Fluctuation-dissipation Theorem Research Articles

FDT for fluctuations of a special perturbated harmonic oscillator and its application in q-deformed Harmonic Oscillator

Abstract In this paper, we will use fluctuation dissipation theorem on the harmonic oscillator with time correlation creation annihilation operator as the perturbated term and through calculation, we find that the exact value of the dissipation associated Imχ of this special perturbation harmonic oscillator is exactly equal to π. Next, we give its potential application in the q-deformed harmonic oscillator, which means connect the q-deformed harmonic oscillator with fluctuation dissipation.

STOCHASTIC MODELING THE TRANSPORT COEFFICIENTS OF LIQUIDS

A method of stochastic molecular modeling (SMM) of liquid transport coefficients has been developed. They are calculated using fluctuation-dissipation theorems, but unlike the molecular dynamics (MD) method, the phase trajectories of the system are simulated stochastically. The force acting on the molecule is determined stochastically using the created database of intermolecular forces. The effectiveness of the method is demonstrated by the example of calculating transport coefficients. It is shown that the SMM method requires much less computational resources than the MD method.

An overdamped Langevin approach to parallel diffusion and energy dissipation in the presence of low-frequency circularly polarized electromagnetic waves

Brownian motion of charged particles in the presence of low-frequency, circularly polarized electromagnetic waves is discussed. By using the wave rest frame, the Langevin system including low-frequency waves can be treated as a periodic system with a constant external force. In an overdamped limit, parallel diffusion is formulated by the Brownian motion whose diffusion coefficient includes wave amplitude. The nonequilibrium work by low-frequency waves satisfies a nonequilibrium relation corresponding to the fluctuation–dissipation relation with the effective temperature defined by wave amplitude.

Noise–dissipation relation for nonlinear electronic circuits

An extension of fluctuation–dissipation theorem is used to derive a “speed limit” theorem for nonlinear electronic devices. This speed limit provides a lower bound on the dissipation that is incurred when transferring a given amount of electric charge in a certain amount of time with a certain noise level (average variance of the current). This bound, which implies a high energy dissipation for fast, low-noise operations (such as switching a bit in a digital memory), brings together recent results of stochastic thermodynamics into a form that is usable for practical nonlinear electronic circuits, as we illustrate on a switching circuit made of an nMOS pass gate in a state-of-the-art industrial technology.

Theory of critical phenomena with long-range temporal interaction

We systematically develop theories of critical phenomena with a prior formed memory of a power-law decaying long-range temporal interaction parameterized by a constant θ > 0 for a space dimension d both below and above an upper critical dimension d c = 6 − 2/θ. We first provide more evidences to confirm the previous theory that a dimensional constant is demanded to rectify a hyperscaling law, to produce correct unique mean-field critical exponents via an effective spatial dimension originating from temporal dimension, and to transform the time and change the dynamic critical exponent. Next, for d < d c , we develop a renormalization-group theory by employing the momentum-shell technique to the leading nontrivial order explicitly and to higher orders formally in ϵ = d c − d but to zero order in ε = 1 − θ and find that more scaling laws besides the hyperscaling law are broken due to the breaking of the fluctuation-dissipation theorem. Moreover, because dynamics and statics are intimately interwoven, even the static critical exponents involve contributions from the dynamics and hence do not restore the short-range exponents even for θ = 1 and the crossover between the short-range and long-range fixed points is discontinuous contrary to the case of long-range spatial interaction. In addition, a new scaling law relating the dynamic critical exponent with the static ones emerges, indicating that the dynamic critical exponent is not independent. However, once is displaced by a series of ϵ and ε such that most values of the critical exponents are changed, all scaling laws are saved again, even though the fluctuation-dissipation theorem keeps violating. Then, for d ≥ d c , we develop an effective-dimension theory by carefully discriminating the corrections of both temporal and spatial dimensions and find three different regions. For d ≥ d c0 = 4, the upper critical dimension of the usual short-range theory, the usual Landau mean-field theory with fluctuations confined to the effective-dimension equal to d c0 correctly describe the critical phenomena with memory, while for d c < d ≤ 4, there exist new universality classes whose critical exponents depend only on the space dimension but not at all on θ. Yet another region consists of d = d c only and the previous theory is retrieved. All these results show that the dimensional constant is the fundamental ingredient of the theories for critical phenomena with memory. However, its value continuously varies with the space dimension and vanishes exactly at d = 4, reflecting the variation of the amount of the temporal dimension that is transferred to the spatial one with the strength of fluctuations. Moreover, special finite-size scaling appears ubiquitously except for d = d c0.

