6.2.2. Definition of Effective Properties 3064 6.3. Response Properties to Magnetic Fields 3066 6.3.1. Nuclear Shielding 3066 6.3.2. Indirect Spin−Spin Coupling 3067 6.3.3. EPR Parameters 3068 6.4. Properties of Chiral Systems 3069 6.4.1. Electronic Circular Dichroism (ECD) 3069 6.4.2. Optical Rotation (OR) 3069 6.4.3. VCD and VROA 3070 7. Continuum and Discrete Models 3071 7.1. Continuum Methods within MD and MC Simulations 3072

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Quantum mechanical continuum solvation models.

The field of phase transitions and critical phenomena continues to be active in research, producing a steady stream of interesting and fruitful results. It has moved into a central place in condensed matter studies. Statistical physics, and more specifically, the theory of transitions between states of matter, more or less defines what we know about 'everyday' matter and its transformations. The major aim of this serial is to provide review articles that can serve as standard references for research workers in the field, and for graduate students and others wishing to obtain reliable information on important recent developments.

Phase Transitions and Critical Phenomena

Electrostatic interaction of a solute with a continuum. A direct utilizaion of AB initio molecular potentials for the prevision of solvent effects

This paper reviews recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near-Feshbach resonances in the BCS-BEC crossover.

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Many-Body Physics with Ultracold Gases

A new definition of order called topological order is proposed for two-dimensional systems in which no long-range order of the conventional type exists. The possibility of a phase transition characterized by a change in the response of the system to an external perturbation is discussed in the context of a mean field type of approximation. The critical behaviour found in this model displays very weak singularities. The application of these ideas to the xy model of magnetism, the solid-liquid transition, and the neutral superfluid are discussed. This type of phase transition cannot occur in a superconductor nor in a Heisenberg ferromagnet.

https://iopscience.iop.org/article/10.1088/0022-3719/6/7/010/pdf

Ordering, metastability and phase transitions in two-dimensional systems

A general reciprocal relation, applicable to transport processes such as the conduction of heat and electricity, and diffusion, is derived from the assumption of microscopic reversibility. In the derivation, certain average products of fluctuations are considered. As a consequence of the general relation $S=k logW$ between entropy and probability, different (coupled) irreversible processes must be compared in terms of entropy changes. If the displacement from thermodynamic equilibrium is described by a set of variables ${\ensuremath{\alpha}}_{1},\ensuremath{\cdots},{\ensuremath{\alpha}}_{n}$, and the relations between the rates ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{1},\ensuremath{\cdots},{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{n}$ and the "forces" $\frac{\ensuremath{\partial}S}{d{\ensuremath{\alpha}}_{1}},\ensuremath{\cdots},\frac{\ensuremath{\partial}S}{d{\ensuremath{\alpha}}_{n}}$ are linear, there exists a quadratic dissipation-function, $2\ensuremath{\Phi}(\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}})\ensuremath{\equiv}\ensuremath{\Sigma}{\ensuremath{\rho}}_{j}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{\mathrm{ij}}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{i}=\frac{\mathrm{dS}}{\mathrm{dt}}=\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{S}(\ensuremath{\alpha},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}})\ensuremath{\equiv}\ensuremath{\Sigma}(\frac{\ensuremath{\partial}S}{d{\ensuremath{\alpha}}_{j}}){\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{j}$ (denoting definition by $\ensuremath{\equiv}$). The symmetry conditions demanded by microscopic reversibility are equivalent to the variation-principle $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{S}(\ensuremath{\alpha},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}})\ensuremath{-}\ensuremath{\Phi}(\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}})=\mathrm{maximum},$ which determines ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{1},\ensuremath{\cdots},{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{n}$ for prescribed ${\ensuremath{\alpha}}_{1},\ensuremath{\cdots},{\ensuremath{\alpha}}_{n}$. The dissipation-function has a statistical significance similar to that of the entropy. External magnetic fields, and also Coriolis forces, destroy the symmetry in past and future; reciprocal relations involving reversal of the field are formulated.

https://link.aps.org/pdf/10.1103/PhysRev.38.2265

Reciprocal Relations in Irreversible Processes. II.

