Dissipation Theorem Research Articles

Decoherence of quantum Brownian particle trapped in a penning potential in a non-commutative space

In this paper, the influence of non-commutativity on quantum systems interacting with the environment in the Penning trap potential is studied. The considered system is a Brownian particle expressed in non-commuting spatial coordinates coupled to a set of harmonic oscillators (environment). The equilibrium state of the total system has been evaluated using the fluctuation dissipation theorem. The effective parameters of the system are evaluated and compared with those of the system in commutative space. We found that non-commutativity effects give rise to an increase in decoherence in the system, and negative heat capacity confirms the presence of self-gravitation effects induced by non-commutativity.

A theory of average response to large jump perturbations.

A key feature of the classical Fluctuation Dissipation theorem is its ability to approximate the average response of a dynamical system to a sufficiently small external perturbation from an appropriate time correlation function of the unperturbed dynamics of this system. In the present work, we examine the situation where the state of a nonlinear dynamical system is perturbed by a finitely large, instantaneous external perturbation (jump), for example, the Earth climate perturbed by an extinction level event. Such jump can be either deterministic or stochastic, and in the case of a stochastic jump its randomness can be spatial, or temporal, or both. We show that, even for large instantaneous jumps, the average response of the system can be expressed in the form of a suitable time correlation function of the corresponding unperturbed dynamics. For stochastic jumps, we consider two situations: one where a single spatially random jump of a system state occurs at a predetermined time, and the other where jumps occur randomly in time with small space-time dependent statistical intensity. For all studied configurations, we compute the corresponding average response formulas in the form of suitable time correlation functions of the unperturbed dynamics. Some efficiently computable approximations are derived for practical modeling scenarios.

Linear response theory for quantum Gaussian processes

Fluctuation dissipation theorems (FDTs) connect the linear response of a physical system to a perturbation to the steady-state correlation functions. Until now, most of these theorems have been derived for finite-dimensional systems. However, many relevant physical processes are described by systems of infinite dimension in the Gaussian regime. In this work, we find a linear response theory for quantum Gaussian systems subject to time dependent Gaussian channels. In particular, we establish a FDT for the covariance matrix that connects its linear response at any time to the steady state two-time correlations. The theorem covers non-equilibrium scenarios as it does not require the steady state to be at thermal equilibrium. We further show how our results simplify the study of Gaussian systems subject to a time dependent Lindbladian master equation. Finally, we illustrate the usage of our new scheme through some examples. Due to broad generality of the Gaussian formalism, we expect our results to find an application in many physical platforms, such as opto-mechanical systems in the presence of external noise or driven quantum heat devices.

Gravity analog model of non-equilibrium thermodynamics

Abstract The non-equilibrium thermodynamics of Onsager and Machlup and of Hashitsume is reformulated as a gravity analog model, in which thermodynamic variables, kinetic coefficients, and generalized forces form, respectively, coordinates and metric tensor and vector fields in a space of thermodynamic variables. The relevant symmetry of the model is the general coordinate transformation. Then, the entropy production is classified into three categories, when a closed path is depicted as a thermodynamic cycle. One category is time-reversal odd, and is attributed to the number of lines of magnetic flux passing through the closed path, having the monopole as a source. There are two time-reversal-even categories, one of which is attributed to the space curvature around the path, having the gravitational instanton as a source, which dominates for a rapid operation of the cycle. The last category is the usual one, which remains even for the quasi-equilibrium operation. It is possible to extend the model to include non-linear responses. In introducing new terms, dimensional counting is important, using two parameters, the temperature and the relaxation time. The effective action, being induced by the non-equilibrium thermodynamics, is derived. This is a candidate for the action that controls the dynamics of kinetic coefficients and thermodynamic forces. An example is given in a chemical oscillatory reaction in a solvent of van der Waals type. The fluctuation–dissipation theorem is examined à la Onsager, and a derivation of the gravity analog thermodynamic model from quantum mechanics is sketched, based on an analogy to the resonance problem.

Callen–Welton fluctuation dissipation theorem and Nyquist theorem as a consequence of detailed balance principle applied to an oscillator

We rederive the Nyquist theorem and Callen–Welton fluctuation dissipation theorem as a consequence of detailed balance principle applied to a harmonic oscillator. The usage of electrical notions in the beginning makes the consideration understandable for every physicists. Perhaps it is the simplest derivation of these well-known theorems in statistical physics. The classical limit is understandable as a consequence of Waterston–Herapath equipartition theorem.

