Space phase

An important concept in computer simulation is that of the phase space. For a system containing N atoms, 6N values are required to define the state of the system (three coordinates per atom and three components of the momentum). Each combination of 3N positions and 3N momenta (usually denoted by T ) defines a point in the 6N-dimensional phase space an ensemble can thus be considered to be a collection of points in phase space. The way in which the system moves through phase space is governed by Hamiltonian s equations  

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 chermcal potential and the entropy itself. The difference between the mechanical and 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 Helmholtz free energy, A. These are related to the partition function by  

Q is given by Equation (6.4) for a system of identical particles. We shall ignore any normalisation constants in our treatment here to enable us to concentrate on the basics, and so it does not matter whether the system consists of identical or distinguishable particles. We also replace the Hamiltonian by the energy, E. The internal energy is obtained via Equation (6.20)  

This probability is written p(p, r ) the internal energy is thus given by 

Let us now consider the problem of calculating the Helmholtz free energy of a molecular liquid. Our aim is to express the free energy in the same functional form as the internal energy, that is as an integral which incorporates the probability of a given state. First, we substitute for the partition function in Equation (6.21)  

This is difficult to accept, since, dear reader, this very moment, during which you are reading these words, was predetermined at the moment of the Big Bang, 13.7 billion years ago. This is patently absurd. Before discussing how quantum mechanics has provided a theoretical framework that resolves these absurdities, we will present the concept of phase space, stressing that classical mechanics is still an eminently powerful and useful approximation of reality. Newtonian mechanics is the first self-contained system of physical causality humans developed and for this, using Einstein s words, Newton deserves our deepest reverence.  

For a system with N particles, consider the 6N-dimensional Cartesian space with 6N mutually orthogonal axes, each representing one of the 6N coordinates q and momenta p of the particles. We call this the phase space of the system, and denote it with F. This is simply a mathematical construct that allows us concisely to define the microscopic states of the system. Each point in phase space X is a microscopic state of the system  

We can now define a phase orbit or phase trajectory as the succession of microstates X(0. with momenta and coordinates (p(r). q(t)) being the solution of forward integration of Hamilton s equations of motion for a set of initial conditions X(0) = (p(0), q(0)). The phase orbit can be simply considered as a geometrical interpretation of Hamilton s equations of motion a system with N mass points, which move in three dimensions, is mathematically equivalent to a single point moving in 6N dimensions (Fig. 3.4). 

As discussed in the previous section, knowledge of the microscopic state of a system X(ti) at a particular time instance ti completely 

Phase spaces, from classical to quantum mechanics, and back 

The derivation of statistical mechanics on the basis of quantum mechanics was done in a simplified version above. What could be simpler than just counting the number of possibilities and taking the logarithm to get manageable numbers for the entropy The important historical question, then, is how Boltzmann derived statistical mechanics the first time, long before quantum mechanics was known. In fact, the original Boltzmann derivation is sometimes much simpler to apply to some problems than the quantum mechanical one therefore, we need to understand how statistical mechanics is derived on the basis of classical mechanics. 

A hint may be obtained from the electron in a box problem. The kinetic energy (pV2m) is given in Equation 1.17. The variation 5p in momentum is between p = hn/L and -hn/L. The variation in distance 5x is L and hence 6p 6x = hn. The space consisting of a p-axis and an x-axis is the phase space in one dimension. In the case of particle in a box, the system has width L on the x-axis. The width along 

In a completely isolated system, we may assume that the total energy is within a certain interval dE. This microcanonical ensemble corresponds quite closely to the treatment that we have already carried out in Sections 5.2-5.4. 

We found that the distances between the energy levels become extremely small when the width of the box or the mass of the particle has macroscopic dimensions. We also commented on the fact that a wave packet, for example, one describing a macroscopic body, obeys Newton s equations and has the same group velocity as the body. In other words, quantum mechanics has an easily reached classical limit when everything behaves as we are used to in the maeroscopic world. 

The microcanonical ensemble is realistic only if the condition of equal energy is replaced by a condition that the total energy has to be in a certain interval dE. This is the only way to do the derivation in the classical case. On the other hand, dE may be chosen arbitrarily small, with still correct end results. 

