Onsagers Results

Consider a medium consisting of elongated, cylindricaUy symmetric hard-core molecules in the form of spherocylinders. For spherocylinders shown in Fig. 6.19b we may introduce parameter 

Onsager used a low-density expansion, that is small packing factor r =pVm-After a cumbersome calculation procedure he has found the excluded volume Vexdi i j) that depends on orientation of the rods. Then, using (6.63) and several approximations concerning averaging, the free energy and the equation of state for hard spherocylinders have been found. 

At that stage a uniaxial, orientational order parameter S is introduced in terms of the mean square projections of molecular vector a  

The order parameter depends on the packing fraction and temperature. With increasing density or decreasing temperature the isotropic phase is substituted by the nematic phase. The equation for S is found in terms of q and y. 

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According to Fig. 3, the Onsager result [Eq. (15)], which shows the escape probability in the diffusion-controlled geminate recombination, gives the lower bound for the simulation results. The simulation results obtained for the lowest value of x (x = 0.05) are... 

In his paper7) Onsager has considered the liquid-crystalline transition in the system of rigid rods using two main assumptions a) the interaction of rods was assumed to be due to the pure steric repulsion (no attraction) b) the virial expansion method was used (for the details of the Onsager method see Sect. 2.4). Thus, the Onsager results... 

A similar treatment for a point dipole of magnitude p, in a sphere of radius a yields the Kirkwood—Onsager result > ... 

Onsager resuir in this limit provided the lattice sums were evaluated for a continuum with Tra /3 s V/H but smaller reductions of the Lorenta result for discrete sums over more realistic cubic lattice points. Extensions to third order dipole interactions for rigid dipoles by Rosenberg and Lax (25) and with harmonic oscillator induced dipoles included by Cole (26) showed further differences from the Onsager result even for a continuum with the conclusion that for the model the true result lies somewhere between the Lorenta and Onsager field expressions. 

The result of the MSM calculation for rigid dipoles is sketched in Figure 2 together with the Lorentz and Onsager results as a function of y. One sees that the Wertheim... 

In more recent theoretical work Hubbard Colonomos and Wolynes (81) developed a molecular theory of the dynamics of the two depolarization processes which predicts specific effects of ion size reducing to the Hubbard-Onsager result for slip boundary conditions in the limit of large ion radii. Numerical results for methanol were later tested by measurements of Winsor and Cole (82) for five salts which... 

In good solvent the polymer dimension of the coil being dy = 5/3, equation 2-36 applies, while for a rod the dimensionality being equal to 1, tiie Onsager result can be used and the virial coefficient is proportional to the square of the rod length (8), i.e the square of the degree of polymerization. 

The Ising model has been solved exactly in one and two dimensions Onsager s solution of the model in two dimensions is only at zero field. Infomiation about the Ising model in tliree dunensions comes from high- and low-temperature expansions pioneered by Domb and Sykes [104] and others. We will discuss tire solution to the 1D Ising model in the presence of a magnetic field and the results of the solution to the 2D Ising model at zero field. 

Onsager s solution to the 2D Ising model in zero field (H= 0) is one of the most celebrated results in theoretical chemistry [105] it is the first example of critical exponents. Also, the solution for the Ising model can be mapped onto the lattice gas, binary alloy and a host of other systems that have Hamiltonians that are isomorphic to the Ising model Hamiltonian. 

The classical treatment of the Ising model makes no distinction between systems of different dimensionality, so, if it fails so badly for d= 2, one might have expected that it would also fail for [Pg.644]

If odd variables, the b, are also included, then a generalization by Casimir [8] results in the Onsager-Casimir relations... 

The Onsager model describes the system as a molecule with a multipole moment inside of a spherical cavity surrounded by a continuum dielectric. In some programs, only a dipole moment is used so the calculation fails for molecules with a zero dipole moment. Results with the Onsager model and HF calculations are usually qualitatively correct. The accuracy increases significantly with the use of MP2 or hybrid DFT functionals. This is not the most accurate method available, but it is stable and fast. This makes the Onsager model a viable alternative when PCM calculations fail. 

Note that systems having a dipole moment of 0 will not exhibit solvent effects for the Onsager SCRF model, and therefore Onsager model (SCRF=Dipole) calculations performed on them will give the same results as for the gas phase. This is an inherent limitation of the Onsager approach. 

Solution The following table lists the energy differences that we computed as well as the original researchers HF/6-31+G(d) Onsager and B3LYP/6-31+G(d) SCRF=IPCM results ... 

These relations are the same as the parity rules obeyed by the second derivative of the second entropy, Eqs. (94) and (95). This effectively is the nonlinear version of Casimir s [24] generalization to the case of mixed parity of Onsager s reciprocal relation [10] for the linear transport coefficients, Eq. (55). The nonlinear result was also asserted by Grabert et al., (Eq. (2.5) of Ref. 25), following the assertion of Onsager s regression hypothesis with a state-dependent transport matrix. 

This result confirms Onsager s regression hypothesis. The most likely velocity in an isolated system following a fluctuation from equilibrium, Eq. (229), is equal to the most likely velocity due to an externally imposed force, Eq. (237), when the internal force is equal to the external force, Ts i =T. ... 

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