On the path integral formulation and the evaluation of quantum statistical averages

Four problems of physical interest have been solved in this thesis using the path integral formalism. Using the trigonometric expansion method of Burton and de Borde (1955), we found the kernel for two interacting one dimensional oscillators• The result is the same as one would obtain using a normal coordinate transformation, We next introduced the method of Papadopolous (1969), which is a systematic perturbation type method specifically geared to finding the partition function Z, or equivalently, the Helmholtz free energy F, of a system of interacting oscillators. We applied this method to the next three problems considered• First, by summing the perturbation expansion, we found F for a system of N interacting Einstein oscillators^ The result obtained is the same as the usual result obtained by Shukla and Muller (1972) • Next, we found F to 0(Xi)f where A is the usual Tan Hove ordering parameter* The results obtained are the same as those of Shukla and Oowley (1971), who have used a diagrammatic procedure, and did the necessary sums in Fourier space* We performed the work in temperature space• Finally, slightly modifying the method of Papadopolous, we found the finite temperature expressions for the Debyecaller factor in Bravais lattices, to 0(AZ) and u(/K/ j,where K is the scattering vector* The high temperature limit of the expressions obtained here, are in complete agreement with the classical results of Maradudin and Flinn (1963) .

Molecular dynamics simulations of proteins: can the explicit water model be varied?

In molecular mechanics simulations of biological systems, the solvation water is typically represented by a default water model which is an integral part of the force field. Indeed, protein nonbonding parameters are chosen in order to obtain a balance between water-water and protein-water interactions and hence a reliable description of protein solvation. However, less attention has been paid to the question of whether the water model provides a reliable description of the water properties under the chosen simulation conditions, for which more accurate water models often exist. Here we consider the case of the CHARMM protein force field, which was parametrized for use with a modified TIP3P model. Using quantum mechanical and molecular mechanical calculations, we investigate whether the CHARMM force field can be used with other water models: TIP4P and TIP5P. Solvation properties of N-methylacetamide (NMA), other small solute molecules, and a small protein are examined. The results indicate differences in binding energies and minimum energy geometries, especially for TIP5P, but the overall description of solvation is found to be similar for all models tested. The results provide an indication that molecular mechanics simulations with the CHARMM force field can be performed with water models other than TIP3P, thus enabling an improved description of the solvent water properties.

Mg/Ca partitioning between aqueous solution and aragonite mineral: a molecular dynamics study

We have calculated the concentrations of Mg in the bulk and surfaces of aragonite CaCO3 in equilibrium with aqueous solution, based on molecular dynamics simulations and grand-canonical statistical mechanics. Mg is incorporated in the surfaces, in particular in the (001) terraces, rather than in the bulk of aragonite particles. However, the total Mg content in the bulk and surface of aragonite particles was found to be too small to account for the measured Mg/Ca ratios in corals. We therefore argue that most Mg in corals is either highly metastable in the aragonite lattice, or is located outside the aragonite phase of the coral skeleton, and we discuss the implications of this finding for Mg/Ca paleothermometry.

Path integral representations in noncommutative quantum mechanics and noncommutative version of Berezin-Marinov action

It is known that the actions of field theories on a noncommutative space-time can be written as some modified (we call them theta-modified) classical actions already on the commutative space-time (introducing a star product). Then the quantization of such modified actions reproduces both space-time noncommutativity and the usual quantum mechanical features of the corresponding field theory. In the present article, we discuss the problem of constructing theta-modified actions for relativistic QM. We construct such actions for relativistic spinless and spinning particles. The key idea is to extract theta-modified actions of the relativistic particles from path-integral representations of the corresponding noncommutative field theory propagators. We consider the Klein-Gordon and Dirac equations for the causal propagators in such theories. Then we construct for the propagators path-integral representations. Effective actions in such representations we treat as theta-modified actions of the relativistic particles. To confirm the interpretation, we canonically quantize these actions. Thus, we obtain the Klein-Gordon and Dirac equations in the noncommutative field theories. The theta-modified action of the relativistic spinning particle is just a generalization of the Berezin-Marinov pseudoclassical action for the noncommutative case.

