Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics

This paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics. We adopt a new point of view which has emerged progressively in recent years, and which takes seriously into account the chaotic character of the microscopic time evolution. The emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of Boltzmann. The nonequilibrium steady states, in presence of a Gaussian thermostat, are described by SRB measures. In terms of these one can prove the Gallavotti–Cohen fluctuation theorem. One can also prove a general linear response formula and study its consequences, which are not restricted to near-equilibrium situations. At equilibrium one recovers in particular the Onsager reciprocity relations. Under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of Dettmann and Morriss. The results just mentioned hold so far only for classical systems; they do not involve large size, i.e., they hold without a thermodynamic limit.     

Strict convexity (“continuity”) of the pressure in lattice systems

It is shown that the pressure is a strictly convex function of the translationally invariant interactions (under certain mild restrictions on the long-range part of these interactions) for classical and quantum lattice systems, by demonstrating that two distinct interactions can never lead to the same translationally invariant equilibrium state. This generalizes a previous result that the pressure is a continuous function of density at fixed temperature.     

Seven competing ways to recover the Michaelis–Menten equation reveal the alternative approaches to steady state modeling

The Michaelis–Menten enzymatic reaction is sufficient to perceive many subtleties of network modeling, including the concentration and time scales separations, the formal equivalence between bulk phase and single-molecule approaches, or the relationships between single-cycle transient probabilities and steady state rates. Seven methods proposed by different authors and yielding the same famous Michaelis–Menten equation, are selected here to illustrate the kinetic and probabilistic use of rate constants and to review basic techniques for handling them. Finally, the general rate of an ordered multistep reaction, of which the Michaelis–Menten reaction is a particular case, is deduced from a Markovian approach.     

Nonlinear dependence of the solubility of water in hydrocarbons on the molar volume of the hydrocarbon

The solubility of water in fifty hydrocarbon solvents at 20°C is estimated by means of the solubility equation derived from the thermodynamics of mobile order in H-bonded liquids. Neglecting the change in nonspecific cohesion forces, and assuming that water is primarily monomeric in solution, the prediction accounts for two main effects: the breaking of the H-bond network linking the water solute molecules together, and the entropy of exchange between water and solvent molecules in solution. The formation of a weak O–H ... hydrogen bond interaction is moreover taken into account according to whether the hydrocarbon is saturated or not. The overall predictive equation foresees a non-linear dependence of the water solubility on the molar volume of the hydrocarbon. Several rules are presented regarding the water solubility-hydrocarbon structure relationship.     

Applications conservant une mesure absolument continue par rapport àdx sur [0, 1]

Sufficient conditions are given such that a differentiable, non invertible, mapg:[0, 1][0, 1] leaves invariant a measure absolutely continuous with respect to the Lebesgue measure. In particular, this is shown to be the case forg(x)=Rx(1–x) whenR=3,6785735 ... .     

A heuristic theory of phase transitions

LetZ be a suitable Banach space of interactions for a lattice spin system. Ifn+1 thermodynamic phases coexist for ₀ Z, it is shown that a manifold of codimensionn of coexistence of (at least)n+1 phases passes through ₀. There are alson+1 manifolds of codimensionn–1 of coexistence of (at least)n phases; these have a common boundary along the manifold of coexistence ofn+1 phases. And so on for coexistence of fewer phases. This theorem is proved under a technical condition (R) which says that the pressure is a differentiable function of the interaction at ₀ when restricted to some codimensionn affine subspace ofZ. The condition (R) has not been checked in any specific instance, and it is possible that our theorem is useless or vacuous. We believe however that the method of proof is physically correct and constitutes at least a heuristic proof of the Gibbs phase rule.     

Theoretical study on the mechanism of the thermal decarboxylation of acrylic and benzoic acids. Models for aqueous solution

The mechanism of acrylic and benzoic acid decarboxylation in aqueous solution has been investigated by ab initio methods using the STO-3G and 3-21G basis sets. In those reactions, the solvent is represented successively by one and two water molecules. Their active participation as a proton relay in the chemical process is demonstrated by the large decrease in the activation energy with respect to the reaction studied in the absence of water. In the absence of any intermediate found along the reaction pathway, the proposed mechanism is the concerted process; the free acid being the species that undergoes decarboxylation via a pseudounimolecular mechanism by interaction with a chain of water molecules. At the transition state, the carboxylic hydrogen transfer to one water molecule, the reorganization of the chain of water molecules through which the proton is transferred and the cleavage of the C? C bond are much more advanced than the proton transfer from the last water molecule to the α-carbon atom of the carboxyl group.     

A mathematical reformulation of Derrida's REM and GREM

The large system limit of the Random Energy Model (REM) and generalized Random Energy Model (GREM) of Derrida is investigated, and found to be universal. This permits systematic calculations of relevance in particular to Parisi's solution of the Sherrington-Kirkpatrick spin-glass model.