# Field-theoretic simulation

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Field-theoretic simulation

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).

Shifted-contour Monte Carlo technique

Mean field representation

To make the field-theoretic methodology amenable for computation, Baeurle proposed to shiftthe contour of integration of the partition function integral through the homogeneous mean field (MF) solution using Cauchy's integral theorem, 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 theory 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).

Alternative techniques

Other promising field-theoretic simulation techniques have been developed recently, but they either still lack the proof of correct statistical convergence, like e.g. the Complex Langevinmethod (Ganesan 2001), and/or still need to prove their effectiveness on systems, where multiple saddle points are important (Moreira 2003).

References

* cite journal |
url = http://prola.aps.org/abstract/PRL/v89/i8/e080602
last = Baeurle
first = S.A.
title = Method of Gaussian Equivalent Representation: A New Technique for Reducing the Sign Problem of Functional Integral Methods
journal = Phys. Rev. Lett.
volume = 89
pages = 080602
year = 2002

* cite journal
url = http://www.mrl.ucsb.edu/~ghf/ghfgroup/pubs/pub156.pdf
last = Fredrickson
first = G.H.
coauthors = Ganesan, V.; Drolet, F.
title = Field-Theoretic Computer Simulation Methods for Polymers and Complex Fluids
journal = Macromolecules
volume = 35
pages = 16
year = 2002

* cite journal
url = http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHY-47MKK8M-2&_user=616165&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000032338&_version=1&_urlVersion=0&_userid=616165&md5=60e4b8712970cf7e07a14b05c5f0e609
last = Baeurle
first = S.A.
title = Computation within the auxiliary field approach
journal = J. Comput. Phys.
volume = 184
pages = 540
year = 2003

* cite journal |
url = http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ5-48XCNNY-3&_user=616165&_coverDate=08%2F01%2F2003&_alid=745709877&_rdoc=6&_fmt=high&_orig=search&_cdi=5301&_sort=d&_docanchor=&view=c&_ct=8&_acct=C000032338&_version=1&_urlVersion=0&_userid=616165&md5=c5a0e9b03b74325c0dc7bf7d88406ecf
last = Baeurle
first = S.A.
title = The stationary phase auxiliary field Monte Carlo method: a new strategy for reducing the sign problem of auxiliary field methodologies
journal = Comput. Phys. Commun.
volume = 154
pages = 111
year = 2003a

* cite journal |
last = Baeurle
first = S.A.
title = Grand canonical auxiliary field Monte Carlo: a new technique for simulating open systems at high density
journal = Comput. Phys. Commun.
volume = 157
pages = 201
year = 2004

* cite journal |
url = http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TFN-3S9T08S-1K&_user=616165&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000032338&_version=1&_urlVersion=0&_userid=616165&md5=afcc30c50e0dd059360e1eb6c9feb448
last = Rom
first = N.
coauthors = Charutz, D.M.; Neuhauser, D.
title = Shifted-contour auxiliary-field Monte Carlo: circumventing the sign difficulty for electronic-structure calculations
journal = Chem. Phys. Lett.
volume = 270
pages = 382
year = 1997

* cite journal |
url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000109000015006219000001&idtype=cvips&gifs=yes
last = Baer
first = R.
coauthors = Head-Gordon, M.; Neuhauser, D.
title = Shifted-contour auxiliary field Monte Carlo for ab initio electronic structure: Straddling the sign problem
journal = J. Chem. Phys.
volume = 109
pages = 6219
year = 1998

* cite journal
url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000117000007003027000001&idtype=cvips&gifs=yes
last = Baeurle
first = S.A.
coauthors = Martonak, R.; Parrinello, M.
title = A field-theoretical approach to simulation in the classical canonical and grand canonical ensemble
journal = J. Chem. Phys.
volume = 117
pages = 3027
year = 2002a

* cite journal
url = http://www.iop.org/EJ/article/0295-5075/55/6/814/epl_55_6_814.pdf?request-id=dfd06fed-53f4-41e8-9835-09aa202962d5
last Ganesan
first = V.
coauthors = Fredrickson, G.H.
title = Field-theoretic polymer simulations
journal = Europhys. Lett.
volume = 55
pages = 814
year = 2001

* cite journal
url = http://prola.aps.org/abstract/PRL/v91/i15/e150201
last Moreira
first = A.G.
coauthors = Baeurle, S.A.; Fredrickson, G.H.
title = Global Stationary Phase and the Sign Problem
journal = Phys. Rev. Lett.
volume = 91
pages = 150201
year = 2003

* [http://www-dick.chemie.uni-regensburg.de/group/stephan_baeurle/index.html Particle and Polymer Field Theories]

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