Long-Time Average of Field Values Measured by a Brownian Wanderer

Abstract

For a given field V(x) on the x-axis, we shall find out the exponent ν such that X[ω] = 1/T^ν∫₀^TV (ω(t))dt has, in the limit T → ∞, a non-trivial statistical distribution, and also the distribution itself, where ω(t) is the Brownian motion. The exponent ν turns out to be 1/2 when ∫_-∞^∞ V(x) dx is nonvanishing, and 1/4 when vanishing. The distributions are independent from the starting point of the Brownian motion. The result is relevant to the physics of chemoreception when the Brownian wanderer is identified with a bacterium or other small organism.