Effects of active fluctuations on energetics of a colloidal particle: Superdiffusion, dissipation and entropy production

We consider a colloidal particle immersed in an active bath and derive a Smoluchowski equation that governs the dynamics of the colloidal particle. We address this as active Smoluchowski equation. Our analysis based on this active Smoluchowski equation shows a short time superdiffusive behavior that strongly depends on the activity. Our model also predicts a non-monotonic dependence of the mean energy dissipation against time, a signature of activity-induced dynamics. By introducing a frequency-dependent effective temperature, we show that the mean rate of entropy production is time-dependent, unlike in a thermal system. The main reason for these anomalies is the absence of any fluctuation–dissipation theorem for the active noise. We also comment on how microscopic details of activity can reverse the trends for the mean energy dissipation and mean rate of entropy production.

Effective Temperature in Active Brownian Particles

In this paper, we perform a numerical analysis of the effective temperature extracted from the deviations from the fluctuation dissipation theorem in a system of active Brownian spherical particles with excluded volume interactions. We show that, in the low density homogeneous phase at fixed Péclet number, the effective temperature decreases when the density of the system is increased. We compare this trend to the one found in the literature with simulations of other active models.

Understanding and Predicting Nonlinear Turbulent Dynamical Systems with Information Theory

Complex nonlinear turbulent dynamical systems are ubiquitous in many areas. Quantifying the model error and model uncertainty plays an important role in understanding and predicting complex dynamical systems. In the first part of this article, a simple information criterion is developed to assess the model error in imperfect models. This effective information criterion takes into account the information in both the equilibrium statistics and the temporal autocorrelation function, where the latter is written in the form of the spectrum density that permits the quantification via information theory. This information criterion facilitates the study of model reduction, stochastic parameterizations, and intermittent events. In the second part of this article, a new efficient method is developed to improve the computation of the linear response via the Fluctuation Dissipation Theorem (FDT). This new approach makes use of a Gaussian Mixture (GM) to describe the unperturbed probability density function in high dimensions and avoids utilizing Gaussian approximations in computing the statistical response, as is widely used in the quasi-Gaussian (qG) FDT. Testing examples show that this GM FDT outperforms qG FDT in various strong non-Gaussian regimes.

Axiomatic construction of quantum Langevin equations

A phenomenological construction of quantum Langevin equations, based on the physical criteria of (i) the canonical equal-time commutators, (ii) the Kubo formula, (iii) the virial theorem and (iv) the quantum fluctuation–dissipation theorem is presented. The case of a single harmonic oscillator coupled to a large external bath is analysed in detail. This allows to distinguish a Markovian semi-classical approach, due to Bedeaux and Mazur, from a non-Markovian full quantum approach, due to to Ford, Kac and Mazur. The quantum-fluctuation–dissipation theorem is seen to be incompatible with a Markovian dynamics. Possible applications to the quantum spherical model are discussed.

Inverse design of radiative thermal meta-sources via discrete dipole approximation model

The radiation from conventional thermal infrared sources is incoherent and unpolarized. It has been recently shown that patterning plasmonic materials into metasurfaces can enhance the coherence of thermal radiation through thermal excitation of localized resonant modes in meta-atoms, enabling realization of thermal emitting metasurfaces—metasources—with high directivity and spectral selectivity. Toward achieving flexible modulation of radiative heat using metasurfaces and realization of complex functionalities, the design rules and modeling tools should be modified to take into account the incoherent nature of thermal radiation. In this paper, we utilize a robust formulation based on discrete dipole approximation within the framework of fluctuation dissipation theorem, which can efficiently characterize the thermal emission from large-area finite thermal metasources consisting of nanostructured materials. We employ an evolutionary lattice approach for the inverse design of radiative thermal metasources through topology optimization without relying on the phase analysis based on the assumption of coherency. To this end, the interaction matrix of dipolar equations is initialized for a lattice of subwavelength blocks and the arrangement of the blocks is subsequently optimized via a genetic algorithm toward achieving a certain spatial emission pattern with minimal computational cost. This approach enables fast and efficient design of large-scale thermal metasources with arbitrary functionalities. In particular, we investigate steering, focusing, and arbitrary shaping of partially coherent thermal radiation via finite silicon carbide gratings supporting surface phonon polaritons. Moreover, we demonstrate that the use of electro-optical materials such as graphene enables active tuning of thermal emission and realization of geometrically fixed multifunctional thermal metasources.

