Stiff quantum systems

Here, we review an adiabatic approximation for the statistical mechanics of a stiff quantum mechanical system, in which vibrations of the hard coordinates are first treated quantum mechanically, while treating the more slowly evolving soft coordinates and momenta for this purpose as parameters, and in which the constrained free energy obtained by summing over vibrational quantum states is then used as a potential energy in a classical treatment of the soft coordinates and momenta. 

Overall, this work highlights how quantum chemical methods can be used to study tribochemical reactions within chemically complex lubricant systems. The results shed light on processes that are responsible for the conversion of loosely connected ZP molecules derived from anti-wear additives into stiff, highly connected anti-wear films, which is consistent with experiments. Additionally, the results explain why these films inhibit wear of hard surfaces, such as iron, yet do not protect soft surface such as aluminum. The simulations also explained a large number of other experimental observations pertaining to ZDDP anti-wear films and additives.103 Perhaps most importantly, the simulations demonstrate the importance of cross-linking within the films, which may aid in the development of new anti-wear additives. 

They relate stiff chains to cases of Brownian motion where the particle s position, velocity, etc., are considered. Connection is also established between stiff polymers and the quantum mechanics of systems which classically would have velocity, acceleration, etc., dependent forces. The excluded volume problem is considered in the next section, and then we discuss problems involving polymers in bulk. 

The primitive approximation contains all the physics involved in the treatment of quantum many-body systems at nonzero temperature. It is simple, intuitive, and highly flexible. Moreover, it reveals clearly the so-called classical isomorphism, Eqs. (25)-(27), a correspondence that has important consequences on a range of different issues (e.g., formal study of structures and computational techniques). Nevertheless, the computational efficiency of the primitive scheme is generally poor as pointed out in earlier applications. A couple of examples will help to understand these drawbacks. First, the kinetic energy, given by the first two terms on the right-hand side of Eq. (31), which shows increasing variances with P, a fact associated with the stiffness of the harmonic links in [63]. Second, the P con-... 

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