Fluctuation-dissipation Theorem Research Articles

Anomalous and ultraslow diffusion of a particle driven by power-law-correlated and distributed-order noises

We study the generalised Langevin equation (GLE) approach to anomalous diffusion for a harmonic oscillator and a free particle driven by different forms of internal noises, such as power-law-correlated and distributed-order noises that fulfil generalised versions of the fluctuation-dissipation theorem. The mean squared displacement and the normalised displacement correlation function are derived for the different forms of the friction memory kernel. The corresponding overdamped GLEs for these cases are also investigated. It is shown that such models can be used to describe anomalous diffusion in complex media, giving rise to subdiffusion, superdiffusion, ultraslow diffusion, strong anomaly, and other complex diffusive behaviours.

Out-of-Equilibrium Fluctuation-Dissipation Bounds.

We prove a general inequality between the charge current and its fluctuations valid for any weakly interacting coherent electronic conductor and for any stationary out-of-equilibrium condition, thereby going beyond established fluctuation-dissipation relations. The developed fluctuation-dissipation bound saturates at large temperature bias and reveals additional insight for heat engines, since it limits the output power by power fluctuations. It is valid when the thermodynamic uncertainty relations break down due to quantum effects and provides stronger constraints close to thermovoltage.

Fluctuation-dissipation relation in cosmic microwave background

We study the fluctuation-dissipation relation for sound waves in the cosmic microwave background (CMB), employing effective field theory (EFT) for fluctuating hydrodynamics.Treating sound waves as the linear response to thermal radiation, we establish the fluctuation-dissipation relation within a cosmological framework.While dissipation is elucidated in established linear cosmological perturbation theory, the standard Boltzmann theory overlooks the associated noise, possibly contributing to inconsistencies in Lambda Cold Dark Matter (ΛCDM) cosmology. This paper employs EFT for fluctuating hydrodynamics in cosmological perturbation theory, deriving sound wave noise.Notably, the long-time limit of the noise spectrum is independent of viscosity details, resembling a Brownian motion bounded in a harmonic potential.The net energy transfer between the sound wave system and the radiation environment reaches a balance within Hubble time, suggesting the thermal equilibrium of the sound waves themselves.The induced density power spectrum is characterized as white noise dependent on the inverse of the entropy density, which is negligibly small on the CMB scale. The energy density of the entire sound wave system scales as a -4, akin to radiation. While the numerical factor is not determined in the present calculation, the back reaction of the sound wave system to the background radiation may not be negligible, serving as a potential source for various fitting issues in ΛCDM cosmology.

Thermodynamic constraints on kinetic perturbations of homogeneous driven diffusions

We analyze the static response to kinetic perturbations of nonequilibrium steady states that can be modeled as diffusions. We demonstrate that kinetic response is purely a nonequilibirum effect, measuring the degree to which the Fluctuation-Dissipation Theorem is violated out of equilibrium. For driven diffusions in a flat landscape, we further demonstrate that such response is constrained by the strength of the nonequilibrium driving via quantitative inequalities.

Energetics of the dissipative quantum oscillator

In this paper, we discuss some aspects of the energetics of a quantum Brownian particle placed in a harmonic trap, also known as the dissipative quantum oscillator. Based on the fluctuation–dissipation theorem, we analyze two distinct notions of thermally-averaged energy that can be ascribed to the oscillator. These energy functions, respectively dubbed hereafter as the mean energy and the internal energy, are found to be inequivalent for arbitrary system–bath coupling strength when the dissipation kernel has a non-zero memory timescale, as in the case of a Drude bath. Remarkably, both the energy functions satisfy the quantum counterpart of energy equipartition theorem, but with different probability distributions on the frequency domain of the heat bath. Moreover, the Gibbs approach to thermodynamics provides us with yet another thermally-averaged energy function. In the weak-coupling limit, all the above-mentioned energy expressions reduce to ϵ=ħω02coth(ħω02kBT), which is the familiar result. We generalize our analysis to the case of the three-dimensional dissipative charged oscillator placed in a spatially-uniform magnetic field.

