Group Theoretical Statistical Mechanics Research Articles

Investigation of artifacts due to periodic boundary conditions

Periodic boundary conditions are routinely employed in molecular dynamics simulations to simulate bulk material and remove surface effects. However, the effect of periodic boundary conditions on simulation results is not well understood. In this paper three distinct approaches are explored in an effort to determine whether artifacts exist due to the introduction of artificial periodicity. The techniques of group theoretical statistical mechanics are applied to provide tests for artifacts due to the cubic symmetry introduced. Autocorrelation functions and generalized squared displacements for large and small system sizes are compared for artifacts and finally transport coefficients for very low density systems are compared with accepted theoretical results. None of these methods show signs of numerically significant artifacts.

The effect of orthogonal electric fields on the molecular dynamics of liquid water, principle three of group theoretical statistical mechanics.

Molecular dynamics computer simulation has been used to investigate the effect of orthogonal electric fields on the molecular dynamics of liquid water. It is found that a combination of a static electric field E X in the x axis and an orthogonal static electric field E Z in the Z axis results in the appearance of symmetric cross correlation functions of the type <A X(O) ZA (t)> = <A Z(O) A X(t)> in the laboratory frame (X, Y, Z). This is in accordance with the third principle of group theoretical statistical mechanics (g.t.s.m.). These cross correlation functions (c.c.f.'s) are observable experimentally at dielectric and far infra red frequencies.

Nonlinear Optical Rotatory Dispersion: Application of Group Theoretical Statistical Mechanics

Abstract The third principle of group theoretical statistical mechanics is used to define the existence of many new nonlinear optical rotatory phenomena caused by the interaction of intense electromagnetic radiation with molecular ensembles. This treatment removes the necessity of having to distinguish between ‘natural’ and ‘magnetic’ optical activity; both processes emerge consistently from the analysis, which is based on an application of the Hellman-Feynman theorem and a double Taylor expansion of the induced electric and magnetic dipole moments.

Symmetry of ensemble averages in smetic liquid crystals

Group-theoretical statistical mechanics is applied to determined the number of non-vanishing ensemble averages in the point groups of smectic liquid crystals, modelled on the 32 possible crystallographic point groups supplemented by linear symmetries. Assuming that the thermodynamic average exists according to the number of irreducible representations in each point group that are totally symmetric—a basic principle of group-theoretical statistical mechanics—it is possible to conclude whether or not that average exists at thermodynamic equilibrium. The conclusion is valid within the point-group symmetry of the liquid crystal. The number of ensemble averages supported by the smectic point group, exemplified by the time correlation functions, decreases from triclinic to monoclinic to orthorhombic to trigonal to tetragonal to hexagonal to cubic. Within each of these major classifications, the pattern of non-vanishing correlation functions and other ensemble averages has its own distinctive signature, based on...

Transient ensemble averages in non-Newtonian flow: Symmetry and simulation.

Transient (nonequilibrium) cross correlation functions (CCF's) are introduced for the characterization of non-Newtonian flow under conditions of shear startup or cessation. Arguments based on the principles of group-theoretical statistical mechanics imply that these are asymmetric for shearing and symmetric for elongational stress. These expectations are confirmed with nonequilibrium molecular dynamics. Nonequilibrium CCF's are evaluated in the rise transient regime, immediately following the application of shear stress, and in the fall transient regime immediately following its removal. The rise and fall transient CCF's of atomic velocity and pressure tensor components are found to have quite different time dependencies, in line with the rise and fall stress transients themselves. They also correlate well with the non-Newtonian characteristics of the fluid, in that, as the response to the stress becomes increasingly more non-Newtonian, the difference between the equilibrium and rise or fall transient CCF's develops an increasingly sharp peak at short time.

Group theoretical statistical mechanics of the Evans effects: Phase changes produced by inhomogeneous electric and magnetic fields

The effects first observed by G.J. Evans are explored in terms of the symmetry of ensemble averages, using the principles of group theoretical statistical mechanics (GTSM). The effects include phase changes brought about by inhomogeneous electric and/or magnetic fields, all of whose gradients are included in the treatment in a multipole expansion of the force differential across the interface between a molecular crystal and a molecular liquid. It is shown by GTSM that the force differential is generated by extra vector components which are sustained in the laboratory frame by the symmetry of the molecular crystal but which vanish in the liquid. In this way the symmetry of the former is continuously imparted to the latter.

On the isolation of possible artifacts due to cubic periodic boundary conditions

Cubic periodic boundary conditions in computer simulations of molecular dynamics may produce artifacts due to imposed O h symmetry. In this communication, group theoretical statistical mechanics is used to isolate ensemble averages which exist under O h symmetry but vanish under the R h(3) symmetry of an ensemble of molecules or atoms at thermodynamic equilibrium. These are artifacts of cubic periodic boundary conditions. This method allows a check on the validity of molecular dynamics computer simulations.

