity increases with applied field. This nonlinear behavior, known phenomenologically for dilute electrolytes as the Onsager-Wien effect, is general and results from the suppression of ionic correlations at large applied fields, as we elucidate through both dynamic and static correlations within nonequilibrium steady states.

On the fluctuation-dissipation relation in non-equilibrium and non-Hamiltonian systems.

We review generalized fluctuation-dissipation relations, which are valid under general conditions even in "nonstandard systems," e.g., out of equilibrium and/or without a Hamiltonian structure. The response functions can be expressed in terms of suitable correlation functions computed in the unperturbed dynamics. In these relations, typically, one has nontrivial contributions due to the form of the stationary probability distribution; such terms take into account the interaction among the relevant degrees of freedom in the system. We illustrate the general formalism with some examples in nonstandard cases, including driven granular media, systems with a multiscale structure, active matter, and systems showing anomalous diffusion.

Entropy and entropy production in Fokker–Plank equation under the generalized fluctuation–dissipation relation

In this paper, we investigate the definition of the entropy in the Fokker–Planck equation under the generalized fluctuation–dissipation relation (FDR), which describes a Brownian particle moving in a complex medium with friction and multiplicative noise. The friction and the noise are related by the generalized FDR. The entropy for such a system is defined first. According to the definition of the entropy, we calculate the entropy production and the entropy flux. Lastly, we make a numerical calculation to display the results in figures.

Effective field theory out of equilibrium: Brownian quantum fields

The emergence of an effective field theory out of equilibrium is studied in the case in which a light field—the system—interacts with very heavy fields in a finite temperature bath. We obtain the reduced density matrix for the light field, its time evolution is determined by an effective action that includes the influence action from correlations of the heavy degrees of freedom. The non-equilibrium effective field theory yields a Langevin equation of motion for the light field in terms of dissipative and noise kernels that obey a generalized fluctuation dissipation relation. These are completely determined by the spectral density of the bath which is analyzed in detail for several cases. At T = 0 we elucidate the effect of thresholds in the renormalization aspects and the asymptotic emergence of a local effective field theory with unitary time evolution. At new ‘anomalous’ thresholds arise, in particular the decay of the environmental heavy fields into the light field leads to dissipative dynamics of the light field. Even when the heavy bath particles are thermally suppressed this dissipative contribution leads to the thermalization of the light field which is confirmed by a quantum kinetics analysis. We obtain the quantum master equation and show explicitly that its solution in the field basis is precisely the influence action that determines the effective non-equilibrium field theory. The Lindblad form of the quantum master equation features time dependent dissipative coefficients. Their time dependence is crucial to extract renormalization effects at asymptotically long time. The dynamics from the quantum master equation is in complete agreement with that of the effective action, Langevin dynamics and quantum kinetics, thus providing a unified framework to effective field theory out of equilibrium.

The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion

We study the time behavior of the Fokker–Planck equation in Zwanzig’s rule (the backward-Ito’s rule) based on the Langevin equation of Brownian motion with an anomalous diffusion in a complex medium. The diffusion coefficient is a function in momentum space and follows a generalized fluctuation–dissipation relation. We obtain the precise time-dependent analytical solution of the Fokker–Planck equation and at long time the solution approaches to a stationary power-law distribution in nonextensive statistics. As a test, numerically we have demonstrated the accuracy and validity of the time-dependent solution.

Are power-law distributions an equilibrium distribution or a stationary nonequilibrium distribution?

We examine whether the principle of detailed balance holds for the power-law distributions that are generated from the well-known two-variable Langevin equation and the associated Fokker–Planck equations. With the detailed balance and the generalized fluctuation–dissipation relation, we derive analytically the stationary power-law distribution from the Ito’s, Stratonovich’s and Zwanzig’s Fokker–Planck equations, and conclude that the power-law distributions can either be a stationary nonequilibrium distribution or an equilibrium distribution, which depend on information about the form of the diffusion coefficient function and the existence and uniqueness of an equilibrium state.

Time asymmetry of the Kramers equation with nonlinear friction: Fluctuation-dissipation relation and ratchet effect

We show by numerical simulations that the presence of nonlinear velocity-dependent friction forces can induce a finite net drift in the stochastic motion of a particle in contact with an equilibrium thermal bath and in an asymmetric periodic spatial potential. In particular, we study the Kramers equation for a particle subjected to Coulomb friction, namely a constant force acting in the direction opposite to the particle's velocity. We characterize the nonequilibrium irreversible dynamics by studying the generalized fluctuation-dissipation relation for this ratchet model driven by Coulomb friction.

Thermal balance and quantum heat transport in nanostructures thermalized by local Langevin heat baths

Modeling of thermal transport in practical nanostructures requires making tradeoffs between the size of the system and the completeness of the model. We study quantum heat transfer in a self-consistent thermal bath setup consisting of two lead regions connected by a center region. Atoms both in the leads and in the center region are coupled to quantum Langevin heat baths that mimic the damping and dephasing of phonon waves by anharmonic scattering. This approach treats the leads and the center region on the same footing and thereby allows for a simple and physically transparent thermalization of the system, enabling also perfect acoustic matching between the leads and the center region. Increasing the strength of the coupling reduces the mean-free path of phonons and gradually shifts phonon transport from ballistic regime to diffusive regime. In the center region, the bath temperatures are determined self-consistently from the requirement of zero net energy exchange between the local heat bath and each atom. By solving the stochastic equations of motion in frequency space and averaging over noise using the general fluctuation-dissipation relation derived by Dhar and Roy [J. Stat. Phys. 125, 801 (2006)], we derive the formula for thermal current, which contains the Caroli formula for phonon transmission function and reduces to the Landauer-Büttiker formula in the limit of vanishing coupling to local heat baths. We prove that the bath temperatures measure local kinetic energy and can, therefore, be interpreted as true atomic temperatures. In a setup where phonon reflections are eliminated, the Boltzmann transport equation under gray approximation with full phonon dispersion is shown to be equivalent to the self-consistent heat bath model. We also study thermal transport through two-dimensional constrictions in square lattice and graphene and discuss the differences between the exact solution and linear approximations.

Fluctuation–dissipation relations. Achievements and misunderstandings

We discuss the ‘generalized fluctuation–dissipation relations (theorems)’ introduced for the first time in our work of 1977–1984 as statistical-thermodynamic consequences of time symmetry (reversibility) of microscopic dynamics. We show, in particular, that various similar relations, including ‘fluctuation theorems’ that have appeared since the 1990s, are, in essence, alternative formulations or special cases of our old results.

Generalized Fluctuation-Dissipation Relation and Effective Temperature Upon Heating a Deeply Supercooled Liquid

We show that a generalized fluctuation-dissipation relation applies upon instantaneously increasing the temperature of a deeply supercooled liquid. This has the same two-step shape of the relation found upon cooling the liquid, but with opposite violation, indicating an effective temperature that is lower than bath temperature. We show that the effective temperature exhibits some sensible time dependence and that it retains its connection with the partitioned phase space visited in aging. We underline the potential relevance of our numerical results for experimental studies of the fluctuation-dissipation relation in glassy systems.