Dressed coordinates: the path-integral approach

The recently introduced dressed coordinates are studied in the path-integral approach. These coordinates are defined in the context of a harmonic oscillator linearly coupled to massless scalar field and it is shown that in this model the dressed coordinates appear as a coordinate transformation preserving the path-integral functional measure. The analysis also generalizes the sum rules established in a previous work. (c) 2006 Elsevier B.V. All rights reserved.

Path integral approach to the full Dicke model

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Structure and thermodynamics of H3O+(H2O)8 clusters: A combined molecular dynamics and quantum mechanics approach

We have studied the structure and stability of (H3O+)(H2O)8 clusters using a combination of molecular dynamics sampling and high-level ab initio calculations. 20 distinct oxygen frameworks are found within 2 kcal/mol of the electronic or standard Gibbs free energy minimum. The impact of quantum zero-point vibrational corrections on the relative stability of these isomers is quite significant. The box-like isomers are favored in terms of electronic energy, but with the inclusion of zero-point vibrational corrections and entropic effects tree-like isomers are favored at higher temperatures. Under conditions from 0 to 298.15 K, the global minimum is predicted to be a tree-like structure with one dangling singly coordinated water molecule. Above 298.15 K, higher entropy tree-like isomers with two or more singly coordinated water molecules are favored. These assignments are generally consistent with experimental IR spectra of (H3O+)(H2O)8 obtained at 150 K.

Modelling the dynamics of genetic algorithms using statistical mechanics

A formalism for modelling the dynamics of Genetic Algorithms (GAs) using methods from statistical mechanics, originally due to Prugel-Bennett and Shapiro, is reviewed, generalized and improved upon. This formalism can be used to predict the averaged trajectory of macroscopic statistics describing the GA's population. These macroscopics are chosen to average well between runs, so that fluctuations from mean behaviour can often be neglected. Where necessary, non-trivial terms are determined by assuming maximum entropy with constraints on known macroscopics. Problems of realistic size are described in compact form and finite population effects are included, often proving to be of fundamental importance. The macroscopics used here are cumulants of an appropriate quantity within the population and the mean correlation (Hamming distance) within the population. Including the correlation as an explicit macroscopic provides a significant improvement over the original formulation. The formalism is applied to a number of simple optimization problems in order to determine its predictive power and to gain insight into GA dynamics. Problems which are most amenable to analysis come from the class where alleles within the genotype contribute additively to the phenotype. This class can be treated with some generality, including problems with inhomogeneous contributions from each site, non-linear or noisy fitness measures, simple diploid representations and temporally varying fitness. The results can also be applied to a simple learning problem, generalization in a binary perceptron, and a limit is identified for which the optimal training batch size can be determined for this problem. The theory is compared to averaged results from a real GA in each case, showing excellent agreement if the maximum entropy principle holds. Some situations where this approximation brakes down are identified. In order to fully test the formalism, an attempt is made on the strong sc np-hard problem of storing random patterns in a binary perceptron. Here, the relationship between the genotype and phenotype (training error) is strongly non-linear. Mutation is modelled under the assumption that perceptron configurations are typical of perceptrons with a given training error. Unfortunately, this assumption does not provide a good approximation in general. It is conjectured that perceptron configurations would have to be constrained by other statistics in order to accurately model mutation for this problem. Issues arising from this study are discussed in conclusion and some possible areas of further research are outlined.