Impedance of nanocapacitors from molecular simulations to understand the dynamics of confined electrolytes

Nanoelectrochemical devices have become a promising candidate technology across various applications, including sensing and energy storage, and provide new platforms for studying fundamental properties of electrode/electrolyte interfaces. In this work, we employ constant-potential molecular dynamics simulations to investigate the impedance of gold-aqueous electrolyte nanocapacitors, exploiting a recently introduced fluctuation-dissipation relation. In particular, we relate the frequency-dependent impedance of these nanocapacitors to the complex conductivity of the bulk electrolyte in different regimes, and use this connection to design simple but accurate equivalent circuit models. We show that the electrode/electrolyte interfacial contribution is essentially capacitive and that the electrolyte response is bulk-like even when the interelectrode distance is only a few nanometers, provided that the latter is sufficiently large compared to the Debye screening length. We extensively compare our simulation results with spectroscopy experiments and predictions from analytical theories. In contrast to experiments, direct access in simulations to the ionic and solvent contributions to the polarization allows us to highlight their significant and persistent anticorrelation and to investigate the microscopic origin of the timescales observed in the impedance spectrum. This work opens avenues for the molecular interpretation of impedance measurements, and offers valuable contributions for future developments of accurate coarse-grained representations of confined electrolytes.

Data-driven dimensionality reduction and causal inference for spatiotemporal climate fields.

We propose a data-driven framework to describe spatiotemporal climate variability in terms of a few entities and their causal linkages. Given a high-dimensional climate field, the methodology first reduces its dimensionality into a set of regionally constrained patterns. Causal relations among such patterns are then inferred in the interventional sense through the fluctuation-response formalism. To distinguish between true and spurious responses, we propose an analytical null model for the fluctuation-dissipation relation, therefore allowing us for uncertainty estimation at a given confidence level. We showcase the methodology on the sea surface temperature field from a state-of-the-art climate model. The usefulness of the proposed framework for spatiotemporal climate data is demonstrated in several ways. First, we focus on the correct identification of known causal relations across tropical basins. Second, we show how the methodology allows us to visualize the cumulative response of the whole system to climate variability in a few selected regions. Finally, each pattern is ranked in terms of its causal strength, quantifying its relative ability to influence the system's dynamics. We argue that the methodology allows us to explore and characterize causal relations in spatiotemporal fields in a rigorous and interpretable way.

Fluctuation-response relation as a probe of long-range correlations in nonequilibrium quantum and classical fluids.

The absence of a simple fluctuation-dissipation theorem is a major obstacle for studying systems that are not in thermodynamic equilibrium. We show that for a fluid in a nonequilibrium steady state characterized by a constant temperature gradient the commutator correlation functions are still related to response functions; however, the relation is to the bilinear response of products of two observables, rather than to a single linear response function as is the case in equilibrium. This modified fluctuation-response relation holds for both quantum and classical systems. It is both motivated and informed by the long-range correlations that exist in such a steady state and allows for probing them via response experiments. This is of particular interest in quantum fluids, where the direct observation of fluctuations by light scattering would be difficult. In classical fluids it is known that the coupling of the temperature gradient to the diffusive shear velocity leads to correlations of various observables, in particular temperature fluctuations, that do not decay as a function of distance, but rather extend over the entire system. We investigate the nature of these correlations in a fermionic quantum fluid and show that the crucial coupling between the temperature gradient and velocity fluctuations is the same as in the classical case. Accordingly, the nature of the long-ranged correlations in the hydrodynamic regime also is the same. However, as one enters the collisionless regime in the low-temperature limit the nature of the velocity fluctuations changes: they become ballistic rather than diffusive. As a result, correlations of the temperature and other observables are still singular in the long-wavelength limit, but the singularity is weaker than in the hydrodynamic regime.

