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Monte Carlo simulation: Drawing a large number of pseudo-random uniform variables from the interval [0,1] at one time, or once at many different times, and assigning values less than or equal to 0.50 as heads and greater than 0.50 as tails, is a Monte Carlo simulation of the behavior of repeatedly tossing a coin.
For example, see the movie, where a Monte Carlo simulation of a pencil beam incident on a semi-infinite medium models both the initial ballistic photon flow and the later diffuse propagation. The Monte Carlo method is necessarily statistical and therefore requires significant computation time to achieve precision.
In 1947, John von Neumann sent a letter to Robert Richtmyer proposing the use of a statistical method to solve neutron diffusion and multiplication problems in fission devices. [5] His letter contained an 81-step pseudo code and was the first formulation of a Monte Carlo computation for an electronic computing machine.
The sign problem is NP-hard, implying that a full and generic solution of the sign problem would also solve all problems in the complexity class NP in polynomial time. [8] If (as is generally suspected) there are no polynomial-time solutions to NP problems (see P versus NP problem), then there is no generic solution to the sign problem. This ...
Another important concept related to the Monte Carlo integration is the importance sampling, a technique that improves the computational time of the simulation. In the following sections, the general implementation of the Monte Carlo integration for solving this kind of problems is discussed.
The Monte Carlo method for electron transport is a semiclassical Monte Carlo (MC) approach of modeling semiconductor transport. Assuming the carrier motion consists of free flights interrupted by scattering mechanisms, a computer is utilized to simulate the trajectories of particles as they move across the device under the influence of an electric field using classical mechanics.
In Transport Monte Carlo simulations, the total scattering rate (), is assumed to only result from ion-water interactions; it is related to ion diffusivity with the expression = where m is the mass of the ion and D is its diffusion constant. As the equation indicates, reduced diffusivity of ions inside the lumen of the channel renders to ...
Photon transport in biological tissue can be equivalently modeled numerically with Monte Carlo simulations or analytically by the radiative transfer equation (RTE). However, the RTE is difficult to solve without introducing approximations. A common approximation summarized here is the diffusion approximation.