Time-correlation function Monte Carlo simulation

For the equihbrium properties and for the kinetics under quasi-equilibrium conditions for the adsorbate, the transfer matrix technique is a convenient and accurate method to obtain not only the chemical potentials, as a function of coverage and temperature, but all other thermodynamic information, e.g., multiparticle correlators. We emphasize the economy of the computational effort required for the application of the technique. In particular, because it is based on an analytic method it does not suffer from the limitations of time and accuracy inherent in statistical methods such as Monte Carlo simulations. The task of variation of Hamiltonian parameters in the process of fitting a set of experimental data (thermodynamic and... 

The approach to the evaluation of vibrational spectra described above is based on classical simulations for which quantum corrections are possible. The incorporation of quantum effects directly in simulations of large molecular systems is one of the most challenging areas in theoretical chemistry today. The development of quantum simulation methods is particularly important in the area of molecular spectroscopy for which quantum effects can be important and where the goal is to use simulations to help understand the structural and dynamical origins of changes in spectral lineshapes with environmental variables such as the temperature. The direct evaluation of quantum time- correlation functions for anharmonic systems is extremely difficult. Our initial approach to the evaluation of finite temperature anharmonic effects on vibrational lineshapes is derived from the fact that the moments of the vibrational lineshape spectrum can be expressed as functions of expectation values of positional and momentum operators. These expectation values can be evaluated using extremely efficient quantum Monte-Carlo techniques. The main points are summarized below. 

Recent years have seen the extensive application of computer simulation techniques to the study of condensed phases of matter. The two techniques of major importance are the Monte Carlo method and the method of molecular dynamics. Monte Carlo methods are ways of evaluating the partition function of a many-particle system through sampling the multidimensional integral that defines it, and can be used only for the study of equilibrium quantities such as thermodynamic properties and average local structure. Molecular dynamics methods solve Newton s classical equations of motion for a system of particles placed in a box with periodic boundary conditions, and can be used to study both equilibrium and nonequilibrium properties such as time correlation functions. 

The limit in front of the ratio means that the time t has to be much longer than the longest relaxation time of the chain. The resulting diffusion coefficients obtained by Monte Carlo simulation of the Evans-Edwards model of entangled polymers are presented in Fig. 9.33(a). The diffusion coefficient decreases with the number of monomers in the chain. Another quantity that can be extracted from the Monte Carlo simulations of the Evans-Edwards model is the relaxation time of the chain. It can be defined as the characteristic decay time of the time correlation function of the end-to-end vector R[t)R 0)) exp( t/Trep). Figure 9.33(b) presents the results of such simulations. 

In Chapters 3, 6 and 7, the two equivalent descriptions of Brownian motion the Langevin and Smoluchowski equations for an entanglement-free system have been studied in the cases where analytic solutions are obtainable the time-correlation function of the end-to-end vector of a Rouse chain and the constitutive equation of the Rouse model. When the Brownian motion of a more complicated model is to be studied, where an analytical solution cannot be obtained, the Monte Carlo simulation becomes a useful tool. Unlike the Monte Carlo simulation that is employed to calculate static properties using the Metropolis criterion, the simulation based on the Langevin equation can be used to calculate both static and dynamic quantities. 

Through Eq. (16.8a) or (16.8b), the time-correlation function C t) = (R(0) R(t)) of the end-to-end vector and the relaxation modulus G t) calculated from Eqs. (3.59) and (7.58), respectively, may be compared with the corresponding simulation results for a Rouse chain. In the simulation of a time-correlation function, an equilibrium state is first established by running a sufficiently large number of Monte Carlo steps. Then, a time window is set up, within which the time-correlation function may be calculated as explained in the following ... 

Fig. 16.1 Comparison of the analytical solutions (solid lines) and the Monte Carlo simulations for the time-correlation functions of the end-to-end vector, C t) = (R(0) -R(t)), of the Rouse chains with two beads (o) and three beads ( ) (b = 10 and d = 0.4 are used in the simulations.).

In order to model the asset s cash flow, it uses a Monte Carlo simulation to generate expected default times for each piece of collateral and utilizes Copula functions and equity indexes to estimate correlation in the default times. The default times allow CDOManager to determine the cash flow expected from each asset over the life of a transaction. Summing up the cash flow from all of the assets generates a picture of the expected future cash flow from the CDO collateral pool. 

This chapter is organized as follows. In section 1.1, we introduce our notation and present the details of the molecular and mesoscale simulations the expanded ensemble-density of states Monte Carlo method,and the evolution equation for the tensor order parameter [5]. The results of both approaches are presented and compared in section 1.2 for the cases of one or two nanoscopic colloids immersed in a confined liquid crystal. Here the emphasis is on the calculation of the effective interaction (i.e. potential of mean force) for the nanoparticles, and also in assessing the agreement between the defect structures found by the two approaches. In section 1.3 we apply the mesoscopic theory to a model LC-based sensor and analyze the domain coarsening process by monitoring the equal-time correlation function for the tensor order parameter, as a function of the concentration of adsorbed nanocolloids. We present our conclusions in Section 1.4. 

Reptation quantum Monte Carlo (RQMC) [15,16] allows pure sampling to be done directly, albeit in common with DMC, with a bias introduced by the time-step (large, but controllable in DMC e.g. [17]) and the fixed-node approach (small, but not controllable e.g. [18]). Property estimation in this manner is free from population-control bias that plagues calculation of properties in diffusion Monte Carlo (e.g. [19]). Inverse Laplace transforms of the imaginary time correlation functions allow simulation of dynamic structure factors and other properties of physical interest. 

The important difference between the microkinetics approach and the kinetic Monte Carlo simulation is that in the former diffusion is not explicitly included. Reaction probabilities are again based on the Eyring transition state rate expression. Its benefit is a substantial reduction in computational time length. Similar as in the kinetic Monte Carlo method, production rates as a function of reaction condition can be computed. These kinetic data can be correlated with changes in surface composition of the adsorbed reactant and intermediate overlayer. Also, rates of reaction intermediate production or removal can be deduced. 

Figure 7-1. Typical autocorrelation function of the energy. In this example it is calculated for the case of benzophenone in water simulated with Monte Carlo Metropolis method. The calculated auto-correlation function (circles) is fitted to the exponential decay (line) and the correlation time t is obtained using Eq. (7-7) applied in the fitted function shown in Eq. (7-8)...

Computer simulations have been applied to studies of the structure of molten salts along two lines one is the fi ee standing application of the computer simulation to obtain the partial pair correlation functions, the other is the refining of x-ray and neutron diffraction and EXAFS measurements by means of a suitable model. In both cases a suitable potential function for the interactions of the ions must be employed, as discussed in Sect. 3.2.4. Such potential functirms are employed in both the Monte Carlo (MC) and the molecular dynamics (MD) simulation methods. A further aspect that has been considered in the case of molten salts is the long range coulombic interaction that exceeds the limits of the periodic simulation boxes usually involved (for 1000 ions altogether), requiring the Ewald summation that is expensive in computation time and is prone to truncation errors if not applied carefully. 

Woodcock and Singer [117] were among the early persons who studied the structure of molten salts, in their case potassium chloride at 772 and 1033 °C, by means of Monte Carlo computer simulations. At about the same time Woodcock [118] reported the partial pair correlation functions for molten lithium chloride at 1000 °C obtained by molecular dynamics simulations. 

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