The Fluctuation-Dissipation Theorem as a Diagnosis and Cure for Zero-Point Energy Leakage in Quantum Thermal Bath Simulations.

Quantum thermal bath (QTB) simulations reproduce statistical nuclear quantum effects via a Langevin equation with a colored random force. Although this approach has proven efficient for a variety of chemical and condensed-matter problems, the QTB, as many other semiclassical methods, suffers from zero-point energy leakage (ZPEL). The absence of a reliable criterion to quantify the ZPEL without resorting to demanding comparisons with path integral-based calculations has so far hindered the use of the QTB for the simulation of real systems. In this work, we establish a quantitative connection between ZPEL in the QTB framework and deviations from the quantum fluctuation-dissipation theorem (FDT) that can be monitored along the simulation. This provides a rigorous general criterion to detect and quantify the ZPEL without any a priori knowledge of the system under study. We then use this criterion to build an adaptive QTB method that strictly enforces the quantum FDT at all frequencies via an on-the-fly, spectrally resolved fine-tuning of the system-bath coupling coefficients. The validity of the adaptive approach is first demonstrated on a simple two-oscillator model. It is then applied to two more realistic problems: the description of the vibrational properties of a model aluminum crystal at low temperature and the simulation of the liquid-solid phase transition in a 13-atom neon cluster. In both systems, the standard QTB results are strongly altered by the ZPEL, which can be essentially eliminated using the adaptive approach.

Quantum analogue of energy equipartition theorem

One of the fundamental laws of classical statistical physics is the energy equipartition theorem which states that for each degree of freedom the mean kinetic energy Ek equals , where is the Boltzmann constant and T is the temperature of the system. Despite the fact that quantum mechanics has already been developed for more than 100 years, still there is no quantum counterpart of this theorem. We attempt to fill this far-reaching gap and consider the simplest system, i.e. the Caldeira–Leggett model for a free quantum Brownian particle in contact with a thermostat consisting of an infinite number of harmonic oscillators. We prove that the mean kinetic energy Ek of the Brownian particle equals the mean kinetic energy per one degree of freedom of the thermostat oscillators, i.e. . We show that this relation can be obtained from the fluctuation–dissipation theorem derived within the linear response theory and is universal in the sense that it holds true for any linear and non-linear systems in contact with a bosonic thermostat.

Near-field thermal radiation between GaAs nanowires under magnetic field

We numerically investigate the near-field radiative heat transfer (NFRHT) between magneto-optical GaAs nanowires using both the local effective medium theory and fluctuational dissipation theorem. Under the action of magnetic field, the atoms or ions within magneto-optical materials will produce magnetic induction phenomenon, resulting in the orderly arrangement of the magnetic moment in NFRHT, which have not been paid enough attention. NFRHT between two aligned GaAs nanowires and plates under different magnetic field intensities is demonstrated, in which the Lorentz–Drude equations of dielectric constant tensor are proposed to describe the dielectric constant as a magnetic field applied on the GaAs nanowires. Moreover, we evaluate the near-field radiation between GaAs nanowires/plates with different filling ratios and gap distance under extent magnetic field. Furthermore, the electric field intensity is also presented for understanding light coupling, propagation, and absorption nature of GaAs grating under variable incidence angles and magnetic field strengths. The relative study is useful to the thermal radiative design in optical instruments.

Ab initio theory of quantum fluctuations and relaxation oscillations in multimode lasers

We present an \emph{ab-initio} semi-analytical solution for the noise spectrum of complex-cavity micro-structured lasers, including central Lorentzian peaks at the multimode lasing frequencies and additional sidepeaks due to relaxation-oscillation (RO) dynamics. In~Ref.~1, we computed the central-peak linewidths by solving generalized laser rate equations, which we derived from the Maxwell--Bloch equations by invoking the fluctuation--dissipation theorem to relate the noise correlations to the steady-state lasing properties; Here, we generalize this approach and obtain the entire laser spectrum, focusing on the RO sidepeaks. Our formulation treats inhomogeneity, cavity openness, nonlinearity, and multimode effects accurately. We find a number of new effects, including new multimode RO sidepeaks and three generalized $\alpha$ factors. Last, we apply our formulas to compute the noise spectrum of single- and multimode photonic-crystal lasers.

