Boltzmann's ergodic hypothesis

1991 ◽  
Vol 42 (1) ◽  
pp. 71-89 ◽  
Author(s):  
Jan von Plato
Keyword(s):  
Ergodic Hypothesis

Relationship between temporal and spatial averages in grid turbulence

2013 ◽  
Vol 730 ◽  
pp. 593-606 ◽  
Author(s):  
L. Djenidi ◽  
S. F. Tardu ◽  
R. A. Antonia
Keyword(s):  
Strong Support ◽  
Statistical Properties ◽  
Grid Turbulence ◽  
Mean Flow ◽  
Lateral Direction ◽  
Ergodic Hypothesis ◽  
Long Time ◽  
The Mean ◽  
Temporal And Spatial ◽  
Boltzmann Method

AbstractA long-time direct numerical simulation (DNS) based on the lattice Boltzmann method is carried out for grid turbulence with the view to compare spatially averaged statistical properties in planes perpendicular to the mean flow with their temporal counterparts. The results show that the two averages become equal a short distance downstream of the grid. This equality indicates that the flow has become homogeneous in a plane perpendicular to the mean flow. This is an important result, since it confirms that hot-wire measurements are appropriate for testing theoretical results based on spatially averaged statistics. It is equally important in the context of DNS of grid turbulence, since it justifies the use of spatial averaging along a lateral direction and over several realizations for determining various statistical properties. Finally, the very good agreement between temporal and spatial averages validates the comparison between temporal (experiments) and spatial (DNS) statistical properties. The results are also interesting because, since the flow is stationary in time and spatially homogeneous along lateral directions, the equality between the two types of averaging provides strong support for the ergodic hypothesis in grid turbulence in planes perpendicular to the mean flow.

BIRKHOFF THEOREM AND ERGOMETER: MEETING OF TWO CULTURES

2008 ◽  
Vol 22 (25n26) ◽  
pp. 4572-4578 ◽  
Author(s):  
M. HOWARD LEE
Keyword(s):  
Statistical Mechanics ◽  
Two Cultures ◽  
Ergodic Hypothesis ◽  
Birkhoff Theorem ◽  
Physical Measurements ◽  
The Two Cultures

There are two approaches to understanding Boltzmann's ergodic hypothesis in statistical mechanics. The first one, purely mathematical, goes by way of theorems while the second one relies on physical measurements. By its own nature the former is universal whereas the latter is specific to a system. By all account they seem orthogonal to each other. But should not they meet at the end? If, for example, both conclude that the hypothesis is not valid in a given system, should not their conclusions be compatible? We illustrate in this work how the two cultures meet in the physics of ergodicity.

scholarly journals Dynamical Simulation and Statistical Analysis of Velocity Fluctuations of a Turbulent Flow behind a Cube

2007 ◽  
Vol 2007 ◽  
pp. 1-28 ◽  
Author(s):  
T. F. Oliveira ◽  
R. B. Miserda ◽  
F. R. Cunha
Keyword(s):  
Turbulent Flow ◽  
Numerical Simulations ◽  
Compressible Flows ◽  
Statistical Approach ◽  
Power Spectra ◽  
Flow Simulation ◽  
Scaling Analysis ◽  
Velocity Fluctuations ◽  
Ergodic Hypothesis ◽  
Time Average

A statistical approach for the treatment of turbulence data generated by computer simulations is presented. A model for compressible flows at large Reynolds numbers and low Mach numbers is used for simulating a backward-facing step airflow. A scaling analysis has justified the commonly used assumption that the internal energy transport due to turbulent velocity fluctuations and the work done by the pressure field are the only relevant mechanisms needed to model subgrid-scale flows. From the numerical simulations, the temporal series of velocities are collected for ten different positions in the flow domain, and are statistically treated. The statistical approach is based on probability averages of the flow quantities evaluated over several realizations of the simulated flow. We look at how long of a time average is necessary to obtain well-converged statistical results. For this end, we evaluate the mean-square difference between the time average and an ensemble average as the measure of convergence. This is an interesting question since the validity of the ergodic hypothesis is implicitly assumed in every turbulent flow simulation and its analysis. The ergodicity deviations from the numerical simulations are compared with theoretical predictions given by scaling arguments. A very good agreement is observed. Results for velocity fluctuations, normalized autocorrelation functions, power spectra, probability density distributions, as well as skewness and flatness coefficients are also presented.

