Kubo theory

Moving downward to the molecular level, a number of lines of research flowed from Onsager s seminal work on the reciprocal relations. The symmetry rule was extended to cases of mixed parity by Casimir [24], and to nonlinear transport by Grabert et al. [25] Onsager, in his second paper [10], expressed the linear transport coefficient as an equilibrium average of the product of the present and future macrostates. Nowadays, this is called a time correlation function, and the expression is called Green-Kubo theory [26-30]. 

For nonequilibrium statistical mechanics, the present development of a phase space probability distribution that properly accounts for exchange with a reservoir, thermal or otherwise, is a significant advance. In the linear limit the probability distribution yielded the Green-Kubo theory. From the computational point of view, the nonequilibrium phase space probability distribution provided the basis for the first nonequilibrium Monte Carlo algorithm, and this proved to be not just feasible but actually efficient. Monte Carlo procedures are inherently more mathematically flexible than molecular dynamics, and the development of such a nonequilibrium algorithm opens up many, previously intractable, systems for study. The transition probabilities that form part of the theory likewise include the influence of the reservoir, and they should provide a fecund basis for future theoretical research. The application of the theory to molecular-level problems answers one of the two questions posed in the first paragraph of this conclusion the nonequilibrium Second Law does indeed provide a quantitative basis for the detailed analysis of nonequilibrium problems. 

Nonequilibrium statistical mechanics Green-Kubo theory, 43-44 microstate transitions, 44-51 adiabatic evolution, 44—46 forward and reverse transitions, 47-51 stationary steady-state probability, 47 stochastic transition, 464-7 steady-state probability distribution, 39—43 Nonequilibrium thermodynamics second law of basic principles, 2-3 future research issues, 81-84 heat flow ... 

An alternative approach involves integrating out the elastic degrees of freedom located above the top layer in the simulation.76 The elimination of the degrees of freedom can be done within the context of Kubo theory, or more precisely the Zwanzig formalism, which leads to effective (potentially time-dependent) interactions between the atoms in the top layer.77-80 These effective interactions include those mediated by the degrees of freedom that have been integrated out. For periodic solids, a description in reciprocal space decouples different wave vectors q at least as far as the static properties are concerned. This description in turn implies that the computational effort also remains in the order of L2 InL, provided that use is made of the fast Fourier transform for the transformation between real and reciprocal space. The description is exact for purely harmonic solids, so that one can mimic the static contact mechanics between a purely elastic lattice and a substrate with one single layer only.81... 

This leads us to express the response on the basis of the perturbed v /(f) and, if the perturbation is very weak, on the basis of the unperturbed v /(f), thereby making us move in a direction different from the path adopted by the conventional approach to the response to external perturbation. If the function /(f) has an inverse power-law form, the external perturbation may have the effect of truncating this inverse power-law form. We notice that a weak perturbation affects the low modes of the system of interest, which are responsible for the long-time property of the function v /(f), if it has an inverse power-law form. Thus, a power-law truncation may well be realized, with a consequent significant departure from the prediction of the Green-Kubo theory. 

Before ending this section, we want to notice that a completely dynamical approach to the noncanonical equilibrium, of the WS form, is still missing. The adoption of the same dynamical models as those used in Ref. 49 to establish the existence of a form of linear response violating the Green-Kubo theory seems to be impractical for numerical reasons. A more accessible dynamic model was studied in Ref. 89, resulting, however, in an unstable form of noncanonical equilibrium, although characterized by an extremely extended lifetime. With all these warnings in mind, we reach the conclusion that a noncanonical form of equilibrium of the WS type cannot be ruled out. 

Lennard-Jones Triple-Point Bulk and Shear Viscosities. Green-Kubo Theory, Hamiltonian Mechanics, and Nonequilihrium Molecular Dynamics. 

Hoover WG, Evans DJ, Hickman RB et al (1980) Lennard-Jones triple-point bulk and shear viscosities. Green-Kubo theory, Hamiltonian mechanics, and nonequilibrium molecular dynamics. Phys Rev A 22 1690-1697... 

