Fluctuation Dissipation Research Articles

Mixing and Bimolecular Reaction Kinetics in a Plane Poisseulle Flow

Mixing and a nonlinear bimolecular chemical reaction (reactant A + reactant B → product; reaction rate r = κc1c2) in laminar shear flow are investigated. It is found that asymptotically the dominant balance between the rates of production and dissipation of the mean-squared concentration fluctuations \((\sigma_{c_1 }^2 ,\sigma_{c_2 }^2)\) and cross-covariance of concentration fluctuations \((\overline {c_1 c_2 })\) occurs under nonreactive and reactive conditions. Longitudinal dispersion of the cross-sectional averages (C1, C2), and variances and the cross-covariance of reactant concentrations can be asymptotically quantified by the classic Taylor dispersion coefficient (D) even under reactive conditions. The characteristic time-scale (τ) over which molecular diffusion dissipates concentration variance and the cross-covariance of reactant concentrations is also shown to be the same under nonreactive and reactive conditions. A variational estimate of τ is shown to be close to the values inferred from detailed numerical simulation. The production-dissipation balance implies that the cross-sectional averaged reaction rate follows \(\overline r =\kappa_{eff} C_1 C_2 \) and \(\kappa _{eff} \approx \kappa \left[ {1+2D\tau \left( {{\partial \ln C_1 } \mathord{\left/ {\vphantom {{\partial \ln C_1 } {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}} \right)\left( {{\partial \ln C_2 } \mathord{\left/ {\vphantom {{\partial \ln C_2 } {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}} \right)} \right]\). The effective reaction rate parameter (κeff) is higher than that of well-mixed batch test reaction rate constant (κ) for initially overlapping species and κeff is smaller than κ for initially non-overlapping species.

Conditional Moment Closure/Large Eddy Simulation of the Delft-III Natural Gas Non-premixed Jet Flame

Large-Eddy Simulation (LES), coupled with the Conditional Moment Closure (CMC) sub-grid model and the GRI3 detailed chemical mechanism, are used to explore the structure of the Delft III piloted turbulent non-premixed flame. The use of a quite refined multi-dimensional CMC grid and the detailed chemistry, together with the capability of LES to follow local fluctuations of the scalar dissipation, allow the prediction of localised extinctions and re-ignitions in locations consistent with experiment. The statistics of velocity, mixture fraction, temperature, mass fractions of the major species and of OH are overall in good agreement with experimental data. Carbon monoxide is captured very well, but NO is overpredicted, perhaps due to inherent limitations of the GRI3 scheme to capture NO emissions.

Statistics of the energy dissipation rate and local enstrophy in turbulent channel flow

Using high-resolution direct numerical simulations, the height and Reynolds number dependence of high-order statistics of the energy dissipation rate and local enstrophy are examined in incompressible, fully developed turbulent channel flow. The statistics are studied over a range of wall distances, spanning the viscous sublayer to the channel flow centerline, for friction Reynolds numbers R e τ = 180 and R e τ = 381 . The high resolution of the simulations allows dissipation and enstrophy moments up to fourth order to be calculated. These moments show a dependence on wall distance, and Reynolds number effects are observed at the edge of the logarithmic layer. Conditional analyses based on locations of intense rotation are also carried out in order to determine the contribution of vortical structures to the dissipation and enstrophy moments. Our analysis shows that, for the simulation at the larger Reynolds number, small-scale fluctuations of both dissipation and enstrophy show relatively small variations for z + ≳ 100 .

The energy and dissipation of turbulent fluctuations of wind velocity and temperature in the boundary layer of the atmosphere

The relationships between the energy of small-scale turbulence and its dissipation rate are studied based on the data of long-term high-frequency measurements of temperature and wind velocity fluctuations in urban area. It is shown that the energy of wind velocity turbulent fluctuations is linearly related to the dissipation rate ɛ. The proportionality coefficient between turbulent kinetic energy (TKE) and ɛ is dimensional and does not depend on the stratification of the atmosphere, the Richardson number, or the Monin-Obukhov scale. Measurements in different seasons show that this coefficient can be related to the mean velocity of adiabatic motions (sound speed or air temperature), which enables one to select a more universal constant, γ. A linear relationship between the temperature fluctuations variance (the characteristic of the inner energy of turbulence) and their dissipation rate is also shown. The revealed proportionality is confirmed by measurements in urban and forest conditions, as well as in the surface layer over a flat desert terrain.