Gibbs fluctuation theory in the context of electrochemical equilibrium noise

2010 ◽  
Vol 83 (2) ◽  
pp. 253-257 ◽  
Author(s):  
Boris M. Grafov
Keyword(s):  
Electrochemical Noise ◽  
Thermal Fluctuations ◽  
Higher Order ◽  
Fluctuation Theory ◽  
Pair Correlations ◽  
Ergodic Hypothesis ◽  
Electrode Processes ◽  
Electrochemical Charge Transfer ◽  
Electrochemical Devices

The electrochemical noise verification of the Gibbs fluctuation theory shows that the Gibbs ergodic idea works perfectly with respect to the pair correlations of the electrode charge thermal fluctuations. At the same time, the Gibbs formulae for the triple- and higher-order correlations of the electrode charge thermal fluctuations are outside of the ergodic hypothesis. This failure of the Gibbs ergodic idea suggests that the noise version of the electrochemical charge-transfer theory should be developed. In the context of nano-electrochemistry, the second- and higher-order correlations of the electrochemical noise processes may be considered as the quantities suitable for the nano-electrochemical characterization of both the electrode processes and electrochemical devices.

scholarly journals Ergodicity test of the eddy-covariance technique

2015 ◽  
Vol 15 (17) ◽  
pp. 9929-9944 ◽  
Author(s):  
J. Chen ◽  
Y. Hu ◽  
Y. Yu ◽  
S. Lü
Keyword(s):  
Eddy Covariance ◽  
Ergodic Theorem ◽  
Atmospheric Turbulence ◽  
Finite Time ◽  
Similarity Theory ◽  
Ensemble Average ◽  
Ergodic Hypothesis ◽  
Eddy Covariance Technique ◽  
Single Station ◽  
Time Average

Abstract. The ergodic hypothesis is a basic hypothesis typically invoked in atmospheric surface layer (ASL) experiments. The ergodic theorem of stationary random processes is introduced to analyse and verify the ergodicity of atmospheric turbulence measured using the eddy-covariance technique with two sets of field observational data. The results show that the ergodicity of atmospheric turbulence in atmospheric boundary layer (ABL) is relative not only to the atmospheric stratification but also to the eddy scale of atmospheric turbulence. The eddies of atmospheric turbulence, of which the scale is smaller than the scale of the ABL (i.e. the spatial scale is less than 1000 m and temporal scale is shorter than 10 min), effectively satisfy the ergodic theorems. Under these restrictions, a finite time average can be used as a substitute for the ensemble average of atmospheric turbulence, whereas eddies that are larger than ABL scale dissatisfy the mean ergodic theorem. Consequently, when a finite time average is used to substitute for the ensemble average, the eddy-covariance technique incurs large errors due to the loss of low-frequency information associated with larger eddies. A multi-station observation is compared with a single-station observation, and then the scope that satisfies the ergodic theorem is extended from scales smaller than the ABL, approximately 1000 m to scales greater than about 2000 m. Therefore, substituting the finite time average for the ensemble average of atmospheric turbulence is more faithfully approximate the actual values. Regardless of vertical velocity or temperature, the variance of eddies at different scales follows Monin–Obukhov similarity theory (MOST) better if the ergodic theorem can be satisfied; if not it deviates from MOST. The exploration of ergodicity in atmospheric turbulence is doubtlessly helpful in understanding the issues in atmospheric turbulent observations and provides a theoretical basis for overcoming related difficulties.

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