Fishing with gorges: Testing a functional hypothesis

The Brain Mechanisms and Functional Hypothesis of Default Mode Network and Its Clinical Application

Advances in Psychological Science ◽

10.3724/sp.j.1042.2014.00234 ◽

2014 ◽

Vol 22 (2) ◽

pp. 234

Author(s):

Yu LI ◽

Hua SHU

Keyword(s):

Clinical Application ◽

Default Mode Network ◽

Default Mode ◽

The Brain ◽

Functional Hypothesis

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On hydrodynamics in the presense of strong external potential field

Journal of Physics and Electronics ◽

10.15421/332103 ◽

2021 ◽

Vol 29 (1) ◽

pp. 21-28

Author(s):

A. I. Sokolovsky ◽

S. A. Sokolovsky

Keyword(s):

Distribution Function ◽

Kinetic Equation ◽

Potential Field ◽

Chemical Potential ◽

Local Equilibrium ◽

Practical Importance ◽

External Potential ◽

Equilibrium Distribution Function ◽

Kinetic Coefficients ◽

Functional Hypothesis

On the base of the Boltzmann kinetic equation, hydrodynamics of a dilute gas in the presence of the strong external potential field is investigated. First of all, a gravitational field is meant, because the consistent development of hydrodynamics in this environment is of great practical importance. In the present paper it is assumed that it is possible to neglect the influence of the field on the particle collisions. The study is based on the Chapman–Enskog method in a Bogolyubov’s formulation, which uses the idea of the functional hypothesis. Consideration is limited to steady gas states, which are subjected to a simpler experimental study. Chemical potential μ0 of the gas at the point where the external field has zero value and its temperature T are selected as the reduced description parameters of the system. In equilibrium, in the presence of the field, these values do not depend on the coordinates. It is assumed that in thehydrodynamic states T and μ0 are weakly dependent on the coordinates and therefore their gradients, considered on the scale of the free path length of the gas, are small. The kinetic equation, accounting for the functional hypothesis, gives an integro-differential equation for a gas distribution function at the hydrodynamic stage of evolution. This equation is solved in perturbation theory in gradients of T and μ0. The main approximation is analyzed for possibility of the system to be in a local equilibrium by means of comparing it with an equilibrium distribution function. Next, the distribution function is calculated in the first approximation in gradients and it is expressed in terms of solutions Ap , Bp of some first kind integral Fredholm equations. An approach to the approximate solution of these equations is discussed. The found distribution function is used to calculate the fluxes of the number of gas particles and their energy in the first order in gradients T and μ0 . Kinetic coefficients, which describe the structure of these fluxes, are introduced. Matrix elements of the operator of the linearized collision integral (integral brackets) are used for their research. It is a question of validity of the principle of symmetry of kinetic coefficients and definition of their signs.

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