Is there any shortcoming of the Langevin equation which is solved by its generalization? The ordinary Langevin equation describing the velocity $v(t)$ of a Brownian particle of mass $M$ in a fluid bath in equilibrium at a fixed temperature reads $$M\frac{dv}{dt}=-M\gamma v(t)+\zeta(t)+F_{\rm ext}(t).\tag{1}$$ This equation is often superseded by the generalized Langevin equation $$M\frac{dv}{dt}=-M\int\limits_{-\infty}^{t}\alpha(t-t') v(t')dt'+\zeta(t)+F_{\rm ext}(t)\tag{2}$$ where $\zeta(t)$ is a delta-correlated stationary Gaussian process with zero-mean. With $\alpha(t-t')=\gamma\delta(t-t')$, Eq.$(2)$ reduces to the ordinary form given by Eq.$(1)$.
Is there any unphysical feature or serious shortcoming of Eq.$(1)$ which is taken care of in Eq.$(2)$? This could motivate the use of Eq.$(2)$ instead of Eq.$(1)$.
 A: The motivation for using the generalized Langevin equation with memory kernel $\alpha(t-t')$ is the desire to include so-called memory effects, in which the evolution of the quantity of interest depends on its past states.  This could be useful in a number of contexts - this recent paper suggests that the dynamics of phase transitions could be understood in terms of the (generalized) Langevin dynamics of the order parameter, and for reference I believe this 1961 paper by Robert Zwanzig is the paper which first generalized the Langevin equation to include temporal non-locality.
This thesis (which, in the interest of full disclosure, I have only skimmed briefly) appears to address the generalized Langevin equation and some contexts in which it is useful for generalized Brownian motion.
A: I agree with the answer by @JMurray, but I would like to stress that Langevin equation (and more generally Brownian motion) is not a physical system, but a mathematical approach, applicable to a multitude of physical situations: chemical reactions, cold atoms, electrons in a metal, nanomechanical pendulums, etc. Claiming that for all these diverse systems the dissipation is always frequency independent and the noise is always white (gaussian and delta correlated) would be a very strong and likely incorrect claim. In fact, this is certainly incorrect at high frequencies.
Just to clarify: by frequency-dependent dissipation I mean that $\alpha(t)$ is not constant, as can be seen, if we Fourier (more precisely Laplace) transform the Langevin equation.
Note also that dissipation is a response of the bath, related to its correlation function via the fluctuation-dissipation theorem - which is more fundamental than Langevin equation. In many cases both are derived from the microscopic properties of the bath, rather than postulated ad-hoc, so there is no guarantee that they behave "nicely".
