Sum-over-all-paths with memory: Some applications from nm to Mm
For a stochastic variable x(s) the probability density function P(x,s;x₀,0) is solved in closed form by summing over all possible paths parametrized by a random white noise variable modulated by a memory function. We apply this white noise functional approach to fluctuating observables in complex systems ranging in scale from nanometer to Megameter. In particular, we investigate: (a) the distribution of nucleotides in bacterial genomes; (b) the motion of tracer particles embedded in an ageing fibrin polymer network; (c) the fluctuating values of protein diffusion coefficients as a function of protein length; and (d) the Great Barrier Reef degradation resulting from increasing atmospheric CO₂ levels and sea surface temperatures. Each case exhibits a particular memory behavior where a close match between empirical and theoretical mean square deviation (MSD) is obtained. For all cases, the probability density function satisfies a modified diffusion equation of the form
∂P/∂s = ½(∂MSD/∂s) (∂²P/∂x²).
The derived probability density function allows us to predict probable values of subsequent data points not yet measured, or to gain insight on the dynamics and memory property of the systems considered.