  Exponential, and not Gaussian, decay of probability density functions was
studied by Laplace in the context of his analysis of errors. Such Laplace
propagators for the diffusive motion of single particles in disordered media
were recently observed in numerous experimental systems. What will happen to
this universality when an external driving force is applied? Using the
ubiquitous continuous time random walk with bias, and the Crooks relation in
conjunction with large deviations theory, we derive two properties of the
positional probability density function $P_F(x,t)$ that hold for a wide
spectrum of random walk models: (I) Universal asymmetric exponential decay of
$P_F(X,t)$ for large $|X|$, and (II) Existence of a time transformation that
for large $|X|$ allows to express $P_F(X,t)$ in terms of the propagator of the
unbiased process (measured at a shorter time). These findings allow us to
establish how the symmetric exponential-like tails, measured in many unbiased
processes, will transform into asymmetric Laplace tails when an external force
is applied.