Mathematical foundation of nonequilibrium fluctuation–dissipation theorems for inhomogeneous diffusion processes with unbounded coefficients

Nonequilibrium fluctuation–dissipation theorems (FDTs) are one of the most important advances in stochastic thermodynamics over the past two decades. Here we provide rigorous mathematical proofs of two types of nonequilibrium FDTs for inhomogeneous diffusion processes with unbounded drift and diffusion coefficients by using the Schauder estimates for partial differential equations of parabolic type and the theory of weakly continuous semigroups. The FDTs proved in this paper apply to any forms of inhomogeneous and nonlinear external perturbations. Furthermore, we prove the uniqueness of the conjugate observables and clarify the precise mathematical conditions and ranges of applicability for the two types of FDTs. Examples are also given to illustrate the main results of this paper.

Finite-energy Lévy-type motion through heterogeneous ensemble of Brownian particles

Complex systems are known to display anomalous diffusion, whose signature is a space/time scaling with in the probability density function (PDF). Anomalous diffusion can emerge jointly with both Gaussian, e.g. fractional Brownian motion, and power-law decaying distributions, e.g. Lévy Flights or Lévy Walks (LWs). Lévy flights get anomalous scaling, but, being jumps of any size allowed even at short times, have infinite position variance, infinite energy and discontinuous paths. LWs, which are based on random trapping events, overcome these limitations: they resemble a Lévy-type power-law distribution that is truncated in the large displacement range and have finite moments, finite energy and, even with discontinuous velocity, they are continuous. However, LWs do not take into account the role of strong heterogeneity in many complex systems, such as biological transport in the crowded cell environment. In this work we propose and discuss a model describing a heterogeneous ensemble of Brownian particles (HEBP). Velocity of each single particle obeys a standard underdamped Langevin equation for the velocity, with linear friction term and additive Gaussian noise. Each particle is characterized by its own relaxation time and velocity diffusivity. We show that, for proper distributions of relaxation time and velocity diffusivity, the HEBP resembles some LW statistical features, in particular power-law decaying PDF, long-range correlations and anomalous diffusion, at the same time keeping finite position moments and finite energy. The main differences between the HEBP model and two different LWs are investigated, finding that, even when both velocity and position PDFs are similar, they differ in four main aspects: (i) LWs are biscaling, while HEBP is monoscaling; (ii) a transition from anomalous () to normal () diffusion in the long-time regime is seen in the HEBP and not in LWs; (iii) the power-law index of the position PDF and the space/time diffusion scaling are independent in the HEBP, while they both depend on the scaling of the inter-event time PDF in LWs; (iv) at variance with LWs, our HEBP model obeys a fluctuation–dissipation theorem.

Fluorescence Spectrum and Thermalization in a Driven Coupled Cavity Array.

We calculate the fluorescence spectra of a driven lattice of coupled cavities. To do this, we extend methods of evaluating two-time correlations in infinite lattices to open quantum systems; this allows access to momentum-resolved fluorescence spectrum. We illustrate this for a driven-dissipative transverse-field anisotropic XY model. By studying the fluctuation-dissipation theorem, we find the emergence of a quasithermalized steady state with a temperature dependent on system parameters; for blue-detuned driving, we show this effective temperature is negative. In the low excitation density limit, we compare these numerical results to analytical spin-wave theory, providing an understanding of the form of the distribution function and the origin of quasithermalization.

