<s>
In	O
statistics	O
and	O
physics	O
,	O
multicanonical	B-Algorithm
ensemble	I-Algorithm
(	O
also	O
called	O
multicanonical	B-Algorithm
sampling	I-Algorithm
or	O
flat	O
histogram	O
)	O
is	O
a	O
Markov	O
chain	O
Monte	O
Carlo	O
sampling	O
technique	O
that	O
uses	O
the	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
to	O
compute	O
integrals	O
where	O
the	O
integrand	O
has	O
a	O
rough	O
landscape	O
with	O
multiple	O
local	O
minima	O
.	O
</s>
<s>
It	O
samples	O
states	O
according	O
to	O
the	O
inverse	O
of	O
the	O
density	O
of	O
states	O
,	O
which	O
has	O
to	O
be	O
known	O
a	O
priori	O
or	O
be	O
computed	O
using	O
other	O
techniques	O
like	O
the	O
Wang	B-Algorithm
and	I-Algorithm
Landau	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Multicanonical	B-Algorithm
sampling	I-Algorithm
is	O
an	O
important	O
technique	O
for	O
spin	O
systems	O
like	O
the	O
Ising	O
model	O
or	O
spin	O
glasses	O
.	O
</s>
<s>
In	O
systems	O
with	O
a	O
large	O
number	O
of	O
degrees	O
of	O
freedom	O
,	O
like	O
spin	O
systems	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
integration	I-Algorithm
is	O
required	O
.	O
</s>
<s>
In	O
this	O
integration	O
,	O
importance	B-Algorithm
sampling	I-Algorithm
and	O
in	O
particular	O
the	O
Metropolis	B-Algorithm
algorithm	I-Algorithm
,	O
is	O
a	O
very	O
important	O
technique	O
.	O
</s>
<s>
However	O
,	O
the	O
Metropolis	B-Algorithm
algorithm	I-Algorithm
samples	O
states	O
according	O
to	O
where	O
beta	O
is	O
the	O
inverse	O
of	O
the	O
temperature	O
.	O
</s>
<s>
Multicanonical	B-Algorithm
ensemble	I-Algorithm
uses	O
the	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
with	O
a	O
sampling	O
distribution	O
given	O
by	O
the	O
inverse	O
of	O
the	O
density	O
of	O
states	O
of	O
the	O
system	O
,	O
contrary	O
to	O
the	O
sampling	O
distribution	O
of	O
the	O
Metropolis	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Another	O
advantage	O
over	O
the	O
Metropolis	B-Algorithm
algorithm	I-Algorithm
is	O
that	O
the	O
sampling	O
is	O
independent	O
of	O
the	O
temperature	O
of	O
the	O
system	O
,	O
which	O
means	O
that	O
one	O
simulation	O
allows	O
the	O
estimation	O
of	O
thermodynamical	O
variables	O
for	O
all	O
temperatures	O
(	O
thus	O
the	O
name	O
"	O
multicanonical	O
"	O
:	O
several	O
temperatures	O
)	O
.	O
</s>
<s>
The	O
biggest	O
problem	O
in	O
performing	O
a	O
multicanonical	B-Algorithm
ensemble	I-Algorithm
is	O
that	O
the	O
density	O
of	O
states	O
has	O
to	O
be	O
known	O
a	O
priori	O
.	O
</s>
<s>
One	O
important	O
contribution	O
to	O
multicanonical	B-Algorithm
sampling	I-Algorithm
was	O
the	O
Wang	B-Algorithm
and	I-Algorithm
Landau	I-Algorithm
algorithm	I-Algorithm
,	O
which	O
asymptotically	O
converges	O
to	O
a	O
multicanonical	B-Algorithm
ensemble	I-Algorithm
while	O
calculating	O
the	O
density	O
of	O
states	O
during	O
the	O
convergence	O
.	O
</s>
<s>
The	O
multicanonical	B-Algorithm
ensemble	I-Algorithm
is	O
not	O
restricted	O
to	O
physical	O
systems	O
.	O
</s>
<s>
When	O
the	O
system	O
has	O
a	O
large	O
number	O
of	O
degrees	O
of	O
freedom	O
,	O
an	O
analytical	O
expression	O
for	O
is	O
often	O
hard	O
to	O
obtain	O
,	O
and	O
Monte	B-Algorithm
Carlo	I-Algorithm
integration	I-Algorithm
is	O
typically	O
employed	O
in	O
the	O
computation	O
of	O
.	O
</s>
<s>
To	O
improve	O
this	O
convergence	O
,	O
the	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
was	O
proposed	O
.	O
</s>
<s>
Generally	O
,	O
Monte	O
Carlo	O
methods	O
 '	O
idea	O
is	O
to	O
use	O
importance	B-Algorithm
sampling	I-Algorithm
to	O
improve	O
the	O
convergence	O
of	O
the	O
estimator	O
by	O
sampling	O
states	O
according	O
to	O
an	O
arbitrary	O
distribution	O
,	O
and	O
use	O
the	O
appropriate	O
estimator	O
:	O
</s>
<s>
Historically	O
,	O
this	O
occurred	O
because	O
the	O
original	O
idea	O
was	O
to	O
use	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
to	O
compute	O
averages	O
on	O
a	O
system	O
in	O
contact	O
with	O
a	O
heat	O
bath	O
where	O
the	O
weight	O
is	O
given	O
by	O
the	O
Boltzmann	O
factor	O
,	O
.	O
</s>
<s>
This	O
is	O
the	O
basis	O
of	O
the	O
multicanonical	B-Algorithm
ensemble	I-Algorithm
.	O
</s>
<s>
This	O
motivates	O
the	O
name	O
"	O
flat-histogram	B-Algorithm
"	O
.	O
</s>
<s>
Because	O
the	O
histogram	O
is	O
flat	O
on	O
the	O
variable	O
F	O
,	O
a	O
multicanonic	B-Algorithm
ensemble	I-Algorithm
can	O
be	O
seen	O
as	O
a	O
diffusion	O
process	O
(	O
i.e.	O
</s>