The method of defining complexions depends on whether we are treating our systems by classical, Newtonian mechanics or by quantum theory. First we shall take up classical mechanics, for that is more familiar. But later, when we describe the methods of quantum theory, we shall observe that that theory is more correct and more fundamental for statistical purposes. In classical mechanics, a system is, df scrihpd by tV 

To find the condition for equilibrium, then, we must investigate the nature of the streamlines. For a periodic motion, a streamline will be closed, the system returning to its original state after a single period. This is a very special case, however most motions of many particles are not periodic and their streamlines never close. Rather, they wind around 

1 For proof, see for example, Slater and Frank, Introduction to Theoretical Physics, pp. 365-366, McGraw-Hill Book Company, Inc., 1933. 

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It is sometimes very usefiil to look at a trajectory such as the synnnetric or antisynnnetric stretch of figure Al.2.5 and figure A1.2.6 not in the physical spatial coordinates (r. . r y), but in the phase space of Hamiltonian mechanics [16, 29], which in addition to the coordinates (r. . r ) also has as additional coordinates the set of conjugate momenta. . pj. ). In phase space, a one-diniensional trajectory such as the aiitisymmetric stretch again appears as a one-diniensional curve, but now the curve closes on itself Such a trajectory is referred to in nonlinear dynamics as a periodic orbit [29]. One says that the aihiamionic nonnal modes of Moser and Weinstein are stable periodic orbits. 

Figure Al.2.7. Trajectory of two coupled stretches, obtained by integrating Hamilton s equations for motion on a PES for the two modes. The system has stable anhamionic synmretric and antisyimnetric stretch modes, like those illustrated in figrne Al.2.6. In this trajectory, semiclassically there is one quantum of energy in each mode, so the trajectory corresponds to a combination state with quantum numbers nj = [1, 1]. The woven pattern shows that the trajectory is regular rather than chaotic, corresponding to motion in phase space on an invariant torus.

Figure Al.2.12. Energy level pattern of a polyad with resonant collective modes. The top and bottom energy levels conespond to overtone motion along the two modes shown in figure Al.2.11. which have a different frequency. The spacing between adjacent levels decreases until it reaches a minimum between the third and fourth levels from the top. This minimum is the hallmark of a separatrix [29, 45] in phase space.

The question of non-classical manifestations is particularly important in view of the chaos that we have seen is present in the classical dynamics of a multimode system, such as a polyatomic molecule, with more than one resonance coupling. Chaotic classical dynamics is expected to introduce its own peculiarities into quantum spectra [29, 77]. In Fl20, we noted that chaotic regions of phase space are readily seen in the classical dynamics corresponding to the spectroscopic Flamiltonian. Flow important are the effects of chaos in the observed spectrum, and in the wavefiinctions of tire molecule In FI2O, there were some states whose wavefiinctions appeared very disordered, in the region of the... 

In the above discussion of relaxation to equilibrium, the density matrix was implicitly cast in the energy representation. However, the density operator can be cast in a variety of representations other than the energy representation. Two of the most connnonly used are the coordinate representation and the Wigner phase space representation. In addition, there is the diagonal representation of the density operator in this representation, the most general fomi of p takes the fomi... 

The set of microstates of a finite system in quantum statistical mechanics is a finite, discrete denumerable set of quantum states each characterized by an appropriate collection of quantum numbers. In classical statistical mechanics, the set of microstates fonn a continuous (and therefore infinite) set of points in f space (also called phase space). 

WIson K G 1971 Renormalization group and critical phenomena. II. Phase space cell analysis of critical behaviour P/rys. Rev. B 4 3184-205... 

A situation that arises from the intramolecular dynamics of A and completely distinct from apparent non-RRKM behaviour is intrinsic non-RRKM behaviour [9], By this, it is meant that A has a non-random P(t) even if the internal vibrational states of A are prepared randomly. This situation arises when transitions between individual molecular vibrational/rotational states are slower than transitions leading to products. As a result, the vibrational states do not have equal dissociation probabilities. In tenns of classical phase space dynamics, slow transitions between the states occur when the reactant phase space is metrically decomposable [13,14] on the timescale of the imimolecular reaction and there is at least one bottleneck [9] in the molecular phase space other than the one defining the transition state. An intrinsic non-RRKM molecule decays non-exponentially with a time-dependent unimolecular rate constant or exponentially with a rate constant different from that of RRKM theory. 

The hypersurface fomied from variations in the system s coordinates and momenta at//(p, q) = /Tis the microcanonical system s phase space, which, for a Hamiltonian with 3n coordinates, has a dimension of 6n -1. The assumption that the system s states are populated statistically means that the population density over the whole surface of the phase space is unifomi. Thus, the ratio of molecules at the dividing surface to the total molecules [dA(qi, p )/A]... 

