Helmholtz free energy statistical mechanics

Statistical mechanics enables one to express the chemical potential i, for an ideal gas phase system in terms of the spectroscopic properties of individual gas phase molecules. The reader is referred to standard statistical mechanics texts (e.g. D. A. McQuarrie Statistical Mechanics , reading list) for the development of the relationship between the system Helmholtz free energy, A , and the corresponding canonical partition function Qi... 

The result of Eq. (5.17) isLfor practical purposes, the most important, result of statistical mechanics. For it gives a perfectly direct and stmiprhtfnrwRrd wav of deriving the Helmholtz free, energy, and henee the equation of state and specific heat, of anv system, if we know its energy as a function of coordinates aufl mniTirntnr- The sum of Eg. 

To carry out any calculations with the equation of state, we wish to approximate it in some analytic way. First, let us consider the most convenient variables to use. The results of experiment are usually expressed by giving the volume as a function oTprcs rcTand temperature. Thus theThermal expansion Is investigated as a function of temperature at atmospheric pressure, and in measurements of compressibility the volume is found as a function of pressure at certain fixed temperatures. On the other hand, for deriving results from statistical mechanics, it is convenient to find the Helmholtz free energy, and hence the pressure, as a function of volume and temperature. We shall express the equation of state in both forms, and shall find the relation between the two. We let V0 be the volume of our solid at no pressure and at the absolute zero of temperature. Then we shall assume... 

The determination of solvation free energy of solutes in solvents is a problem of primary importance, since all thermodynamic quantities can be derived from the free energies. For a system of N particles located at r/, r2, . ., rN, the statistical mechanical expression for the Helmholtz free energy, A, reads as [2]... 

The osmotic pressure calculation remains simple for all dilute solutions. Statistical mechanics gives the Helmholtz free-energy... 

In this section we study closed systems (closed to mass transport but not energy transfer) held at constant temperature. In statistical mechanics these systems are referred to as NVT systems (because the thermodynamic variables N, V, and T are held fixed). We shall see that the Helmholtz free energy represents the driving force for NVT systems. Just as an isolated system (an NVE system) evolves to increase its entropy, an NVT system evolves to decrease its Helmholtz free energy. 

Statistical mechanics of assemblies of axially symmetric molecules.— To illustrate the way in which the contributions of the directional forces to the thermodynamic functions can be estimated by statistical mechanics, we shall limit ourselves to pure substances in this section. The extension to mixtures will be discussed in the last section. The Helmholtz free energy of an assembly of N identical molecules occupying a volume V at temperature T is given by... 

To further speed up this approach, one can replace the expensive explicit-solvent simulations with implicit ones. Statistical mechanical theory gives the Helmholtz free energy A, apart from the scaling constant of the classical partition function that cancels out in binding energy calculations, as... 

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... 

Once the dispersion relation is known, thermodynamic functions can be calculated on the basis of statistical mechanics equations. As an example, the Helmholtz free energy, P, can be obtained as ... 

The statistical mechanical expression for the pseudo-chemical potential can be expressed, similarly to (3.53), as a ratio between two partition functions corresponding to the difference in the Helmholtz free energies in (3.88), i.e.,... 

In the case of two fluids 1 and 2, three different nearest-neighbor interaaions n, 22, and 12 are present. Since incompressibility is assumed from the outset, the Helmholtz free energy F is the relevant thermodynamic function. According to statistical mechanics it follows from the partition function Qby... 

The connection of statistical mechanics to thermodynamics is most commonly made through the canonical ensemble. This is the procedure we will adopt. Further, the connection must be made through one and only one equation, which we will take to be the Helmholtz free energy F ... 

It is well known that direct calculation of Helmholtz free energy is a nontrivial problem, which is readily seen from a classical statistical mechanical formula ... 

Comparing the vibrational branches and electronic bands calculations we note that in the former case the equations for different k values are solved independently while in the latter case the self-consistent calculation is necessary due to the BZ summation in the HP or KS Hamiltonian (see Chapters 4 and 7). Once the phonon dispersion in a crystal is known, thermodynamic functions can be calculated on the basis of statistical mechanics equations. As an example, the Helmholtz free energy, F, can be obtained as ... 

Ideas that go back to van der Waals [217, 218] and Lord Rayleigh [219] on inhomogeneous systems were applied by Cahn and Hilliard [216] to the interface problem. In inhomogeneous fluids, the Helmholtz free energy is a functional of the component density distributions. Although exact formal expressions for this functional have been derived [220,221] from statistical mechanics, they are impractical without approximation [222]. In the gradient approximation, this functional has been expressed as the sum of two contributions one is a function of the local composition and the other is a function of the local composition derivatives [216, 223, 224]. The free energy for a binary system is postulated to have the form ... 

Secondly, in the theory of irreversible processes, variation principles may be expected to help establish a general statistical method for a system which is not far from equilibrium, just as the extremal property of entropy is quite important for establishing the statistical mechanics of matter in equilibrium. The distribution functions are determined so as to make thermod5mamic probability, the logarithm of which is the entropy, be a maximum under the imposed constraints. However, such methods for determining the statistical distribution of the s retem are confined to the case of a system in thermodynamic equilibrium. To deal with a system out of equilibrium, we must use a different device for each case, in contrast to the method of statistical thermodynamics, which is based on the general relation between the Helmholtz free energy and the partition function of the system. 

In statistical mechanics, for a system at constant volume and temperature, the (Helmholtz) free energy can be calculated from MD simulations through ... 

Because the partition function can be expressed in terms of phase space probabilities, the fundamental classical thermodynamic relationships provide the link between statistical mechanics and thermodynamics. The fundamental thermodynamic function for the canonical ensemble is the Helmholtz free energy, F ... 

Other approaches for computing the surface tension start from the statistical mechanical expression for the Helmholtz free energy or for the pressure. The Kirkwood-Buff formula for the surface tension of a liquid/vapor interface of an atomic liquid described by the pair potential approximation is ... 

The Helmholtz free energy is of more theoretical importance, used often in statistical mechanics. 

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