A Non-local Quasi-Equilibrium State in the Bhatnagar–Gross–Krook Boltzmann Equation for Thermo-hydrodynamics: Conservation Laws, The Boltzmann H-Theorem, and the Fluctuation-Dissipation Theorem

A Non-local Quasi-Equilibrium State in the Bhatnagar–Gross–Krook Boltzmann Equation for Thermo-hydrodynamics: Conservation Laws, The Boltzmann H-Theorem, and the Fluctuation-Dissipation Theorem

Fluctuation-dissipation relations in the imbalanced Wilson-Cowan model.

The relation between spontaneous and stimulated brain activity is a fundamental question in neuroscience which has received wide attention in experimental studies. Recently, it has been suggested that the evoked response to external stimuli can be predicted from temporal correlations of spontaneous activity. Previous theoretical results, confirmed by the comparison with magnetoencephalography data for human brains, were obtained for the Wilson-Cowan model in the condition of balance of excitation and inhibition, a signature of a healthy brain. Here we extend previous studies to imbalanced conditions by examining a region of parameter space around the balanced fixed point. Analytical results are compared to numerical simulations of Wilson-Cowan networks. We evidence that in imbalanced conditions the functional form of the time correlation and response functions can show several behaviors, exhibiting also an oscillating regime caused by the emergence of complex eigenvalues. The analytical predictions are fully in agreement with numerical simulations, validating the role of cross-correlations in the response function. Furthermore, we identify the leading role of inhibitory neurons in controlling the overall activity of the system, tuning the level of excitability and imbalance.

Unification of Thermal and Quantum Noises in Gravitational-Wave Detectors.

Contemporary gravitational-wave detectors are fundamentally limited by thermal noise-due to dissipation in the mechanical elements of the test mass-and quantum noise-from the vacuum fluctuations of the optical field used to probe the test-mass position. Two other fundamental noises can in principle also limit sensitivity: test-mass quantization noise due to the zero-point fluctuation of its mechanical modes and thermal excitation of the optical field. We use the quantum fluctuation-dissipation theorem to unify all four noises. This unified picture shows precisely when test-mass quantization noise and optical thermal noise can be ignored.

Generalized Langevin equationwith shear flow and its fluctuation-dissipation theorems derived from a Caldeira-Leggett Hamiltonian.

We provide a first-principles derivation of the Langevin equation with shear flow and its corresponding fluctuation-dissipation theorems. Shear flow of simple fluids has been widely investigated by numerical simulations. Most studies postulate a Markovian Langevin equationwith a simple shear drag term in the manner of Stokes. However, this choice has never been justified from first principles. We start from a particle-bath system described by a classical Caldeira-Leggett Hamiltonian modified by adding a term proportional to the strain-rate tensor according to Hoover's DOLLS method, and we derive a generalized Langevin equationfor the sheared system. We then compute, analytically, the noise time-correlation functions in different regimes. Based on the intensity of the shear rate, we can distinguish between close-to-equilibrium and far-from-equilibrium states. According to the results presented here, the standard, simple, and Markovian form of the Langevin equationwith shear flow postulated in the literature is valid only in the limit of extremely weak shear rates compared to the effective vibrational temperature of the bath. For even marginally higher shear rates, the (generalized) Langevin equationis strongly non-Markovian, and nontrivial fluctuation-dissipation theorems are derived.

Long-time correlations in a binary mixture: analysis of the nonlinearities of fluctuating-hydrodynamic equations

The equations of fluctuating nonlinear hydrodynamics (FNH) following the proper conservation laws are considered for a binary mixture. We focus on the density (ρ) correlations’ renormalization due to the FNH equations’ nonlinearities. The consequence of density nonlinearities treated in simplest approximations gives rise to the well-studied form of the mode coupling theory (MCT) for a two-component system. The MCT predicts a sharp ergodicity–nonergodicity transition similar to the one-component fluid in this idealized form. In the first part of the present paper, we compare the predictions of the idealized MCT model with the computer simulation results for a hard sphere mixture. We show that there is clear disagreement in long-time dynamic behaviour. Next, we consider the full set of nonlinearities in the FNH equations using a Martin–Siggia–Rose field theory. From the time reversal properties of the correlation and response function of the associated field theory, a set of fluctuation–dissipation relations (FDR) are obtained. These FDRs impose constraints on the long-time behaviour of the correlation functions. Our non-perturbative analysis considers the viability of freezing the time correlations for the two-component fluid over the longest time scale. Due to the FDR constraints arising from the nonlinearity in the FNH equations, a sharp ergodicity–nonergodicity transition for the binary mixture is not supported. If the are replaced as in terms of the average density ρ 0, ad hoc, while the density-nonlinearities in the pressure term of the corresponding FNH equations are kept, the ideal transition model of the simplified MCT is recovered.

