Fluctuation free energy

Figure 6. Computed vibration-rotation contributions to the Gibbs free energy and 4th-order polynomial curve fits for H20, HDO, and D20 as a function of temperature. AH computations were carried out using the AOSS-U Monte Carlo method in mass-weighted Jacobi coordinates. Three hundred Fourier coefficients were used per degree of freedom and 106 Monte Carlo samples were used for each calculation. Error bars at the 95% confidence level, as weU as all free energy fluctuations, are smaller than the width of the lines showing the curve fits. An increment of 10 was used over the temperature interval (1000-4000 K).

It would be of interest to extend the LS method and the hypothetical scanning method to MC and MD samples of continuum models of fluids (e.g., argon and water) and peptides in aqueous solutions. These methods enable one to calculate differences in free energy between significantly different wide microstates directly—that is, from two samples—and in principle, can handle even small differences in free energy of such microstates because the free energy fluctuations decrease with enhancement of the quality of the approximation. 

Eqs. 21,22 can be used to calculate 0, the free energy fluctuation exponent. 

This is trivially v did for the Gaussian pure polymer problem but gives a relation between the free energy fluctuation and the size of the polymer. This is borne out by the intuitive picture we develop below. This relation gives the size exponent v = 2/3 in d = 1. 

Powered by the quantitative estimates of the free energy fluctuation and size exponents, we now try to generate a physical picture. 

The appearance and decay of concentration fluctuations results in energy dissipation, which can be determined by the rate of change of the corresponding free energy fluctuations. 

A general prerequisite for the existence of a stable interface between two phases is that the free energy of formation of the interface be positive were it negative or zero, fluctuations would lead to complete dispersion of one phase in another. As implied, thermodynamics constitutes an important discipline within the general subject. It is one in which surface area joins the usual extensive quantities of mass and volume and in which surface tension and surface composition join the usual intensive quantities of pressure, temperature, and bulk composition. The thermodynamic functions of free energy, enthalpy and entropy can be defined for an interface as well as for a bulk portion of matter. Chapters II and ni are based on a rich history of thermodynamic studies of the liquid interface. The phase behavior of liquid films enters in Chapter IV, and the electrical potential and charge are added as thermodynamic variables in Chapter V. 

Figure A3.8.3 Quantum activation free energy curves calculated for the model A-H-A proton transfer reaction described 45. The frill line is for the classical limit of the proton transfer solute in isolation, while the other curves are for different fully quantized cases. The rigid curves were calculated by keeping the A-A distance fixed. An important feature here is the direct effect of the solvent activation process on both the solvated rigid and flexible solute curves. Another feature is the effect of a fluctuating A-A distance which both lowers the activation free energy and reduces the influence of the solvent. The latter feature enliances the rate by a factor of 20 over the rigid case.

The fluctuations of the local interfacial position increase the effective area. This increase in area is associated with an increase of free energy Wwhich is proportional to the interfacial tension y. The free energy of a specific interface configuration u(r,) can be described by the capillary wave Hamiltonian ... 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

C and I account for gradients of the smectic order parameter the fifth tenn also allows for director fluctuations, n. The tenn is the elastic free-energy density of the nematic phase, given by equation (02.2.9). In the smectic... 

Levy, R. M., Belhadj, M., Kitchen, D. B. Gaussian fluctuation formula for electrostatic free energy changes. J. Chem. Phys. 95 (1991) 3627-3633... 

The free energy differences obtained from our constrained simulations refer to strictly specified states, defined by single points in the 14-dimensional dihedral space. Standard concepts of a molecular conformation include some region, or volume in that space, explored by thermal fluctuations around a transient equilibrium structure. To obtain the free energy differences between conformers of the unconstrained peptide, a correction for the thermodynamic state is needed. The volume of explored conformational space may be estimated from the covariance matrix of the coordinates of interest, = ((Ci [13, lOj. For each of the four selected conform-... 

As the simulation proceeds, the values of A fluctuate, subject to the constraint in Equatic 11.38). The free energy difference between two molecules i and j can be determined t identifying the probability that each molecule occupies the state A, = 1 or A = 1, respe lively. Thus ... 

Next we consider how to evaluate the factor 6p. We recognize that there is a local variation in the Gibbs free energy associated with a fluctuation in density, and examine how this value of G can be related to the value at equilibrium, Gq. We shall use the subscript 0 to indicate the equilibrium value of free energy and other thermodynamic quantities. For small deviations from the equilibrium value, G can be expanded about Gq in terms of a Taylor series ... 

The first step in studying phenomenological theories (Ginzburg-Landau theories and membrane theories) has usually been to minimize the free energy functional of the model. Fluctuations are then included at a later stage, e.g., using Monte Carlo simulations. The latter will be discussed in Sec. V and Chapter 14. 

Random interface models for ternary systems share the feature with the Widom model and the one-order-parameter Ginzburg-Landau theory (19) that the density of amphiphiles is not allowed to fluctuate independently, but is entirely determined by the distribution of oil and water. However, in contrast to the Ginzburg-Landau approach, they concentrate on the amphiphilic sheets. Self-assembly of amphiphiles into monolayers of given optimal density is premised, and the free energy of the system is reduced to effective free energies of its internal interfaces. In the same spirit, random interface models for binary systems postulate self-assembly into bilayers and intro-... 

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