Conjugate field

Define the order parameter V with conjugate field P and suppose that (3) is obtained by minimization of the Landau free energy ip(V, P,T), i.e. such that (3) is the equation that corresponds to dtp/dV = 0. Thus by integration of (3) at constant P and T,... 

Landau and Lifshitz, 1980), where J is the conjugate field of the order parameter, coefficient 4 is a function of temperature ... 

It reduces to the familiar form with the macrovariables M and M2 chosen as respectively the enthalpy and the volume whose conjugate fields are 1/T and P/T. 

Here, a is the lattice unit dimension, M is the number of lattice units per clay platelet (so that the product Ma2 = A is the total area of the platelet), a is the grafting density of "surfactants," Xa/3 are the Flory-Huggins parameters between species a and (3, //, is the chemical potential of the zth component, and 0, is the excess amount of the zth component in the system. The density profiles of various species, a(z), and conjugate fields, ua(z), are calculated as described below. Note that we introduce a separate species and component—voids (denoted as subscript v)—to account for density variation within the gallery. 

We illustrate the behavior for a first order transition between a vapor and a dense liquid in the framework of a simple Lennard-Jones model. The condensation of a vapor into a dense liquid upon cooling is a prototype of a phase transition that is characterized by a single scalar order parameter - the density, p. The thermodynamically conjugated field is the chemical potential, p. The qualitative features, however, are general and carry over to other types of phase coexistence, e.g., Sect. 3.4. 

Note that the conjugated field, Ap, does not have a direct interpretation in a multicanonical simulation. 

Another critical exponent (5) is defined considering the variation of the order parameter at Tc as a function of the conjugate field H, D being the associated critical amplitude,... 

Another choice of independently vuiable densities might be the c molar densities and the energy density The conjugate fields v are now the ratios iii/T, and -1/T. In these variables the Clapeyron equation is... 

Alternatively, it enters in products of direct and conjugated field pairs in the Schrodinger Lagrangian, see the Volume I of the present five volume work... 

According to the fluctuation-dissipation theorem the equilibrium mean square deviation of thermal composition fluctuations (5 ) is related to the first derivative of the order parameter with respect to the chemical potential [3,4]. If the order parameter is defined as the composition

chemical potentials Afi(= - mb) so the degree of thermal fluctuations is related to dAfi idA/x = d/dA/x) which is equivalent to 1/9 AG, where AG represents the Gibbs free energy of mixing with its natural parameters temperature T, pressure P, and composition This shows how the thermodynamic parameters can be determined from measurements of thermal fluctuations. 

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