The Quantum Potential Distribution Theorem

As mentioned above, there are multiple ways to derive the PDT for the chemical potential. Here we utilize the older method in the canonical ensemble which says that 3/j,0 is just minus the logarithm of the ratio of two partition functions, one for the system with the distinguished atom or molecule present, and the other for the system with no solute. Using (11.7) we obtain [9, 48,49] 

The energetic factors are slightly more complicated with differing endpoints x and x.  

We now use a trick to partition this exact expression for the chemical potential into classical and quantum correction parts [29]. To do this we multiply and divide inside the logarithm of the excess term by the classical average 

The Fourier coefficient average in the denominator of the last term is added to make the numerator and denominator symmetrical. It has no effect on the classical average. The classical factor AUa(x) signifies that the potential is evaluated at the centroid of the path 

This is helpful in deriving approximations later. Intuitively it makes sense since we would like to evaluate fluctuations about the center of mass of the path. 

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Beck, T. L. Marchioro, T. L., The quantum potential distribution theorem, in Path Integrals from meV to MeV Tutzing 1992, Grabert, H. Inomata, A. Schulman, L. Weiss, U., Eds., World Scientific Singapore, 1993, pp. 238-243... 

Xq is the origin of the path corresponding to t = 0, or 1, and aj is the th Fourier variable, t describes the evolution along the path and progresses from 0 to 1. Then the quantum potential distribution theorem (Beck, 1992 Beck and Marchioro, 1993 Wang et al, 1997 Beck, 2006) for a quantum solute with no internal structure is... 

This will prove useful in deriving the approximations. Then we can write the quantum potential distribution theorem as... 

In this section we discuss quantum mechanical models that can be brought to bear on evaluation of the potential distribution theorem. These models could be tried and tested in practical calculations, but the basics of these models should be studied elsewhere - the present discussion is not about quantum mechanics for its own sake. The remainder of this section then gives a more technical discussion of current ideas for inclusion of nonexchange quantum mechanical effects. 

As a preliminary point, we note that the decoupled averaging discussed here in classical views of the potential distribution theorem derives from the denominator of Eq. (3.17), p. 40. This is unchanged in the present quantum mechanical discussion, and thus the sampling of the separated subsystems could be highly quantum mechanical without changing those formalities. 

For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian function, centered at the equilibrium position. For the classical harmonic oscillator, this follows directly from the expression of a Boltzmann distribution in a quadratic potential. The result for the quantum-mechanical harmonic oscillator, referred to as Bloch s theorem, is less obvious, as a population-weighted average over all discrete levels must be evaluated (see, e.g., Prince 1982). 

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