scholarly journals Linear non-equilibrium thermodynamics of human voluntary behavior: a canonical-dissipative Fokker-Planck equation approach involving potentials beyond the harmonic oscillator case

2016 ◽  
Vol 19 (3) ◽  
pp. 34001 ◽  
Author(s):  
Gordon ◽  
Kim ◽  
Frank
Keyword(s):  
Harmonic Oscillator ◽  
Planck Equation ◽  
Equation Approach ◽  
Equilibrium Thermodynamics ◽  
Fokker Planck Equation ◽  
Voluntary Behavior ◽  
Non Equilibrium ◽  
Fokker Planck

scholarly journals On universal features of the turbulent cascade in terms of non-equilibrium thermodynamics

2018 ◽  
Vol 848 ◽  
pp. 117-153 ◽  
Author(s):  
Nico Reinke ◽  
André Fuchs ◽  
Daniel Nickelsen ◽  
Joachim Peinke
Keyword(s):  
Probability Density ◽  
Entropy Production ◽  
Turbulent Flows ◽  
Planck Equation ◽  
Small Scale ◽  
Equilibrium Thermodynamics ◽  
Fokker Planck Equation ◽  
Velocity Increments ◽  
Turbulent Cascade ◽  
Fokker Planck

Features of the turbulent cascade are investigated for various datasets from three different turbulent flows, namely free jets as well as wake flows of a regular grid and a cylinder. The analysis is focused on the question as to whether fully developed turbulent flows show universal small-scale features. Two approaches are used to answer this question. First, two-point statistics, namely structure functions of longitudinal velocity increments, and, second, joint multiscale statistics of these velocity increments are analysed. The joint multiscale characterisation encompasses the whole cascade in one joint probability density function. On the basis of the datasets, evidence of the Markov property for the turbulent cascade is shown, which corresponds to a three-point closure that reduces the joint multiscale statistics to simple conditional probability density functions (cPDFs). The cPDFs are described by the Fokker–Planck equation in scale and its Kramers–Moyal coefficients (KMCs). The KMCs are obtained by a self-consistent optimisation procedure from the measured data and result in a Fokker–Planck equation for each dataset. Knowledge of these stochastic cascade equations enables one to make use of the concepts of non-equilibrium thermodynamics and thus to determine the entropy production along individual cascade trajectories. In addition to this new concept, it is shown that the local entropy production is nearly perfectly balanced for all datasets by the integral fluctuation theorem (IFT). Thus, the validity of the IFT can be taken as a new law of the turbulent cascade and at the same time independently confirms that the physics of the turbulent cascade is a memoryless Markov process in scale. The IFT is taken as a new tool to prove the optimal functional form of the Fokker–Planck equations and subsequently to investigate the question of universality of small-scale turbulence in the datasets. The results of our analysis show that the turbulent cascade contains universal and non-universal features. We identify small-scale intermittency as a universality breaking feature. We conclude that specific turbulent flows have their own particular multiscale cascades, in other words, their own stochastic fingerprints.

Dynamic Snow Distribution Modeling using the Fokker-Planck Equation Approach

2021 ◽  
Author(s):  
Noriaki Ohara
Keyword(s):  
Time Series ◽  
Snow Depth ◽  
Planck Equation ◽  
Snow Accumulation ◽  
Airborne Lidar ◽  
Equation Approach ◽  
Diffusion Type ◽  
Fokker Planck Equation ◽  
Time Space ◽  
Fokker Planck

<p>The Fokker-Planck equation (FPE) describes the time evolution of the distribution function of fluctuating macroscopic variables.  Although the FPE was originally derived for the Brownian motion, this framework can be applied to various physical processes.  In this presentation, applications in the snow accumulation and thaw process, which attributes to considerable spatial and temporal variations, are discussed. It is well known that snow process is a major source of heterogeneity in hydrological systems in high altitude or latitude regions; therefore, better treatment of the snow sub-grid variability is desirable. The main advantage of the FPE approach is that it can dynamically compute the probability density function (PDF) governed by an advection-diffusion type FPE without a prescribed PDF.</p><p>First, a bivariate FPE was derived from point scale process-based governing equations (Ohara et al., 2008). This FPE can express the evolution of the PDF of snow depth and temperature within a finite space, possibly a computational cell or small basin, whose shape is irrelevant. This conceptual model was proven to be effective through comparing to the corresponding Monte-Carlo simulation.  Then, the more realistic single variated FPE model for snow depth was implemented with the snow redistribution and snowmelt rate as the main sources of stochasticity. In this study, several realistic approximations were proposed to compute the time-space covariances describing effects induced by uneven snowmelt and snow redistribution.</p><p>Meanwhile, observed high-resolution snow depth data was analyzed using statistical methods to characterize the sub-grid variability of snow depth, which is essential to validate the FPE model for representing such sub-grid variability.  Airborne light detection and ranging (Lidar) provided the snow depth measurements at 0.5 m resolution over two mountainous areas in southwestern Wyoming, Snowy Range and Laramie Range (He et al., 2019). It was found that PDFs of snow depth tend to be Gaussian distributions in the forest areas. However, due to the no-snow areas effect, mainly caused by snow redistribution and uneven snowmelt, the PDFs are eventually skewed as non-Gaussian distribution.</p><p>The simulated results of the FPE model were validated using the measured time series of snow depth at one site and the spatial distributions of snow depth measured by ground penetrating radar (GPR) and airborne Lidar. The modeled and observed time series of the mean snow depth agreed very well while the simulated PDFs of snow depth within the study area were comparable to the observed PDFs of snow depth by GPR and Lidar (He and Ohara, 2019). Accordingly, the FPE model is capable to capture the main characteristics of the snow sub-grid variability in the nature.</p><p><strong>References</strong></p><p>Ohara, N., Kavvas, M. L., & Chen, Z. Q. (2008). Stochastic upscaling for snow accumulation and melt processes with PDF approach. Journal of Hydrologic Engineering, 13(12), 1103-1118.</p><p>He, S., Ohara, N., & Miller, S. N. (2019). Understanding subgrid variability of snow depth at 1‐km scale using Lidar measurements. Hydrological Processes, 33(11), 1525-1537.</p><p>He, S., & Ohara, N. (2019). Modeling subgrid variability of snow depth using the Fokker‐Planck equation approach. Water Resources Research, 55(4), 3137-3155.</p>

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