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\date{\color{blue} Singapore MCMC Meeting, March 2004}
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\begin{document}
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\foilhead{\r Generalized Ensembles: Multicanonical Simulations} \b 
\vskip -180pt

\begin{enumerate}
\item Multicanonical Ensemble
\item How to get the Weights?
\item Example Runs and Re-Weighting to the Canonical Ensemble
\item Energy and Specific Heat Calculation
\item Free Energy and Entropy Calculation
\item Complex Systems with Rugged Free Energy Landscapes
\item Summary
\end{enumerate}

% \noindent
% Reference: Berg, Comp. Phys. Commun. 153 (2003) [cond-mat/0205377].
 
\foilhead{\r Multicanonical Ensemble} 
\vskip -30pt

One of the questions which ought to be addressed before performing
a large scale computer simulation is {\r ``What are suitable weight
factors for the problem at hand?''} So far we used the Boltzmann 
weights as this appears natural for simulating the
Gibbs ensemble. However, a broader view of the issue
is appropriate.  

Conventional canonical simulations can by re-weighting techniques only 
be extrapolated to a vicinity of this temperature. For 
multicanonical simulations this is different. A single simulation 
allows to obtain equilibrium properties of the Gibbs ensemble
over a range of temperatures. Of particular interest are two 
situations for which canonical simulations do not provide the 
appropriate implementation of importance sampling: 
\begin{enumerate}
\item The physically important configurations are rare in the 
      canonical ensemble.
\item A rugged free energy landscape makes the physically important 
      configurations difficult to reach. 
\end{enumerate}
\newpage

Multicanonical simulations sample, in an appropriate energy range, with 
an {\r approximation} to the weights 
$$\s \widehat{w}_{mu}(k)=w_{mu}(E^{(k)}) = 
   e^{-b(E^{(k)})\, E^{(k)} + a (E^{(k)})} = {1\over n(E^{(k)})} $$
where $\s n(E)$ is the {spectral density}. The function $\s b(E)$ 
defines the inverse {\r microcanonical temperature} and $\s a(E)$ the 
{\r dimensionless, microcanonical free energy}. The function $\s b(E)$ 
has a relatively smooth dependence on its arguments, which makes it a 
useful quantity when dealing with the weight factors.  
The multicanonical method requires {\r two steps:}

\begin{enumerate}

\item Obtain a {\r working estimate} the weights. Working estimate means
      that the approximation has to be good enough so that the simulation
      covers the desired eneryg or temperature range. 

\item Perform a Markov chain MC simulation with the {\r fixed} weights.
      The thus generated configurations constitute the {\r multicanonical 
      ensemble}.

\end{enumerate}

\newpage

\foilhead{\r How to get the Weights?}\vskip -30pt

To get the weights is at the heart of the method (and a stumbling block 
for beginners). Some approaches used are listed in the following:

\begin{enumerate}
\item Overlapping constrained (microcanonical) MC simulations.
      Potential problem: ergodicity.
\item {\r Finite Size Scaling (FSS) Estimates.} Best when it works!
      Problem: There may 
      be no FSS theory or the prediction may be so inaccurate that the 
      initial simulation will not cover the target region.
\item General Purpose Recursions. Problem: Tend to deteriorate with
      increasing lattice size (large lattices). 
\end{enumerate}

\vfil
\newpage

\noindent
{\large\r Multicanonical Recursion} {\b\small (A variant of Berg, 
J. Stat. Phys. 82 (1996) 323.)}

\noindent
{\b Multicanonical parameterization of the weights}
({\s $\epsilon$} smallest stepsize): \s
$$ w(a) = e^{-S(E_a)} = e^{-b(E_a)\, E_a + a(E_a)}\ , $$
$$ {\rm where}~~~ b(E) = 
   \left[ S(E+\epsilon) - S(E) \right]/\epsilon~~~{\rm and}$$
$$ a(E-\epsilon) =  a(E) 
  + \left[ b(E-\epsilon)-b(E) \right]\, E\ . $$
{\r {\large Recursion:}}
$$ {\r b^{n+1}(E) = b^n (E) + \hat{g}^n_0(E)\,
 [ \ln H^n(E+\epsilon)-\ln H^n(E)] / \epsilon} $$
$$ \hat{g}^n_0 (E) = g^n_0(E)\, /\, [g^n(E) + \hat{g}^n_0 (E)]\, ,$$
$$ g^n_0 (E) = H^n (E+\epsilon)\, H^n (E)\, /\,
   [H^n (E+\epsilon) + H^n (E)]\, ,$$
$$ g^{n+1} (E) = g^n(E) + g^n_0(E),\ g^0(E)=0\, .$$ \b

