  Multiplicative random processes in (not necessaryly equilibrium or steady
state) stochastic systems with many degrees of freedom lead to Boltzmann
distributions when the dynamics is expressed in terms of the logarithm of the
normalized elementary variables. In terms of the original variables this gives
a power-law distribution. This mechanism implies certain relations between the
constraints of the system, the power of the distribution and the dispersion law
of the fluctuations. These predictions are validated by Monte Carlo simulations
and experimental data. We speculate that stochastic multiplicative dynamics
might be the natural origin for the emergence of criticality and scale
hierarchies without fine-tuning.
