Superadiabatic Forces in Brownian Many-Body Dynamics

Theoretical approaches to nonequilibrium many-body dynamics generally rest upon an adiabatic assumption, whereby the true dynamics is represented as a sequence of equilibrium states. Going beyond this simple approximation is a notoriously difficult problem. For the case of classical Brownian many-body dynamics, we present a simulation method that allows us to isolate and precisely evaluate superadiabatic correlations and the resulting forces. Application of the method to a system of one-dimensional hard particles reveals the importance for the dynamics, as well as the complexity, of these nontrivial out-of-equilibrium contributions. Our findings help clarify the status of dynamical density functional theory and provide a rational basis for the development of improved theories.

Figure 1

(a) Schematic representation of a system of 10 hard particles confined between hard walls. (b) Density profiles calculated from BD simulations (black-solid line) and DDFT (red-dashed line) at reduced time $t^{*}=t_s/{\tau }_B=0.5$ (top), and 1.0 (bottom) for a system initialized in a parabolic trap. (c) Density profiles calculated from BD simulations (black-solid line) and DDFT (red-dashed line) at reduced time $t^{*}=0.1$ (top), and 0.2 (bottom) for a system initialized in a crystal structure (for a comparison of the adiabatic forces see the Supplemental Material [19]).

Figure 2

System relaxing following release from a parabolic trap ($t_s=0.5{\tau }_B$). (a) One-body density ${\rho }^{(1)}(x,t_s)$. (b) Total current $\mathrm{J}(x)$ from BD (continuous line) and DDFT (dotted line). (c) Total force integral $I^{*}(x)=I(x){\sigma }^2/k_{B}T$. (d) Adiabatic force $I_{\text{ad}}^{*}(x)=I_{\text{ad}}(x){\sigma }^2/k_{B}T$. (e) Superadiabatic force $I_{\text{sad}}^{*}(x)=I_{\text{sad}}(x){\sigma }^2/k_{B}T$. The vertical dashed lines serve as a guide for the eye.

Figure 3

Same as Fig. 2 but for the system initialized in a crystal state and sampled at time $t_s=0.2{\tau }_B$.

Figure 4

Total $I^{*}(x)=I(x){\sigma }^2/k_{B}T$ (thick line), adiabatic $I_{\text{ad}}^{*}(x)=I_{\text{ad}}(x){\sigma }^2/k_{B}T$ (dashed line), and superadiabatic $I_{\text{sad}}^{*}(x)=I_{\text{sad}}(x){\sigma }^2/k_{B}T$ (thin line) force integrals at different densities for the system initialized in a crystal state and sampled at time $t_s=0.2{\tau }_B$. The vertical dashed lines serve as a guide for the eye. (a) $\rho \sigma =0.25$ ($L_x/\sigma =40$). (b) $\rho \sigma =0.5$ ($L_x/\sigma =20$). (c) $\rho \sigma =0.67$ ($L_x/\sigma =15$).