The group theory of shear induced pair radial distribution functions in atomic liquids

Angular resolution of the pair radial distribution function in computer simulations of shear induced thickenening in atomic (Lennard-Jones) liquids has revealed the presence of anisotropic local structure which is explained on the basis of group theoretical statistical mechanics. This explains the observed structural asymmetry from the basic assumption that the symmetry of allowed ensemble averages in the steady state in the presence of shear is the same as that of the applied strain rate. The computer simulation results are reproduced from group theory by assuming that the crystal like lattice arrangement of atoms which appear in the simulations under shear can be described by some of the thirty two crystallographic point groups, namely hexagonal, trigonal, and triclinic. The hexagonal lattice symmetries C3h and C6h; the trigonal symmetry S6; and the triclinic symmetry Ci(S2) are found to support the crystal-like structures necessary to explain the observed angular resolution of the pair radial distribution function.

Applications of group-theoretical statistical mechanics: Interaction of electromagnetic fields with molecular ensembles.

The techniques of group-theoretical statistical mechanics (GTSM) are applied to various aspects of the interaction of electromagnetic radiation with molecular ensembles. Six nonlinear optical effects are examined in terms of the three principles of GTSM, using a classical series expansion of the induced electric dipole moment in terms of the electric and magnetic components of the electromagnetic field. Classical-quantum equivalence relations are derived at each stage. Symmetry arguments are applied to the chiral properties of some of the new nonlinear optical effects using the third principle of GTSM. A careful distinction is made between natural and magnetic optical activity, and our analysis attempts to resolve some recent disagreements in the literature.

Chirality of field induced natural and magnetic optical activity

The symmetry and chirality of field induced natural and magnetic optical activity are investigated with Barron's definition of chirality and the third principle of group theoretical statistical mechanics (g.t.s.m.). The results for several different external field symmetries are given for the point group R h(3) of achiral ensembles and for R(3) of chiral ensembles. The distinction is made between natural and magnetic optical rotation/dichoroism and the field symmetries necessary for these to appear are defined in both point groups. The analysis resolves some recent literature confusion and suggests new field induced dichroic phenomena.

Group theory and the diffusion of molecules, dynamics and structure

Point group theory is used in the laboratory and molecule fixed frames (X, Y, Z) and (x, y, z) respectively to develop three principles upon which to base the analysis of diffusional processes in N particle atomic and molecular ensembles. The symmetry signatures of ensemble averages at field free equilibrium, and in the steady state under applied external fields are given in the point group Rh (3) and in thirty-six molecular point groups. Physical examples of "group theoretical statistical mechanics" (g.t.s.m.) are given in terms of a new interpretation of the Weissenberg effect and of a theoretical prediction of shear induced depolarized light scattering. New fluctuation-dissipation theorems are developed from the principles and applications of g.t.s.m. for an applied electric field of arbitrary strength, and some corroborative computer simulations reported for liquid water and methyl chloride. Structural applications are discussed finally in the form of angularly resolved pair radial distribution functions.

New non-linear optical effects: Application of group theoretical statistical mechanics

Abstract The interaction of intense electromagnetic radiation with molecular matter is considered in terms of group theoretical statistical mechanics (gtsm). Field induced dipole moments are expanded in a double Taylor series in powers of the electric and magnetic field components multiplied by complex molecular property tensors (susceptibilities). The third principle of gtsm leads to several new non-linear optical effects, together with a symmetry classification of known effects. One new effect is detailed in terms of the Maxwell equation, and an estimate made of its order of magnitude.

Computer simulation and group theoretical statistical mechanics of liquid water and methyl chloride: Part 2, field effects.

The new methods of group theoretical statistical mechanics (g.t.s.m.) are used to determine field effects on cross correlation functions (c.c.f.'s) in molecular liquids, on the basis that the ensemble average reduces as the totally symmetric irreducible representation (t.s.r.) in the laboratory frame (X, Y, Z) and the molecule fixed frame (x, y, z) of the point group character tables. It is shown that a Z axis electric field promotes the existence of two ymmetric elements of the cross correlation function (c.c.f.) heyween molecular linear and angular velocity in frame (X, Y, Z) as predicted by g.t.s.m. These results are fully in agreement with computer simulations of water and liquid methyl chloride. The methods of g.t.s.m. are extended to predict the possible effects of electric and magnetic fields on c.c.f.'s in both frames.

