Probability current

The first paper that was devoted to the escape problem in the context of the kinetics of chemical reactions and that presented approximate, but complete, analytic results was the paper by Kramers [11]. Kramers considered the mechanism of the transition process as noise-assisted reaction and used the Fokker-Planck equation for the probability density of Brownian particles to obtain several approximate expressions for the desired transition rates. The main approach of the Kramers method is the assumption that the probability current over a potential barrier is small and thus constant. This condition is valid only if a potential barrier is sufficiently high in comparison with the noise intensity. For obtaining exact timescales and probability densities, it is necessary to solve the Fokker-Planck equation, which is the main difficulty of the problem of investigating diffusion transition processes. 

Reflecting Boundary. The reflecting boundary may be represented as an infinitely high potential wall. Use of the reflecting boundary assumes that there is no probability current behind the boundary. Mathematically, the reflecting boundary condition is written as... 

Let us consider the case when the diffusion coefficient is small, or, more precisely, when the barrier height A is much larger than kT. As it turns out, one can obtain an analytic expression for the mean escape time in this limiting case, since then the probability current G over the barrier top near xmax is very small, so the probability density W(x,t) almost does not vary in time, representing quasi-stationary distribution. For this quasi-stationary state the small probability current G must be approximately independent of coordinate x and can be presented in the form... 

The escape time is introduced as the probability P divided by the probability current G. Then, using (3.4) and (3.6), we can obtain the following expression for the escape time ... 

Integrating Eq. (3.16) and taking into account the reflecting boundary conditions (probability current is equal to zero at the points d), we get... 

We introduce into consideration the Laplace transformations of the probability density and the probability current... 

It is obvious that the steady-state quantities of the probability density Wst (x) and the probability current Gst(x) may be obtained without any difficulties at appropriate boundary conditions directly from Eq. (5.72). 

Here it is taken into account that for all profiles in question the steady-state probability current H0(x) equals 0 for any finite x. 

If there is the symmetrical potential profile initial distribution is located at the origin (i.e., xo = 0), then all results concerning the relaxation times will be the same as for the potential profile in which the reflecting wall is at x = 0 and at xo = +0. This coincidence of the relaxation times may be proven in a common case taking into account that the probability current at x = 0 is equal to zero at any instant of time. 

This is concerned with the fact that in the case of the relaxation time, roughly speaking only half of all Brownian particles should leave the initial potential minimum to reach the equilibrium state, while for the profile of the decay time case all particles should leave the initial minimum. Expression (5.120), of course, is true only in the case of the sufficiently large potential barrier, separating the stable states of the bistable system, when the inverse probability current from the second minimum to the initial one may be neglected (see Ref. 33). 

Note that the probability current in Laplace transform terms is... 

Here W[z] = U(z) - v(< ) is Wronskian> and C] and Cl are arbitrary constants that may be found from the continuity condition of the probability density and the probability current at the origin ... 

Calculating from (9.4) the values of arbitrary constants and putting them into (9.3), one can obtain the following value for the probability current Laplace transform G(z, s) (9.2) at the point of symmetry z — 0. 

Before finding the Laplace-transformed probability density wj(s, zo) of FPT for the potential, depicted in Fig. A 1(b), let us obtain the Laplace-transformed probability density wx s, zo) of transition time for the system whose potential is depicted in Fig. Al(c). This potential is transformed from the original profile [Fig. Al(a)] by the vertical shift of the right-hand part of the profile by step p which is arbitrary in value and sign. So far as in this case the derivative dpoints except z = 0, we can use again linear-independent solutions U(z) and V(z), and the potential jump that equals p at the point z = 0 may be taken into account by the new joint condition at z = 0. The probability current at this point is continuous as before, but the probability density W(z, t) has now the step, so the second condition of (9.4) is the same, but instead of the first one we should write Y (0) + v1 (0) = YiiOje f1. It gives new values of arbitrary constants C and C2 and a new value of the probability current at the point z = 0. Now the Laplace transformation of the probability current is... 

For the mean transition time, this fact may be explained in the following way If the transition process is going from up to down, then the probability current is large, but it is necessary to fill the lower minimum by the larger part of the probability to reach the steady state if the transition process is going from down to up, then the probability current is small, and it is necessary to fill the upper minimum by the smaller part of the probability to reach the steady state. 

The probability current is conserved for E>V. When penetrating the barrier, E — V becomes negative and decays as g-C/ dv i2m(v(x)-E)]8x as particle moves from position x + <5x to position x. The total probability of... 

Equation (7) is an exact analogue of the continuity equation (1.7) of hydrodynamics, and this allows definition of a probability current density... 

If the vector j(r,t) = idca I is interpreted as a probability current density, one has the continuity equation... 

In the standard overdamped version of the Kramers problem, the escape of a particle subject to a Gaussian white noise over a potential barrier is considered in the limit of low diffusivity—that is, where the barrier height AV is large in comparison to the diffusion constant K [14] (compare Fig.6). Then, the probability current over the potential barrier top near xmax is small, and the time change of the pdf is equally small. In this quasi-stationary situation, the probability current is approximately position independent. The temporal decay of the probability to find the particle within the potential well is then given by the exponential function [14, 22]... 

In order to further elucidate the choice of boundary conditions let us consider the probability current J which is defined for a wavefunction ip(R) by (Cohen-Tannoudji, Diu, and Laloe 1977 ch.III)... 

Since the total wavefunction is stationary, the incoming and the outgoing probability currents must cancel, i.e., J t +X)n = 0) which imposes... 

In passing we note that the scattering wavefunctions appropriate for full collisions, A + BC(n ) —> A + BC(n), are defined such that the incoming and outgoing probability currents are... 

In Section 5.2, we used an expression for the flux operator. In most quantum mechanics textbooks expressions for the probability current density, or probability flux density, are given in terms of the wave function in the coordinate representation. We need an expression for the flux density operator without reference to any particular representation, and since it is rarely found in the textbooks, let us in the following derive this expression. 

The scattering matrix S is used in quantum discussions of scattering theory Sjf is equal to the ratio (total probability current scattered in channel j)/(total probability current incident in channel i). S is a unitary matrix SS = 1. Pji is the probability that collision partners incident in channel i will emerge in channel j. 

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