Thermodynamics from ensembles

We can now determine the probability density at each point in a continuum phase space as follows  

This simply means that the NVE probability density is uniform everywhere in the available area of the phase space, or that all microscopic state points of an A V ensemble are of equal probability. 

In a discrete phase space the probability of each state is determined as follows  

This means that all possible phase space points X that correspond to a particular NVE macroscopic state have the same probability of occurring. 

The size of the available phase space, E, and the number of microscopic states, corresponding to the particular NVE state can be determined in principle for any system with known Hamiltonian. We determine them for an ideal gas later in this chapter. 

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The average values of physical quantities calculated through ensemble theory should obey the laws of thermodynamics. In order to derive thermodynamics from ensemble theory we have to make the proper identification between the usual thermodynamic variables and quantities that can be obtained directly from the ensemble. In the microcanonical ensemble the basic equation that relates thermodynamic variables to ensemble quantities is... 

The link to the molecular level of description is provided by statistical thermodynamics whore our focus in Chapter 2 will be on specialized statistical physical ensembles designed spc cifically few capturing features that make confined fluids distinct among other soft condensed matter systems. We develop statistical thermodynamics from a quantum-mechanical femndation, which has at its core the existence of a discrete spectrum of energj eigenstates of the Hamiltonian operator. However, we quickly turn to the classic limit of (quantum) statistical thermodynamics. The classic limit provides an adequate framework for the subsequent discussion because of the region of thermodynamic state space in which most confined fluids exist. 

When we want to calculate thermodynamics from MD, the first question which arises is, how are thermodynamic functions to be defined in the MD ensemble It is natural to define thermodynamic functions in the MD ensemble in terms of averages of dynamic variables, exactly as In equations (16)-(21) for the canonical ensemble. However, in different ensembles, the averages of the same dynamical variable are different, the difference being of... 

Configurational Thermodynamics from Grand-Canonical Ensembles.313... 

CONFIGURATIONAL THERMODYNAMICS FROM GRAND-CANONICAL ENSEMBLES... 

Using statistical thermodynamics [3,4] entropy and enthalpy can be calculated from ensemble averages over similar systems of identical molecules. In an ensemble of systems of identical energy (micro-canonical ensemble), the entropy may be calculated using Boltzmann s equation,... 

It can be seen that function (2) is continuous, can be rapidly evaluated for any configuration, and is readily differentiable. Two widely used methods are available to generate a thermodynamically relevant ensemble from a function such as equation (2) molecular dynamics (MD) and Monte Carlo (MC). MD is an integration of the analytic force field according to Newton s equation... 

Again, aU that is needed to derive macroscopic thermodynamics from microscopic relations is the functional dependence of the partition function on ensemble constraints. 

About 1902, J. W. Gibbs (1839-1903) introduced statistical mechanics with which he demonstrated how average values of the properties of a system could be predicted from an analysis of the most probable values of these properties found from a large number of identical systems (called an ensemble). Again, in the statistical mechanical interpretation of thermodynamics, the key parameter is identified with a temperature, which can be directly linked to the thermodynamic temperature, with the temperature of Maxwell s distribution, and with the perfect gas law. 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

An evaluation of the retardation effects of surfactants on the steady velocity of a single drop (or bubble) under the influence of gravity has been made by Levich (L3) and extended recently by Newman (Nl). A further generalization to the domain of flow around an ensemble of many drops or bubbles in the presence of surfactants has been completed most recently by Waslo and Gal-Or (Wl). The terminal velocity of the ensemble is expressed in terms of the dispersed-phase holdup fraction and reduces to Levich s solution for a single particle when approaches zero. The basic theoretical principles governing these retardation effects will be demonstrated here for the case of a single drop or bubble. Thermodynamically, this is a case where coupling effects between the diffusion of surfactants (first-order tensorial transfer) and viscous flow (second-order tensorial transfer) takes place. Subject to the Curie principle, it demonstrates that this retardation effect occurs on a nonisotropic interface. Therefore, it is necessary to express the concentration of surfactants T, as it varies from point to point on the interface, in terms of the coordinates of the interface, i.e.,... 

The electrochemical potential of single ionic species cannot be determined. In systems with charged components, all energy effects and all thermodynamic properties are associated not with ions of a single type but with combinations of different ions. Hence, the electrochemical potential of an individual ionic species is an experimentally undefined parameter, in contrast to the chemical potential of uncharged species. From the experimental data, only the combined values for electroneutral ensembles of ions can be found. Equally inaccessible to measurements is the electrochemical potential, of free electrons in metals, whereas the chemical potential, p, of the electrons coincides with the Fermi energy and can be calculated very approximately. 

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