Partial Differential Equation for the Probability Density of Quantum Statistical Thermodynamics

1967 ◽  
Vol 47 (12) ◽  
pp. 5450-5451 ◽  
Author(s):  
M. D. Kostin
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Probability Density ◽  
Statistical Thermodynamics ◽  
Partial Differential ◽  
Quantum Statistical

scholarly journals A description of stochastic systems using chaotic maps

2004 ◽  
Vol 2004 (2) ◽  
pp. 137-141 ◽  
Author(s):  
Abraham Boyarsky ◽  
Pawel Góra
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Probability Density ◽  
Stochastic Systems ◽  
Chaotic Maps ◽  
Probability Density Functions ◽  
Density Functions ◽  
Partial Differential ◽  
The Family ◽  
Point Transformations

Let ρ(x,t) denote a family of probability density functions parameterized by time t. We show the existence of a family {τ1:t>0} of deterministic nonlinear (chaotic) point transformations whose invariant probability density functions are precisely ρ(x,t). In particular, we are interested in the densities that arise from the diffusions. We derive a partial differential equation whose solution yields the family of chaotic maps whose density functions are precisely those of the diffusion.

scholarly journals A partial differential equation for pseudocontact shift

2014 ◽  
Vol 16 (37) ◽  
pp. 20184-20189 ◽  
Author(s):  
G. T. P. Charnock ◽  
Ilya Kuprov
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Probability Density ◽  
Unpaired Electron ◽  
Tensor Field ◽  
Source Term ◽  
Elliptic Partial Differential Equation ◽  
Three Dimensions ◽  
Pseudocontact Shift ◽  
Partial Differential

It is demonstrated that pseudocontact shift (PCS), viewed as a scalar or a tensor field in three dimensions, obeys an elliptic partial differential equation with a source term that depends on the Hessian of the unpaired electron probability density.

Qxygen transfer in gravity flow sewers

2000 ◽  
Vol 42 (3-4) ◽  
pp. 417-422 ◽  
Author(s):  
T.Y. Pai ◽  
C.F. Ouyang ◽  
Y.C. Liao ◽  
H.G. Leu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Dissolved Oxygen ◽  
Flow Velocity ◽  
Molecular Diffusion ◽  
Eddy Diffusion ◽  
Partial Differential ◽  
Diffusion Term ◽  
Sewer Pipes ◽  
Gravity Sewer

Oxygen diffused to water in gravity sewer pipes was studied in a 21 m long, 0.15 m diameter model sewer. At first, the sodium sulfide was added into the clean water to deoxygenate, then the pump was started to recirculate the water and the deoxygenated water was reaerated. The dissolved oxygen microelectrode was installed to measure the dissolved oxygen concentrations varied with flow velocity, time and depth. The dissolved oxygen concentration profiles were constructed and observed. The partial differential equation diffusion model that considered Fick's law including the molecular diffusion term and eddy diffusion term were derived. The analytic solution of the partial differential equation was used to determine the diffusivities by the method of nonlinear regression. The diffusivity values for the oxygen transfer was found to be a function of molecular diffusion, eddy diffusion and flow velocity.

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