Chandrasekhar equation

In the case of mean friction, equation (2) is no longer valid, and one has to use the so-called Chandrasekhar equation (Kramers 1940, ChamdraLsekhair 1943). From a method of derivation similar to the one used above, a more general equation is obtained for the rate constant, named KKAM ... 

The dynamical behavior of particles whose mass and size are much larger than those of the solvent particles can be explained by the theory of Brownian motion. Two approaches, Fokker-Planck and Chandrasekhar, have generally been used to solve the Langevin equation to describe Brownian motion. The Fokker-Planck differential equation is the diffusion equation in velocity space, while the Chandrasekhar equation is... 

Phosphinic Acid Reactions. Reaction of n-butylstannoic acid with diphenylphosphate instead of a carboxylic acid also results in the formation of a drum composition [ n-BuSn(0) 02P (OPh)2] g (Chandrasekhar, V. Holmes, J. M. Day, R. 0. Holmes, R. R., unpublished work). However, when diphenylphosphinic acid is reacted with n-butylstannoic acid under reflux in toluene, a new structural form of tin is obtained (7 ). The reaction proceeds according to Equation 4 giving the stable oxide composition in 90% yield, mp 198-208°C dec. 

The equations describing linear, adiabatic stellar oscillations are known to be Hermitian (Chandrasekhar 1964). This property of the equations is used to relate the differences between the structure of the Sun and a known reference solar model to the differences in the frequencies of the Sun and the model by known kernels. Thus by determining the differences between solar models and the Sun by inverting the frequency differences between the models and the Sun we can determine whether or not mixing took place in the Sun. 

Figure 1. The relation between central density and the mass of various degenerate star models. Chandrasekhar s curve is for white dwarfs with a mean molecular weight 2 of atomic mass units. Rudkjobing s curve is the same except for inclusion of the relativistic spin-orbit effects Rudkjobing (1952). The curve labeled Oppenheimer and Volkoff is for a set of neutron star models. The solid line marked Wheeler is a set of models computed with a generalized equation of state, from Cameron (1959).

This effectively states that the probability of the final state (left-hand side) is equal to that of all the initial states transforming to the final state (with probability P). Chandrasekhar expanded out the infinitesimal velocity and time changes of these quantities as Taylor series and used the Langevin equation to relate 5u and 5f. He showed that if the probability of changing velocity and position is given by a Gaussian distribution, then the probability, W(u, r, t) that a Brownian particle has a velocity u at a position r and at time t is... 

Dyson, in a letter to Chandrasekhar which the latter quotes, suggests that Dirac had arrived at that position when later work shewed that his physical logic had been flawed, but that his final equations were left intact, leading him to conclude that one should not trust the physical logic but should only trust the mathematical beauty (Chandrasekhar, 1994). Such, at any rate, is the analysis of Freeman Dyson whose reading of the matter is surely the most cogent and informed in the world today. 

Equations (3) and (4) are formally identical with the earlier Kubelka s hyperbolic solutions of differential equations for forward and backward fluxes (11), although the Chandrasekhar-Klier and Kubelka s theories start from different sets of assumptions and employ different definitions of constants characterizing the scattering and absorption properties of the medium. In Kubelka s theory, the constants a, b, and Y are related to the Schuster-Kubelka-Munk (SIM) absorption K and scattering S coefficients as... 

For an isotropic medium, by substituting the Gaussian quadrature formula for the integral in Eq. (4.78), the integral-differential equation may be reduced into a system of ordinary linear differential equations. Specifically, the integral can be treated as [Chandrasekhar, 1960]... 

The random force Q in the dynamic equations (2.29) is determined by its average moments and is specified from the condition that the equilibrium moments of the co-ordinates and velocities are known beforehand (Chandrasekhar 1943). In the linearised version, with ipn -C 1, this requirement determines the relation... 

The properties of the stochastic forces in the system of equations (3.31)-(3.35) are determined by the corresponding correlation functions which, usually (Chandrasekhar 1943), are found from the requirement that, at equilibrium, the set of equations must lead to well-known results. This condition leads to connection of the coefficients of friction with random-force correlation functions - the dissipation-fluctuation theorem. In the case under consideration, when matrixes f7 -7 and G 7 depend on the co-ordinates but not on the velocities of particles, the correlation functions of the stochastic forces in the system of equations (3.31) can be easily determined, according to the general rule (Diinweg 2003), as... 

If inertia is not negligible, the problem must l>e solved by using the more general Fokker-Plank equation (Chandrasekhar, 1943, p. 65). 

The Brownian motion of a particle under the influence of an external force field, and its consequent escape over a potential barrier has to be treated, in general, using the Fokker-Planck equation. This equation gives the distribution function W governing the probability that a particle will be after time t at a point x with velocity u (Chandrasekhar, 1943). In one dimension it has the form ... 

It is important to notice that both the original and the modified Fokker-Planck equations give the probability distribution of a particle as a function of time, position and velocity. However, if we are interested in time intervals large enough compared to jS 1, the Fokker-Planck equation, equation (3), can be reduced to a diffusional equation for the distribution function w, frequently called the Smoluchowski equation (Chandrasekhar, 1943) ... 

In order to establish the validity condition of a diffusion like equation for the probability of escape of a particle over a potential barrier, the solution of the modified Fokker-Planck equation is compared to the solution of the modified Smoluchowski equation. Since the main contribution to the determination of the escape probability comes from the neighborhood of the maximum in the potential energy (x = x J, the potential energy function was approximated by a parabolic function and the original Fokker-Planck equation was approximated at the vicinity of xmax by (Chandrasekhar, 1943) ... 

Following the mathematical steps described by Chandrasekhar (1943), the probability per unit time, P, that a Brownian particle will escape over a potential barier A< is derived from equation (13) as ... 

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