Equilibrium external fields

The motion of particles in a fluid is best approached tlirough tire Boltzmaim transport equation, provided that the combination of internal and external perturbations does not substantially disturb the equilibrium. In otlier words, our starting point will be the statistical themiodynamic treatment above, and we will consider the effect of botli the internal and external fields. Let the chemical species in our fluid be distinguished by the Greek subscripts a,(3,.. . and let f (r, c,f)AV A be the number of molecules of type a located m... 

Note that while the power-law distribution is reminiscent of that observed in equilibrium thermodynamic systems near a second-order phase transition, the mechanism behind it is quite different. Here the critical state is effectively an attractor of the system, and no external fields are involved. 

Formal Theory A small neutral particle at equilibrium in a static electric field experiences a net force due to DEP that can be written as F = (p V)E, where p is the dipole moment vector and E is the external electric field. If the particle is a simple dielectric and is isotropically, linearly, and homogeneously polarizable, then the dipole moment can be written as p = auE, where a is the (scalar) polarizability, V is the volume of the particle, and E is the external field. The force can then be written as ... 

Two other attempts, without the use of a distribution function, are worth mentioning, as these are operationally related to experiments and serve to give a rough estimate of the thermalization time. Christophorou et al. (1975) note that in the presence of a relatively weak external field E, the rate of energy input to an electron by that field is (0 = eEvd, where vd is the drift velocity in the stationary state. Under equilibrium, it must be equal to the difference between the energy loss and gain rates by an electron s interaction with the medium. The mean electron energy is now approximated as (E) = (3eD )/(2p), where fl = vd /E is the drift mobility and D is the perpendicular diffusion coefficient (this approximation is actually valid for a Maxwellian distribution). Thus, from measurements of fl and D the thermalization time is estimated to be... 

When a strong static electric field is applied across a medium, its dielectric and optical properties become anisotropic. When a low frequency analyzing electric field is used to probe the anisotropy, it is called the nonlinear dielectric effect (NLDE) or dielectric saturation (17). It is the low frequency analogue of the Kerr effect. The interactions which cause the NLDE are similar to those of EFLS. For a single flexible polar molecule, the external field will influence the molecule in two ways firstly, it will interact with the total dipole moment and orient it, secondly, it will perturb the equilibrium conformation of the molecule to favor the conformations with the larger dipole moment. Thus, the orientation by the field will cause a decrease while the polarization of the molecule will cause an... 

The reorganization of the solvent molecules can be expressed through the change in the slow polarization. Consider a small volume element AC of the solvent in the vicinity of the reactant it has a dipole moment m = Ps AC caused by the slow polarization, and its energy of interaction with the external field Eex caused by the reacting ion is —Ps Eex AC = —Ps D AC/eo, since Eex = D/eo- We take the polarization Ps as the relevant outer-sphere coordinate, and require an expression for the contribution AU of the volume element to the potential energy of the system. In the harmonic approximation this must be a second-order polynomial in Ps, and the linear term is the interaction with the external field, so that the equilibrium values of Ps in the absence of a field vanishes ... 

Equilibrium properties are surprisingly accurately predicted by molecular-level SCF calculations. MC simulations help us to understand why the SCF theory works so well for these densely packed layers. In effect, the high density screens the correlations for chain packing and chain conformation effects to such a large extent that the properties of a single chain in an external field are rather accurate. Cooperative fluctuations, such as undulations, are not included in the SCF approach. Also, undulations cannot easily develop in an MD box. To see undulations, one needs to perform molecularly realistic simulations on very large membrane systems, which are extremely expensive in terms of computation time. 

Section II deals with the general formalism of Prigogine and his co-workers. Starting from the Liouville equation, we derive an exact transport equation for the one-particle distribution function of an arbitrary fluid subject to a weak external field. This equation is valid in the so-called "thermodynamic limit , i.e. when the number of particles N —> oo, the volume of the system 2-> oo, with Nj 2 = C finite. As a by-product, we obtain very easily a formulation for the equilibrium pair distribution function of the fluid as well as a general expression for the conductivity tensor. 

These two expressions are exact they allow us in principle to calculate the N-particle distribution function at time t (to the first order in the external field) if its initial value is known. This will be our starting point for analyzing electrolytes both at equilibrium and out of equilibrium. 

