The Chandrasekhar equation

The idea in Kramers theory is to describe the motion in the reaction coordinate as that of a one-dimensional Brownian particle and in that way include the effects of the solvent on the rate constants. Above we have seen how the probability density for the velocity of a Brownian particle satisfies the Fokker-Planck equation that must be solved. Before we do that, it will be useful to generalize the equation slightly to include two variables explicitly, namely both the coordinate r and the velocity v, since both are needed in order to determine the rate constant in transition-state theory. 

For the Markovian random process (r,v), the integral equation Eq. (H.15) can be written 

As before, we now expand the left-hand side about t and the integrand in a Taylor series about W2(r, v r, v t)w2(Av v r), and obtain 

From this, the version of the Fokker-Planck equation for the transition probability density with two variables r and v is seen to be 

The probability density P(r,v,t) also satisfies a Fokker Planck equation. This was shown in Eq. (H.24) with one stochastic variable, and similarly we find in this case 

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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... 

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... 

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... 

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... 

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) ... 

The excellent review of Chandrasekhar provides a detailed account of the history of the subject, to which both Smoluchowski and Einstein made fundamental contributions. It is worth mentioning the well-known paper of Kramers, who provided a rigorous derivation of the Smoluchowski equation from the complete Fokker-Planck equation of a Brownian particle in an external potential. This problem allows us to explain what we mean by a systematic version of the AEP. We can state the problem as follows. Let us consider the motion of a free Brownian particle described by the one-dimensional counterpart of Eq. (1.2),... 

As said in Sect. 2, the iron core left over following Si burning suffers a dynamical instability as a result of endothermic electron captures and Fe photodid-integration. To a first approximation, this gravitational instability sets in near the classical Chandrasekhar mass limit for cold white dwarfs, Mch = 5.83 T) , Ye being the electron mole fraction. In the real situation of a hot stellar core, collapse may start at masses that differ somewhat from this value, depending on the details of the core equation of state. The reader is referred to [18] (especially Chaps. 12 and 13) for a detailed discussion of the implosion mechanism and for its theoretical outcome and observable consequences. Here, we just briefly summarise the situation. 

Upon inserting these last equations into Eq. 20, the continuity equation becomes formally identical in form with Eq. 19 obtained by Chandrasekhar. However, it differs in that an expression for C has been obtained which is explicit in the intermolecular forces in contrast to the older Brownian motion theory in which C is a phenomenological constant only. 

The discrete ordinates (DO) approximation is also a multiflux model. The discrete ordinates approximation was originally suggested by Chandrasekhar [19] for astrophysical applications, and a detailed derivation of the related equations was discussed by several researchers for application to neutron transport problems [33, 57-61], During the last two decades the method has been applied to various heat transfer problems [62-81]. 

The linear stability of inviscid jets has been presented in the previous Section, and it is straightforward to extend the results to the viscous equations of motion and boundary conditions. Of relevance in this Section are the temporal results since we want to describe pinching in the absence of a convective fiow - this can arise either from a Galilean transformation to remove the convective uniform component of the fiow, or we can think of examples such as the breakup of liquid bridges. The viscous dispersion relation has been given by Rayleigh and further discussion can be found in the texts of Chandrasekhar [8], Lamb [39] and Middleman [48], for example. 

This is known as Chandrasekhar s equation. For those of you familiar with the kinetic theory of gases, note the identity of the LHS with the streaming part of the Liouville equation for the reduced single particle distribution function, whereas the RHS can be viewed as the Brownian dynamics analogue of the collision terms in the Boltzmann 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 ... 

As seen above, a solution of the Langevin equation (Equation 6.50) (which is a nonlinear partial differential equation with random noise) consists of constructing the correlation functions of f (t) from the equation and then averaging the expressions with the help of the properties of the noise r(t). An alternative method of solution is to find the probability distribution function P(x, t) for realizing a situation in which the random variable f (t) has the particular value X at time t. P(x, t) is an equivalent description of the stochastic process f (0 and is given by the Fokker-Planck equation (Chandrasekhar 1943, Gardiner 1985, Risken 1989, Redner 2001, Mazo 2002)... 

Consider the simplest situation of Figure 9.4a, where an absorbing spherical sink of radius R is at the center of the coordinate system and polymer chains are present in the solution around the sink. Following the classical theory of Smoluchowski (Chandrasekhar 1943), we assume that the sink absorbs the polymer chains as soon as the center-of-mass of a polymer chain approaches the surface of the sink. We identify the capture rate of polymer chains as the steady-state net flux of polymer chains into the absorbing sink. Let the initial number concentration of the polymer chains be cq. The polymer concenuation is continuously maintained as co at distances far from the sink. The polymer chains undergo only diffusion and there are no other convective contributions. The continuity equation for the number concentration of polymer chains in three... 

Our aim in this chapter is to develop the mathematical formalism that serves as the foundation for all our analyses involving the radiation field in sufficient depth to be essentially autonomous, though our indebtedness to some of the procedures developed by Chandrasekhar (1950) is obvious. The equation of transfer is derived in Section 2.1, and formal solutions are found in Section 2.2. Very general techniques for solving the transfer equation numerically are developed in Section 2.3. Though... 

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).

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