Mesoscopic Dynamics in Phospholipid Membranes under Osmotic Stress

Phospholipid membranes are biological liquid crystals exhibiting long-range molecular order [1]. Intermolecular interactions at mesoscopic length scales are key for explaining lipid-protein interactions and biomembrane functioning. Elastic deformations in such materials are manifested as director fluctuations with timescales spanning picoseconds to seconds [2]. Here we examine effects of osmotic stress on liquid-crystalline properties of lipid membranes using NMR relaxation methods. Membrane osmotic stress was monitored by addition of osmolytes and gravimetric dehydration. Solid-state 2H longitudinal (R1Z) and transverse quadrupolar-echo decay (R2QE) rates were measured for acyl-chain-perdeuterated lipid multilamellar dispersions. Plots of R1Z rates versus squared segmental order parameters (SCD2) follow an empirical linear function (square-law) suggesting the emergence of 3-D director fluctuations [2]. Slopes of these plots are sensitive to osmotic stress and invariant under temperature variation. Linear trends were also observed for R2QE rates but limited to segments deeper in the bilayer. Enhanced R2QE rates and their temperature dependence indicate additional relaxation contributions from slower dynamics. The dependence of R2QEon SCD2 near the headgroup suggests the onset of 2-D surface undulations. Membrane composition at the transition of 3-D to 2-D collective fluctuations has important implications for biomembrane functioning. The observed slow dynamics are due to mesoscopic phenomena that explain bulk material properties, where the connection between mechanical properties and NMR relaxation involves the fluctuation dissipation theorem. Restricted water accessibility into bilayer may suppress slow dynamic modes at high osmotic stress. Analogous studies with cholesterol in raft-like lipid mixtures [3] report on membrane compositions relevant for biomembrane function. Osmotic stress-mediated lipid dynamics thus opens a new window to exploring their coupling to biomembrane function. [1] T.R. Molugu et al. (2017) Chem. Rev. 117, 12087. [2] Leftin, A. et al. (2011) BBA 1808, 818. [3] T.R. Molugu et al. (2016) Chem. Phys. Lipids 199, 39.

Fluctuation Dissipation Theorem and Electrical Noise Revisited

The fluctuation dissipation theorem (FDT) is the basis for a microscopic description of the interaction between electromagnetic radiation and matter. By assuming the electromagnetic radiation in thermal equilibrium and the interaction in the linear-response regime, the theorem interrelates the macroscopic spontaneous fluctuations of an observable with the kinetic coefficients that are responsible for energy dissipation in the linear response to an applied perturbation. In the quantum form provided by Callen and Welton in their pioneering paper of 1951 for the case of conductors [H. B. Callen and T. A. Welton, Irreversibility and generalized noise, Phys. Rev. 83 (1951) 34], electrical noise in terms of the spectral density of voltage fluctuations, [Formula: see text], detected at the terminals of a conductor was related to the real part of its impedance, [Formula: see text], by the simple relation [Formula: see text] where [Formula: see text] is the Boltzmann constant, [Formula: see text] is the absolute temperature, [Formula: see text] is the reduced Planck constant and [Formula: see text] is the angular frequency. The drawbacks of this relation concern with: (i) the appearance of a zero-point contribution which implies a divergence of the spectrum at increasing frequencies; (ii) the lack of detailing the appropriate equivalent-circuit of the impedance, (iii) the neglect of the Casimir effect associated with the quantum interaction between zero-point energy and boundaries of the considered physical system; (iv) the lack of identification of the microscopic noise sources beyond the temperature model. These drawbacks do not allow to validate the relation with experiments, apart from the limiting conditions when [Formula: see text]. By revisiting the FDT within a brief historical survey of its formulation, since the announcement of Stefan–Boltzmann law dated in the period 1879–1884, we shed new light on the existing drawbacks by providing further properties of the theorem with particular attention to problems related with the electrical noise of a two-terminals sample under equilibrium conditions. Accordingly, among others, we will discuss the duality and reciprocity properties of the theorem, the role played by different statistical ensembles, its applications to the ballistic transport-regime, to the case of vacuum and to the case of a photon gas.

Turbulent Flow with the Inner Cylinder Rotating

The dissipation theorem does a respectable job of describing turbulent mass transfer in the rotating-cylinder system, although parameters still need to be obtained by fitting against experimental data. Unfortunately the available experimental data show discrepancies that hamper theoretical development. This is part of a long-range program to compare the predictions of the dissipation theorem to systems ranging from pipe flow and rotating cylinders to developing flows on a rotating disk and on a flat plate at zero incidence, where extensive experimental data exist. One learns that values of the eddy viscosity should superpose after dividing by the stress parameter R+ in order to obtain coherent behavior at very large Reynolds numbers. One innovation here is to use the mass-transfer data as a proxy for the absence of torque data. One finds that the data of Eisenberg and of Mohr do not constitute a single coherent data set even though they were obtained on the same apparatus and with the same chemical system. New experimental data are needed to resolve these discrepancies. The dissipation theorem is supposed to enhance our understanding of turbulence and permit prediction of the behavior of turbulence in a variety of systems. The next step is to apply it to developing flows on a flat plate and a rotating disk.