Decoding the hydrodynamic properties of microscale helical propellers from Brownian fluctuations

The complex motility of bacteria, ranging from single-swimmer behaviors such as chemotaxis to collective dynamics, including biofilm formation and active matter phenomena, is driven by their microscale propellers. Despite extensive study of swimming flagellated bacteria, the hydrodynamic properties of their helical-shaped propellers have never been directly measured. The primary challenges to directly studying microscale propellers are 1) their small size and fast, correlated motion, 2) the necessity of controlling fluid flow at the microscale, and 3) isolating the influence of a single propeller from a propeller bundle. To solve the outstanding problem of characterizing the hydrodynamic properties of these propellers, we adopt a dual statistical viewpoint that connects to the hydrodynamics through the fluctuation-dissipation theorem (FDT). We regard the propellers as colloidal particles and characterize their Brownian fluctuations, described by 21 diffusion coefficients for translation, rotation, and correlated translation-rotation in a static fluid. To perform this measurement, we applied recent advances in high-resolution oblique plane microscopy to generate high-speed volumetric movies of fluorophore-labeled, freely diffusing Escherichia coli flagella. Analyzing these movies with a bespoke helical single-particle tracking algorithm, we extracted trajectories, calculated the full set of diffusion coefficients, and inferred the average propulsion matrix using a generalized Einstein relation. Our results provide a direct measurement of a microhelix's propulsion matrix and validate proposals that the flagella are highly inefficient propellers, with a maximum propulsion efficiency of less than 3%. Our approach opens broad avenues for studying the motility of particles in complex environments where direct hydrodynamic approaches are not feasible.

Threshold Pressure in Non-Darcian Flow Derived from the Langevin Equation and Fluctuation Dissipation Theorem: Generalized Darcy’s Law

Threshold Pressure in Non-Darcian Flow Derived from the Langevin Equation and Fluctuation Dissipation Theorem: Generalized Darcy’s Law

On the Application of Non-Gaussian Noise in Stochastic Langevin Simulations

In light of recent advances in time-step independent stochastic integrators for Langevin equations, we revisit the considerations for using non-Gaussian distributions for the thermal noise term in discrete-time thermostats. We find that the desirable time-step invariance of the modern methods is rooted in the Gaussian noise, and that deviations from this distribution will distort the Boltzmann statistics arising from the fluctuation-dissipation balance of the integrators. We use the GJ stochastic Verlet methods as the focus of our investigation since these methods are the ones that contain the most accurate thermodynamic measures of existing methods. Within this set of methods we find that any distribution of applied noise, which satisfies the two first moments given by the fluctuation-dissipation theorem, will result in correct, time-step independent results that are generated by the first two moments of the system coordinates. However, if non-Gaussian noise is applied, undesired deviations in higher moments of the system coordinates will appear to the detriment of several important thermodynamic measures that depend especially on the fourth moments. The deviations, induced by non-Gaussian noise, become significant with the one-time-step velocity attenuation, thereby inhibiting the benefits of the methods. Thus, we conclude that the application of Gaussian noise is necessary for reliable thermodynamic results when using modern stochastic thermostats with large time steps.

Primitive chain network simulations of the nonlinear rheology of polystyrene melts: Friction reduction and fluctuation-dissipation theorem

Primitive chain network simulations of the nonlinear rheology of polystyrene melts: Friction reduction and fluctuation-dissipation theorem

Critical phenomena in compressible polar active fluids: Dynamical and functional renormalization group studies

Active matter is not only relevant to living matter and diverse nonequilibrium systems, but also constitutes a fertile ground for novel physics. Indeed, dynamic renormalization group (DRG) analyses have uncovered many new universality classes (UCs) in polar active fluids (PAFs)---an archetype of active matter systems. However, due to the inherent technical difficulties in the DRG methodology, almost all previous studies have been restricted to polar active fluids in the incompressible or infinitely compressible (i.e., Malthusian) limits, and, when the $\ensuremath{\epsilon}$ expansion was used in conjunction, to the one-loop level. Here, we use functional renormalization group (FRG) methods to overcome some of these difficulties and unveil critical behavior in compressible polar active fluids, and calculate the corresponding critical exponents beyond the one-loop level. Specifically, we investigate the multicritical point of compressible PAFs, where the critical order-disorder transition coincides with critical phase separation. We first study the critical phenomenon using a DRG analysis and find that it is insufficient since two-loop effects are important to obtain a nontrivial correction to the scaling exponents. We then remedy this defect by using a FRG analysis. We find three universality classes and obtain their critical exponents, which we then use to show that at least two of these universality classes are out of equilibrium because they violate the fluctuation-dissipation relation.

