Canonical classical mechanics

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

The foregoing unitary transformations may be interpreted as the analogues of canonical transformations in classical mechanics. 

In 1933, J.G. Kirkwood explicitly showed that the canonical partition function Q for a system of N monatomic particles reduces to an integral over phase space in the limit of high temperature (Equation 4.81). The result corresponds to classical mechanics (i.e. the spacing between energy levels is small compared to kT)... 

Of course, one is not really interested in classical mechanical calculations. Thus in normal practice the partition functions used in TST, as discussed in Chapter 4, are evaluated using quantum partition functions for harmonic frequencies (extension to anharmonicity is straightforward). On the other hand rotations and translations are handled classically both in TST and in VTST, which is a standard approximation except at very low temperatures. Later, by introducing canonical partition functions one can direct the discussion towards canonical variational transition state theory (CVTST) where the statistical mechanics involves ensembles defined in terms of temperature and volume. There is also a form of variational transition state theory based on microcanonical ensembles referred to by the symbol p,. Discussion of VTST based on microcanonical ensembles pVTST is beyond the scope of the discussion here. It is only mentioned that in pVTST the dividing surface is... 

Recall that the canonical equations of classical mechanics can be used to derive the Hamiltonian expression for the total energy of a system from the momenta pk and positional coordinates qk ... 

To quantize the dynamics of the particles first requires that we express the velocities of the particles in terms of canonical momenta. In the presence of electromagnetic fields, the canonical momenta are not merely m dx-Jdt). Rather, in order to incorporate Lorentz s velocity-dependent forces into Hamilton s formulation of classical mechanics, the canonical momenta are given by [2]... 

The concept of a canonical transformation is fundamental to the Hamiltonian formulation of classical mechanics, the formulation which... 

It is within the Hamiltonian formulation of classical mechanics that one introduces the concept of a canonical transformation. This is a transformation from some initial set of ps and qs, which satisfy the canonical equations of motion for H(p, q, t) as given in eqn (8.57), to a new set Q and P, which depend upon both the old coordinates and momenta with defining equations. 

There remains but one important concept to complete our summary of the role of canonical transformations in classical mechanics, that of the Poisson bracket. Let F p,q) and G p,q) denote two mechanical properties of the system. Their Poisson bracket is defined as... 

An alternative approach to describe steady-state thermodynamics for shear flow was formulated by Taniguchi and Morriss.192 Their method involves the development of a canonical distribution for shear flow by a Lagrangian formalism of classical mechanics. They then derive the Evans-Hanley thermodynamics, i.e. 

The above systematics of substitution reaction mechanisms are automatically deducibie from the complete chemical set of entities for any atom (constructed by us above). To demonstrate this, it is sufficient to note that the universal operator for all the ligand-electron networks described in the preceding section precisely corresponds to all the classical mechanisms for mono- and bimolecular substitution at a saturated atom. To facilitate understanding of this operator, and the canonical designations of the corresponding mechanisms, we illustrate it in Fig. 4.23 for a typical stable hydrocarbon (the methane molecule). Naturally, the conclusions arrived at can be readily extended to include any other E—X bonds and D reagents. 

Earlier in this section it was commented on how the minimal-coupling QED Hamiltonian is obtained from fhe classical Lagrangian function. A few words are in order regarding the derivation of the multipolar Hamiltonian (6). One method involves the application of a canonical transformation to the minimal-coupling Hamiltonian [32]. In classical mechanics, such a transformation renders the Poisson bracket and Hamilton s canonical equations of motion invariant. In quantum mechanics, a canonical transformation preserves both the commutator and Heisenberg s operator equation of motion. The appropriate generating function that converts H uit is propor-... 

It is generally thought that these four canonical mechanisms of IgE-mediated activation of human FceRI + cells are responsible for the pathophysiological involvement of these cells in the majority of patients with allergic disorders [34], However, there is evidence that a significant percentage of allergic diseases (e.g. certain cases of intrinsic asthma and chronic idiopathic urticaria) cannot be explained by the four classical mechanisms of FceRI + cell activation. 

