Exceptionally Large Fluctuations in Orientational Order: The Lessons of Large-Deviation Theory for Liquid Crystalline Systems

TL;DR

This paper applies large-deviation theory to liquid crystalline systems, enabling more efficient sampling of rare large fluctuations in orientational order and providing insights into their thermodynamic behavior.

Contribution

It introduces a simulation-guided approach using large-deviation theory to accurately estimate free energies related to orientational order in liquid crystals.

Findings

Efficiently captures large orientational fluctuations in simulations.

Connects order parameters with thermodynamic fields via the equation of state.

Improves sampling efficiency by focusing on most probable configurations.

Abstract

How condensed-matter simulations depend on the number of molecules being simulated ($N$) is sometimes itself a valuable piece of information. Liquid crystals provide a case in point. Light scattering and $2d$-IR experiments on isotropic-phase samples display increasingly large orientational fluctuations ("pseudo-nematic domains") as the samples approach their nematic phase. The growing length scale of those locally ordered domains is readily seen in simulation as an ever-slower convergence of the distribution of orientational order parameters with $N$. But the rare-event character and exceptionally slow time scales of the largest fluctuations make them difficult to sample accurately. We show in this paper how taking a large-deviation-theory perspective enables us to leverage simulation-derived information more effectively. A key insight of the theory is that finding quantities such as orientational order parameters (extensive variables), is completely equivalent to deducing the conjugate (intensive) thermodynamic field required to equilibrate that amount of order - and that knowing the relationship between the two (the "equation of state") can easily be turned into knowing the relative free energy of that degree of order. A variety of well-known thermodynamic integration strategies are already founded on this idea, but instead of applying an artificially imposed external field, we use a priori statistical mechanical insights into the small and large-field limits to construct a simulation-guided, interpolated, equation of state. The free energies that result mostly need information from the most probable configurations, making the simulation process far more efficient than waiting for (or artificially generating) large fluctuations.