Ab initio generalized Langevin equation

We introduce a machine learning-based approach called ab initio generalized Langevin equation (AIGLE) to model the dynamics of slow collective variables (CVs) in materials and molecules. In this scheme, the parameters are learned from atomistic simulations based on ab initio quantum mechanical models. Force field, memory kernel, and noise generator are constructed in the context of the Mori-Zwanzig formalism, under the constraint of the fluctuation-dissipation theorem. Combined with deep potential molecular dynamics and electronic density functional theory, this approach opens the way to multiscale modeling in a variety of situations. Here, we demonstrate this capability with a study of two mesoscale processes in crystalline lead titanate, namely the field-driven dynamics of a planar ferroelectric domain wall, and the dynamics of an extensive lattice of coarse-grained electric dipoles. In the first case, AIGLE extends the reach of ab initio simulations to a regime of noise-driven motions not accessible to molecular dynamics. In the second case, AIGLE deals with an extensive set of CVs by adopting a local approximation for the memory kernel and retaining only short-range noise correlations. The scheme is computationally more efficient than molecular dynamics by several orders of magnitude and mimics the microscopic dynamics at low frequencies where it reproduces accurately the dominant far-infrared absorption frequency.

Thermal radiation transferred to the guided modes of optical interference coatings.

An electromagnetic model is developed to predict the thermal radiation which is trapped in a multilayer structure and transferred to its guided modes. The theory is based on the electromagnetic power supplied by the thermal currents given by the fluctuation-dissipation theorem. The source of the radiation is the ambient temperature or that caused by the optical absorption of the component subjected to spatio-temporal illumination. A numerical example is given for a multi-dielectric mirror at thermodynamic equilibrium. It is shown that the thermal radiation transferred to the guided modes of the multilayer can be much larger or lower than the radiation emerging in free space outside the component.

Restoring the Fluctuation-Dissipation Theorem in Kardar-Parisi-Zhang Universality Class through a New Emergent Fractal Dimension.

The Kardar-Parisi-Zhang (KPZ) equation describes a wide range of growth-like phenomena, with applications in physics, chemistry and biology. There are three central questions in the study of KPZ growth: the determination of height probability distributions; the search for ever more precise universal growth exponents; and the apparent absence of a fluctuation-dissipation theorem (FDT) for spatial dimension d>1. Notably, these questions were answered exactly only for 1+1 dimensions. In this work, we propose a new FDT valid for the KPZ problem in d+1 dimensions. This is achieved by rearranging terms and identifying a new correlated noise which we argue to be characterized by a fractal dimension dn. We present relations between the KPZ exponents and two emergent fractal dimensions, namely df, of the rough interface, and dn. Also, we simulate KPZ growth to obtain values for transient versions of the roughness exponent α, the surface fractal dimension df and, through our relations, the noise fractal dimension dn. Our results indicate that KPZ may have at least two fractal dimensions and that, within this proposal, an FDT is restored. Finally, we provide new insights into the old question about the upper critical dimension of the KPZ universality class.

Basis Set Requirements of σ-Functionals for Gaussian- and Slater-Type Basis Functions and Comparison with Range-Separated Hybrid and Double Hybrid Functionals.

σ-Functionals belong to the class of Kohn-Sham (KS) correlation functionals based on the adiabatic-connection fluctuation-dissipation theorem and are technically closely related to the random phase approximation (RPA). They have the same computational demand as the latter, with the computational effort of an energy evaluation for both methods being lower than that of a preceding hybrid DFT calculation for typical systems but yield much higher accuracy, reaching chemical accuracy of 1 kcal/mol for quantities such as reactions and transition energies in main group chemistry. In previous work on σ-functionals, rather large Gaussian basis sets have been used. Here, we investigate the actual basis set requirements of σ-functionals and present three setups that employ smaller Gaussian basis sets ranging from quadruple-ζ (QZ) to triple-ζ (TZ) quality and represent a good compromise between accuracy and computational efficiency. Furthermore, we introduce an implementation of σ-functionals based on Slater-type basis sets and present two setups of QZ and TZ quality for this implementation. We test the accuracy of these setups on a large database of various physical properties and types of reactions, as well as equilibrium geometries and vibrational frequencies. As expected, the accuracy of σ-functional calculations becomes somewhat lower with a decreasing basis set size. However, for all setups considered here, calculations with σ-functionals are clearly more accurate than those within the RPA and even more so than those of the conventional KS methods. For the smallest setup using Gaussian-type basis functions and Slater-type basis functions, we introduce a reparametrization that reduces the loss in accuracy due to the basis set error to some extent. A comparison with the range-separated hybrid ωB97X-V and the double hybrid DSD-BLYP-D3 shows that σ functionals outperform in accuracy both of these accurate and, for their class, representative functionals.