\noindent {\r ANIMATION:} {\b 2d 10-state $80\times 80$ Potts model
(Iterations of 100 sweeps each).}

\newpage

\noindent {\r\large F. Wang and Landau:}
{\b Update are performed with estimators} $g(E)$ {\b of the
density of states} \s
$$ p(E_1\to E_2) = \min \left[ {g(E_1)\over g(E_2)}, 1\right]\ .$$
{\b Each time an energy level is visited, they update the estimator} 
$$ g(E) \to g(E)\,f $$
{\b where, initially,} $g(E)=1$ and $f=f_0=e^1$. {\b Once the
desired energy range is covered, the factor} $f$ {\b is refined:}
$$ f_1=\sqrt{f},\ f_{n+1}=\sqrt{f_{n+1}} $$
{\b until some small value like} $f=e^{-8}=1.00000001$ {\b is
reached. 

Afterwards the usual, MUCA production runs may be carried out.}

\foilhead{\r Example Runs ($2d$ Ising and Potts models)}\vskip -30pt

\b \noindent 
Ising model on a $20\times 20$ lattice: The multicanonical 
recursion is run in the range
$$\s {\tt namin} = 400 \le {\tt iact} \le 800 = {\tt namax}\ . $$
The recursion is terminated after a number of so called tunneling 
events. A {\bf tunneling event} is defined as an updating process 
which finds its way from
$$\s {\tt iact} = {\tt namin} ~~~{\rm to}~~~ {\tt iact} = {\tt namax}
      ~~~{\rm and}~~{\rm back}\ . $$
This notation comes from applications to first order phase transitions. 
An alternative notation for tunneling event is {\bf random walk cycle}.  
For most applications 10 tunneling events lead to acceptable weights.

For an example run of the Ising model we find the requested 10 tunneling 
events after 787 recursions and 64,138 sweeps (assignment {\tt a0501\_01},
see assignment {\tt a0501\_02} for the $2d$ 10-state Potts model).

\newpage
\centerline{\r\Large Performance}

If the multicanonical weighting would remove all relevant {free energy 
barriers}, the behavior of the updating process would become that 
of a free {\bf random walk}.  Therefore, the theoretically optimal 
performance for the second part of the multicanonical simulation is
\begin{equation} \label{muca_RW}
{\s \tau_{\rm tun}\ \sim\ V^2\ .}
\end{equation}
Recent work about first order transitions by Neuhaus and Hager shows 
that that the multicanonical procedure removes only the leading free 
energy barrier, while at least one subleading barrier causes still a 
residual supercritical slowing done. Up to certain medium sized lattices 
the behavior {\s $V^{2+\epsilon}$} gives a rather good effective 
description. For large lattices supercritical slowing down of the form
\begin{equation} \label{muca_supercritical} {\s
\tau_{\rm tun}\ \sim\ \exp \left( + {\rm const}\, L^{d-1} \right)}
\end{equation}
dominates again. The slowing down of the weight recursion with the 
volume size is expected to be even (slightly) worse than that of the 
second part of the simulation. 

\foilhead{\r Re-Weighting to the Canonical Ensemble}\vskip -30pt \b

Given the {multicanonical time series}, where $i=1,\dots,n$ labels 
the generated configurations. The formula 
$$\s \overline{\mathcal{O}} = { \sum_{i=1}^n \mathcal{O}^{(i)}\, \exp 
 \left[ -\beta\,E^{(i)} + b(E^{(i)})\,E^{(i)}-a(E^{(i)}) \right] \over 
 \sum_{i=1}^n \exp \left[ -\beta\,E^{(i)} + 
 b(E^{(i)})\,E^{(i)}-a(E^{(i)}) \right] }\ .$$
replaces the multicanonical weighting of the simulation by the Boltzmann 
factor. The denominator differs from the partition function $Z$ by a 
constant factor which drops out (for discrete systems this simplifies
for functions of the energy using histograms). 
The computer implementation of these equations requires care and a {\r 
Jackknife analysis} with {\r logarithmic coding} relying on the formula
$$\s \ln C = \max \left( \ln A,\ln B \right) + 
   \ln \{ 1 + \exp \left[ - |\ln A - \ln B| \right] \} $$
ought to be used.