Of course, in many situations, these equations may be simplified. For instance, we may consider the system in the absence of an external field (E == 0) or for uncharged particles (Zf — 0) in these cases, only the first equation remains. On the contrary, we may look at a system which is initially in its equilibrium state pe< - in this case, we have ... 

A quantitative evaluation of the relaxivities as a function of the magnetic field Bo requires extensive numerical calculations because of the presence of two different axes (the anisotropy and the external field axis), resulting in non-zero off-diagonal elements in the Hamiltonian matrix (15). Furthermore, the anisotropy energy has to be included in the thermal equilibrium density matrix. Figures 7 and 8 show the attenuation of the low field dispersion of the calculated NMRD profile when either the crystal size or the anisotropy field increases. 

Retaining the approximations of an incompressible liquid phase, a discontinuous density profile and curvature independent surface tension the conditions are those studied by Rao, Berne and Kalos (2). The essential physics was unchanged from the usual treatment in an open system, except that a minimum in the free energy of formation is found which corresponds to the unique equilibrium phase separated state whose symmetry, in the absence of an external field, is spherical. 

By definition, polymer brushes are made up of polymer chains grafted (tethered) by one end to a surface or an interface (Fig. 1) [ 1 - 3]. The density can be small or high in the latter case, the polymer chains are crowded and forced to stretch in order to avoid other chains. This results eventually in an equilibrium condition where no external field is necessary to force the chains into this geometry. 

The energy dissipation of a system containing free charges subjected to electric fields Is well known but this Indicates a non-equilibrium situation and as a result a thermodyanmlc description of the FDE Is Impossible. Within the framework of interionic attraction theory Onsager was able to derive the effect of an electric field on the Ionic dissociation from the transport properties of the Ions In the combined coulomb and external fields (2). It is not improper to mention here the notorious mathematical difficulty of Onsager s paper on the second Wien effect. 

The power dissipation is linearly related to <7BB(k, co) which is called, for obvious reasons, the power spectrum of the random process Bk. It should be noted that the energy dissipated by a system when it is exposed to an external field is related to a time-correlation function CBB(k, t) which describes the detailed way in which spontaneous fluctuations regress in the equilibrium state. This result, embodied in Eq. (51), is called the fluctuation... 

This equation expresses the well-known condition of detailed balance according to which every transition out of a microscopic state of a system in equilibrium is balanced on the average by a transition into that state. This condition is sufficient for the maintenance of thermodynamic equilibrium. Equation (60) demonstrates that the system absorbs more energy per unit time than it emits. It can be concluded that there is a net energy dissipation from the external field with a consequent production of heat. 

When sodium chloride is dissolved in water at ordinary temperatures, it is practically completely dissociated into sodium and chloride ions which, under the action of an external field, move in opposite directions and independently of each other subject to coulombic interactions. If, however, sodium chloride is dissolved in a solvent of lower dielectric constant, and if the solution is sufficiently dilute, there is an equilibrium between ions and a coulombic compound of the two ions which are commonly termed 4 ion pairs. This equilibrium conforms to the law of mass action when the interaction of the ions with the surrounding ion atmosphere is taken into account. In solvents of very low dielectric constant, such as the hydrocarbons, sodium chloride is not soluble however, many quaternary ammonium salts are quite soluble, and their conductance has been measured. Here at very low concentrations, there also is an equilibrium between ions and ion pairs which conforms to the law of mass action but at higher concentration, in the neighborhood of 1 X 10 W, or below, a minimum occurs in the conductance. Thereafter, it may be shown that the conductance increases continuously up to the molten electrolyte, provided that a suitable electrolyte and solvent are employed which are miscible above the melting point of the electrolyte. 

Stable states can be found, for example, by graphical solution of the equation 1 /x(4> — 4>o) = 7(potential minima [42,65], and it can be shown immediately that OB arises only if the system is biased by a sufficiently strong external field, that is, when it is far away from thermal equilibrium. If the noise intensity is weak, the system, when placed initially in an arbitrary state, will, with an overwhelming probability, approach the nearest potential minimum and will fluctuate near this minimum. Both the fluctuations and relaxation... 

In all of the studies of thermodynamic equilibrium that have been presented in the previous chapters, we have neglected the effects of an external field on the equilibrium properties of a system. This has been justified because the field may be present only in specific cases, the effect of the field may be negligible, or the position of the system in the field may be unchanged. The conditions of equilibrium in the presence of a gravitational or centrifugal field, an electrostatic field, and a magnetic field are developed in this chapter. 

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