Statistical mechanics grand-canonical ensemble

In the density functional theoiy (DFT) the statistical mechanical grand canonical ensemble is utilized. The appropriate free energy quantity is the grand Helmholtz free energy, or grand potential functional, 2(r. This free energy functional is expressed in terms of the density... 

The grand canonical ensemble is a set of systems each with the same volume V, the same temperature T and the same chemical potential p (or if there is more than one substance present, the same set of p. s). This corresponds to a set of systems separated by diathennic and penneable walls and allowed to equilibrate. In classical thennodynamics, the appropriate fimction for fixed p, V, and Tis the productpV(see equation (A2.1.3 7)1 and statistical mechanics relates pV directly to the grand canonical partition function... 

No system is exactly unifomi even a crystal lattice will have fluctuations in density, and even the Ising model must pemiit fluctuations in the configuration of spins around a given spin. Moreover, even the classical treatment allows for fluctuations the statistical mechanics of the grand canonical ensemble yields an exact relation between the isothemial compressibility K j,and the number of molecules Ain volume V ... 

Chesnut D A and Salsburg Z W 1963 Monte Carlo procedure for statistical mechanical calculation in a grand canonical ensemble of lattice systems J. Chem. Phys. 38 2861-75... 

In quantum statistical mechanics where a density operator replaces the classical phase density the statistics of the grand canonical ensemble becomes feasible. The problem with the classical formulation is not entirely unexpected in view of the fact that even the classical canonical ensemble that predicts equipartitioning of molecular energies, is not supported by observation. 

Limitation to ensembles that allow exchange of energy, but not of matter, with their environment is unnecessarily restrictive and unrealistic. What is required is an ensemble for which the particle numbers, Nj also appear as random variables. As pointed out before, the probability that a system has variable particle numbers N and occurs in a mechanical state (p, q) can not be interpreted as a classical phase density. In quantum statistics the situation is different. Because of second quantization the grand canonical ensemble, like the microcanonical and canonical ensembles, can be represented by means of a density operator in Hilbert space. 

Obviously Eq. (11) can also be derived directly from equilibrium thermodynamic considerations involving equilibrium constants. A statistical-mechanical derivation4 of Eq. (11), utilizing an equilibrium (actually grand canonical) ensemble, points up the analogy between this transition and a special case of helix-coil transition, the latter being most usually treated with equilibrium ensembles. 

The statistical mechanics of solutions at nonvamshing concentration is best-formulated within the grand canonical ensemble, where the number of solute molecules contained in volume Si is allowed to fluctuate. We only control the average concentration cp via the chemical potential fi.p This has great technical advantages, allowing for a very simple, analysis of the thermodynamic limit. In contrast, in the canonical ensemble, where the particle number M = Qc.p is taken as basic variable, an analysis of the thermodynamic limit is more tricky. A short discussion is given in the appendix. 

This statistical mechanical expression for surface tension depends explicitly on the potentials of inteimolecular force and molecular distribution functions. Upon recognition that the two-phase system under consideration is thermodynamically open, it follows that the distribution functions must be represented in the grand canonical ensemble. Thus, the dependence of y on temperature, T, and chemical potentials, enters through the implicit dependence of the distribution func-... 

Other lattice models are noteworthy as well. Roe (1974), for instance, developed a statistical mechanical formulation for an adsorbed layer capable of exchanging polymer and solvent with the bulk solution. The grand canonical ensemble, first introduced by DiMarzio and Rubin (1971),... 

Recently Haymet and Oxtoby and Klupsch ° independently develojjed related density functional theories of the liquid-solid interface. These are statistical mechanical theories that work with the grand canonical ensemble free energy Cl, which is a functional of the one-particle density p r),... 

See properties of the grand canonical ensemble, for example, in T.L. Hill, Statistical Mechanics, McGraw-Hill, New York, 1956, p. 72. 

In a grand canonical ensemble, the numbers of molecules of species. vt are fixed only on average, and statistical mechanics shows that the weight... 

We are particularly interested at this point in the density fluctuations present on a macroscopic scale. As stated in any textbook on statistical mechanics, the fluctuation in the number N of atoms, contained in an open system under constant volume and temperature, can be calculated by means of the grand canonical ensemble formalism. The result shows that the mean square fluctuation ((AN)2) in N is on the order of N itself and is related, in a system of macroscopic size, to the isothermal compressibility fa by... 

It is seen that inappropriate statistical mechanical treatments do exist and lead to artifacts. If we use the grand canonical ensemble 3 to calculate the above plots, we readily find that the artifacts still existi Therefore, the question is what the proper statistical mechanical treatment for this case is. The answer has been given in and it is the use of the generalized ensemble A. The solid lines in Figures 17 and 18 were calculated using this ensemble. Alternatively, we may use the equations of the canonical ensemble, i.e., the equations derived in the present chapter, as follows. ... 

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. 

A third form of the fundamental equation will be useful when we come to discuss the statistical mechanics in Chapter 4. The free energy F is the proper thermodynamic function to associate with the (petit) canonical ensemUe, but it is often more convenient to work with die grand canonical ensemble, for whidi the appropriate potential is O, where... 

The grand canonical ensemble represents an open system with a macrostate specified by values of T, V, and fi (the chemical potential) for each substance present. Each system of the ensemble is open to the other systems of the ensemble and can exchange matter and energy with the other systems. Discussions of the grand canonical ensemble can be found in statistical mechanics textbooks, but we will not discuss it. 

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