Fluctuation-dissipation relation, Maxwell-Boltzmann statistics, equipartition theorem, and stochastic calculus

We derive the fluctuation-dissipation relation and explore its connection with the equipartition theorem and Maxwell-Boltzmann statistics through the use of different stochastic analytical techniques. Our first approach is the theory of backward stochastic differential equations, which arises naturally in this context, and facilitates the understanding of the interplay between these classical results of statistical mechanics. Moreover, it allows to generalize the classical form of the fluctuation-dissipation relation. The second approach consists in deriving forward stochastic differential equations for the energy of an electric system according to both Itô and Stratonovich stochastic calculus rules. While the Itô equation possesses a unique solution, which is the physically relevant one, the Stratonovich equation admits this solution along with infinitely many more, none of which has a physical nature. Despite of this fact, some, but not all of them, obey the fluctuation-dissipation relation and the equipartition of energy.

Time-reversal symmetries and equilibriumlike Langevin equations.

Graham has shown [Z. Phys. B 26, 397 (1977)0340-224X10.1007/BF01570750] that a fluctuation-dissipation relation can be imposed on a class of nonequilibrium Markovian Langevin equationsthat admit a stationary solution of the corresponding Fokker-Planck equation. The resulting equilibrium form of the Langevin equationis associated with a nonequilibrium Hamiltonian. Here we provide some explicit insight into how this Hamiltonian may lose its time-reversal invariance and how the "reactive" and "dissipative" fluxes loose their distinct time-reversal symmetries. The antisymmetric coupling matrix between forces and fluxes no longer originates from Poisson brackets and the "reactive" fluxes contribute to the ("housekeeping") entropy production, in the steady state. The time-reversal even and odd parts of the nonequilibrium Hamiltonian contribute in qualitatively different but physically instructive ways to the entropy. We find instances where fluctuations due to noise are solely responsible for the dissipation. Finally, this structure gives rise to a new, physically pertinent instance of frenesy.

First-principles study of electronic and optical properties of novel 2D TiOS monolayer and bilayer—Dimensionality reduction opens up a band gap in TiOS

The finding of advanced functional materials with superior properties to existing ones to develop cutting-edge technologies for societal advancement is indispensable. We identify a new two-dimensional (2D) TiOS, which opens up a band gap due to the lowering of the dimensionality employing first-principles computations within the framework of density functional theory (DFT) and beyond. Electronic structure estimations reveal that bulk TiOS is metal whereas the 2D-TiOS possesses a band gap of ∼4.5 eV within GW approximation, which is higher than the band gap of 2D-InOF and 2D transition metal dichalcogenides. The computed phonon dispersion curves show that the 2D-TiOS is dynamically stable. The 2D-TiOS has a very high absorption coefficient in the 0–50 eV. Third-order elastic constants (TOECs) of this 2D-TiOS are achieved using DFT and within the adiabatic-connection fluctuation–dissipation theorem in the random phase approximation. Surprisingly, we find a band overlap, indicating that TiOS/TiOS bilayer is a metal. If the incident light frequency surpasses the plasma frequency (60.00 eV), and then the bilayer turns out to be transparent. Our findings suggest that this 2D sheet is a promising alternative for nanotechnology and optoelectronic devices.

An undisclosed restriction of the Fluctuation-Dissipation Theorem for electromagnetic fields

An undisclosed restriction of the Fluctuation-Dissipation Theorem for electromagnetic fields

On the link between mechanics and thermal properties: mechanothermics

We report on the theoretical derivation of macroscopic thermal properties (specific heat, thermal conductivity) of an electrically insulating rod connected to two reservoirs, from the linear superposition of its mechanical mode Brownian motions. The calculation is performed for a weak thermal gradient, in the classical limit (high temperature). The development is kept basic as far as geometry and experimental conditions are concerned, enabling an almost fully analytic treatment. In the modeling, each of the modes is subject to a specific Langevin force, which enables to produce the required temperature profile along the rod. The theory is predictive: the temperature gradient (and therefore energy transport) is linked to motion amplitude cross-correlations between nearby mechanical modes. This arises because energy transport is actually mediated by mixing between the modal waves, and not by the modes themselves. This result can be tested on experiments, and shall extend the concepts underlying equipartition and fluctuation–dissipation theorems. The theory links intimately the macroscopic size of the clamping region where the mixing occurs to the microscopic lengthscale of the problem at hand: the phonon mean-free-path. This clamping region, which is key, has received recently a renewed attention in the field of nanomechanics with topical works on ‘phonon shields’ and ‘soft clamping’. We believe that our work should impact the domain of thermal transport in nanostructures, with future developments of the theory toward the quantum regime.