ABSTRACT
The quantum Wigner function and non-equilibrium equation for a microscopic particle in one spatial dimension (1D) subject to a potential and a heat bath at thermal equilibrium are considered by non-trivially extending a previous analysis. The non-equilibrium equation yields a general hierarchy for suitable non-equilibrium moments. A new non-trivial solution of the hierarchy combining the continued fractions and infinite series thereof is obtained and analyzed. In a short thermal wavelength regime (keeping quantum features adequate for chemical reactions), the hierarchy is approximated by a three-term one. For long times, in turn, the three-term hierarchy is replaced by a Smoluchovski equation. By extending that 1D analysis, a new model of the growth (polymerization) of a molecular chain (template or te) by binding an individual unit (an atom) and activation by a catalyst is developed in three spatial dimensions (3D). The atom, te, and catalyst move randomly as solutions in a fluid at rest in thermal equilibrium. Classical statistical mechanics describe the te and catalyst approximately. Atoms and bindings are treated quantum-mechanically. A mixed non-equilibrium quantum-classical Wigner-Liouville function and dynamical equations for the atom and for the te and catalyst, respectively, are employed. By integrating over the degrees of freedom of te and with the catalyst assumed to be near equilibrium, an approximate Smoluchowski equation is obtained for the unit. The mean first passage time (MFPT) for the atom to become bound to the te, facilitated by the catalyst, is considered. The resulting MFPT is consistent with the Arrhenius formula for rate constants in chemical reactions.
ABSTRACT
We review and improve previous work on non-equilibrium classical and quantum statistical systems, subject to potentials, without ab initio dissipation. We treat classical closed three-dimensional many-particle interacting systems without any "heat bath" (h b), evolving through the Liouville equation for the non-equilibrium classical distribution W c, with initial states describing thermal equilibrium at large distances but non-equilibrium at finite distances. We use Boltzmann's Gaussian classical equilibrium distribution W c , e q, as weight function to generate orthogonal polynomials (H n's) in momenta. The moments of W c, implied by the H n's, fulfill a non-equilibrium hierarchy. Under long-term approximations, the lowest moment dominates the evolution towards thermal equilibrium. A non-increasing Liapunov function characterizes the long-term evolution towards equilibrium. Non-equilibrium chemical reactions involving two and three particles in a h b are studied classically and quantum-mechanically (by using Wigner functions W). Difficulties related to the non-positivity of W are bypassed. Equilibrium Wigner functions W e q generate orthogonal polynomials, which yield non-equilibrium moments of W and hierarchies. In regimes typical of chemical reactions (short thermal wavelength and long times), non-equilibrium hierarchies yield approximate Smoluchowski-like equations displaying dissipation and quantum effects. The study of three-particle chemical reactions is new.
ABSTRACT
We summarize previous researches regarding neutron guides of small transverse cross-section (neutron fibres), smaller than those of the standard hollow guides and collimators employed currently. Those studies may not be widely known in the neutron capture therapy (NCT) community, but they may be interesting for it. Such neutron fibres could allow to deliver and concentrate neutron beams selectively in regions of size smaller than 1mm. We present new estimates and point out and discuss some new possible specific applications of those neutron fibres, which would not replace standard NCT but could supplement it. Thus, we entertain the possibility that neutron fibres could be useful for additional therapies (in typical NCT durations) of: (i) rather small tumours, (ii) thin borders of tumours. The use of these neutron fibres could reduce the undesirable delivery of radiation to healthy tissue around regions with malignant tissue.
