Towards Rigorous Derivation of Quantum Kinetic Equations

We develop a rigorous formalism for the description of the evolution of states of quantum many-particle systems in terms of a one-particle density operator. For initial states which are specified in terms of a one-particle density operator the equivalence of the description of the evolution of quantum many-particle states by the Cauchy problem of the quantum BBGKY hierarchy and by the Cauchy problem of the generalized quantum kinetic equation together with a sequence of explicitly defined functionals of a solution of stated kinetic equation is established in the space of trace class operators. The links of the specific quantum kinetic equations with the generalized quantum kinetic equation are discussed.Comment: 25 page

Corrections to the Law of Mass Action and Properties of the Asymptotic t=∞t = \infty State for Reversible Diffusion-Limited Reactions

On example of diffusion-limited reversible $A+A \rightleftharpoons B$ reactions we re-examine two fundamental concepts of classical chemical kinetics - the notion of "Chemical Equilibrium" and the "Law of Mass Action". We consider a general model with distance-dependent reaction rates, such that any pair of $A$ particles, performing standard random walks on sites of a $d$-dimensional lattice and being at a distance $\mu$ apart of each other at time moment $t$, may associate forming a $B$ particle at the rate $k_+(\mu)$. In turn, any randomly moving $B$ particle may spontaneously dissociate at the rate $k_-(\lambda)$ into a geminate pair of $A$s "born" at a distance $\lambda$ apart of each other. Within a formally exact approach based on Gardiner's Poisson representation method we show that the asymptotic $t = \infty$ state attained by such diffusion-limited reactions is generally \textit{not a true thermodynamic equilibrium}, but rather a non-equilibrium steady-state, and that the Law of Mass Action is invalid. The classical concepts hold \text{only} in case when the ratio $k_+(\mu)/k_-(\mu)$ does not depend on $\mu$ for any $\mu$.Comment: 30 pages, 2 figure

Commensurate-Incommensurate Phase Transitions for Multichain Quantum Spin Models: Exact Results

The behavior in an external magnetic field is studied for a wide class of multichain quantum spin models. It is shown that the magnetic field together with the interchain couplings cause commensurate-incommensurate phase transitions between the gapless phases in the ground state. The conformal limit of these models is studied and it is shown that the low-lying excitations for the incommensurate phases are not independent. A scenario for the transition from one to two space dimensions for the integrable multichain models is proposed. The similarities in the external field behavior for the quantum multichain spin models and a wide class of quantum field theories are discussed. The exponents for the gaps caused by relevant perturbations of the models are calculated.Comment: 23 pages, LaTeX, typos correcte

On Rigorous Derivation of the Enskog Kinetic Equation

We develop a rigorous formalism for the description of the kinetic evolution of infinitely many hard spheres. On the basis of the kinetic cluster expansions of cumulants of groups of operators of finitely many hard spheres the nonlinear kinetic Enskog equation and its generalizations are justified. It is established that for initial states which are specified in terms of one-particle distribution functions the description of the evolution by the Cauchy problem of the BBGKY hierarchy and by the Cauchy problem of the generalized Enskog kinetic equation together with a sequence of explicitly defined functionals of a solution of stated kinetic equation is an equivalent. For the initial-value problem of the generalized Enskog equation the existence theorem is proved in the space of integrable functions.Comment: 28 page