Interpreting pore-scale fluctuations: Predicting transport coefficients in multiphase flow through porous media using the Green–Kubo formulation—An experimental investigation

Flow fluctuations that are commonly associated with multiphase flow in porous media are studied using concepts from non-equilibrium thermodynamic and statistical mechanics. We investigate how the Green–Kubo formulation of the fluctuation dissipation theorem can be used to predict the transport coefficient from the two-phase extension of Darcy's law. Flow rate-time series data are recorded at the millisecond timescale using a novel experimental setup that allows for the determination of flow fluctuation statistics. By using Green–Kubo relations, a transport coefficient is predicted based on the integrated autocorrelation function. Notably, this coefficient aligned closely with the total effective phase mobility computed using Darcy's equation for multiphase flow, particularly in scenarios where a linear relationship between flow rate and pressure gradient was observed. Our results open a new field of coefficient explorations where microscale fluctuations during multiphase flow are directly linked to macroscale parameters.

Introduction to dynamical mean-field theory of randomly connected neural networks with bidirectionally correlated couplings

Dynamical mean-field theory is a powerful physics tool used to analyze the typical behavior of neural networks, where neurons can be recurrently connected, or multiple layers of neurons can be stacked. However, it is not easy for beginners to access the essence of this tool and the underlying physics. Here, we give a pedagogical introduction of this method in a particular example of random neural networks, where neurons are randomly and fully connected by correlated synapses and therefore the network exhibits rich emergent collective dynamics. We also review related past and recent important works applying this tool. In addition, a physically transparent and alternative method, namely the dynamical cavity method, is also introduced to derive exactly the same results. The numerical implementation of solving the integro-differential mean-field equations is also detailed, with an illustration of exploring the fluctuation dissipation theorem.

Schwinger-Keldysh effective field theory for stable and causal relativistic hydrodynamics

We construct stable and causal effective field theories (EFTs) for describing statistical fluctuations in relativistic diffusion and relativistic hydrodynamics. These EFTs are fully non-linear, including couplings to background sources, and enable us to compute n-point time-ordered correlation functions including the effects of statistical fluctuations. The EFTs we construct are inspired by the Maxwell-Cattaneo model of relativistic diffusion and Müller-Israel-Stewart model of relativistic hydrodynamics respectively, and have been derived using both the Martin-Siggia-Rose and Schwinger-Keldysh formalisms. The EFTs non-linearly realise the dynamical Kubo-Martin-Schwinger (KMS) symmetry, which ensures that n-point correlation functions and interactions in the theory satisfy the appropriate fluctuation-dissipation theorems. Since these EFTs typically admit ultraviolet sectors that are not fixed by the low-energy infrared symmetries, we find that they simultaneously admit multiple realisations of the dynamical KMS symmetry. We also comment on certain obstructions to including statistical fluctuations in the recently-proposed stable and causal Bemfica-Disconzi-Noronha-Kovtun model of relativistic hydrodynamics.