Kubo R, Yokota M and Nakajima S 1957 Statistical-mechanical theory of irreversible processes. Response to thermal disturbance J. Phys. Soc. Japan 12 1203... 

Kubo R and Tomita K 1954 A generai theory of magnetio resonanoe absorption J. Phys. Soc. Japan 9 888-919... 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

The stochastic theory of lineshape has been developed by Anderson and Weiss [157], by Kubo [158], and by Kubo and Tomita [159] in order to treat the narrowing of spectral lines by exchange or motion, a generalized formulation having been subsequently presented by Blume [31]. We consider below an application of the theory of Blume to the specific problem of relaxation between LS and HS states in Mossbauer spectra of powder materials which is based on the formulation by Blume and Tjon [32, 33], Accordingly, the probability of emission of a photon of wave vector Ik and frequency m is given as [160] ... 

Gallicchio, E. Kubo, M. M. Levy, R. M., Enthalpy-entropy and cavity decomposition of alkane hydration free energies numerical results and implications for theories of hydrophobic solvation, J. Phys. Chem. B 2000,104, 6271-6285... 

The theory of Robertson and Yarwood [84] is very similar in its spirit to the initial one of Bratos [83], but it considers, following Kubo [97], the angular... 

Caspers has developed a theory of spin-spin relaxation starting from the general expressions established by Kubo and Tomita.23 His results however differ from those of Hartmann and Anderson and have been criticized recently by Tjon.11 We shall first outline the main points of Caspers method and then examine the basis on which it may be criticized. 

Another approach to ET reactions originated in the work of Weiss and Libby, who suggested that activation energy for ET reactions in solution does not arise from the collisional-vibrational or electrostatic interaction of the solvent in the first and second solvation sphere, but rather from continuum solvent polarization fluctuation far out in solution. This approach, called continuum theory, was further developed by Kubo and Toyozawa, " Marcus, Platzmann and Erank, and by Levich and Dogo-nadze. ... 

Lindner presented an important paper on this type of systems already in mid-1960s (102). She considered the whole system as an ensemble of microcrystallites with different orientations of the principal axis of the ZFS tensor with respect to the magnetic field and applied the linear response theory of Kubo and Tomita (103). The expression for the nuclear Ti for the case of S = 1 could be written as ... 

A similar approach, also based on the Kubo-Tomita theory (103), has been proposed in a series of papers by Sharp and co-workers (109-114), summarized nicely in a recent review (14). Briefly, Sharp also expressed the PRE in terms of a power density function (or spectral density) of the dipolar interaction taken at the nuclear Larmor frequency. The power density was related to the Fourier-Laplace transform of the time correlation functions (14) ... 

The important step of identifying the explicit dynamical motivation for employing centroid variables has thus been accomplished. It has proven possible to formally define their time evolution ( trajectories ) and to establish that the time correlations ofthese trajectories are exactly related to the Kubo-transformed time correlation function in the case that the operator 6 is a linear function of position and momentum. (Note that A may be a general operator.) The generalization of this concept to the case of nonlinear operators B has also recently been accomplished, but this topic is more complicated so the reader is left to study that work if so desired. Furthermore, by a generalization of linear response theory it is also possible to extract certain observables such as rate constants even if the operator 6 is linear. 

The relation between the osmotic pressure II and the polymer concentration, referred to as the equation of state for the solution, is often used for a critical comparison between theory and experiment (or simulation). Kubo and Ogino... 

Hentschke [45] and DuPre and Yang [46] compared Kubo and Ogino s data with Lee s theory [53] extended to a (monodisperse) wormlike chain system by using ct(N) (cf. Sect. 2.3). Hentschke took d to be 1.6 nm, which is consistent with the value estimated from the partial specific volume (cf. Table 2). Though good for the middle and highest molecular weight samples in the... 

---