Brownian dynamics: From glassy to trivial

We endow a system of interacting particles with two distinct, local, Markovian, and reversible microscopic dynamics that both converge to the Boltzmann-Gibbs equilibrium of standard liquids. While the first, standard, one leads to glassy dynamics, we use field-theoretical techniques to show that the latter displays no sign of glassiness. The approximations we use, akin to the mode-coupling approximation, are famous for magnifying glassy aspects of the dynamics, supposedly through the neglect of activated events. Despite this, the modified dynamics seem to stick to standard liquid relaxation. This finding singles out as applying to a realistic system of interacting particles in low dimensions and questions the role of the dynamical rules used to explore a given static free-energy landscape. Moreover, our peculiar choice of dynamical rules offers the possibility of a direct connection with replica theory, and our findings therefore call for a clarification of the interplay between replica theory and the underlying dynamics of the system.

Figure 1

Two watermelon diagrams that involve two and six propagator lines. The value of these diagrams are therefore (in our case) $\mathrm{Cte}\times G_{\rho \rho }{(\mathbf{\text{k}},t)}^2$ and $\mathrm{Cte}\times G_{\rho \rho }{(\mathbf{\text{k}},t)}^6$, where the constants are given by the interacting vertices in (10).

Figure 2

An arbitrary diagram contributing to ${\Sigma }_{\overline{\rho }\overline{\rho }}\left(t\right)$ involving 10 internal vertices. Time increases along an arrowed-leg because of causality in (15). A simple leg carries a correlation function and a causal arrowed one carries its time derivative due to the FDT. In the large-$t$ limit one would start with integrating $s_1$, then $s_3,s_2,s_9,s_{10},s_8,s_5,s_4,s_6$, and, finally, $s_7$.