In deriving the RRKM rate constant in section A3.12.3.1. it is assumed that the rate at which reactant molecules cross the transition state, in the direction of products, is the same rate at which the reactants fonn products. Thus, if any of the trajectories which cross the transition state in the product direction return to the reactant phase space, i.e. recross the transition state, the actual unimolecular rate constant will be smaller than that predicted by RRKM theory. This one-way crossing of the transition state, witii no recrossmg, is a fiindamental assumption of transition state theory [21]. Because it is incorporated in RRKM theory, this theory is also known as microcanonical transition state theory. 

The intennolecular Hamiltonian of the product fragments is used to calculate the sum of states of the transitional modes, when they are treated as rotations. The resulting model [28] is nearly identical to phase space theory [29],... 

The bulk of the infomiation about anhannonicity has come from classical mechanical calculations. As described above, the aidiannonic RRKM rate constant for an analytic potential energy fiinction may be detemiined from either equation (A3.12.4) [13] or equation (A3.12.24) [46] by sampling a microcanonical ensemble. This rate constant and the one calculated from the hamionic frequencies for the analytic potential give the aidiannonic correctiony j ( , J) in equation (A3.12.41). The transition state s aidiannonic classical sum of states is found from the phase space integral... 

As discussed in section A3.12.2. intrinsic non-RRKM behaviour occurs when there is at least one bottleneck for transitions between the reactant molecule s vibrational states, so drat IVR is slow and a microcanonical ensemble over the reactant s phase space is not maintained during the unimolecular reaction. The above discussion of mode-specific decomposition illustrates that there are unimolecular reactions which are intrinsically non-RRKM. Many van der Waals molecules behave in this maimer [4,82]. For example, in an initial microcanonical ensemble for the ( 211 )2 van der Waals molecule both the C2H4—C2H4 intennolecular modes and C2H4 intramolecular modes are excited with equal probabilities. However, this microcanonical ensemble is not maintained as the dimer dissociates. States with energy in the intermolecular modes react more rapidly than do those with the C2H4 intramolecular modes excited [85]. 

In other words, if we look at any phase-space volume element, the rate of incoming state points should equal the rate of outflow. This requires that be a fiinction of the constants of the motion, and especially Q=Q i). Equilibrium also implies d(/)/dt = 0 for any /. The extension of the above equations to nonequilibriiim ensembles requires a consideration of entropy production, the method of controlling energy dissipation (diennostatting) and the consequent non-Liouville nature of the time evolution [35]. 

The velocity Verlet algorithm may be derived by considering a standard approximate decomposition of the Liouville operator which preserves reversibility and is symplectic (which implies that volume in phase space is conserved). This approach [47] has had several beneficial consequences. 

Hoover W G 1985 Canonical dynamics equilibrium phase-space distributions Phys. Rev. A 31 1695-7... 

Even expression ( B3.4.31), altiiough numerically preferable, is not the end of the story as it does not fiilly account for the fact diat nearby classical trajectories (those with similar initial conditions) should be averaged over. One simple methodology for that averaging has been tln-ough the division of phase space into parts, each of which is covered by a set of Gaussians [159, 160]. This is done by recasting the initial wavefunction as... 

Hamiltonian) trajectory in the phase space of the model from which infonnation about the equilibrium dyuamics cau readily be extracted. The application to uou-equilibrium pheuomeua (e.g., the kinetics of phase separation) is, in principle, straightforward. 

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

F after transients have decayed. This final set of phase-space points is tire attractor, and tire set of all initial conditions tliat eventually reaches tire attractor is called its basin of attraction. 

For certain parameter values tliis chemical system can exlribit fixed point, periodic or chaotic attractors in tire tliree-dimensional concentration phase space. We consider tire parameter set... 

The description of chemical reactions as trajectories in phase space requires that the concentrations of all chemical species be measured as a function of time, something that is rarely done in reaction kinetics studies. In addition, the underlying set of reaction intennediates is often unknown and the number of these may be very large. Usually, experimental data on the time variation of the concentration of a single chemical species or a small number of species are collected. (Some experiments focus on the simultaneous measurement of the concentrations of many chemical species and correlations in such data can be used to deduce the chemical mechanism [7].)... 

The trajectory description problem of chemical reactions is resolved by using phase-space reconstmction from a single time series [8] this method uses delayed data at times t, t+ip t+X2,.. ., for an -dimensional attractor,... 

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