Real Monte-Carlo method

The basic scheme of this algorithm is similar to cell-to-cell mapping techniques [14] but differs substantially In one important aspect If applied to larger problems, a direct cell-to-cell approach quickly leads to tremendous computational effort. Only a proper exploitation of the multi-level structure of the subdivision algorithm (also for the eigenvalue problem) may allow for application to molecules of real chemical interest. But even this more sophisticated approach suffers from combinatorial explosion already for moderate size molecules. In a next stage of development [19] this restriction will be circumvented using certain hybrid Monte-Carlo methods. 

Accdg to Hammersley Handscomb (Addnl Ref N, p 8), S. Ulam, J. von Neumann and E. Fermi independently rediscovered Monte Carlo methods ca 1944 and started its systematic development. They also ensured that their scientific colleagues should become aware of the possibilities, potentialities and physical applications. The real use of Monte Carlo methods as research tools is attributed to von Neumann Ulam who applied them to random neutron diffusion in fissile material... 

The Monte Carlo method permits simulation, in a mathematical model, of stochastic variation in a real system. Many industrial problems involve variables which are not fixed in value, but which tend to fluctuate according to a definite pattern. For example, the demand for a given product may be fairly stable over a long time period, but vary considerably about its mean value on a day-to-day basis. Sometimes this variation is an essential element of the problem and cannot be ignored. 

Napier began by using Monte Carlo methods to establish that an essential precondition for a rigorous analysis of the type proposed was the availability of a sufficiently large sample of redshifts, each with formal accuracy better than 5 km/s anything less would result in even a real signal at 36 km/s being washed out by measurement errors. 

Fig. 4. Model of the real crystalline surface using the Monte-Carlo-method [after Ref. 75]...

The force-bias Monte Carlo method [Pangali et al. 1978 Rao and Berne 1979] biases the movement according to the direction of the forces on it. Having chosen an atom or a molecule to move, the force on it is calculated. The force corresponds to the direction in which a real atom or molecule would move. In the force-bias Monte Carlo method the random displacement is chosen from a probability distribution function that peaks in the direction of this force. The smart Monte Carlo method [Rossky et al. 1978] also requires the forces on the moving atom to be calculated. The displacement of an atom or molecule in this method has two components one component is the force, and the other is a random vector... 

Simulation of a mixing process by using a computer provides a better alternative. Monte Carlo simulation techniques are often utilized. Monte Carlo techniques are numerical methods that involve sampling from statistical distributions, either theoretical or empirical, to approximate the real physical phenomena without reference to the actual physical systems. For the general discussion of Monte Carlo methods, readers are referred to Hammersley and Handscomb [19] and Tocher [20]. 

The great interest in this topic was further demonstrated by the paper of Kattawar et al. [23], Several models of the vertical stractme of the cloud layer were analyzed using the Monte-Carlo method and polarimetric observations. Using homogeneous and inhomogeneous atmosphere models yielded the same real part of the refractive index and the same particle size. This fact seems to confirm Sobolev s hypothesis that linear polarization is mostly formed by single scattering within an optically thin top atmospheric layer. 

MONTE CARLO METHODS FOR REAL-TIME PATH INTEGRATION... 

MONTE CARLO METHODS FOR REAL-TIME PATH INTEGRATION which can be evaluated exactly. Thus we arrive at... 

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