Non-Equilibrium Wigner Function and Application to Model of Catalyzed Polymerization

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.

Keywords: approximate Smoluchovski equation; catalyzed polymerization; non-equilibrium Wigner function and hierarchy for moments; short thermal wavelength and long-time regimes.

Figure 1

Catalyst (circle), individual unit atom (box), and freely jointed template formed by 11 units (small boxes). Dashed lines represent binding (bonds). The successive numbering is: catalyst (unit 0), individual atom 1, and, in the template, atom 2, atom 3,…, atom 11. After polymerization, atom 1 becomes bound to atom 2, and the catalyst is bound (weakly) to atoms 1 and 2. The term ${\mathbf{y}}_1$ is the vector (not displayed) along the dashed line from atom 1 (box) to atom 2 (small box).