Brownian Particle in a Poisson‐Shot‐Noise Active Bath: Exact Statistics, Effective Temperature, and Inference

AbstractThe dynamics of an overdamped Brownian particle in a thermal bath that contains a dilute solution of active particles is studied. The particle moves in a harmonic potential and experiences Poisson shot‐noise kicks with specified amplitude distribution due to moving active particles in the bath. From the Fokker–Planck equation for the particle dynamics, the stationary solution for the displacement distribution is derived along with the moments characterizing mean, variance, skewness, and kurtosis, as well as finite‐time first and second moments. An effective temperature is also computed through the fluctuation–dissipation theorem and show that equipartition theorem holds for all zero‐mean kick distributions, including those leading to non‐Gaussian stationary statistics. For the case of Gaussian‐distributed active kicks, a re‐entrant behavior from non‐Gaussian to Gaussian stationary states and a heavy‐tailed leptokurtic distribution across a wide range of parameters are found as seen in recent experimental studies. Further analysis reveals statistical signatures of the irreversible dynamics of the particle displacement in terms of the time asymmetry of cross‐correlation functions. Fruits of the work is the development of an compact inference scheme that may allow experimentalists to extract the rate and moments of underlying shot‐noise solely from the statistics the particle position.

Kinetic theory of inhomogeneous systems with long-range interactions and fluctuation–dissipation theorem

Kinetic theory of inhomogeneous systems with long-range interactions and fluctuation–dissipation theorem

MD Simulation of Water Using a Rigid Body Description Requires a Small Time Step to Ensure Equipartition.

In simulations of aqueous systems, it is common to freeze the bond vibration and angle bending modes in water to allow for a longer time step δt for integrating the equations of motion. Thus, δt = 2 fs is often used in simulating rigid models of water. We simulate the SPC/E model of water using δt from 0.5 to 3.0 fs and up to 4 fs using hydrogen mass repartitioning. In these simulations, we find that for all but δt = 0.5 fs, equipartition is not obtained between translational and rotational modes, with the rotational modes exhibiting a lower temperature than the translation modes. To probe the reasons for the lack of equipartition, we study the autocorrelation of the translational velocity of the center of mass and the angular velocity of the rigid water molecule, respectively. We find that the rotational relaxation occurs on a timescale comparable to vibrational periods, calling into question the original motivations for freezing the vibrations. Furthermore, a time step with δt ≥ 1 fs is not able to capture accurately the fast rotational relaxation, which reveals its impact as an effective slowing-down of rotational relaxation. The fluctuation-dissipation relation then leads to the conclusion that the rotational temperature should be cooler for δt greater than the reference value of 0.5 fs. Consideration of fluctuation-dissipation in equilibrium molecular dynamics simulations also emphasizes the need to capture the temporal evolution of fluctuations with fidelity and the role of δt in this regard. The time step also influences the solution thermodynamic properties: both the mean system potential energies and the excess entropy of hydration of a soft repulsive cavity are sensitive to δt.

Avoiding spin contamination and spatial symmetry breaking by exact-exchange-only optimized-effective-potential methods within the symmetrized Kohn-Sham framework.

For open-shell atoms and molecules, Kohn-Sham (KS) methods typically resort to spin-polarized approaches that exhibit spin-contamination and often break spatial symmetries. As a result, the KS Hamiltonian operator and the KS orbitals do not exhibit the space and spin symmetry of the physical electron system. The KS formalism can be symmetrized in a rigorous way only in real space, only in spin space, or both in real and spin space. Within such symmetrized KS frameworks, we present exact-exchange-only optimized-effective-potential (OEP) methods that are free of spin contamination and/or spatial symmetry breaking. The effect of symmetrizations on the total energy and its parts and on the exchange potential is analyzed. The presented exact-exchange-only OEP methods may serve as a starting point for high-level symmetrized KS methods based, e.g., on the adiabatic-connection fluctuation-dissipation theorem.

Violations of the fluctuation-dissipation theorem reveal distinct nonequilibrium dynamics of brain states.

The brain is a nonequilibrium system whose dynamics change in different brain states, such as wakefulness and deep sleep. Thermodynamics provides the tools for revealing these nonequilibrium dynamics. We used violations of the fluctuation-dissipation theorem to describe the hierarchy of nonequilibrium dynamics associated with different brain states. Together with a whole-brain model fitted to empirical human neuroimaging data, and deriving the appropriate analytical expressions, we were able to capture the deviation from equilibrium in different brain states that arises from asymmetric interactions and hierarchical organization.