Ensemble Properties and Basic Statistical Mechanics

Statistical mechanics is, obviously, a course unto itself in the standard chemistry/physics curriculum, and no attempt will be made here to introduce concepts in a formal and rigorous fashion. Instead, some prior exposure to the field is assumed, or at least to its thermodynamical consequences, and the fundamental equations describing the relationships between key thermodynamic variables are presented without derivation. From a computational-chemistry standpoint, many simplifying assumptions make most of the details fairly easy to follow, so readers who have had minimal experience in this area should not be adversely affected. 

In order to deal with collections of molecules in statistical mechanics, one typically requires that certain macroscopic conditions be held constant by external influence. The enumeration of these conditions defines an ensemble . We will confine ourselves in this chapter to the so-called canonical ensemble , where the constants are the total number of particles N (molecules, and, for our purposes, identical molecules), the volume V, and the temperature 

This ensemble is also sometimes referred to as the (N, V, T) ensemble. 

Just as there is a fundamental function that characterizes the microscopic system in quantum mechanics, i.e., the wave function, so too in statistical mechanics there is a fundamental function having equivalent status, and this is called the partition function. For the canonical ensemble, it is written as 

Within the canonical ensemble, and using established thermodynamic definitions, all of the following are true 

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We will delay a more detailed discussion of ensemble thermodynamics until Chapter 10 indeed, in this chapter we will make use of ensembles designed to render the operative equations as transparent as possible without much discussion of extensions to other ensembles. The point to be re-emphasized here is that the vast majority of experimental techniques measure molecular properties as averages - either time averages or ensemble averages or, most typically, both. Thus, we seek computational techniques capable of accurately reproducing these aspects of molecular behavior. In this chapter, we will consider Monte Carlo (MC) and molecular dynamics (MD) techniques for the simulation of real systems. Prior to discussing the details of computational algorithms, however, we need to briefly review some basic concepts from statistical mechanics. 

Since percolation is a property of macroscopic many-particle systems, it can be analyzed in terms of statistical mechanics. The basic idea of statistical mechanics is the relaxation of the perturbed system to the equilibrium state. In general the distribution function p(p,q t) of a statistical ensemble depends on the generalized coordinates q, momentum p, and time t. However, in the equilibrium state it does not depend explicitly on time [226-230] and obeys the equation... 

Surface tension is one of the most basic thermodynamic properties of the system, and its calculation has been used as a standard test for the accuracy of the intermolecular potential used in the simulation. It is defined as the derivative of the system s free energy with respect to the area of the interface[30]. It can be expressed using several different statistical mechanical ensemble averages[30], and thus we can use the molecular dynamics simulations to directly compute it. An example for such an expression is ... 

Abstract Fluctuation Theory of Solutions or Fluctuation Solution Theory (FST) combines aspects of statistical mechanics and solution thermodynamics, with an emphasis on the grand canonical ensemble of the former. To understand the most common applications of FST one needs to relate fluctuations observed for a grand canonical system, on which FST is based, to properties of an isothermal-isobaric system, which is the most common type of system studied experimentally. Alternatively, one can invert the whole process to provide experimental information concerning particle number (density) fluctuations, or the local composition, from the available thermodynamic data. In this chapter, we provide the basic background material required to formulate and apply FST to a variety of applications. The major aims of this section are (i) to provide a brief introduction or recap of the relevant thermodynamics and statistical thermodynamics behind the formulation and primary uses of the Fluctuation Theory of Solutions (ii) to establish a consistent notation which helps to emphasize the similarities between apparently different applications of FST and (iii) to provide the working expressions for some of the potential applications of FST. 

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