Stochastic Model of a Quantum Diffusion-Reaction Process

Francisco Castilho Alcaraz, V. Rittenberg

C. V. Heer

Chris Greenman

Abstract Reaction diffusion systems describe the behaviour of dynamic, interacting, particulate systems. Quantum stochastic processes generalise Brownian motion and Poisson processes, having operator valued Itô calculus machinery. Here it is shown that the three standard noises of quantum stochastic processes can be extended to model reaction diffusion systems, the methods being exemplified with spatial birth–death processes. The usual approach for these systems are master equations, or Doi-Peliti path integration techniques. The machinery described here provide efficient analyses for many systems of interest, and offer an alternative set of tools to investigate such problems.

Gunter Markus Schutz

We study a 12-parameter stochastic process involving particles with two-site interaction and hardcore repulsion on a d-dimensional lattice. In this model, which includes the asymmetric exclusion process, contact processes and other processes, the stochastic variables are particle occupation numbers taking values n⃗x = 0, 1. We show that on a 10-parameter submanifold the k-point equaltime correlation functions 〈n⃗x1 · · · n⃗xk 〉 satisfy linear differential-difference equations involving no higher correlators. In particular, the average density 〈n⃗x 〉 satisfies an integrable diffusion-type equation. These properties are explained in terms of dual processes and various duality relations are derived. By defining the time evolution of the stochastic process in terms of a quantum Hamiltonian H, the model becomes equivalent to a lattice model in thermal equilibrium in d + 1 dimensions. We show that the spectrum of H is identical to the spectrum of the quantum Hamiltonian of a d-dimensional, anisotropic spin-1/2 Heisenberg model. In one dimension our results hint at some new algebraic structure behind the integrability of the system. Keywords: Reaction-diffusion processes, correlation functions, integrable models An interesting class of non-equilibrium problems with a rich dynamical behaviour are

Abel Klein

Mareš, Jiří J., P. (Pavel) Hubík, J. (Jaroslav) Šesták

Periodic chemical reactions typical for biological objects controlled by diffusion in which the diffusion action attains the value of Planck’s quantum are explained in terms of stochastic theory where the random force is identified with temperature independent molecular quantum jiggling.