Ensemble grand

Lattice homopolymers (pure V-L M-T Gibbs ensemble/grand canonical 71... 

The potentials used to describe these interfaces are those used to describe bulk hquids. The differences stay in the choice of the thermodynamical ensemble (grand canonical ensembles are often necessary), in the boundary conditions to be used in calculations, and in the explicit introduction in the model of some properties and concepts not used for bulk liquids, like the surface tension. Much could be said in this preliminary presentation of liquid/gas interfaces, but we postpone the few aspects we have decided to mention, because they may be treated in comparison with the other kind of surfaces. 

Keywords Configuration interaction Thermodynamics Partition function Temperature Canonical ensemble Grand canonical ensemble Fermi-Dirac statistics... 

Once the force field is chosen, a proper simulation method needs to be selected. Molecular dynamics simulations are applied to determine the solvation behaviour of ionic liquids by means of solving the Newtonian equations of motion for all molecules in the presence of a gradient in potential energy. Ionic liquid phase equilibria are determined by using Monte Carlo simulations in the isothermal isobaric Gibbs ensemble, grand canonical ensemble or osmotic ensemble with clever sampling schemes. 

The MC simulations discussed above all used the canonical (T, N) ensemble. Grand canonical Monte Carlo (GCMC) simulations offer a powerful means of assessing the effects of ionic activity coefficients on the counterion atmosphere of DNA. The grand canonical ensemble is a constant (T, p.)... 

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... 

With this result and arguments similar to those used in the last seetion, one finds the grand eanonieal ensemble distribution as (quantum meohanieally)... 

The eombination e oeeurs frequently. It is ealled the fiigaeity and is denoted by z. The grand eanonieal ensemble is also known as J - p ensemble. 

The T-P ensemble distribution is obtained in a maimer similar to the grand canonical distribution as (quantum mechanically)... 

In a canonical ensemble, the system is held at fixed (V, T, N). In a grand canonical ensemble the (V, T p) of the system are fixed. The change from to p as an independent variable is made by a Legendre transfomiation in which the dependent variable, the Flelmlioltz free energy, is replaced by the grand potential... 

The coimection between the grand canonical ensemble and thennodynamics of fixed (V, T, p) systems is provided by the identification... 

In the grand canonical ensemble, the number of particles flucPiates. By differentiating log E, equation (A2.2.121) with respect to Pp at fixed V and p, one obtains... 

Thennodynamics of ideal quantum gases is typically obtained using a grand canonical ensemble. In principle this can also be done using a canonical ensemble partition function, Q =. exp(-p E ). For the photon and... 

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation fimction of a system with a pairwise additive potential detemrines all of its themiodynamic properties. It also detemrines the compressibility of systems witir even more complex tliree-body and higher-order interactions. The pair correlation fiinctions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally detennined correlation fiinctions. We discuss the basic relations for the correlation fiinctions in the canonical and grand canonical ensembles before considering applications to model systems. 

The grand canonical ensemble is a collection of open systems of given chemical potential p, volume V and temperature T, in which the number of particles or the density in each system can fluctuate. It leads to an important expression for the compressibility Kj, of a one-component fluid ... 

It was shown in section A2.3.3.2 that the grand canonical ensemble (GCE) PF is a generating fiinction for the canonical ensemble PF, from which it follows that correlation fiinctions in the GCF are just averages of the fluctuating numbers N and N - 1... 

Removing the restriction on fixed [i], by considering the grand ensemble which sums over [ ], one has... 

Orkoulas G and Panagiotopoulos A Z 1999 Phase behavior of the restricted primitive model and square-well fluids from Monte Carlo simulations in the grand canonical ensemble J. Chem. Phys. 110 1581... 

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 ... 

The grand canonical ensemble corresponds to a system whose number of particles and energy can fluctuate, in exchange with its surroundings at specified p VT. The relevant themiodynamic quantity is the grand potential n = A - p A. The configurational distribution is conveniently written... 

Another triek is applieable to, say, a two-eomponent mixture, in whieh one of the speeies. A, is smaller than the other, B. From figure B3.3.8 for hard spheres, we ean see that A need not be particularly small in order for the test partiele insertion probability to elimb to aeeeptable levels, even when insertion of B would almost always fail. In these eireumstanees, the ehemieal potential of A may be detemiined direetly, while that of B is evaluated indireetly, relative to that of A. The related semi-grand ensemble has been diseussed in some detail by Kofke and Glandt [110]. 

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... 

Yoon K, Chae D G, Ree T and Ree F H 1981 Computer simulation of a grand canonical ensemble of rodlike molecules J. Chem. Phys. 74 1412-23... 

Esoobedo F A and de Pablo J J 1996 Expanded grand oanonioal and Gibbs ensemble Monte Carlo simulation of polymers J. Chem. Phys. 105 4391-4... 

In Ihc canonical, microcanonical and isothermal-isobaric ensembles the number of particles is constant but in a grand canonical simulation the composition can change (i.e. the number of particles can increase or decrease). The equilibrium states of each of these ensembles are cha racterised as follows ... 

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