\foilhead{\r Energy and Specific Heat Calculation}\vskip -30pt

We are now ready to produce multicanonical data for the energy per spin 
(with jackknife errors) of the $2d$ Ising model on a $20\times 20$ 
lattice and compared them with the exact results of Ferdinand and 
Fisher (assignment
{\tt a0501\_03}):

\begin{figure} \begin{center}
\includegraphics[height=0.64\textheight]{figures/2dI_es.pdf}
\end{center} \end{figure}

\newpage\noindent
The same numerical data allow to calculate the {\bf specific heat} 
defined by
$$\s  C\ =\ {d\,\widehat{E}\over d\,T}\ =\ \beta^2\, \left( \,
      \langle E^2 \rangle - \langle E \rangle^2 \right)\ . $$
\begin{figure} \begin{center}
\includegraphics[height=0.64\textheight]{figures/2dI_C.pdf}
\end{center} \end{figure}

\newpage The energy histogram of the multicanonical simulation 
together its canonically re-weighted descendants at $\beta = 0$, 
$\beta = 0.2$ and $\beta =0.4$ (assignment {\tt a0501\_04}): 
\begin{figure} \begin{center}
\includegraphics[height=0.64\textheight]{figures/2dI_muh.pdf}
\end{center} \end{figure}
The normalization of the multicanonical histogram is adjusted so that
it fits into the same figure with the three re-weighted histograms.

\newpage\noindent Similar data for the $2d$ 10-state Potts model, 
action variable {\tt actm} data (assignment {\tt a0501\_05}):
\begin{figure} \begin{center}
\includegraphics[height=0.74\textheight]{figures/2d10q_act.pdf}
\end{center} \end{figure}

\newpage
The canonically re-weighted histogram at $\beta=0.71$ together with
the multicanonical histogram using suitable normalizations
(the ordinate is on a logarithmic scale):
\begin{figure} \begin{center}
\includegraphics[height=0.74\textheight]{figures/2d10q_muh.pdf}
\end{center} \end{figure}

\newpage \b
The multicanonical method allows to estimate the interface 
tension of the transition by following the minimum to maximum 
ratio over many orders of magnitude {\s [Berg--Neuhaus, Phys. 
Rev. Lett. 68 (1992) 9]:
\vskip -132pt
\begin{figure} \begin{center}
\includegraphics[height=1.10\textheight]{figures/potts2d10q_ln.pdf}
\end{center} \end{figure}
\vfill

\newpage
\begin{figure} \begin{center}
\includegraphics[height=0.84\textheight]{figures/92bn01.pdf}
\end{center} \end{figure}

\newpage
\vskip -132pt
\begin{figure} \begin{center} \vskip -20pt
\includegraphics[width=0.7\textwidth]{figures/color_potts3.jpg}
\end{center} \end{figure}

\newpage
\vskip -32pt
\begin{figure} \begin{center} \vskip -20pt
\includegraphics[width=0.7\textwidth]{figures/color_potts2.jpg}
\end{center} \end{figure}

\newpage
\begin{figure} \begin{center}
\includegraphics[height=0.84\textheight]{figures/92bn04.pdf}
\caption{Interface tension fit (Berg and Neuhaus 1992).  }
\end{center} \end{figure}

\foilhead{\r Free Energy and Entropy Calculation}\vskip -30pt \b

At $\beta=0$ the Potts partition function is $Z=q^N$. Therefore, 
multicanonical simulations allow for proper normalization of the 
partition function, if $\beta=0$ is included in the temperature 
range.
The properly normalized partition function allows to calculate 
the\index{Helmholtz free energy: see free energy}\index{free energy|textbf} 
\index{free energy!canonical}\index{ensemble!canonical!free energy}{\bf 
Helmholtz free energy}
$$\s F\ =\ - \beta^{-1}\, \ln ( Z ) $$
and the\index{entropy|textbf}\index{ensemble!canonical!entropy}
\index{entropy!canonical}{\bf entropy}
$$\s S\ =\ {F - {E}\over T}\ =\ \beta\, (F - {E}) $$
of the canonical ensemble. Here ${E}$ is the expectation 
value of the internal energy and the last equal sign holds because of 
our choice of units for the temperature.
For the $2d$ Ising model as well as for the $2d$ 10-state Potts model, 
we show in the next figure multicanonical estimates of the free energy 
density per site
$$\s f\ =\ F/N\ . $$
\vskip -30pt

\begin{figure} \begin{center}
\includegraphics[height=0.7\textheight]{figures/free_energy.pdf}
\caption{Free energies from multicanonical simulations of the 
Ising and 10-state Potts models on an $20\times 20$ lattice
(assignments {\tt a0501\_03} and {\tt a0501\_05}). The full lines 
are the exact results of Ferdinand and Fischer for the Ising model 
and the asymptotic behavior $\s f_{\rm as} = {2\,d\over q} - 2\,d 
- \beta^{-1}\, {\ln(q)\over N}$ for the 10-state Potts model.}
\end{center} \end{figure}

