# Theoretical Methods in the Physical Sciences: An introduction to problem solving using Maple V For use in a standard one-term course, in which both discrete and continuous probability is covered, students should have taken as a prerequisite two terms of calculus, including an introduction to multiple integrals. In order to cover Chap- ter 11, which contains material on Markov chains, some knowledge of matrix theory is necessary. The text can also be used in a discrete probability course. The material has been organized in such a way that the discrete and continuous probability discussions are presented in a separate, but parallel, manner.

For use in a discrete probability course, students should have taken one term of calculus as a prerequisite. The computer programs, solutions to the odd-numbered exercises, and current errata are also available at this site. We have tried not to spoil its beauty by presenting too much formal mathematics. Rather, we have tried to develop the key ideas in a somewhat leisurely style, to provide a variety of interesting applications to probability, and to show some of the nonintuitive examples that make probability such a lively sub ject.

Exercises: There are over exercises in the text providing plenty of oppor- tunity for practicing skills and developing a sound understanding of the ideas. In the exercise sets are routine exercises to be done with and without the use of a computer and more theoretical exercises to improve the understanding of basic con- cepts. A solution manual for all of the exercises is available to instructors. Historical remarks: Introductory probability is a sub ject in which the funda- mental ideas are still closely tied to those of the founders of the sub ject.

For this reason, there are numerous historical comments in the text, especially as they deal with the development of discrete probability. Pedagogical use of computer programs: Probability theory makes predictions about experiments whose outcomes depend upon chance. Consequently, it lends itself beautifully to the use of computers as a mathematical tool to simulate and analyze chance experiments.

In the text the computer is utilized in several ways. First, it provides a labora- tory where chance experiments can be simulated and the students can get a feeling for the variety of such experiments. This use of the computer in probability has been already beautifully illustrated by William Feller in the second edition of his famous text An Introduction to Probability Theory and Its Applications New York: Wiley, The record of a simulated experiment is therefore included. For example, the graphical illustration of the approximation of the standardized binomial distributions to the normal curve is a more convincing demonstration of the Central Limit Theorem than many of the formal proofs of this fundamental result.

Finally, the computer allows the student to solve problems that do not lend themselves to closed-form formulas such as waiting times in queues. Indeed, the introduction of the computer changes the way in which we look at many problems in probability. For example, being able to calculate exact binomial probabilities for experiments up to trials changes the way we view the normal and Poisson approximations.

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There is an equivalent representation of quantum mechanics in terms of Hamilton's equations Gray and Verosky, that makes possible the use of integrators for the quantum dynamics studies that are used for classical dynamics. Area-preserving mapping are also of interest in their own right in studies of dynamical systems Meiss, Symplectic integrators may be implicit or explicit. In explicit methods, the solution at the end of the time step is obtained by performing operations on the variables at the beginning of each time step.

The explicit versions generally involve simple algorithms that for propagation only use modest memory, while implicit methods involve more complex algorithms but are often more powerful for treating systems with disparate time scale dynamics, as discussed below. The development of symplectic integrators has involved significant interplay among mathematicians, physicists, and chemists. Seminal work on symplectic integrators was done by both physicists and mathematicians Ruth, ; Feng, ; Candy and Rozmus, ; McLachlan and Atela, ; Okunbor and Skeel, ; Calvo and Sanz-Serna, based on second-and third-order explicit approaches and Runge-Kutta methods.

Recently, these ideas have found their way into the chemistry community Gray et al. The Verlet integrator Verlet, , already in common use, was found to be symplectic, thereby explaining the good associated stability observed in practice. However, symplectic integrators that improve on previously available methods have also been developed Gray et al. Initial applications using these methods suggest that they may become favored for simulations of polymer dynamics and related problems with small time steps. Although standard explicit schemes, such as the Verlet and related methods, are simple to formulate and fast to propagate, they impose a severe constraint on the maximum time step possible.

Instability—uncontrollable growth of coordinates and velocities—occurs for step sizes much larger than 1 femtosecond 10 second. This step size is determined by the period associated with high-frequency modes present in all macromolecules, and it contrasts with the much longer time scales up to 10 2 seconds that govern key conformational changes e.

