Monte Carlo simulation partition function

The computation of quantum many-body effects requires additional effort compared to classical cases. This holds in particular if strong collective phenomena such as phase transitions are considered. The path integral approach to critical phenomena allows the computation of collective phenomena at constant temperature — a condition which is preferred experimentally. Due to the link of path integrals to the partition function in statistical physics, methods from the latter — such as Monte Carlo simulation techniques — can be used for efficient computation of quantum effects. 

The partition function Z is given in the large-P limit, Z = limp co Zp, and expectation values of an observable are given as averages of corresponding estimators with the canonical measure in Eq. (19). The variables and R ( ) can be used as classical variables and classical Monte Carlo simulation techniques can be applied for the computation of averages. Note that if we formally put P = 1 in Eq. (19) we recover classical statistical mechanics, of course. 

Figure 4.11 Monte-Carlo simulation (100 trials) of error propagation for La/Yb fractionation in residual melts by clinopyroxene-garnet removal from a basaltic parent magma (see text for parameter description and distributions used). Top mineral-liquid partition coefficients for La and Yb. Bottom variations of the La/Yb ratio as a function of the fraction F of residual melt.

The Hamiltonian is transformed exactly to one having planar symmetry, and defined on the (u, 4>) strip, where it = log(r/a) and varies between zero and L = log (R/a), and 4> is between zero and 2n [56]. The canonical partition function is explicitly evaluated in the strip. In the mean-field approximation, the threshold for countercondensation is the same as that predicted by Manning [56]. As the Manning parameter E, is increased, transitions that reflect condensation due to a single ion is observed [56]. The unique feature has been verified by Monte Carlo simulations [57]. 

The variational method proposed earlier by the authors relied on Monte Carlo simulations to select an intermembrane distance distribution function.8 The purpose of this paper is to present a new approach, in which the interaction between two membranes, in the presence of thermal fluctuations, is calculated directly by employing a suitable approximate partition function. Thus, the asymmetry of the distance distributions results in a natural manner from the calculation. 

Random displacement of molecules witluii each box. These are the usual moves of Monte Carlo simulation, insuring internal equilibrium and generating the ensemble upon wlucli die partition function is based, thus leading to a set of thermodynamic properties for the molecules of each box. 

The basic problem for Monte Carlo simulations of quantum systems is that the partition function is no longer a simple sum over classical configurations as in (16) but an operator expression... 

While this allows Monte Carlo simulations to be performed, the errors increase exponentially with the particle number N and the inverse temperature [3. To see this, consider the mean value of the sign (s) = Z/Z, which is just the ratio of the partition functions of the frustrated system Z = W(C) with weights W C) and the unfrustrated system used for sampling with... 

Molecular partition function, 200 Monolayer, 240 Monte Carlo simulation, 542 Moore, J.W., 195, 197, 219 Morse function, 196 Motooka, T., 11 Moulder, J.F., 513 Multiple-reaction system, 151 Murray, W., 581 Myers, AX., 265... 

In computer simulations, we are particularly interested in the properties of a system comprising a number of particles. An ensemble is a collection of such systems, as might be generated using a molecular d)mamics or a Monte Carlo simulation. Each member of the ensemble has an energy, and the distribution of the system within the ensemble follows the Boltzmann distribution. This leads to the concept of the ensemble partition function, Q. 

Monte Carlo simulations An alternative strategy to calculate thermodynamic properties is to explicitly follow the trajectory of a magnetic system by a computer simulation of the system. Along such trajectory, the system will adopt many conformations with different energy, magnetization and other microscopic observables. If the sampling of the conformational space is done correctly, a good estimate of the partition function can be made and with this all type of thermodynamic functions can be calculated. 

In Monte Carlo simulation, the choice of polymer model is governed by the choice of attempted moves. Typically kinetic energy is integrated analytically over all modes, and the partition function for the flexible model in the limit of infinite stiffness results [105]. In molecular dynamics, constraints (SHAKE, etc.) freeze kinetic energy contributions and the partition function for the rigid model results [105,197]. To achieve sampling from the desired partition function, it is necessary to add a pseudopotential based on the covariant metric tensor a [198]. 

In the case of general chain architectures, however, the mean fleld problem of a single chain in an external fleld cannot be cast in the form of a modified diflusion equation, and the density that a single chain creates in the external field and the concomitant single-chain partition function have to be estimated by partial enumeration [30-34], This methodology has been successfully applied to study the packing of short hydrocarbon chains in the hydrophobic interior of lipid bilayers [31,32,34] and polymer brushes [33] and to quantitatively compare the results of Monte Carlo simulations to the predictions of the mean field theory without adjustable parameters [30], The latter application is illustrated in Figure 5.2. 

The thermodynamic properties that we have considered so far, such as the internal energy, the pressure and the heat capacity are collectively known as the mechanical properties and can be routinely obtained from a Monte Carlo or molecular dynamics simulation. Other thermodynamic properties are difficult to determine accurately without resorting to special techniques. These are the so-called entropic or thermal properties the free energy, the chemical potential and the entropy itself. The difference between the mechanical emd thermal properties is that the mechanical properties are related to the derivative of the partition function whereas the thermal properties are directly related to the partition function itself. To illustrate the difference between these two classes of properties, let us consider the internal energy, U, and the Fielmholtz free energy, A. These are related to the partition function by ... 

There are two basic approaches to the computer simulation of liquid crystals, the Monte Carlo method and the method known as molecular dynamics. We will first discuss the basis of the Monte Carlo method. As is the case with both these methods, a small number (of the order hundreds) of molecules is considered and the difficulties introduced by this restriction are, at least in part, removed by the use of artful boundary conditions which will be discussed below. This relatively small assembly of molecules is treated by a method based on the canonical partition function approach. That is to say, the energy which appears in the Boltzman factor is the total energy of the assembly and such factors are assumed summed over an ensemble of assemblies. The summation ranges over all the coordinates and momenta which describe the assemblies. As a classical approach is taken to the problem, the summation is replaced by an integration over all these coordinates though, in the final computation, a return to a summation has to be made. If one wishes to find the probable value of some particular physical quantity, A, which is a function of the coordinates just referred to, then statistical mechanics teaches that this quantity is given by... 

The grand canonical ensemble is appropriate for adsorption systems, in which the adsorbed phase is in equilibrium with the gas at some specified temperature. The use of a computer simulation allows us to calculate average macroscopic properties directly without having to explicitly calculate the partition function. The grand canonical Monte Carlo (GCMC) method as applied in this work has been described in detail earlier (55). The aspects involving binary fluid mixtures have been described previously in our Xe-Ar work (30). 

The quantum partition function, eq.(35), of a system can be obtained through either Monte Carlo or molecular dynamics simulations. In PIMD simulation, the corresponding Lagrangian for such a classical isomorphic system can be given as ... 

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. 

---