Ensemble theory

Josiah Willard Gibbs (1839-1903) developed the theory that provides an insightful, mechanistic understanding of thermodynamics. The theory accomplishes this by introducing the idea of a macroscopic thermodynamic state as an ensemble of microscopic states. Gibbs connected this ensemble of microscopic states to macroscopic properties by answering the following question what is the observable difference between two different microscopic states, X i and X2, that represent systems that are macroscopically identical The answer is at the same time simple and profound there is no observable difference. 

Indeed in classical mechanics with a continuum phase space, there exists an inhnitely large collection of microscopic systems that correspond to a particular macroscopic state. Gibbs named this collection of points in phase space an ensemble of systems. Gibbs then shifted the attention from trajectories, i.e., a succession of microscopic states in time, to all possible available state points in phase space that conform to given macroscopic, thermodynamic constraints. He then defined the probability of each member of an ensemble and determined thermodynamic properties as averages over the entire ensemble. In this chapter, we present the important elements of Gibbs ensemble theory, setting the foundation for the rest of the book. 

As mentioned earlier, constructing an average of a physical quantity (such as the energy of the system) is very difficult if the system has many particles and we insist on evolving the system between states until we have included enough states to obtain an accurate estimate. 

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Elucidating the origin of magic numbers has been a problem of long-standing interest, made accessible through the use of the laser-based reflectron TOF technique and evaporative ensemble theory. Three test cases are considered, first protonated ammonia clusters where (NH3)4 NHj has been found to be especially prominent, and then two other cases are considered, one involving water cluster ions and another rare gas clusters. 

The implimentation of quantum statistical ensemble theory applied to physically real systems presents the same problems as in the classical case. The fundamental questions of how to define macroscopic equilibrium and how to construct the density matrix remain. The ergodic theory and the hypothesis of equal a priori probabilities again serve to forge some link between the theory and working models. 

By the way, through ensemble theory with unequal weights, Ref. [68] identifies an effective potential derivative discontinuity that links physical excitation energies to excited Kohn-Sham orbital energies from a ground-state calculation.)... 

Most investigations on nonequilibrium systems were initially carried out in the NESS. It is widely believed that NESSs are among the best candidate nonequilibrium systems to possibly extend the Boltzmann-Gibbs ensemble theory beyond equilibrium [50, 51]. 

There are several reasons for this unsatisfactory state of affairs. Most important is perhars the different conceptual demands on theories of chemistry and physics respectively. In this instance there has been no effort to re-interpret mathematical quantum theory to satisfy the needs of chemistry. The physical, or Copenhagen, interpretation, which is essentially an ensemble theory, is simply not able to handle the individual elementary units needed to formulate a successful theory of chemical cohesion and interaction. Computational dexterity without some mechanistic basis does not constitute a theory. Equally unfortunate has been the dogmatic insistence of theoretical chemists to drag their outdated phenomenological notions into the formulation of a hybrid theory, neither classical nor quantum even to the point of discarding... 

For classical systems the microstates are not discrete and the number of possible states for a fixed NVE ensemble is in general not finite. To see this imagine a system of a single particle (N = 1) traveling in an otherwise empty box of volume V. There are no external force fields acting on the particle so its total energy is E = mv2. The particle could be found in any location within the box, and its velocity could be directed in any direction without changing the thermodynamic macrostate defined by the fixed values of N, V, and E. To apply ensemble theory to classical systems Q(N, V, E) is defined as the (appropriately scaled) total volume accessible by the state variables of position and momentum accessible by the particles in the system. 

While the NVE (microcanonical) ensemble theory is sound and useful, the NVT (canonical) ensemble (which fixes the number of particles, volume, and temperature while allowing the energy to vary) proves more convenient than the NVE for numerous applications. 

A second approach to the NVT ensemble found in Feynman s lecture notes on statistical mechanics [55] is also based on the central idea from NVE ensemble theory that the probability of a microstate is proportional to the number of microstates available to the system. Thus... 

The NMR data can give only some average values for the ) LDOS on surface sites, and in the averaging all information required by ensemble theories is lost. It is also true that the data cannot give values for the partial... 

M.Tegmark. Is the Theory of Everything merely the Ultimate Ensemble Theory Annals of Physics (NY), 270, 1 (1998). 

According to the literature, the former results on the influence of modifications of the metal function on coking can be interpreted with the ensemble theory on a uniform metal surface or with the assumption of a heterogeneous surface or by the presence of electronic effects. The first interpretation correlates better with experimental results.In the case of the ensemble theory,... 

This chapter describes statistical-mechanical tools for the study of ionic solutions at equilibrium. We have attempted to cover those topics which are essential to prospective workers in the held, assuming only that the reader is familiar with the main features of grand ensemble theory, including spatial distribution functions. 

We next recall one of the general results of grand-ensemble theory. 

The virial theorem was also derived for ensanbles of excited states (Nagy 2002a). In the ground-state theory, several forms of the virial theoran were derived. The local and differential forms proved to be especially useful. In this chapter, the local virial theorem is derived for ensembles of excited states. In Section 7.2, the ensemble theory of excited states is summarized. The ensemble local virial theoran is derived in Section 7.3. Extension of the differential virial theorem of Holas and March (1995) to ensembles is presented in Section 7.4. Finally, Section 7.5 is devoted to discussion. 

Since this is not a textbook for thermodynamics, only a short derivation will be given, with the aim to obtain quick and useful results. Toward the end of the chapter, the problems in the derivation will be pointed out and an ensemble theory is mentioned. Here, we use simplified versions of the microcanonical ensemble. 

If we look at a small portion of a macroscopic system or study a mesoscopic system, we must study fluctuations. The probability of fluctuations is phenomenologically described by the thermodynamic theory of fluctuations (5). From the ensemble theory point of view, the fluctuation theory is the study of large deviations from the expectation value. This is the reason why large deviation theory is becoming increasingly important in statistical thermodynamics. Standard works on large deviation theory are References 16 and 17 perhaps as accessible introduction to the topic may be found in Reference 18. 

Jaroniec presented a statistical thermodynamic derivation of the Jovanovic equation using generalized ensemble theory. The final result may be written in the form ... 

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