Escorted Free Energy Simulations: Improving Convergence by Reducing Dissipation

Nonequilibrium, “fast switching” estimates of equilibrium free energy differences $\Delta F$ are often plagued by poor convergence due to dissipation. We propose a method to improve these estimates by generating trajectories with reduced dissipation. Introducing an artificial flow field that couples the system coordinates to the external parameter driving the simulation, we derive an identity for $\Delta F$ in terms of the resulting trajectories. When the flow field effectively escorts the system along a near-equilibrium path, the free energy estimate converges efficiently and accurately. We illustrate our method on a model system and discuss the general applicability of our approach.

Figure 1

The potential energy landscape for $\lambda =0$ (solid line) and $\lambda =1$ (dashed line). Also depicted are the equilibrium distribution and the flow field, at $\lambda =0$.

Figure 2

Comparison of estimates of $\Delta F$ using Eqs. (1, 13). We performed simulations for switching times ranging from $\tau =0.01$ to $\tau =1.0$. Each $\Delta F_{\mathrm{est}}$ was obtained using ${10}^6$ trajectories, evolving under either Eq. (4) (open circles) or Eqs. (6, 19) (filled circles). Error bars were computed using the bootstrap method; for the filled circles these were smaller than the symbols, and are not shown.