Direct Quadrature Conditional Moment Closure for Modelling of Turbulent Combustion

We present a method of direct quadrature conditional moment closure (DQCMC) for the treatment of realistic turbulence-chemistry interaction in computational fluid dynamics (CFD) software. The method which is based on the direct quadrature method of moments (DQMOM) coupled with the conditional moment closure (CMC) equations is in simplified form and easily implementable in existing CMC formulation for CFD. The observed fluctuations of scalar dissipation around the conditional mean values are captured by the treatment of a set of mixing environments, each with its pre-defined weight. Unlike the early versions of the DQCMC method the resulting equations are similar to that of the first-order CMC, and the “diffusion” term is strictly positive and no correction factors are used. We present results for two mixing environments where the resulting matrices of the DQCMC can be inverted analytically. We have performed this analysis for a simple hydrogen flame using a multi species chemical scheme containing nine species. The effects of the fluctuations around the conditional means are captured accurately and the predicted results are in very good agreement with observed trends from direct numerical simulations. Furthermore, the differences between the first order CMC and DQCMC are discussed.

Thermal fluctuation field for current-induced domain wall motion

Current-induced domain wall motion in magnetic nanowires is affected by thermal fluctuation. In order to account for this effect, the Landau-Lifshitz-Gilbert equation includes a thermal fluctuation field and literature often utilizes the fluctuation-dissipation theorem to characterize statistical properties of the thermal fluctuation field. However, the theorem is not applicable to the system under finite current since it is not in equilibrium. To examine the effect of finite current on the thermal fluctuation, we adopt the influence functional formalism developed by Feynman and Vernon, which is known to be a useful tool to analyze effects of dissipation and thermal fluctuation. For this purpose, we construct a quantum-mechanical effective Hamiltonian describing current-induced domain wall motion by generalizing the Caldeira-Leggett description of quantum dissipation. We find that even for the current-induced domain wall motion, the statistical properties of the thermal noise is still described by the fluctuation-dissipation theorem if the current density is sufficiently lower than the intrinsic critical current density and thus the domain wall tilting angle is sufficiently lower than $\ensuremath{\pi}/4$. The relation between our result and a recent result [R. A. Duine, A. S. N\'u\~nez, J. Sinova, and A. H. MacDonald, Phys. Rev. B 75, 214420 (2007)], which also addresses the thermal fluctuation, is discussed. We also find interesting physical meanings of the Gilbert damping $\ensuremath{\alpha}$ and the nonadiabaticy parameter $\ensuremath{\beta}$; while $\ensuremath{\alpha}$ characterizes the coupling strength between the magnetization dynamics (the domain wall motion in this paper) and the thermal reservoir (or environment), $\ensuremath{\beta}$ characterizes the coupling strength between the spin current and the thermal reservoir.

Noise in the wire: The real impact of wire resistance for the Johnson(-like) noise based secure communicator

We re-evaluate the impact of wire resistance on the noise voltage and current in the Johnson(-like) noise based secure communicator, correcting the result presented in [J. Scheuer, A. Yariv, Phys. Lett. A 359 (2006) 737]. The analysis shown here is based on the fluctuation–dissipation and the linear response theorems. The results indicate that the impact of wire resistance in practical communicators is significantly lower than the previous estimation.