The thermodynamics and statistical mechanics of protein folding

The physics of self-organization and complexity is manifested on a variety of biological scales, from large ecosystems to the molecular level. Protein molecules exhibit characteristics of complex systems in terms of their structure, dynamics, and function. Proteins have the extraordinary ability to fold to a specific functional three-dimensional shape, starting from a random coil, in a biologically relevant time. How they accomplish this is one of the secrets of life. In this work, theoretical research into understanding this remarkable behavior is discussed. Thermodynamic and statistical mechanical tools are used in order to investigate the protein folding dynamics and stability. Theoretical analyses of the results from computer simulation of the dynamics of a four-helix bundle show that the excluded volume entropic effects are very important in protein dynamics and crucial for protein stability. The dramatic effects of changing the size of sidechains imply that a strategic placement of amino acid residues with a particular size may be an important consideration in protein engineering. Another investigation deals with modeling protein structural transitions as a phase transition. Using finite size scaling theory, the nature of unfolding transition of a four-helix bundle protein was investigated and critical exponents for the transition were calculated for various hydrophobic strengths in the core. It is found that the order of the transition changes from first to higher order as the strength of the hydrophobic interaction in the core region is significantly increased. Finally, a detailed kinetic and thermodynamic analysis was carried out in a model two-helix bundle. The connection between the structural free-energy landscape and folding kinetics was quantified. I show how simple protein engineering, by changing the hydropathy of a small number of amino acids, can enhance protein folding by significantly changing the free energy landscape so that kinetic traps are removed. The results have general applicability in protein engineering as well as understanding the underlying physical mechanisms of protein folding. ^

A coupled meshless technique/molecular dynamics approach for deformation characterization of monocrystalline metal

This paper presents a multiscale study using the coupled Meshless technique/Molecular Dynamics (M2) for exploring the deformation mechanism of mono-crystalline metal (focus on copper) under uniaxial tension. In M2, an advanced transition algorithm using transition particles is employed to ensure the compatibility of both displacements and their gradients, and an effective local quasi-continuum approach is also applied to obtain the equivalent continuum strain energy density based on the atomistic poentials and Cauchy-Born rule. The key parameters used in M2 are firstly investigated using a benchmark problem. Then M2 is applied to the multiscale simulation for a mono-crystalline copper bar. It has found that the mono-crystalline copper has very good elongation property, and the ultimate strength and Young's modulus are much higher than those obtained in macro-scale.

Molecular dynamics simulation of fracture strength and morphology of defective graphene

Different types of defects can be introduced into graphene during material synthesis, and significantly influence the properties of graphene. In this work, we investigated the effects of structural defects, edge functionalisation and reconstruction on the fracture strength and morphology of graphene by molecular dynamics simulations. The minimum energy path analysis was conducted to investigate the formation of Stone-Wales defects. We also employed out-of-plane perturbation and energy minimization principle to study the possi-ble morphology of graphene nanoribbons with edge-termination. Our numerical results show that the fracture strength of graphene is dependent on defects and environmental temperature. However, pre-existing defects may be healed, resulting in strength recovery. Edge functionalization can induce compressive stress and ripples in the edge areas of gra-phene nanoribbons. On the other hand, edge reconstruction contributed to the tensile stress and curved shape in the graphene nanoribbons.

Molecular dynamics simulation of fracture strength and morphology of defective graphene

Different types of defects can be introduced into graphene during material synthesis, and significantly influence the properties of graphene. In this work, we investigated the effects of structural defects, edge functionalisation and reconstruction on the fracture strength and morphology of graphene by molecular dynamics simulations. The minimum energy path analysis was conducted to investigate the formation of Stone-Wales defects. We also employed out-of-plane perturbation and energy minimization principle to study the possible morphology of graphene nanoribbons with edge-termination. Our numerical results show that the fracture strength of graphene is dependent on defects and environmental temperature. However, pre-existing defects may be healed, resulting in strength recovery. Edge functionalization can induce compressive stress and ripples in the edge areas of graphene nanoribbons. On the other hand, edge reconstruction contributed to the tensile stress and curved shape in the graphene nanoribbons.