This disparity in time scales urges the development of methods that increase the time step for biomolecular simulations. Even if the stability of the numerical formulation can be ensured, an important issue concerning the reliability of the results arises, since vibrational modes in molecular systems are.

Standard techniques of effectively freezing the fast vibrational modes by a constrained formulation Ryckaert et al. The multiple time step approaches for updating the slow and fast forces provide additional speedup Streett et al. Various implicit formulations are available that balance stability, accuracy, and complexity. However, the standard implicit techniques used by numerical analysts Kreiss, have not been directly applicable to MD simulations of macromolecules, for the following reasons.

First, such implicit schemes are often designed to suppress the rapidly decaying component of the motion. However, this situation does not hold for biomolecular systems because of the intricate vibrational coupling. It is well recognized that concerted conformational transitions e. Thus, although the absence of the positional fluctuations associated with these high-frequency modes may not by itself be a severe problem, the absence of the energies associated with these modes may be undesirable for proteins and nucleic acids, since cooperative motions among the correlated vibrational modes may rely on energy transfer from these high-frequency modes.

Second, implicit schemes with known high stability e. This has prompted the application of such implicit schemes to the Langevin dynamics formulation, which involves frictional and Gaussian random forces in addition to the systematic force to mimic molecular collisions and therefore a thermal reservoir.

This stabilizes implicit discretizations and can be used to quench high-frequency vibrational modes Peskin and Schlick, ; Schlick and Peskin, , but unphysical increased rigidity can result Zhang and Schlick, Therefore, more rigorous approaches are required to resolve the subdynamics correctly, such as by combining normal-mode techniques with implicit integration Zhang and Schlick, ; significant linear algebra work in the spectral decomposition is necessary for feasibility for macromolecular systems.

For example, banded structures for the Hessian approximation see related discussion in the section on multivariate minimization beginning on page 68 can be exploited in the linearized equations of motion. There has also been some work on implicit schemes that do not have inherent damping, but preliminary experience suggests that for nonlinear systems, desirable energy conservation properties can be obtained only up to moderate time steps Simo et al.

In particular, serious resonance problems have been noted Mandziuk and Schlick, Third, implicit schemes for multiple time scale problems increase complexity, since they involve solution of a nonlinear system or minimization of a nonlinear function at each time step. Therefore, very efficient implementations of these additional computations are necessary, and even then, computational gain with respect to standard ''brute-force" integrations at small time steps can be realized only at very large time steps.

The preceding subsections have described several recent accomplishments in the development of integration methods in MD simulations and have also outlined several important challenges for the future. What makes these integration problems particularly challenging is the fact that solutions demand much more than straightforward application of standard mathematical techniques. At this point it appears that the optimal algorithms for MD will require a combination of methods and strategies discussed above, including symplectic and implicit numerical integration schemes that have minimal intrinsic damping, and correct resolution of the subdynamics of the system by some other technique e.

Undoubtedly, high-performance implementations will make possible a gain of several orders of magnitude in the simulation times, and there are certainly additional gains to be achieved by clever programming strategies. Allen, M. Biesiadecki, J. Skeel, , Dangers of multiple-time-step methods, J. Calvo, M. Candy, J. Rozmus, , A symplectic integration algorithm for separable Hamiltonian functions, J.

Channell, P. Scovel, , A symplectic integration of Hamiltonian systems, Nonlinearity Dahlquist, G. De Frutos, J.

## Electrodynamics: A Modern Geometric Approach

Sanz-Serna, , An easily implementable fourth-order method for the time integration of wave problems, J. Feng, K. Gear, C. Gray, S. Verosky, , Classical Hamiltonian structures in wave packet dynamics, J. Noid, and B. Sumpter, , Symplectic integrators for large scale molecular dynamics simulations: A comparison of several explicit methods, J. Heller, A. Windemuth, and K. Schulten, , Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions, Mol. Kreiss, H. Mandziuk, M. Schlick, , Resonance in chemical-system dynamics simulated by the implicit-midpoint scheme, Chem.

