Field-theoretic simulation
Encyclopedia
A field-theoretic simulation is a numerical strategy to calculate structure and physical properties of a many-particle system within the framework of a statistical field theory
, like e.g. a polymer field theory
. A convenient possibility is to use Monte Carlo
(MC) algorithms, to sample the full partition function integral expressed in field-theoretic representation. The
procedure is then called the auxiliary field Monte Carlo
method. However, it is well-known that MC sampling in conjunction with the basic field-theoretic representation of the partition function integral, directly obtained via the Hubbard-Stratonovich transformation
, is impracticable, due to the so-called numerical sign problem
(Baeurle 2002, Fredrickson 2002). The
difficulty is related to the complex and oscillatory nature of the resulting distribution function, which causes a bad statistical convergence of the ensemble averages of the desired structural and thermodynamic quantities. In such cases special analytical and numerical techniques are required to accelerate the statistical convergence of the field-theoretic simulation (Baeurle 2003, Baeurle 2003a, Baeurle 2004).
, which provides its so-called mean-field representation. This strategy was previously successfully employed in field-theoretic electronic structure calculations (Rom 1997, Baer 1998). Baeurle could demonstrate that this technique provides a significant acceleration of the statistical convergence of the ensemble averages in the MC sampling procedure (Baeurle 2002).
and leads to the Gaussian equivalent representation of the partition function integral, in conjunction with advanced MC techniques in the grand canonical ensemble. They could convincingly demonstrate that this strategy provides an additional boost in the statistical convergence of the desired ensemble averages (Baeurle 2002).
Statistical field theory
A statistical field theory is any model in statistical mechanics where the degrees of freedom comprise a field or fields. In other words, the microstates of the system are expressed through field configurations...
, like e.g. a polymer field theory
Polymer field theory
A polymer field theory is a statistical field theory describing the statistical behavior of a neutral or charged polymer system. It can be derived by transforming the partition function from its standard many-dimensional integral representation over the particle degrees of freedom in a functional...
. A convenient possibility is to use Monte Carlo
Monte Carlo
Monte Carlo is an administrative area of the Principality of Monaco....
(MC) algorithms, to sample the full partition function integral expressed in field-theoretic representation. The
procedure is then called the auxiliary field Monte Carlo
Auxiliary field Monte Carlo
Auxiliary field Monte Carlo is a method that allows the calculation, by use of Monte Carlo techniques, of averages of operators in many-body quantum mechanical or classical problems .-Reweighting procedure and numerical sign problem:The distinctive ingredient of "auxiliary field Monte Carlo" is...
method. However, it is well-known that MC sampling in conjunction with the basic field-theoretic representation of the partition function integral, directly obtained via the Hubbard-Stratonovich transformation
Hubbard-Stratonovich transformation
The Hubbard–Stratonovich transformation is an exact mathematical transformation invented by Russian physicist Ruslan L. Stratonovich and popularized by British physicist John Hubbard...
, is impracticable, due to the so-called numerical sign problem
Numerical sign problem
The numerical sign problem refers to the difficulty of numerically evaluating the integral of a highly oscillatory function of a large number of variables. Numerical methods fail because of the near-cancellation of the positive and negative contributions to the integral...
(Baeurle 2002, Fredrickson 2002). The
difficulty is related to the complex and oscillatory nature of the resulting distribution function, which causes a bad statistical convergence of the ensemble averages of the desired structural and thermodynamic quantities. In such cases special analytical and numerical techniques are required to accelerate the statistical convergence of the field-theoretic simulation (Baeurle 2003, Baeurle 2003a, Baeurle 2004).
Mean field representation
To make the field-theoretic methodology amenable for computation, Baeurle proposed to shift the contour of integration of the partition function integral through the homogeneous mean field (MF) solution using Cauchy's integral theoremCauchy's integral theorem
In mathematics, the Cauchy integral theorem in complex analysis, named after Augustin-Louis Cauchy, is an important statement about line integrals for holomorphic functions in the complex plane...
, which provides its so-called mean-field representation. This strategy was previously successfully employed in field-theoretic electronic structure calculations (Rom 1997, Baer 1998). Baeurle could demonstrate that this technique provides a significant acceleration of the statistical convergence of the ensemble averages in the MC sampling procedure (Baeurle 2002).
Gaussian equivalent representation
In subsequent works Baeurle et al. (Baeurle 2002, Baeurle 2002a) applied the concept of tadpole renormalization, which originates from quantum field theoryQuantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
and leads to the Gaussian equivalent representation of the partition function integral, in conjunction with advanced MC techniques in the grand canonical ensemble. They could convincingly demonstrate that this strategy provides an additional boost in the statistical convergence of the desired ensemble averages (Baeurle 2002).