Temperature Integration: an efficient procedure for calculation of free energy differences

We propose a method, Temperature Integration, which allows an efficient calculation of free energy differences between two systems of interest, with the same degrees of freedom, which may have rough energy landscapes. The method is based on calculating, for each single system, the difference between the values of lnZ at two temperatures, using a Parallel Tempering procedure. If our two systems of interest have the same phase space volume, they have the same values of lnZ at high-T, and we can obtain the free energy difference between them, using the two single-system calculations described above. If the phase space volume of a system is known, our method can be used to calculate its absolute (versus relative) free energy as well. We apply our method and demonstrate its efficiency on a toy model of hard rods on a 1-dimensional ring.

💡 Research Summary

The paper introduces a novel technique called Temperature Integration (TI) for computing free‑energy differences (ΔF) between two systems that share the same degrees of freedom but may possess rugged energy landscapes. Traditional approaches such as Thermodynamic Integration (TI), Free‑Energy Perturbation (FEP), or Bennett Acceptance Ratio (BAR) rely on a continuous coupling parameter λ that morphs one Hamiltonian into another. When the underlying potential energy surface contains many minima or high barriers, sampling across λ becomes inefficient, leading to large statistical errors or systematic bias. Temperature Integration circumvents the λ‑path entirely by exploiting the temperature dependence of the partition function Z(T).

The central identity used is
 ln Z(T₁) − ln Z(T₂) = ∫_{β₂}^{β₁}⟨E⟩_β dβ, β = 1/(k_BT),
where ⟨E⟩_β denotes the ensemble‑averaged internal energy at inverse temperature β. By measuring ⟨E⟩ at a set of temperatures spanning a high‑temperature reference point (T_high) down to the temperature of interest (T_low), one can directly evaluate the logarithmic change in the partition function.

A crucial observation is that if the two systems A and B occupy the same phase‑space volume V (i.e., they have identical configurational degrees of freedom), then at sufficiently high temperature the Boltzmann weights become uniform and ln Z_A(T_high) ≈ ln Z_B(T_high) ≈ ln V. Consequently, the free‑energy difference at the target temperature reduces to
 ΔF = F_B − F_A = −k_BT_low