McCammon, J. McLachlan R. Atela, , The accuracy of symplectic integrators, Nonlinearity Meiss, J. Miyamoto, S. Okunbor, D. Skeel, , Canonical numerical methods for molecular dynamics simulations, Math. Peskin, C. Schlick, , Molecular dynamics by the backward-Euler method, Commun. Pure Appl. Ruth, R. Ryckaert, J. Ciccotti, and H. Berendsen, , Numerical integration of the cartesian equations of motion of a system with constraints: Molecular dynamics of n -alkanes, J. Schlick, T. Peskin, , Can classical equations simulate quantum-mechanical behavior? A molecular dynamics investigation of a diatomic molecule with a Morse potential, Commun.

Simo, J. Tarnow, and K. K Wong, , Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics, Computer Methods in Applied Mechanics and Engineering — Streett, W. Tildesley , and G. Saville , , Multiple time step methods in molecular dynamics, Mol. Stuart, A. Tuckerman, M. Berne, , Molecular dynamics in systems with multiple time scales: systems with stiff and soft degrees of freedom and with short and long range forces, J.

Berendsen, , Algorithms for macromolecular dynamics and constraint dynamics, Mol.

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Verlet, L. Thermodynamical properties of Lennard-Jones molecules, Phys. Watanabe, M. Karplus, , Dynamics of molecules with internal degrees of freedom by multiple time-step methods, J. Zhang, G. Schlick, , LIN: A new algorithm to simulate the dynamics of biomolecules by combining implicit-integration and normal mode techniques, J. Schlick, , Implicit discretization schemes for Langevin dynamics, Mol. Classical equilibrium statistical mechanics presents a class of unsolved N -representability problems analogous to those in the quantum mechanical regime discussed earlier in this chapter.

In this case, N refers to the number of particles atoms or molecules present, rather than the number of electrons The most straightforward version of this classical problem concerns a single-species monatomic system i. This nonnegative function of interparticle distance r is defined by the occurrence probability of particle pairs at r , relative to random expectation, Consequently, deviations of g r greater than 1 indicate that interparticle interactions have biased the distance distribution to a greater-than-random expectation, while deviations less than 1 indicate the Opposite.

For many cases of interest, the interparticle potential energy function V can be regarded as a sum of terms arising from each pair of particles present:. The pair potentials v r typically are taken to satisfy the following criteria:. Under these circumstances, g r plays a special rule in the thermodynamic properties of the N -particle system Hansen and McDonald, This fundamental quantity appears in closed-form expressions giving the pressure and mean energy at the prevailing temperature and density. Furthermore, it appears in expressions for the X-ray and neutron diffraction patterns for the substance; consequently, these diffraction measurements constitute an experimental means for measuring g r for real substances.

It should be added that g r is also one of the traditional results reported from computer simulations of N -body systems Ciccotti et al. The experimentally, or computationally, adjustable parameters are temperature; particle number density; container size, shape, and boundary conditions; and number N of particles. For most cases of interest, one focuses on the infinite-system limit, where the container size and N diverge, while temperature, number density, and container shape are held constant. The central problem then concerns the mapping between the pair of functions v r and g r , where the latter is interpreted as the infinite-system limit function.

Historically, the fundamental theory of classical systems particularly in the liquid state concentrated heavily on prediction of g r for a given v r , that is, the mapping from v to g. This has generated several well-known approximate integral equation predictive theories for g r , including those conventionally identified in the theoretical chemistry literature by the names Kirkwood , Bogoliubov-Born-Green-Yvon Born and Green, , Percus and Yevick , and hypernetted chain van Leeuwen et al.

However, in all cases the respective approximations invoked have, strictly speaking, been uncontrolled. Consequently, the local structure and thermodynamic property predictions based on these various integral equations have had only modest success in describing the dense liquid state,. BOX 4. A large part of computational chemistry is concerned with the properties of systems at or near thermal equilibrium. The statistics of configurations at thermal equilibrium therefore dominate many of the questions studied by chemists.

The principles of the statistical mechanics of equilibrium systems are quite simple to state but are profound and sometimes surprising in their results. A fundamental postulate of equilibrium statistical mechanics is that in the long run, all states of an isolated system that are consistent with conservation of energy will be observed with equal probability. Here k B is Boltzmann's constant. Thus, combinatorial and various counting problems play a special rule in our thinking about the thermodynamics of chemical systems. Although each of the states of an isolated system is equally probable, this is not the case when we consider only a part of a system.

