Phase space functions

It is also useful to define the transformed operator L whose operation on a function f is L f = L[Peqf). This operator coincides with the time reversed backward operator, further details on these relationships may be found in Refs. 43,44. L operates in the Hilbert space of phase space functions which have finite second moments with respect to the equilibrium distribution. The scalar product of two functions in this space is defined as (f, g) = (fgi q. It is the phase space integrated product of the two functions, weighted by the equilibrium distribution P The operator L is not Hermitian, its spectrum is in principle complex, contained in the left half of the complex plane. 

This derived expression satisfies conditions a-d mentioned above and based on numerical computatiotf 6-2 seems to bound the exact result from above. It is similar but not identical to Wigner s original guess. The quantum phase space function which appears in Eq. 52 is that of the symmetrized thermal flux operator, instead of the quantum density. 

Because S is a linear operator and WQq is a non-negative function, the expression on the right can be shown to have all the necessary properties of a scalar product, provided that Weq and the phase-space functions F, G appropriate to each thermodynamic variable can be defined and the integral exists. Equation (13.53) is then treated as the equilibrium dot product inherited from the phase-space vectors and dot product in (13.54). 

Within the Wigner function framework the operator exp iaq + ibp) is associated with the phase-space function exp iaq + ibp). This particular association between a phase-space function and a function of noncommuting operators is an example of WeyTs rule [69]. Indeed, by generalizing Eq. (1.317) one obtains the Weyl transform Aw(q,p) of an arbitrary operator... 

Furthermore, inverting Eq. (319) yields WeyTs rule for quantizing a classical phase-space function/(, p), that is. 

After the system has reached equilibrium, in order to compute the canonical ensemble average of any phase space function, one may continue integrating the trajectories using the constant temperature algorithm. Details about technical issues involved in this calculation can be found in the standard texts mentioned above. Here we briefly discuss the computation methodology of properties that are relevant to simulating interfacial systems. 

G. J. Martyna, J. Chem. Phys. (in press, 1996). In this paper, an effective set of molecular dynamics equations are specified that provide an alternative path-integral approach to the calculation of position and velocity time correlation functions. This approach is essentially based on the Wigner phase-space function. For general nonlinear systems, the appropriate MD mass in this approach is not the physical mass, but it must instead be a position-dependent effective mass. 

The map H H(q, p fieff) leading to Eq. (A.25) is also called the Wigner map. It is the inverse of the transformation which yields a Hamilton operator H from the Weyl quantization, Op[H], of a phase space function H (the Weyl map) which, using Dirac notation, is given by... 

It is always defined with respect to the given set of canonical variables which is sometimes symbolized by suitable indices. The result, i.e., the Poisson bracket itself is, in general, another phase space function. In anticipation of the next section 2.3.3, all Poisson brackets are invariant under canonical transformations, i.e., their value does not depend on the choice of canonical variables. The indices q and p attached to the Poisson brackets are therefore often suppressed. 

Since also the canonical variables q and p are phase space functions, we may calculate their Poisson brackets. They are given by... 

Poisson brackets can be employed to cast the time dependence of an arbitrary phase space function u = u q,p,t) in a more compact form, which may be useful for the exhibition of conservation laws. Since the total time derivative of u is given as... 

Note that we have introduced the symbol 7t for linear momentum here in order to better distinguish it from canonical momentum. This notational rigor is only needed for the discussion of this section and will thus be dropped elsewhere in the book. In most cases it will become obvious from the context to which kind of momentum we are referring. Obviously, it is rather the linear than the canonical momentum which is gauge invariant. The canonical momentum p satisfies the fundamental Poisson brackets of Eq. (2.81), of course, whereas the components of linear momentum, interpreted as phase space functions tt, = Ki r,p,t), feature nonvanishing but gauge invariant Poisson brackets. 

The ensemble average of any phase space function / can now be defined by the weighted average... 

Suppose we have a normalized autocorrelation function of a phase space function A(p, q) where, as before, A implicitly depends on time through p and q (McQuarrie, 1976 Hansen and McDonald, 1976) ... 

Now define a projection operator P so that P projects any phase space function G(p, q) onto A(p, q) ... 

The operator (1 - P) is usually called the orthogonal projector since P(1 - P) annihilates a phase-space function. Equation (35) is useful because it expresses the memory function explicitly as a projected correlation function of the phase-space variables. 

The starting point of the statistical mechanics is the Liouville equation, named after French mathematician Joseph Liouville. It describes the time evolution of the phase space function as... 

For every microstate z of the system, the instantaneous values of the relevant variables are defined by a set of phase space functions II(z). The functions II(z) cannot generally be identified with x they are rather connected with x through x = (II(z)), that is, as averages based on a suitable probability density Qx(z) at the microscopic phase space F. Thus, the coarse-grained energy (x) is obtained from the microscopic Hamiltonian H(z) by straightforward averaging. 

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