Statistical mechanics Field theory

As the foundation of quantum statistical mechanics, the theory of open quantum systems has remained an active topic of research since about the middle of the last century [1-40]. Its development has involved scientists working in fields as diversified as nuclear magnetic resonance, quantum optics and nonlinear spectroscopy, solid-state physics, material science, chemical physics, biophysics, and quantum information. The key quantity in quantum dissipation theory (QDT) is the reduced system density operator, defined formally as the partial trace of the total composite density operator over the stochastic surroundings (bath) degrees of freedom. 

A.A.Abrikosov,L.P.Gor kov, and I.E.Dzyaloshinskii Methods of Quantum Field Theory in Statistical Mechanics, (Dover, New York,1975). 

L, Pehti. In F. David, P. Ginsparg, J. Zinn-Justin, eds. Fluctuating Geometries in Statistical Mechanics and Field Theory. Amsterdam North-Holland, 1996, pp. 195-285. 

In this section we introduce the basic ingredients of a field-theoretic approach to electrified interfaces and compare it with both the standard method of statistical mechanics and the density functional theory. 

For general rules, a first-order statistical approximation for limiting densities Pi t —> oo) can be obtained by a method akin to the mean-field theory in statistical mechanics (more sophisticated approaches will be introduced in chapter 4). 

If fondly recall the first day of an introductory graduate statistical mechanics class. As our instructor walked into the class saying something entirely appropriate like So, are we all ready for a lesson in quantum field theory today , he was of course met with a room-full of blank stares (even a few - later embarrassed - behind-the-back giggles). As first-year graduate students we had unfortunately not yet developed the requisite maturity to appreciate the profound link that exists between statistical mechanics and modern field theory. I resolved to never again be as quick to dismiss any obvious disparity or seeming disconnectedness between two subjects. 

Because the focus is on a single, albeit rather general, theory, only a limited historical review of the nonequilibrium field is given (see Section IA). That is not to say that other work is not mentioned in context in other parts of this chapter. An effort has been made to identify where results of the present theory have been obtained by others, and in these cases some discussion of the similarities and differences is made, using the nomenclature and perspective of the present author. In particular, the notion and notation of constraints and exchange with a reservoir that form the basis of the author s approach to equilibrium thermodynamics and statistical mechanics [9] are used as well for the present nonequilibrium theory. 

The present theory can be placed in some sort of perspective by dividing the nonequilibrium field into thermodynamics and statistical mechanics. As will become clearer later, the division between the two is fuzzy, but for the present purposes nonequilibrium thermodynamics will be considered that phenomenological theory that takes the existence of the transport coefficients and laws as axiomatic. Nonequilibrium statistical mechanics will be taken to be that field that deals with molecular-level (i.e., phase space) quantities such as probabilities and time correlation functions. The probability, fluctuations, and evolution of macrostates belong to the overlap of the two fields. 

Perhaps the best starting point in a review of the nonequilibrium field, and certainly the work that most directly influenced the present theory, is Onsager s celebrated 1931 paper on the reciprocal relations [10]. This showed that the symmetry of the linear hydrodynamic transport matrix was a consequence of the time reversibility of Hamilton s equations of motion. This is an early example of the overlap between macroscopic thermodynamics and microscopic statistical mechanics. The consequences of time reversibility play an essential role in the present nonequilibrium theory, and in various fluctuation and work theorems to be discussed shortly. 

Equation (4-5) can be directly utilized in statistical mechanical Monte Carlo and molecular dynamics simulations by choosing an appropriate QM model, balancing computational efficiency and accuracy, and MM force fields for biomacromolecules and the solvent water. Our group has extensively explored various QM/MM methods using different quantum models, ranging from semiempirical methods to ab initio molecular orbital and valence bond theories to density functional theory, applied to a wide range of applications in chemistry and biology. Some of these studies have been discussed before and they are not emphasized in this article. We focus on developments that have not been often discussed. 

Quantum mechanical calculations are appropriate for the electrons in a metal, and, for the electrolyte, modern statistical mechanical theories may be used instead of the traditional Gouy-Chapman plus orienting dipoles description. The potential and electric field at any point in the interface can then be calculated, and all measurable electrical properties can be evaluated for comparison with experiment. 

Gibbs found the solution of the fundamental Equation 9.1 only for the case of moderate surfaces, for which application of the classic capillary laws was not a problem. But, the importance of the world of nanoscale objects was not as pronounced during that period as now. The problem of surface curvature has become very important for the theory of capillary phenomena after Gibbs. R.C. Tolman, F.P. Buff, J.G. Kirkwood, S. Kondo, A.I. Rusanov, RA. Kralchevski, A.W. Neimann, and many other outstanding researchers devoted their work to this field. This problem is directly related to the development of the general theory of condensed state and molecular interactions in the systems of numerous particles. The methods of statistical mechanics, thermodynamics, and other approaches of modem molecular physics were applied [11,22,23],... 

A. L. Fetter, J. D. Walecka, Quantum Theory ofMany-Particle systems (McGraw-Hill, New York, 1971) A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski, Methods of Quantum field Theory in Statistical Mechanics (Dover, New York, 1975). 

It is important to point out here, in an early chapter, that the Born-Oppenheimer approximation leads to several of the major applications of isotope effect theory. For example the measurement of isotope effects on vapor pressures of isotopomers leads to an understanding of the differences in the isotope independent force fields of liquids (or solids) and the corresponding vapor molecules with which they are in equilibrium through use of statistical mechanical theories which involve vibrational motions on isotope independent potential functions. Similarly, when one goes on to the consideration of isotope effects on rate constants, one can obtain information about the isotope independent force constants which characterize the transition state, and how they compare with those of the reactants. 

M = (Mx,My,M ) is the dipole moment of the system. Moreover, the indices i, j designate the Cartesian components x, y, z of these vectors, ()q realizes an averaging over all possible realizations of the optical field E, and () realizes an averaging over the states of the nonperturbed liquid sample. Two three-time correlation functions are present in Eq. (4) the correlation function of E(t) and the correlation function of the variables/(q, t), M(t). Such objects are typical for statistical mechanisms of systems out of equilibrium, and they are well known in time-resolved optical spectroscopy [4]. The above expression for A5 (q, t) is an exact second-order perturbation theory result. 

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