Water-ethanol mixtures by molecular dynamics and x-ray Compton scattering

Water-ethanol mixtures are commonly used in industry and house holds. However, quite surprisingly their molecular-level structure is still not completely understood. In particular, there is evidence that the local intermolecular geometries depend significantly on the concentration. The aim of this study was to gain information on the molecular-level structures of water-ethanol mixtures by two computational methods. The methods are classical molecular dynamics (MD), where the movement of molecules can be studied, and x-ray Compton scattering, in which the scattering cross section is sensitive to the electron momentum density. Firstly, the water-ethanol mixtures were studied with MD simulations, with the mixture concentration ranging from 0 to 100%. For the simulations well-established force fields were used for the water and ethanol molecules (TIP4P and OPLS-AA, respectively). Moreover, two models were used for ethanol, rigid and non-rigid. In the rigid model the intramolecular bond lengths are fixed, whereas in the non-rigid model the lengths are determined by harmonic potentials. Secondly, mixtures with three different concentrations employing both ethanol models were studied by calculating the experimentally observable x-ray quantity, the Compton profile. In the MD simulations a slight underestimation in the density was observed as compared to experiment. Furthermore, a positive excess of hydrogen bonding with water molecules and a negative one with ethanol was quantified. Also, the mixture was found more structured when the ethanol concentration was higher. Negligible differences in the results were found between the two ethanol models. In contrast, in the Compton scattering results a notable difference between the ethanol models was observed. For the rigid model the Compton profiles were similar for all the concentrations, but for the non-rigid model they were distinct. This leads to two possibilities of how the mixing occurs. Either the mixing is similar in all concentrations (as suggested by the rigid model) or the mixing changes for different concentrations (as suggested by the non-rigid model). Either way, this study shows that the choice of the force field is essential in the microscopic structure formation in the MD simulations. When the sources of uncertainty in the calculated Compton profiles were analyzed, it was found that more statistics needs to be collected to reduce the statistical uncertainty in the final results. The obtained Compton scattering results can be considered somewhat preliminary, but clearly indicative of the behaviour of the water-ethanol mixtures when the force field is modified. The next step is to collect more statistics and compare the results with experimental data to decide which ethanol model describes the mixture better. This way, valuable information on the microscopic structure of water-ethanol mixtures can be found. In addition, information on the force fields in the MD simulations and on the ability of the MD simulations to reproduce the microscopic structure of binary liquids is obtained.

Can current force fields reproduce ring puckering in 2-O-sulfo-α-l-iduronic acid? A molecular dynamics simulation study

The monosaccharide 2-O-sulfo-α-l-iduronic acid (IdoA2S) is one of the major components of glycosaminoglycans. The ability of molecular mechanics force fields to reproduce ring-puckering conformational equilibrium is important for the successful prediction of the free energies of interaction of these carbohydrates with proteins. Here we report unconstrained molecular dynamics simulations of IdoA2S monosaccharide that were carried out to investigate the ability of commonly used force fields to reproduce its ring conformational flexibility in aqueous solution. In particular, the distribution of ring conformer populations of IdoA2S was determined. The GROMOS96 force field with the SPC/E water potential can predict successfully the dominant skew-boat to chair conformational transition of the IdoA2S monosaccharide in aqueous solution. On the other hand, the GLYCAM06 force field with the TIP3P water potential sampled transitional conformations between the boat and chair forms. Simulations using the GROMOS96 force field showed no pseudorotational equilibrium fluctuations and hence no inter-conversion between the boat and twist boat ring conformers. Calculations of theoretical proton NMR coupling constants showed that the GROMOS96 force field can predict the skew-boat to chair conformational ratio in good agreement with the experiment, whereas GLYCAM06 shows worse agreement. The omega rotamer distribution about the C5–C6 bond was predicted by both force fields to have torsions around 10°, 190°, and 360°.

Exact path-integral evaluation of the heat distribution function of a trapped Brownian oscillator

Using path integrals, we derive an exact expression-valid at all times t-for the distribution P(Q,t) of the heat fluctuations Q of a Brownian particle trapped in a stationary harmonic well. We find that P(Q, t) can be expressed in terms of a modified Bessel function of zeroth order that in the limit t > infinity exactly recovers the heat distribution function obtained recently by Imparato et al. Phys. Rev. E 76, 050101(R) (2007)] from the approximate solution to a Fokker-Planck equation. This long-time result is in very good agreement with experimental measurements carried out by the same group on the heat effects produced by single micron-sized polystyrene beads in a stationary optical trap. An earlier exact calculation of the heat distribution function of a trapped particle moving at a constant speed v was carried out by van Zon and Cohen Phys. Rev. E 69, 056121 (2004)]; however, this calculation does not provide an expression for P(Q, t) itself, but only its Fourier transform (which cannot be analytically inverted), nor can it be used to obtain P(Q, t) for the case v=0.