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The neutron transport equation is a balance statement that conserves neutrons. Each term represents a gain or a loss of a neutron, and the balance, in essence, claims that neutrons gained equals neutrons lost. It is formulated as follows: [1]
The Monte Carlo method for radiation particle transport has its origins at LANL dates back to 1946. [3] The creators of these methods were Stanislaw Ulam, John von Neumann, Robert Richtmyer, and Nicholas Metropolis. [4] Monte Carlo for radiation transport was conceived by Stanislaw Ulam in 1946 while playing Solitaire while recovering from an ...
Theory, computation and applications, Lecture Notes in Computational Science and Engineering, 11. Springer-Verlag, Berlin, 2000. P. Lesaint, and P. A. Raviart. "On a finite element method for solving the neutron transport equation." Mathematical aspects of finite elements in partial differential equations 33 (1974): 89–123.
In the case of time-independent monochromatic radiation in an elastically scattering medium, the RTE is [1] (,) = (,) + (,) (, ′) ′where the first term on the RHS is the contribution of emission, the second term the contribution of absorption and the last term is the contribution from scattering in the medium.
A continuity equation is the mathematical way to express this kind of statement. For example, the continuity equation for electric charge states that the amount of electric charge in any volume of space can only change by the amount of electric current flowing into or out of that volume through its boundaries.
The convection–diffusion equation can be derived in a straightforward way [4] from the continuity equation, which states that the rate of change for a scalar quantity in a differential control volume is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume: + =, where j is the total flux and R is a net ...
Macromolecules, such as RNA and proteins, are actively transported across the nuclear membrane in a process called the Ran-GTP nuclear transport cycle. G-proteins are GTPase enzymes that bind to a molecule called guanosine triphosphate (GTP) which they then hydrolyze to create guanosine diphosphate (GDP) and release energy.
The updating step is given by Bayes' rule, and the prediction step is a Chapman-Kolmogorov transport equation. The mean field particle interpretation of these nonlinear filtering equations is a genetic type selection-mutation particle algorithm [ 48 ] During the mutation step, the particles evolve independently of one another according to the ...