Onsager

Excluded volume interactions will further decrease the density This is a strong argument suggesting that overlapping coils (semi-dilute solutions are necessary to obtain liquid crystalline order in solutions of worm-like macromolecules. 

We assume, for this model, that only excluded volume type interactions are important. We then approximate the rigid rod Onsager interaction— by the lowest angular dependent term in its multipole expansion which is of quadrupolar symmetry  

In order to simplify the algebra, let us specialize to the case of sharp helix-coil transitions, i.e., E 0 as is common for the synthetic polypeptides. (The general situation will be discussed in a later publication.) For this limit, a given chain is either a flexible coil or a rigid helix, i.e., V = 0 or N (Fig. 

The free energy is separable into two contributions (1) the intra-molecular interactions associated with the helix-coil transition (2) the intermolecular excluded volume interactions which are responsible for any nematic order. The difference in free energy per chain between a helical and randomly coiled molecule arising from intramolecular interactions is -N n s. The intermolecular excluded volume interactions in the mean field Meier-Saupe like approximation [Eqs. (III.2) - (III.6) with V = N] gives a contribution to the free energy difference, (AF) which is sketched in 

For s 1, a phase transition may only occur if the gain in free energy of ordering exceeds the entropic cost in loss of flexibility of the individual chains. In the (s,c) plane, the phase boundary will then be given by 

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We conclude this section by discussing an expression for the excess chemical potential in temrs of the pair correlation fimction and a parameter X, which couples the interactions of one particle with the rest. The idea of a coupling parameter was mtrodiiced by Onsager [20] and Kirkwood [Hj. The choice of X depends on the system considered. In an electrolyte solution it could be the charge, but in general it is some variable that characterizes the pair potential. The potential energy of the system... 

The reference free energy in this case is an upper bound for tlie free energy of the electrolyte. A lower bound for the free energy difference A A between the charged and uncharged RPM system was derived by Onsager... 

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. 

By a deft application of the transfer matrix teclmique, Onsager showed that the free energy is given by... 

Onsager L 1939 Electrostatic interaction of molecules J. Phys. Chem. 43 189... 

Onsager L 1944 Orystal statistics I. A two-dimensional model with an order-disorder transition Phys. Rev. 65 117... 

Onsager L and Kaufman B 1949 Orystal statistics III. Short range order in a binary Ising lattice Phys. Rev. 65 1244... 

That analyticity was the source of the problem should have been obvious from the work of Onsager (1944) [16] who obtained an exact solution for the two-dimensional Ising model in zero field and found that the heat capacity goes to infinity at the transition, a logarithmic singularity tiiat yields a = 0, but not the a = 0 of the analytic theory, which corresponds to a finite discontinuity. (Wliile diverging at the critical point, the heat capacity is synnnetrical without an actual discontinuity, so perhaps should be called third-order.)... 

Subsequently Onsager (1948) reported other exponents, and Yang (1952) completed the derivation. The exponents are rational numbers, but not the classical ones. 

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]

Onsager postulates [4, 5] the phenomenological equations for irreversible processes given by... 

Inserting the definition of G gives the celebrated Onsager reciprocal relations [4, 5]... 

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

Wherein the definition of the thennodynamic fluxes and forces of (A3.2,131 and (A3.2.141 have been used. Onsager defined [5] the analogue of the Rayleigh dissipation fiinction by... 

Onsager relation implies that measurement of one of these effects is sufficient to detemiine the coupling for both. The coefficient L is proportional to the heat conductivity coefficient and is a single scalar quantity in... 

These are just a few of the standard examples of explicit applications of the Onsager theory to concrete cases. 

A byproduct of the preceding analysis is that the Onsager theory innnediately detennines the fonn of the fluctuations that should be added to the difhision equation. Suppose that a solute is dissolved in a solvent with concentration c. The difhision equation for this is... 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

A proposal based on Onsager s theory was made by Landau and Lifshitz [27] for the fluctuations that should be added to the Navier-Stokes hydrodynamic equations. Fluctuating stress tensor and heat flux temis were postulated in analogy with the Onsager theory. Flowever, since this is a case where the variables are of mixed time reversal character, tlie derivation was not fiilly rigorous. This situation was remedied by tlie derivation by Fox and Ulilenbeck [13, H, 18] based on general stationary Gaussian-Markov processes [12]. The precise fomi of the Landau proposal is confimied by this approach [14]. 

Onsager s theory can also be used to detemiine the fomi of the flucUiations for the Boltzmaim equation [15]. Since hydrodynamics can be derived from the Boltzmaim equation as a contracted description, a contraction of the flucUiating Boltzmann equation detemiines fluctuations for hydrodynamics. In general, a contraction of the description creates a new description which is non-Markovian, i.e. has memory. The Markov... 

Linear response theory is an example of a microscopic approach to the foundations of non-equilibrium thennodynamics. It requires knowledge of tire Hamiltonian for the underlying microscopic description. In principle, it produces explicit fomuilae for the relaxation parameters that make up the Onsager coefficients. In reality, these expressions are extremely difficult to evaluate and approximation methods are necessary. Nevertheless, they provide a deeper insight into the physics. 

Onsager L 1931 Reciprocal relations in irreversible processes. I Rhys. Rev. 37 405... 

Onsager L and Machlup S 1953 Fluctuations and irreversible processes Rhys. Rev. 91 1505... 

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