Group theoretical statistical mechanics of the evans effects: Phase changes produced by inhomogeneous electric and magnetic fields

The effects first observed by G. J. Evans are explored in terms of the symmetry of ensemble averages, using the principles of group theoretical statistical mechanics (g.t.s.m.). The effects include phase changes brought about by inhomogeneous electric and/or magnetic fields, all of whose gradients are included in the treatment in a multipole expansion of the force differential across the interface between a molecular crystal and a molecular liquid. It is shown by g.t.s.m. that the force differential is generated by extra vector components which are sustained in the laboratory frame by the symmetry of the molecular crystal but which vanish in the liquid. In this way the symmetry of the former is continuously imparted to the latter.

Computer simulation and group theoretical statistical mechanics of liquid water and methyl chloride: Part 1, Neumann's Principle.

Neumann's Principle is applied to the molecular dynamics of liquids in the laboratory frame (X, Y, Z) and molecule fixed frame (x, y, z). In the absence of symmetry breaking influences, and in an isotropic medium of diffusing molecules, it states that the totally symmetric irreducible representation is the key to whether a thermodynamic ensemble average exists in either frame. In liquid water, with molecules of C 2ν symmetry, and liquid methyl chloride, with molecules of C 3ν symmetry, the Neumann Principle, evaluated with the help of group theory, is found to be in agreement with computer simulations of components of linear and angular momentum auto correlation functions in both frames.

Symmetry analysis of non-equilibrium time correlation functions

A symmetry analysis of non-equilibrium time correlation functions is introduced using the principles of group theoretical statistical mechanics and the new fluctuation dissipation theorem of Morriss and D. J. Evans. This approach permits a generalization of the customary Neumann or Curie theorem, providing a new microscopic approach to symmetry rules valid outside the linear regime.

Group theoretical statistical mechanics of nematogenic and cholesteric liquid crystals

The principles of group theoretical statistical mechanics have been applied to the molecular dynamics of nematogens and cholesterics, with point groups C ∞v, D ∞h, C ∞ and D ∞. The effect of alignment with an external static electric field is discussed in terms of new ensemble averages that take the symmetry of the applied field and make swarm averages directly visible in the laboratory frame ( X, Y, Z). The symmetry arguments lead to experimentally observable effects and to characteristic ensemble averages of swarm dynamics which should be reproducible numerically by computer simulation.

Principles of group-theoretical statistical mechanics.

Three principles are presented as the basis for group-theoretical statistical mechanics (GTSM), incorporating Neumann's (or Curie's) principle as the first. The second extends Neumann's principle to molecule fixed axes, (x,y,z), as defined in molecular point-group theory, and the third deals with the symmetry of applied fields and their effect on molecular ensembles in the field-on steady state. The power of GTSM is illustrated with reference to the set of field-induced n-time correlation functions in the laboratory frame (X,Y,Z) and frame (x,y,z).

Group theoretical statistical mechanics applied to couette flow

In group theoretical statistical mechanics (gtsm) the ensemble average is treated with the rules of group theory in the laboratory frame ( X, Y, Z) and the molecule-fixed frame ( x, y, z) of the point group character tables. This allows the determination of non-vanishing averages, such as time correlation functions, at the field-on equilibrium. The effect of couette flow is evaluated on a range of ensemble averages by establishing the symmetry of the strain rate tensor in terms of the irreducible representations of the R h(3) rotation-reflection group in frame ( X, Y, Z). This symmetry, which is also that of the pressure tensor, is D g (0) + D g (1) + D g (2), consisting of an antisymmetric vorticity of symmetry D g (1) and a symmetric strain rate component of general symmetry D g (0) + D g (2). Thus, in addition to ensemble averages of symmetry D g (0) in frame ( X, Y, Z), the applied strain rate tensor allows new ensemble averages of D g (1) symmetry and of D g (0) + D g (2) symmetry. Depending on the number of off-diagonal elements in the strain rate tensor, up to six new elements of time correlation functions of the general type < A ( t) A T(0)> may appear by gtsm in frame ( X, Y, Z), together with new bulk properties whose symmetry is included in D g (0) + D g (1) + D g (2). The latter include thermal conductivity, thermal expansion and electric polarisability. The treatment is extended to the molecule-fixed frame ( x, y, z) using the mapping rules of group theory.

Three principles of group theoretical statistical mechanics

Three new principles are introduced from the application of group theory to statistical mechanics at thermodynamical equilibrium and in the field-on steady state. They are illustrated with reference to computer simulations in the laboratory frame ( X, Y, Z) and in the molecule fixed frame ( x,y,z) of the point group character tables. The principle governing the effect of applied external fields is illustrated with reference to an applied vortex and shear in conventional and non-equilibrium molecular dynamics computer simulations.