 
Summary: NUMERICAL SOLUTIONS OF THE COMPLEX LANGEVIN
EQUATIONS IN POLYMER FIELD THEORY
ERIN M. LENNON, GEORGE O. MOHLER, HECTOR D. CENICEROS§, CARLOS J.
GARC´IACERVERA¶, AND GLENN H. FREDRICKSON
Abstract. Using a diblock copolymer melt as a model system, we show that complex Langevin
(CL) simulations constitute a practical method for sampling the complex weights in field theory
models of polymeric fluids. Prior work has primarily focused on numerical methods for obtaining
meanfield solutionsthe deterministic limit of the theory. This study is the first to go beyond Euler
Maruyama integration of the full stochastic CL equations. Specifically, we use analytic expressions
for the linearized forces to develop improved time integration schemes for solving the nonlinear,
nonlocal stochastic CL equations. These methods can decrease the computation time required by
orders of magnitude. Further, we show that the spatial and temporal multiscale nature of the system
can be addressed by the use of Fourier acceleration.
Key words. diblock copolymer, Langevin equation, stochastic simulation
AMS subject classifications. 65Z05, 60H35
1. Introduction. Systems with mesoscopic ordering on scales of 1 nm to 1 µm
have proven vital to the development of novel polymeric materials. As modeling on
these length scales is not tractable with either molecular or macroscopic simulations,
there has been much interest in using techniques of statistical field theory to build
coarsegrained models of polymers that selfassemble on the mesoscale. In this ap
