Gibbs ensemble theory

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]. 

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. 

The acceptance criteria for the Gibbs ensemble were originally derived from fluctuation theory [17]. An approximation was implicitly made in the derivation that resulted in a difference in the acceptance criterion for particle transfers proportional to 1/N relative to the exact expressions given subsequently [18]. A full development of the statistical mechanics of the ensemble was given by Smit et al. [19] and Smit and Frenkel [20], which we follow here. A one-component system at constant temperature T, total volume V, and total number of particles N is divided into two regions, with volumes Vj and Vu = V - V, and number of particles Aq and Nu = N - N. The partition function, Q NVt is... 

The lattice cluster theory (LCT) for glass formation in polymers focuses on the evaluation of the system s configurational entropy Sc T). Following Gibbs-DiMarzio theory [47, 60], Sc is defined in terms of the logarithm of the microcanonical ensemble (fixed N, V, and U) density of states 0( 7),... 

I ic. 11. Relationship between the pore filling pressure and the pore width predicted by the modified Kelvin equation (MK). the Horvath-Kawazoe method (HK), density functional theory (DFT). and Gibbs ensemble Monte Carlo simulation (points) for nitrogen adsorption in carbon slit pores at 77 K [11]. 

Figure 3. The equilibrium vapor and liquid densities of an associating fluid with one square-well bonding site. The circles are data from RCMC-Gibbs ensemble simulations, and the lines are calculations from three different implementations of a theory for associating fluids. The solid line uses exact values of the reference fluid radial distribution function the dashed and long dashed-short dashed lines use the WCA and modified WCA approximations to the radial distribution function, respectively. (Reprinted with permission from Muller et al. [43]. Copyright 1995 American Institute of Physics.)...

A number of alternative adsorption simulation methods have been proposed and tested, including algorithms for other ensembles sueh as the isobarie. isothermal [19], and alternatives to the GCMC. One sueh method is the Gibbs ensemble [20-22]. For example, adsorption isotherms have been evaluated using the potential distribution theory [23-25], which is based on Eq. (3) for the local chemical potential. Simulations of the Gibbs ensemble (which in adsorption simulations consists of two systems, one a dilute gas and one the actual dense adsorbed fluid, that are in equilibrium due to exchange of molecules between the two) have proved useful in studies of adsorbed fluids [26]. Variants on this method include those in Refs. 27 and 28. 

Smith WR, Triska B (1994) The reaction ensemble method for the computer simulation of chemical and phase equilibria I. Theory and basic examples. J Chem Phys 100 3019-3027 Kiyohara K, Spyriouni T, Gubbins KE et al (1996) Thermodynamic scaling Gibbs ensemble Monte Carlo a new method for determination of phase coexistence properties of fluids. Mol Phys 89 965-974... 

The acceptance criteria for the Gibbs ensemble were originally derived from fluctuation theory [19]. An approximation was implicitly made in the derivation that resulted in a difference in the acceptance criterion for particle transfers proportional to 1/N relative to the exact expressions given subsequently [20]. A full development of the statistical mechanics of the... 

McCabe, C., Gfl-VHegas, A., and Jackson, G., 1999. Gibbs ensemble computer simulation and SAFT-VR theory of non-conformal square-well monomer-dimer mixtures. Chem. Phys. Lett., 303 27. 

Ensemble theory is the crowning achievement of Josiah Willard Gibbs. With ensemble theory he was able to provide for the first time a lucid connection between microscopic states and macroscopic observables. In this section we describe how. Consider any observable, thermodynamic or mechanical property M of a system with N particles. We have argued that any such property is a function of positions and momenta of the particles in the system. 

Consider a system of N particles in volume V. Assume that the temperature T of the system is constant. The ensemble of microscopic points in phase space that correspond to this NVT macroscopic state is called the ATT or canonical ensemble. The canonical ensemble is called that simply because it was the first ensemble tackled by Gibbs. Indeed, ensemble theory as described in Gibbs seminal text (see Further reading) was formulated entirely for the constant temperature ensemble. The reason, as will become apparent in this chapter, is the ease in determining the canonical partition function. 

As indicated, Gibbs warily averted molecular dynamic assumptions to formulate an alternative ensemble-based reformulation of statistical mechanics that was able to seamlessly survive the revolutionary changes of 20th-century quantum theory, much to the approval of Einstein (see Sidebar 5.1) and others in the forefront of that revolution (see, e.g., Schrodinger s statement, Sidebar 13.4). 

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