Geometrical statistics and vortex structures in helical and nonhelical turbulences

In this paper we conduct an analysis of the geometrical and vortical statistics in the small scales of helical and nonhelical turbulences generated with direct numerical simulations. Using a filtering approach, the helicity flux from large scales to small scales is represented by the subgrid-scale (SGS) helicity dissipation. The SGS helicity dissipation is proportional to the product between the SGS stress tensor and the symmetric part of the filtered vorticity gradient, a tensor we refer to as the vorticity strain rate. We document the statistics of the vorticity strain rate, the vorticity gradient, and the dual vector corresponding to the antisymmetric part of the vorticity gradient. These results provide new insights into the local structures of the vorticity field. We also study the relations between these quantities and vorticity, SGS helicity dissipation, SGS stress tensor, and other quantities. We observe the following in both helical and nonhelical turbulences: (1) there is a high probability to find the dual vector aligned with the intermediate eigenvector of the vorticity strain rate tensor; (2) vorticity tends to make an angle of 45° with both the most contractive and the most extensive eigendirections of the vorticity strain rate tensor; (3) the vorticity strain rate shows a preferred alignment configuration with the SGS stress tensor; (4) in regions with strong straining of the vortex lines, there is a negative correlation between the third order invariant of the vorticity gradient tensor and SGS helicity dissipation fluctuations. The correlation is qualitatively explained in terms of the self-induced motions of local vortex structures, which tend to wind up the vortex lines and generate SGS helicity dissipation. In helical turbulence, we observe that the joint probability density function of the second and third tensor invariants of the vorticity gradient displays skewed distributions, with the direction of skewness depending on the sign of helicity input. We also observe that the intermediate eigenvalue of the vorticity strain rate tensor is more probable to take negative values. These interesting observations, reported for the first time, call for further studies into their dynamical origins and implications.

Buckling, stiffening, and negative dissipation in the dynamics of a biopolymer in an active medium

We present a generic theory for the dynamics of a stiff filament under tension, in an active medium with orientational correlations, such as a microtubule in contractile actin. In sharp contrast to the case of a passive medium, we find the filament can stiffen, and possibly oscillate or buckle, depending on both the contractile or tensile nature of the activity and the filament-medium anchoring interaction. We also demonstrate a strong violation of the fluctuation-dissipation (FD) relation in the effective dynamics of the filament, including a negative FD ratio. Our approach is also of relevance to the dynamics of axons, and our model equations bear a remarkable formal similarity to those in recent work [Martin P, Hudspeth AJ, Juelicher F (2001) Proc Natl Acad Sci USA 98:14380-14385] on auditory hair cells. Detailed tests of our predictions can be made by using a single filament in actomyosin extracts or bacterial suspensions.

Multiple mapping conditioning for flames with partial premixing

Fully closed multiple mapping conditioning (MMC) is used to model partially premixed flames in homogeneous, isotropic decaying turbulence where the partial premixing is caused by local extinction and reignition phenomena. Two reference variables that represent mixing and reaction progress, such as mixture fraction and sensible enthalpy, are used to emulate turbulent scalar fluctuations. Local extinction is achieved by a priori coupling between scalar dissipation and temperature fluctuations via a correlation function that is based on the conditionally averaged sensible enthalpy at stoichiometric composition. The proposed model provides closures for the joint PDF of mixture fraction and sensible enthalpy, for the conditional variance equation of a reactive scalar, and for the doubly conditioned dissipation terms. Model results are compared with DNS in three flame cases with varying levels of local extinction, up to global extinction. The joint PDF predicted by MMC is in fair agreement with DNS. It constitutes, however, a clear improvement over conventional models using preassumed distribution functions for the PDFs. The doubly conditioned dissipation terms are modeled well and the results for all major chemical species are in good agreement with DNS. Predictions for intermediate species are also satisfactory.

A stochastic mean-field approach for nuclear dynamics

We propose a microscopic stochastic approach to improve description of nuclear dynamics beyond the mean-field approximation at low energies. It is shown that, for small amplitude fluctuations, the proposed model gives a result for the dispersion of a one-body observable that is identical to the result obtained previously through a variational approach. Furthermore, by projecting the proposed stochastic mean-field evolution on a collective path, a generalized Langevin equation is derived for collective variable, which incorporates one-body dissipation and one-body fluctuation mechanism in accordance with quantal fluctuation–dissipation relation.

