Dissipative Structures and Entropy Export in Self-Organizing Systems

Abstract

Prigogine's theory of dissipative structures predicts that far-from-equilibrium open systems can maintain ordered,
low-entropy internal states only by exporting entropy to their surroundings at a rate exceeding their internal entropy
production, yet direct, simultaneous measurement of internal entropy production and boundary entropy export
across a controlled range of driving strength is reported comparatively rarely for a single experimental system. We
measured entropy production and export in a continuously stirred Belousov-Zhabotinsky (BZ) reactor across 42
steady-state operating points spanning inflow (Damkohler number) rates from 0.008 to 0.095 s^-1, quantifying
internal entropy production from measured reaction rates and driving forces via nonequilibrium thermodynamics,
and boundary entropy export from measured heat and mass exchange with the reservoir. Below a critical inflow rate
of 0.031 s^-1, the reactor remained in a spatially uniform, near-equilibrium steady state with internal entropy
production of 0.8-2.1 mW/K per liter; above this threshold, the system transitioned to self-organized spatiotemporal
target and spiral wave patterns, and internal entropy production rose sharply to 14.6-38.2 mW/K per liter across the
patterned regime, a 12- to 18-fold increase. Entropy export exceeded internal entropy production at every
steady-state operating point tested (mean export-to-production ratio 1.34 ± 0.09 in the uniform regime and 1.19 ±
0.05 in the patterned regime, both p < 0.001 vs. unity by one-sample t-test), consistent with the requirement for a
nonequilibrium steady state, and this ratio declined systematically as driving strength increased across the
patterned regime (from 1.28 at the transition to 1.09 at the highest inflow rate tested), indicating that the margin
between export and production narrows as the system is driven harder even while both quantities grow in absolute
terms. Pattern wavelength, measured optically, decreased from 4.8 mm near the transition to 1.6 mm at the highest
inflow rate, scaling inversely with the square root of internal entropy production density (R^2 = 0.91), consistent with
reaction-diffusion length-scale theory. These results provide a direct, quantitative demonstration that self-organized
pattern formation in this system is accompanied by both a large increase in internal entropy production and a
systematically narrowing, but never inverted, margin of entropy export over production as the driving force
increases.