\foilhead{\r Entropy density $\s s\ =\ S/N\ :$} \vskip -30pt

\begin{figure} \begin{center}
\includegraphics[height=0.67\textheight]{figures/entropy.pdf}
\caption{Entropies from multicanonical simulations of the Ising 
and 10-state Potts models on an $20\times 20$ lattice 
(assignments {\tt a0501\_03} and {\tt a0501\_05}). The full 
line is the exact result of Ferdinand and Fischer for the 
Ising model.}  
\end{center} \end{figure}

\newpage
For the $2d$ Ising model one may also compare directly with the number
of states using an algebraic computational method of Beale. One finds 
no real advantages over using other physical quantities.

\begin{center}
{\Large\bf\r Time series analysis}
\end{center}

Typically, one prefers in continuous systems time series data over 
keeping histograms, because one avoids then dealing with the 
discretization errors.
Even in discrete systems time series data are of importance, as
one often wants to measure more physical quantities than just the 
energy. Then the RAM storage requirements may require to use
a time series instead of histograms. To illustrate this point,
we use the {\r Potts magnetization} (assignmentss {\tt a0501\_08} 
and {\tt a0501\_09}).

For zero magnetic field, $\s H=0$, the expectation value of the Potts 
magnetization on a finite lattice is is simply
$$\s M_{q0} = \langle\,\delta_{q_i,q_0}\,\rangle = {1\over q}\ ,$$
independently of the temperature. For the multicanonical simulation 
it is quite obvious that even at low temperatures each Potts state 
is visited with probability $\s 1/q$. In contrast to this, the 
expectation value of the magnetization squared 
$$\s M_{q0}^2\ =\ q\, \left\langle \left( {1\over N}\,\sum_{i=1}^N 
     \delta_{q_i,q_0} \right)^2 \right\rangle $$
is a non-trivial quantity. At $\s \beta=0$ its $\s N\to\infty$ value 
is $\s M^2_{q0}=q\,(1/q)^2=1/q$, whereas it approaches $\s 1$ for 
$\s \beta\to\infty$. For $\s q=2$ and $\s q=10$ the next figure shows 
our numerical results and we see that the crossover of $\s M^2_{q0}$ 
from $\s 1/q$ to $\s 1$ happens in the neighborhood of the critical 
temperature. A FSS analysis would reveal that, a \r singularity 
develops at $\s \beta_c$, which is in the derivative of $\s M^2_{q0}$ 
for the {second order phase transitions} ($\s q\le 4$) and in 
$\s M^2_{q0}$ itself for the {first order transitions} ($\s q\ge 5$).

\newpage
\begin{figure} \begin{center}
\includegraphics[height=0.7\textheight]{figures/Potts_M2.pdf}
\caption{The Potts magnetization per lattice site squared for the $q=2$ 
and $q=10$ Potts models on an $20\times 20$ lattice (assignments
{\tt a0501\_08} and {\tt a0501\_09}). }
\end{center} \end{figure}

\foilhead{\r\Large\bf Complex Systems with Rugged Free
                       Energy Landscapes } \b
\begin{enumerate}
\item Spin Glasses 

\s (first paper Berg-Celik, Phys.$\!$ Rev.$\!$ Lett.  69 (1992) 2292.) \b

\item Biomolecules 

\s (first paper Hansmann-Okamoto, J.$\!$ Comp.$\!$ Chem. 14 (1993) 1333).  
% Hao and Scheraga).
\end{enumerate}

\noindent {\r\bf Basic Idea:} Bridge free energy barriers through 
          excursions into the disordered phase of the system.

\newpage
\begin{figure} \begin{center} \vskip -20pt
\includegraphics[width=0.7\textwidth]{figures/color_spg.jpg}
\end{center} \end{figure}

\newpage
\begin{figure} \begin{center} \vskip -20pt 
\includegraphics[width=0.8\textwidth]{figures/color_alpha_helix.jpg}
\end{center} \end{figure}

\foilhead{\r\Large\bf Summary}

\begin{itemize}

\item It is a strength of computer simulations that one can generate
      artificial (not realized by nature) ensembles, which enhance the 
      probabilities of rare events on may be interested in, or speed up 
      the dynamics.

\item Nowadays {\r Generalized Ensembles} (umbrella, multicanonical, 1/k, 
      ...) have found many applications. Besides for first order phase 
      transitions in particular for {\r Complex Systems} such as 
      biomolecules, where they accelerate the dynamics.
\end{itemize}

\end{document}