Fluctuation theorems for harmonic oscillators

We study experimentally the thermal fluctuations of energy input and dissipation in aharmonic oscillator driven out of equilibrium, and search for fluctuation relations. Boththe transient evolution from the equilibrium state, and non-equilibrium steadystates are analyzed. Fluctuation relations are obtained experimentally for boththe work and the heat, for the stationary and transient evolutions. A stationarystate fluctuation theorem is verified for various time dependences of the imposedexternal torque. The transient fluctuation theorem is satisfied for the work given tothe system but not for the heat dissipated by the system in the case of linearforcing. Experimental observations on the statistical and dynamical properties ofthe position fluctuations of the torsion pendulum allow us to derive analyticalexpressions for the probability density functions of the work and the heat. We obtainfor the first time an analytic expression for the probability density function ofthe heat. The agreement between experiments and our predictions is excellent.

Lagrangian conditional statistics, acceleration and local relative motion in numerically simulated isotropic turbulence

Lagrangian statistics of fluid-particle velocity and acceleration conditioned on fluctuations of dissipation, enstrophy and pseudo-dissipation representing different characteristics of local relative motion are extracted from a direct numerical simulation database of stationary (forced) homogeneous isotropic turbulence. The grid resolution in the simulations is up to 20483, and the Taylor-scale Reynolds number ranges from about 40 to 650, where characteristics of small-scale intermittency in the Eulerian flow field are well developed. A key joint statistic of the conditioning variables is the dissipation-enstrophy cross-correlation, which is asymmetric, but becomes less so at high Reynolds number. Conditional velocity autocorrelations are consistent with rapid changes in the velocity of fluid particles moving in regions of large velocity gradients. Examination of statistics conditioned upon enstrophy, especially in a local coordinate frame moving with the vorticity vector, and of the centripetal acceleration suggests the presence of vortex-trapping effects which persist for several Kolmogorov time scales. Further results on acceleration statistics and joint velocity-acceleration autocorrelations are also presented to help characterize in detail the properties of a joint stochastic process of velocity, acceleration and the pseudo-dissipation. Together with recent work on Eulerian conditional acceleration and Reynolds-number dependence of basic Lagrangian quantities, the present results are directly useful for the development of a new stochastic model formulated to account for intermittency and Reynolds-number effects as described in detail in a companion paper.

From dispersed nano-objects to solutions—A thermodynamic approach

Nanotechnologies take now a wide interest both in the applied and fundamental concerns. The interfacial properties of nano-particles at the fluid–fluid interfaces question the physicist on the extension of the Gibbs surface thermodynamics to interfaces in which nano-particles are entrapped as dispersion stabilizers [K.D. Danov, P.A. Kralchevsky, J. Colloid Interface Sci. 298 (2006) 213] (foams or emulsions). An attempt is proposed to give the physical meaning of the measured surface properties of nano-particles attached to fluid–fluid interfaces like the so-called dynamic surface tension and surface dilational elasticity of suspensions in terms of wetting of the particles at the interface. The Hill [T.L. Hill, Thermodynamics of Small Systems, W.A. Benjamin, New York, 1963] theory for small systems linked to the dissipation fluctuation theorem should give a deeper insight in the statistical mechanics understanding of the interfacial region in which nano-particles are imbedded.

The status of the quantum dissipation–fluctuation relation and the Langevinequation

I examine the arguments which have been given for quantum fluctuation–dissipationtheorems. I distinguish between a weak form of the theorem, which is true under rathergeneral conditions, and a strong form which requires a Langevin equation forits statement. I argue that the latter has not been reliably derived in general.