When only part of a system is being examined, we must ask the question, How many states of the entire system are accessible when a subsystem is in a given configuration? The answer is given by. A very interesting and powerful special case of his formula is used constantly in equilibrium statistical mechanics. If the subsystem considered is only weakly coupled to the rest of a much larger system, we can decompose the total energy of the entire system into parts:.

The energetic coupling is small and can be neglected if we are considering a system that is itself fairly large and therefore has a relatively small surface of interaction with the remainder of the system. In this case the counting problem can be solved since we know that the entropy of the environment is a smooth function of its total energy. This then gives a count of states expressed by.

The probability then of an exactly specified state of a subsystem that is part of a larger one is proportional to this number of states. It is given by the Boltzmana distribution law. The temperature entering here is the thermodynamic derivative of the entropy and is proportional to the average kinetic energy of each particle in the system.

This distribution law contains within it many of the great phenomena of chemistry and physics. First we see that the most important states are those that have the lowest energy. If the energy then is a continuous function of the coordinates of part of the system, the most probable configurations are those that give minima this potential. Chemistry is usually a low-temperature phenomenon—most chemical reactions are studied around room temperature, although, of course, many do occur under greater extremes of conditions—and room temperature corresponds to only one-fortieth of the typical energy scale of chemical bonds.

Thus, the Boltzmann distribution law tells us that chemistry will mostly concern itself with the specific configurations that minimize the energy. Of course, if molecular systems remained entirely at their energy minima, little would go on. Occasionally, a molecular system must make a transition between one minimum on the energy surface and another. To do this, the system must occasionally find itself in an intermediate high-energy configuration, which the Boltzmann distribution law tells us is rather improbable.

If we ask which of the relatively improbable intermediate states between two minima are the most probable, it is clear that these should correspond to saddlepoints of the energy. These saddlepoint configurations are known as transition states to chemists. The probability of a system being found at a transition state determines the rate of a chemical transformation.

We see, therefore, that the geometry of minima and saddlepoints of potential energy surfaces is extremely important in determining the chemical properties of a molecular system. Sometimes only certain aspects of a system are considered explicitly. For example, when we study the shapes, structures, and motions of a biological molecule e. A special case of these geometrical problems arises when the subsystem being considered is itself rather large and involves strong interactions between its molecular subunits.

In this case, it sometimes happens that the minimum-energy saddlepoint actually possesses an extremely high energy. We then say that the transformation between two minima has a large barrier and the transformation will be extremely slow. Sometimes as the subsystem studied grows larger and larger, the transformation barrier itself also grows larger and larger.

Thus, for a macroscopic system, certain transformations may actually take place effectively only on infinite time scales. We can then treat each part of the configuration space very nearly as separate regions.

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• This situation arises when a phase transition occurs. The theory of phase transitions is then concerned with the problem of how a many-dimensional configurational space gets fragmented into parts that are separated by very high energy barriers. The Boltzmann distribution law applies only to a completely specified subsystem that is interacting weakly with its environment. The biological macromolecule is interacting strongly with its solvent environment, and so the Boltzmann distribution law using the energy alone is inappropriate for describing its configurations.

On the other hand, for different configurations of the biomolecule, we can in principle compute the number of configurations of the surrounding solvent that are compatible with that configuration of the biomolecule. Thus, the probability of a particular configuration of the biomolecule would have the form. For this reason, the geometry of free energy surfaces is often also of great interest to chemists and physicists.

Occasionally the distinction between energies and free energies is blurred in offhand writing by chemists and physicists, and the uninitiated reader must be careful about these distinctions when applications are made. Perhaps as a result of these shortcomings, the recent trend in classical statistical mechanics has been to rely heavily on direct computer simulation of condensed-phase phenomena. Because these simulations often require massive computational resources, a case can be made that revival of analytic predictive theory for g r would be favorable from the point of view of the ''productivity issue" in theoretical and computational chemistry.

In some respects, the inverse mapping of g to v is even more subtle, intriguing, and mathematically challenging. At the outset, one encounters the obvious matter of defining the space of functions g r that in fact can be generated by a pairwise additive potential energy function V. A few necessary conditions are straightforward; as already remarked, g r cannot be negative. It is generally accepted but not rigorously demonstrated that g must approach unity as r diverges if the temperature is positive, even though the system itself may be in a spatially periodic crystalline state.