Let (p, q) be one set of canonically conjugate coordinates and momenta (the old variables) and (P,Q) be another such set (the new variables).13 (P, Q, p and q are IV-dimensional vectors for a system with N degrees of freedom, but for the sake of clarity multidimensional notation will not be used the explicitly multidimensional expressions are in most cases obvious.) In classical mechanics P and Q may be considered as functions of p and q, or inversely, P and Q may be chosen as the independent variables with p and q being functions of them. To carry out the canonical transformation between these two sets of variables, however, one must rather choose one old variable and one new variable as the independent variables, the remaining two variables then being considered as functions of them. The canonical transformation is then carried out with the aid of a generating function, or generator, which is some function of the two independent variables, and two equations which express the dependent variables in terms of the independent variables.13... 

There are, just as in classical mechanics, four ways of choosing one old variable and one new variable , so there are four equivalent sets of unitary transformation elements connecting the old and new representations << Q>, (q Py,

and

. The fundamental correspondence relations express the classical limit of these unitary transformation elements in terms of the classical generating functions for the related classical canonical transformation ... 

For a clear and concise discussion of canonical transformations see H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Mass., 1950, pp. 237-247. 

In the time concept of the pre-relativistic mechanics, the observable quantities, time t and energy E, have to be considered as another canonically conjugate pair, as in classical mechanics. The dynamic law (time-dependent energy term) of the Schrodinger equation will then completely disappear [19]. A good occasion for Weyl to introduce the relativistic view would have been his contributions to Dirac s electron theory. His other colleagues developed the method of the so-called second quantization that seemed easier for the entire community of physicists and chemists to accept. 

The first step in the study of collision dynamics is to assume that nuclear motion obeys tha laws of classical mechanics. This approximation is expected to give, at least qualitatively, a correct description of the collision between heavy particles at high (relative) velocities. The most appropriate formalism for such a description is based on the Hamilton canonical equations of motion... 

In classical mechanics, if there is about a function that depends on the conjugate canonical variables and explicitly on time, F = F q, p, t), its... 

The third equivalent formulation of classical mechanics to be briefly discussed here is the Hamiltonian formalism. Its main practical importance especially for molecular simulations lies in the solution of practical problems for processes that can be adequately described by classical mechanics despite their intrinsically quantum mechanical character (such as protein folding processes). However, more important for our purposes here is that it can serve as a useful starting point for the transition to quantum theory. The basic idea of the Hamiltonian formalism is to eliminate the / generalized velocities in favor of the canonical momenta defined by Eq. (2.54). This is achieved by a Legendre transformation of the Lagrangian with respect to the velocities. 

With the above Hamiltonian, Eq. (6.10), an analogy to purely classical mechanics brings about the canonical equations of motion for nuclear classical variables (R, P) as [492]... 

From classical mechanics (e.g., Goldstein 1950), we can show that the presence of a vector potential requires that the Hamiltonian function must be constructed using the kinetic momentum (or mechanical momentum), which is the momentum that is given in nonrelativistic theory by m. We must express this momentum in terms of the canonical momentum of Lagrangian mechanics, because it is the canonical momentum to which the quantization rule p —ihV applies. Here (and hereafter) we will use p for the canonical momentum and n for the kinetic momentum. The relation between the two is... 

Applications of the canonical ensemble to quantum mechanical systems other than dilute gases are beyond the scope of this book, and we omit them. These applications are discussed in some of the statistical mechanics books listed at the end of this volume. We will make some comments on the application of the canonical ensemble to systems obeying classical mechanics in the next section. 

Classical mechanical formulas must agree with those obtained by taking the limit of quantum mechanical formulas as masses and energies become large (the correspondence limit). This limit does not affect the formula representing the equilibrium canonical probability density, so it must therefore be the same function of the energy as that of quantum statistical mechanics. For a one-component monatomic gas or liquid of N molecules without electronic excitation but with intermolecular forces, the classical energy (classical Hamiltonian function Jf) is expressed in terms of momentum components and coordinates ... 

Statistical mechanics can also be based on classical mechanics, and a brief introduction to this subject was included in the chapter, based on the canonical ensemble. Since classical states are specified by values of coordinates and momentum components, the probability distribution for classical statistical mechanics is a probability density... 

---