Off-equilibrium fluctuation dissipation relation in binary mixtures

In this note we present numerical simulations of binary mixtures. We study the diffusion of particles and the response to an external driving force. We find evidence for the validity of the Cugliandolo Kurchan off-equilibrium fluctuation dissipation relation. The results are in agreement with the predictions of one step replica symmetry breaking and the dependance of the breakpoint parameter on the temperature coincides with that found in simple generalized spin glass models. The behaviour of an Hamiltonian system (with dissipative dynamics) approaching equilibrium is well understood in a mean field approach for infinite range disordered systems [1, 2, 3]. In this case we must distinguish an high and a low temperature region. In the low temperature phase the correlation and response functions satisfy some simple relations derived by Cugliandolo and Kurchan [1]. In a previous note [4, 5] we have found some preliminary evidence for the validity of these relations. Here we perform a different and more accurate numerical experiment and we are able to study the temperature dependance of the phenomenon. Let us define our notations. We concentrate our attention on a quantity A(t), which depends on the dynamical variables x(t). Later on we will make a precise choice of the function A. We suppose that the system starts at time t = 0 from a given initial condition and subsequently it is at a given temperature T . If the initial configuration is at equilibrium at a temperature T ′ > T , we observe an off-equilibrium behaviour when we change the temperature. In this note we will consider only the case T ′ >> T (in particular we will study the case T ′ = ∞). We can define a correlation function C(t, tw) ≡< A(tw)A(t+ tw) > (1) and the response function G(t, tw) ≡ δA(t+ tw) δǫ(tw) ∣