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Statistical MechanicsEnsemblesPostulatesOther ensemblesPartition functions and bridge equationsEnsemble averagingTime Averaging and Ergodicity (A brief aside)Simple Thermodynamic AveragesInternal energyTemperaturePressureEntropy and free energySecond-derivative propertiesFluctuationsStatistical MechanicsA lot can be accomplished without ever acknowledging the existence of molecules. Indeed, much of thermodynamics exists for just this purpose. Thermodynamics permits us to explain and predict phenomena that depend crucially on the fact that our world comprises countless molecules, and it does this without ever recognizing their existence. In fact, establishment of the core ideas of thermodynamics predates the general acceptance of the atomic theory of matter. Thermodynamics is a formalism with which we can organize and analyze macroscopic experimental observations, so that we have an intelligent basis for making predictions from limited data. Thermodynamics was developed to solve practical problems, and it is a marvelous feat of science and engineering.Of course, to fully understand and manipulate the world we must deal with the molecules.But this does not require us to discard thermodynamics. On the contrary, thermodynamics provides the right framework for constructing a molecular understanding of macroscopic behavior. Thermodynamics identifies the interesting macroscopic features of a system. Statistical mechanics is the formalism that connects thermodynamics to the microscopic world. Remember that a statistic is a quantitative measure of some collection of objects. An observation of the macroscopic world is necessarily an observation of some statistic of the molecular behaviors. The laws of thermodynamics derive largely from laws of statistics, in particular the simplifications found in the statistics of large numbers of objects. These objects—molecules—obey mechanical laws that govern their behaviors; these laws, through the filter of statistics, manifest themselves as macroscopic observables such as the equation of state, heat capacity, vapor pressure, and so on. The correct mechanics of molecules is of course quantum mechanics, but in a large number of situations a classical treatment is completely satisfactory.A principal aim of molecular simulation is to permit calculation of the macroscopic behaviors of a system that is defined in terms of a microscopic model, a model for the mechanical interactions between the molecules. Clearly then, statistical mechanics provides the appropriate theoretical framework for conducting molecular simulations. In this section we summarize from statistical mechanics the principal ideas and results that are needed to design, conduct, and interpret molecular simulations. Our aim is not to be rigorous or comprehensive in our presentation. The reader needing a more detailed justification for the results given here is referred to one of the many excellent texts on thetopic. Our focus at present is with thermodynamic behaviors of equilibrium systems, so we will not at this point go into the ideas needed to understand the microscopic origins oftransport properties, such as viscosity, thermal conductivity and diffusivity.EnsemblesA key concept in statistical mechanics is the ensemble. An ensemble is a collection of microstates of system of molecules, all having in common one or more extensive properties. Additionally, an ensemble defines a probability distribution  accords aweight to each element (microstate) of the ensemble. These statements require some elaboration. A microstate of a system of molecules is a complete specification of all positions and momenta of all molecules (i.e., all atoms in all molecules, but for brevity we will leave this implied). This is to be distinguished from a thermodynamic state, which entails specification of very few features, e.g. just the temperature, density and total mass. An extensive quantity is used here in the same sense it is known in thermodynamics—it is a property that relates to the total amount of material in the system. Most frequently we encounter the total energy, the total volume, and/or the total number of molecules (of one or more species, if a mixture) as extensive properties. Thus an ensemble could be a collection of all the ways that a set of N molecules could be arranged (specifying the location and momentum of each) in a system of fixed volume. As an example, in Illustration 1 we show a few elements of an ensemble of five molecules.If a particular extensive variable is not selected as one that all elements of the ensemble have in common, then all physically possible values of that variable are represented in thecollection. For example, Illustration 2 presents some of the elements of an ensemble in which only the total number of molecules is fixed. The elements are not constrained to have the same volume, so all possible volumes from zero to infinity are represented. Likewise in both Illustrations 1 and 2 the energy is not selected as one of the common extensive variables. So we see among the displayed elements configurations in which molecules overlap. These high-energy states are included in the ensemble, even though we do not expect them to arise in the real system. The likelihood of observing a given element of an ensemble—its physical relevance—comes into play with the probability distribution  that forms part of the definition of the ensemble.Any extensive property omitted from the specification of the ensemble is replaced by its conjugate intensive property. So, for example, if the energy is not specified to becommon to all ensemble elements, then there is a temperature variable associated with the ensemble. These intensive properties enter into the weighting distribution  in a way that will be discussed shortly. It is common to refer to an ensemble by the set of independent variables that make up its definition. Thus the TVN ensemble collects all microstates of the same volume and molecular number, and has temperature as the third independent variable. The more important ensembles have specific names given to them.These are Microcanonical ensemble (EVN) Canonical ensemble (TVN) Isothermal-isobaric ensemble (TPN) Grand-canonical ensemble (TV)These are summarized in Illustration 3, with a schematic of the elements presented for each ensemble.PostulatesStatistical mechanics rests on two postulates:N a m e A l l s t a t


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UB CE 530 - Statistical Mechanics

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