In addition, the Fourier transform of g r — 1,. These generic conditions can be supplemented by others that are necessary if v r has an infinitely repelling hard core, that is,. This hard-core property prevents neighbors from clustering too densely around any given particle, and from the geometry of hard-sphere close packings it is possible to bound the integral of r 2 g r over finite intervals of r.

A primary challenge concerns formulation of sufficient conditions on g r , given that V possesses the pairwise-additive form displayed above. At present we have no rational criterion for deciding whether a given g r , however "reasonable" it may appear to be by conventional physical standards, corresponds to the thermal-equilibrium short-range order for any pairwise additive V. It is not even clear at present how to construct a counterexample, namely, a g r meeting the necessary conditions above that cannot map to a v r of the class described.

In any case, formulation of sufficient conditions would likely improve prospects for more satisfactory integral equation or other analytical predictive techniques for g r. Several directions of generalization exist for this classical V -Representability problem; these include the following matters:. Properties of triplet and higher-order correlation functions g n for occurrence probabilities of particle n -tuples;. Properties of correlation functions for particles molecules with internal degrees of freedom rotation, vibration, conformational flexibility ;.

Effects of specific nonadditive potentials, which would be the case when including. Multicomponent several species, or mixture systems, in particular the important case of electrostatically charged particles ions with their long-ranged Coulombic interactions. Born, M. Ciccotti, G. Frankel, and I. Kirkwood, , eds. Hansen, J. Kirkwood, J. Percus, J. Frisch and J. Lebowitz, eds. Benjamin, New York, pp. Yevick, , Analysis of classical statistical mechanics by means of collective coordinates, Phys. Groeneveld, and J. De Boer, , New method for the calculation of the pair correlation function, Physica — Widom, B.

The Born-Oppenheimer approximation dates from the s, and the entire notion of molecular structure can be based upon it. It is thus a surprise that significant qualitative physics has been ignored by most chemical physicists in applying the Born-Oppenheimer approximation to systems with degenerate electronic states. The basic idea behind the Born-Oppenheimer approximation is that nuclei move much more slowly than electrons.

This has proved valid for systems that do not have significant electronic degeneracy. A serious mathematical problem is the uniqueness of the wave function for the nuclei. The Born-Oppenheimer approximation really assumes a single path for the slowly moving nuclei. If there is an electronic degeneracy, topologically distinct paths may connect two different positions on the same electronic surface. Thus, in addition to the phases that one develops for the quantum dynamics through the simple scalar potential dynamics, there is an additional topological phase.

The existence of this topological phase, which depends on the path between two points, has been known since at least the p, when Longuet-Higgins studied it in the context of Jahn-Teller distortions. Only in recent years has its significance been truly appreciated, however. One of the leaders in bringing out the significance of topology in quantum molecular dynamics was M. However, it was appreciated somewhat earlier by Truhlar and Mead that this topological phase plays a rule in chemical reactions.

Very recently, the discrepancy. Many phenomena in chemistry are at the border of applicability of classical mechanics. Quantum mechanical phenomena, such as tunneling and interference, certainly are relevant to many chemical reactions. Thus, in addition to purely classical dynamical methods, semiclassical approximations are used quite commonly in chemical physics. Semiclassical methods work fairly well for low-dimensional systems such as those encountered in gasphase chemical reactions, because the collisions that act as randomizers are infrequent and the chaotic character of the processes may often be neglected.

On the other hand, in attempting to extend semiclassical methods to condensed-phase systems, one is immediately faced with the problem of the underlying classical chaos, No completely adequate semiclassical quantization of chaotic systems yet exists.

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• Most of the effort of theoretical chemists working in this area has been devoted to understanding simple themes that may give some qualitative insight to phenomena that occur in the quantum mechanics of systems that are classically chaotic. Several important themes have been developed, one of the most notable being the connection of quantum chaos to random matrix theory. The notions behind this have their roots in Wigner's use of random matrices to describe nuclear systems, but the application of these ideas to molecules has been equally rewarding. One can examine the evolution of the energy levels of a system under perturbation in order to understand its quantum chaotic nature.