The partition function of a two-dimensional "ferromagnetic" with scalar "spins" (Ising model) is computed rigorously for the case of vanishing field. The eigenwert problem involved in the corresponding computation for a long strip crystal of finite width ($n$ atoms), joined straight to itself around a cylinder, is solved by direct product decomposition; in the special case $n=\ensuremath{\infty}$ an integral replaces a sum. The choice of different interaction energies ($\ifmmode\pm\else\textpm\fi{}J,\ifmmode\pm\else\textpm\fi{}{J}^{\ensuremath{'}}$) in the (0 1) and (1 0) directions does not complicate the problem. The two-way infinite crystal has an order-disorder transition at a temperature $T={T}_{c}$ given by the condition $sinh(\frac{2J}{k{T}_{c}}) sinh(\frac{2{J}^{\ensuremath{'}}}{k{T}_{c}})=1.$ The energy is a continuous function of $T$; but the specific heat becomes infinite as $\ensuremath{-}log |T\ensuremath{-}{T}_{c}|$. For strips of finite width, the maximum of the specific heat increases linearly with $log n$. The order-converting dual transformation invented by Kramers and Wannier effects a simple automorphism of the basis of the quaternion algebra which is natural to the problem in hand. In addition to the thermodynamic properties of the massive crystal, the free energy of a (0 1) boundary between areas of opposite order is computed; on this basis the mean ordered length of a strip crystal is ${(\mathrm{exp} (\frac{2J}{\mathrm{kT}}) tanh(\frac{2{J}^{\ensuremath{'}}}{\mathrm{kT}}))}^{n}.$

Crystal statistics. I. A two-dimensional model with an order-disorder transition

Electric Moments of Molecules in Liquids

Introdzution. The shapes of colloidal particles are often reasonably compact, so that no diameter greatly exceeds the cube root of the volume of the particle. On the other hand, we know many coiloids whose particles are greatly extended into sheets (bentonite), rods (tobacco virus), or flexible chains (myosin, various Iinear polymers). In some instances, a t least, solutions of such highly anisometric particles are known to exhibit remarkably great deviations from Raoult’s law, even to the extent that an anisotropic phase may separate from a solution in which the particles themselves occupy but one or two per cent of the total volume (tobacco virus, bentonite). We shall show in what follows how such results may arise from electrostatic repulsion between highly anisometric particles. Most colloids in aqueous solution owe their stability more or less to electric charges, so that each particle will repel others before they come into actual contact, and effectively claim for itself a greater volume than what it actuaily occupies. Thus, we can understand that colloids in general are apt to exhibit considerable deviations from Raoult’s law and that crystalline phases retaining a fair proportion of solvent may separate from concentrated solutions. However, if we tentatively increase the known size of the particles by the known range of the electric forces and multiply the resulting volume by four in order to compute the effective van der Waal’s co-volume, we have not nearly enough to explain why a solution of 2 per cent tobacco virus in 0.005 normal NaCZ forms two phases.

The effects of shape on the interaction of colloidal particles

The probability that a pair of ions of given initial separation will recombine with each other is computed from the laws of Brownian motion, which is the proper procedure whenever the Langevin factor equals unity, as in gases at high pressures. In the absence of forces other than the Coulomb attraction, the probability of escape equals the reciprocal of the Boltzmann factor. This result includes the correlation between temperature and pressure coefficients of the ionization by light particles previously predicted by Compton, Bennett and Stearns, if one allows their basic hypothesis about the laws which govern the initial separation of the ions. The effect of an electric field is to increase the fraction of escaping ions by a factor which in the incipient stage of the effect is proportional to the field intensity and independent of the initial distance, although it depends on the orientation of an ion pair. The predicted increase of the ionization current is a little more than one percent for every 100 volts/cm, which accounts for the observed effects of fields exceeding 1500 volts/cm. A reasonable amount of columnar recombination would help to explain the proportionately greater effects of weak fields. The inferred initial separations of the ions are apparently compatible with present knowledge of electron scattering and attachment.

Lars Onsager

Papers

Reciprocal Relations in Irreversible Processes. II.

Crystal statistics. I. A two-dimensional model with an order-disorder transition

Electric Moments of Molecules in Liquids

The effects of shape on the interaction of colloidal particles

Initial Recombination of Ions