PHYSICAL REVIEW A VOLUME 45, NUMBER 8 15 APRIL 1992 Fluctuations and correlations in a diffusion-reaction system: Unified description of internal fluctuations and external noise Werner Horsthemke Department of Chemistry and Center for Nonequilibrium Structures, Southern Methodist University, Dallas, Texas 75275-0314 Charles R. Doering* and Tane S. Ray Department of Physics, Clarkson University, Potsdam, New York 13699-5820 Martin A. Burschka Institut fii 'r Theoretische Physik (IV), Universitat Du sseld'orf, 4000 Diisseldorf I, Germany (Received 7 October 1991) The one-dimensional single-species diffusion-limited-coagulation process with irreversible random particle input (A ~A + A reversibly and B ~ A irreversibly), under the influence of external fluctuations in the system parameters, is formulated in terms of a closed and linear partial-differential equation. Our theoretical treatment includes both internal fluctuations and external noise simultaneously and without approximation, allowing investigation of the interplay of their effects on the macroscopic behavior of this diffusion-reaction system. For the reversible model with the rate of the A ~ A+ A reaction fluc- tuating between two values as a Markov stochastic process, we solve the system exactly. We observe that spatially homogeneous macroscopic fluctuations in the system parameters can induce microscopic spatial correlations in the nonequilibrium steady state. Direct Monte Carlo simulations of the micro- scopic dynamics are presented, confirming the theoretical analysis and directly illustrating the external- noise-induced spatial correlations. PACS number(s): 05. 40. +j, 82.20. Mj, 02.50. + s, 05.70.Ln I. INTRODUCTION Fluctuations in macroscopic many-body systems arise from two sources. The discrete nature of the elementary constituents produces the so-called internal fluctuations, while random variations in the environment introduce external noise into the system. Theoretical treatment of reaction-diffusion systems often disregards both kinds of fluctuations. Internal fluctuations are neglected because they occur on microscopic length and time scales and are thus deemed unimportant for macroscopic phenomena (except for phase-transition phenomena), while external noise is usually excluded because its amplitude can be controlled in laboratory systems — it is usually considered more of a nuisance than an essential factor in the dynam- ics of the system. Results obtained during the last decade have demonstrated that internal fluctuations and external noise can each be a crucial factor in the qualitative mac- roscopic behavior of nonlinear interacting particle sys- tems. External noise can postpone or advance instabili- ties, and may even give rise to transitions to states that cannot occur if the surroundings are free from random fluctuations [1, 2]. Internal fluctuations can give rise to strong particle-particle correlations in transport-limited diffusion-reaction systems dominating the macroscopic dynamics [3 — 5]. Most theoretical studies that include effects of fluctua- tions consider either only internal fluctuations or only external noise, although a few authors have attempted to develop a unified description of both sources of stochastic behavior [6]. External noise is usually treated on a mac- roscopic level of description where mean-field rate equa- tions are converted into stochastic differential equations by including random terms. For diffusion-reaction sys- tems, inclusion of internal fluctuations requires a mesos- copic or microscopic description in terms of a random process for the particle numbers of the chemical species. These different levels of description lead to conceptual difficulties for a unified treatment of internal and external fluctuations, and so far no wholly satisfactory one exists. In this paper we show that for a specific model system, namely the reversible coagulation-growth process ( A + A ~ A ) with irreversible input (B ~ A ) in one spa- tial dimension, the above-mentioned difficulties can be overcome, and a unified description of internal fluctua- tions and external noise can be naturally formulated in terms of a quantity satisfying a closed kinetic equation without any approximation. We consider the coagulation reaction A + A ~ A in the diffusion controlled lim-it where the particles coalesce immediately upon contact. These new results are generalizations of methods previ- ously introduced for this diffusion-limited-reaction pro- cess [7,8], and we are able to obtain some exact results. Direct Monte Carlo simulation of the interacting particle system with externally imposed noise, also reported here, confirms our analysis quantitatively. We observe external-noise-induced spatial correlations in the steady- state microscopic particle positions, even in the case of spatially homogeneous external noise with no intrinsic spatial length scale: Environmental fluctuations can in- troduce new microscopic length scales into the system. This model provides an example of far-from-equilibrium 45 5492 1992 The American Physical Society