It has been shown that a random matrix description could arise from multiple level crossings. This approach has also been shown to be related to some exactly integrahie systems of particles moving in one dimension. The great irony is that the random matrix description arises from a problem that, in another guise, leads to exactly integrable equations. Very interesting connections exist to the theory of solutions of nonlinear partial differential equations.

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Classical systems in many dimensions, when they are chaotic, often exhibit diffusive dynamics. Arnold has shown how weakly coupled systems of dimension higher than 2 exhibit such diffusion. It has recently been argued that a phenomenon analogous to Arnold diffusion in the classical limit arises in quantum problems and that weakly coupled systems of quantized oscillators are analogous to local random matrix models.

These local random matrix models are closely tied to the problem of Anderson localization, which concerns itself with the nature of eigenfunctions of random differential operators. A most enticing development in the understanding of quantum chaos has been the connection of problems in quantum chaos with deep problems in number theory. For chaotic systems these classical paths are extremely numerous and the Green's function is indeed the sum of a statistically fluctuating quantity that itself presents interesting mathematical problems to be discussed later.

One model for the classical paths represents them as repetitions of some fundamental periodic orbits. Some very simple special models of the actions of these orbits lead to a Green's function that is closely tied to the Riemann zeta function. The prime numbers represent the fundamental periodic orbits. This has suggested that the zeros of the Riemann zeta function are related to the quantum mechanical eigenvalues of some Hamiltonian that is classically chaotic.

This ansatz has led to interesting predictions about the spacing of zeros of the Riemann zeta function and other statistics that seem very much in keeping with the random matrix theories being used to describe quantum chaos. Thus, it seems that problems in quantum chaos might be clarified considerably by considerations from probabilistic number theory and, conversely, that deep number theoretic questions might be addressed by using ideas from the quantum mechanics of chaotic systems.

Although significant progress has already been made in developing conjectures based on these ideas, there is still a tremendous amount to do and many deep mysteries remain. For problems with small amounts of degeneracy, the topological phase is easy to handle with little mathematical sophistication. Either a trajectory encircles a conical intersection of Born-Oppenheimer energy surfaces or it does not, leading to two values of the phase.

This encircling of singularities can be described by using the idea of a gauge potential. With higher degeneracies, however, very difficult topological problems may be encountered since many surfaces can make avoided crossings in many locations. The paradigm of such complicated topology problems may well be metal clusters. For metals in the thermodynamic limit, there are numerous energy levels corresponding to the excitation of electrons just below the Fermi sea to just above it. Since the electronic levels are highly delocalized, these energy changes are quite small and the energy surfaces are close together.

The actual dynamics of the nuclei must involve the coupling of several surfaces. There are many possible interchanges of the metallic ionic cores, and complicated topologies can result.

## Engineering Physics (E P)

Another place in which topology enters is when an underlying approximate wave function is built up out of many degenerate electronic wave functions and the dynamics of electronic excitations is studied. The paradigm for this is the recent interest in resonating valence bond descriptions of metallic and superconducting materials. Here, reorganization of the different valence bond structures as an excited electron or hole moves around gives rise to topological phases and gauge fields. It has been argued that these effects are at the heart of the new high-temperature superconductors and represent a real breakdown of the traditional band structure picture of metals.

Most models studied by physicists, however, have been very simple, and it will be necessary to understand how the topological phases arise in completely realistic electronic structure calculations if one is to make predictions of new high-temperature superconductors on the basis of these ideas. A major mathematical landmark in the eighteenth century was Euler's introduction and exploitation of the famous gamma function. One of its basic and striking properties is that it provides a natural "smooth" extension of the factorials n!

The pervasive appearance of the Euler gamma function throughout classical mathematical analysis constitutes a powerful paradigm suggesting that analogous extensions from the discrete positive integers to the complex plane in other contexts might generate analogous intellectual benefits. During roughly the last two decades, simultaneous developments in several distinct areas of physical science appear to point to the necessity or at least the desirability of just such an extension.