Linear response, susceptibility and resonances in chaotic toy models

We consider simple examples illustrating some new features of the linear response theory developed by Ruelle for dissipative and chaotic systems [D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics, J. Stat. Phys. 95 (1999) 393–468]. In this theory the concepts of linear response, susceptibility and resonance, which are familiar to physicists, have been revisited due to the dynamical contraction of the whole phase space onto attractors. In particular the standard framework of the “fluctuation–dissipation” theorem breaks down and new resonances can show up outside the power spectrum. In previous papers we proposed and used new numerical methods to demonstrate the presence of the new resonances predicted by Ruelle in a model of chaotic neural networks. In this article we deal with simpler models which can be worked out analytically in order to gain more insights into the genesis of the “stable” resonances and their consequences on the linear response of the system. We consider a class of two-dimensional time-discrete maps describing simple rotator models with a contracting radial dynamics onto the unit circle and a chaotic angular dynamics θ t + 1 = 2 θ t ( mod 2 π ) . A generalisation of this system to a network of interconnected rotators is also analysed and related with our previous studies [B. Cessac, J.-A. Sepulchre, Stable resonances and signal propagation in a chaotic network of coupled units, Phys. Rev. E 70 (2004) 056111; B. Cessac, J.-A. Sepulchre, Transmitting a signal by amplitude modulation in a chaotic network, Chaos 16 (2006) 013104-1–13104-12]. These models permit us to classify the different types of resonances in the susceptibility and to discuss in particular the relation between the relaxation time of the system to equilibrium with the mixing time given by the decay of the correlation functions. Also it enables one to propose some general mechanisms responsible for the creation of stable resonances with arbitrary frequencies, widths, and dependence on the pair of perturbed/observed variables.

Similarity relationships in the unstable aquatic surface layer

[1] Turbulence mixing in aquatic surface layers controls the exchange of energy, momentum, heat, and mass between the atmosphere and the water beneath it. In many instances, due to a lack of suitable observations, scaling laws from atmospheric surface layers over land have been “borrowed” to model turbulence in oceans and lakes. However, it is unclear to what extent, and under what forcing conditions, atmospheric scaling laws are valid in aquatic surface layers. Here we present unique observations of the vertical structure of two key turbulence parameters, the dissipation of turbulence kinetic energy and the dissipation of temperature fluctuations, when the aquatic surface layer was forced by a combination of wind stirring and heat loss to the atmosphere. The good agreement found with scaling laws in atmospheric surface layers, under similar forcing and stability conditions, provides new evidence linking the structure of turbulence in aquatic and atmospheric surface layers.

Stability of Decoherence-Free Subspaces under Stochastic PhaseFluctuations

We study the stability of decoherence-free subspaces understochastic phase fluctuations by analytically and numerically evaluatingthe fidelity of the corresponding decoherence-free subspace bases withstochastic phase fluctuations under the evolution of environment.The environment is modeled by a bath of oscillators with infinite degreesof freedom and the register-bath coupling is chosen to be a generaldissipation-decoherence form. It is found that thedecoherence-free subspaces take on good stabilityin the case of small dissipation and small phase fluctuations.

STEADY STATE FLUCTUATIONS OF THE DISSIPATED HEAT FOR A QUANTUM STOCHASTIC MODEL

We introduce a quantum stochastic dynamics for heat conduction. A multi-level subsystem is coupled to reservoirs at different temperatures. Energy quanta are detected in the reservoirs allowing the study of steady state fluctuations of the entropy dissipation. Our main result states a symmetry in its large deviation rate function.

Acceleration and dissipation statistics of numerically simulated isotropic turbulence

Direct numerical simulation (DNS) data at grid resolution up to 20483 in isotropic turbulence are used to investigate the statistics of acceleration in a Eulerian frame. A major emphasis is on the use of conditional averaging to relate the intermittency of acceleration to fluctuations of dissipation, enstrophy, and pseudodissipation representing local relative motion in the flow. Pseudodissipation (the second invariant of the velocity gradient tensor) has the same intermittency exponent as dissipation and is closest to log-normal. Conditional acceleration variances increase with each conditioning variable, consistent with the scenario of rapid changes in velocity for fluid particles moving in local regions of large velocity gradient, but in a manner departing from Kolmogorov’s refined similarity theory. Acceleration conditioned on the pseudodissipation is closest to Gaussian, and well represented by a novel “cubic Gaussian” distribution. Overall the simulation data suggest that, with the aid of appropriate parameterizations, Lagrangian stochastic modeling with pseudodissipation as the conditioning variable is likely to produce superior results. Reduced intermittency of conditional acceleration also makes the present results less sensitive to resolution concerns in DNS.