Numerical sign problem: Difference between revisions

Revision as of 12:14, 5 June 2008

Revision as of 00:02, 7 June 2008

Line 9:

Baeurle 2007, Schmid 1998, Baer 1998), as well as in real-time

Feynman-path integral quantum dynamics (Makri 1987, Miller 2005).

== Sign problem of statistical field theories ==

Line 22 ⟶ 21:

physical property <math>A</math> can be expressed in the general form

(Baeurle 2003)

:<math>

< \;langle A \; >rangle = \frac{\int D \sigma \; A

\left[ \; \sigma \; \right] \; \rho \left[ \;

\sigma \; \right]}{\int D \sigma

\; \rho \left[ \; \sigma \; \right]},

</math>

where <math>D \sigma</math> represents the field integration measure and

<math>A \left[ \; \sigma \; \right]</math> the estimator belonging to the

Line 45 ⟶ 46:

<math>A</math> can then be calculated with a standard simulation approach,

like Metropolis Monte Carlo, by evaluating the following expression

:<math>

< \;langle A \; >rangle = \frac{

\int D \sigma \; A \left[ \; \sigma \; \right]

\; \frac{\rho \left[ \; \sigma \; \right]}{\rho^\text{ref}

\left[ \; \sigma \; \right]}

\; \rho^\text{ref} \left[ \; \sigma \; \right]}{

\int D \sigma \; \frac{\rho \left[ \; \sigma \; \right]}{\rho^\text{ref}

\left[ \; \sigma \; \right]} \; \rho^\text{ref} \left[ \; \sigma \; \right]}

= \frac{< \;langle A \left[ \; \sigma \; \right]

\; \frac{\rho \left[ \; \sigma \; \right]}{\rho^\text{ref}

\left[ \; \sigma \; \right]} \; >rangle^\text{ref}}{

< \;langle \frac{\rho \left[ \; \sigma \; \right]}{\rho^\text{ref}

\left[ \; \sigma \; \right]} \; >rangle^\text{ref}},

</math>

where the brackets <math>< \;langle \cdots \; >rangle^\text{ref}</math> denote averaging

with respect to the real and positive definite reference distribution

<math>\rho^\text{ref} \left[ \; \sigma \; \right]</math>. In practice, the

averages in the numerator and denominator are approximated by their

respective discrete sum (Baeurle 2003)

:<math>

< \;langle A \; >rangle \approx \lim_{\tau_{run} \longrightarrow \infty}

\frac{\sum\limits_{i=1}^{\tau_{run}}

\; A \left[ \; \sigma_i \; \right] \; \frac{\rho

\left[ \; \sigma_i \; \right]}{\rho^\text{ref} \left[ \; \sigma_i \; \right]}}{

\sum\limits_{i=1}^{\tau_{run}} \; \frac{\rho \left[ \; \sigma_i \;

\right]}{\rho^\text{ref} \left[ \; \sigma_i \; \right]}},

</math>

where <math>\tau_\text{run}</math> defines the total number of simulation steps.

=== Choice of reference system and sign problem ===

A crucial issue for the effective evaluation of the ensemble average

<math>< \;langle A \; >rangle</math> is to choose a reference distribution, which

minimizes the standard deviations of the averages of the numerator and

denominator, as well as is independent of the estimator

Line 83 ⟶ 88:

distribution is obtained through the application of the variational method

(Kieu 1994), which provides

:<math>

\rho^\text{ref} \left[ \; \sigma \; \right] = \frac{\left| \;

\rho \left[ \; \sigma \; \right] \; \right|}{\int D \sigma

\left| \; \rho \left[ \; \sigma \; \right] \; \right|},

Line 90 ⟶ 96:

with the optimal standard deviation for the denominator average

:<math>

\sigma \left( < \;langle {\rm sign} \left[ \; \sigma \;

\right] \; >rangle^\text{ref} \right) = \sqrt{1 \; - \; < \;langle

{\rm sign} \left[ \; \sigma \; \right] \; >rangle^\text{ref \; ^2}},

</math>

where

:<math>

< \;langle {\rm sign} \left[ \; \sigma \;

\right] \; >rangle^\text{ref} = \frac{\int D \sigma \; \rho

\left[ \; \sigma \; \right]}{\int D \sigma \; \left|

\; \rho \left[ \; \sigma \; \right] \; \right|}

</math>

is the average of the sign function. The sign problem now occurs when

is the average of the sign function. The sign problem now occurs when the average of the sign

:<math>

< \;langle {\rm sign} \left[ \; \sigma \;

\right] \; >rangle^\text{ref} \longrightarrow 0,

</math>

which causes that, unless a huge number of configurations are sampled,

which causes that, unless a huge number of configurations are sampled, the large statistical fluctuations of the quantity render the calculation meaningless.

meaningless.

== Methods for reducing the sign problem ==