Specifically, this involves generalizing the familiar notion of Euclidean D -dimensional spaces from positive integer D at least to the positive reals, if not to the complex D -plane. This is not an empty pedantic exercise; at least one serious proposal has been published Zeilinger and Svozil, claiming that accurate spectroscopic measurements of the electron "g-factor" indicate that the space dimension of our world is less than 3 by approximately 5 x 10 Furthermore, in various theoretical applications that have so far been suggested for the continuous- D concept, D itself or its inverse appears to be a natural expansion parameter for various fundamental quantities of interest.

However, most of the work along these lines thus far has been ad hoc, lacking rigorous mathematical underpinning. Naturally this calls into question the validity of claimed results. Three physical science research areas deserve mention in this context. The first is quantum field theory; dimension D has been treated as a continuously variable "regularizing parameter" whose.

The third area holds perhaps the greatest promise for chemical progress, namely, the development of atomic and molecular quantum mechanics with useful computational algorithms in spaces of arbitrary D Goodson et al. Knowledge of the nodes of the many-fermion wavefunction would make possible exact calculation of the properties of fermion systems by Monte Carlo methods. Little is known about nodes of many-body fermion systems even though the one-dimensional case is ubiquitous in textbooks on quantum mechanics. The nodes referred to here are the nodes of the exact many-body wavefunction and are very different from the nodes of orbitals.

In the absence of a rigorous simulation method for fermion systems, the fixed-node approximation has been found to be a useful and powerful approach. One assumes knowledge of where the exact wavefunction is positive and negative based on the nodes of a trial wavefunction. For the ground state, Ceperley has proved that ground state nodal cells have the tiling property i.

The tiling property is the generalization to fermions of the theorem that a bosonic ground state is nodeless. The nodal hypervolumes of a series of atomic N -body Hartree-Fock level electronic wavefunctions have been mapped by using a Monte Carlo simulation in 3 N -dimensional configuration space Glauser et al.

The basic structural elements of the domain of atomic and molecular wavefunctions have been identified as nodal regions and permutational cells identical building blocks. The results of this study on lithium-carbon indicate that Hartree-Fock wavefunctions generally consist of four equivalent nodal regions two positive and two negative , each constructed from one or more permutational cells. A generalization of the fixed-node method has been proposed that could solve the fermion problem at finite temperature if only the nodes of the fermion density matrix were known Ceperley, Glauser, W.

Brown, W. Lester, Jr. Bressanini, B. Hammond, and M. Koszykowski, , J. These latter afford convenient fixed points for refining the series summation attempts. The presumption that spaces with noninteger dimension were available as analytic tools for atomic and molecular quantum mechanics rests largely on simple observations such as the fact that the D-dimensional hyper spherical volume element,.

The implicit assumption in the various applications to date, quantum mechanical and otherwise, seems to have been that the same expression can be invested with mathematical legitimacy for noninteger D , in the sense that it is an attribute of a family of precisely defined spaces. The published literature reveals some attempts to axiomatize spaces of noninteger dimension Wilson, ; Stillinger, , but it is clear that the subject requires deeper mathematical insight than it has thus far experienced.

In particular, it is desirable to determine the extent to which arbitrary- D spaces are uniquely definable as uniform and isotropic metric spaces and what their relation to conventional vector spaces might be. It has been suggested Wilson, that noninteger- D spaces can be viewed as embedded in an infinite-dimensional vector space, but whether this is uniquely possible or even necessary to perform calculations remains open.

It is important to stress the distinction between the general- D spaces that may be obtained by interpolation between the familiar Euclidian spaces for integer D on the one hand and the so-called fractal sets to which a generally noninteger Hausdorff-Besicovitch dimension can be assigned Mandelbrot, The latter are normally viewed as point sets contained in a Euclidean host space; furthermore, they fail to display translational and rotational invariance, and are therefore not uniform and isotropic.

Ashmore, J. Bollini, C. Giambiagi, , Dimensional renormalization: The number of dimensions as a regularizing parameter, Nuovo Cimento B Goodson, D. Lopez-Cabrera, D. Herschbach, and J. Gorishny, S. Larin, and F. Tkachov, , Phys. Herrick, D. Herschbach, D. Avery, and O. Goscinski, eds. Research Methods and Communication in the Social Sciences. The read theoretical methods in the physical sciences an introduction stands Do Multiple Trials fall main papers?

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