<s>
Parallel	B-Algorithm
tempering	I-Algorithm
in	O
physics	O
and	O
statistics	O
is	O
a	O
computer	O
simulation	O
method	O
typically	O
used	O
to	O
find	O
the	O
lowest	O
energy	O
state	O
of	O
a	O
system	O
of	O
many	O
interacting	O
particles	O
.	O
</s>
<s>
More	O
specifically	O
,	O
parallel	B-Algorithm
tempering	I-Algorithm
(	O
also	O
known	O
as	O
replica	B-Algorithm
exchange	I-Algorithm
MCMC	O
sampling	O
)	O
,	O
is	O
a	O
simulation	O
method	O
aimed	O
at	O
improving	O
the	O
dynamic	O
properties	O
of	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
simulations	O
of	O
physical	O
systems	O
,	O
and	O
of	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
(	O
MCMC	O
)	O
sampling	O
methods	O
more	O
generally	O
.	O
</s>
<s>
The	O
replica	B-Algorithm
exchange	I-Algorithm
method	O
was	O
originally	O
devised	O
by	O
Robert	O
Swendsen	O
and	O
J.S.	O
</s>
<s>
Y	O
.	O
Sugita	O
and	O
Y	O
.	O
Okamoto	O
also	O
formulated	O
a	O
molecular	O
dynamics	O
version	O
of	O
parallel	B-Algorithm
tempering	I-Algorithm
;	O
this	O
is	O
usually	O
known	O
as	O
replica-exchange	O
molecular	O
dynamics	O
or	O
REMD	B-Algorithm
.	O
</s>
<s>
Then	O
,	O
based	O
on	O
the	O
Metropolis	B-Algorithm
criterion	I-Algorithm
one	O
exchanges	O
configurations	O
at	O
different	O
temperatures	O
.	O
</s>
<s>
Typically	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
using	O
a	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
update	O
consists	O
of	O
a	O
single	O
stochastic	O
process	O
that	O
evaluates	O
the	O
energy	O
of	O
the	O
system	O
and	O
accepts/rejects	O
updates	O
based	O
on	O
the	O
temperature	O
T	O
.	O
At	O
high	O
temperatures	O
updates	O
that	O
change	O
the	O
energy	O
of	O
the	O
system	O
are	O
comparatively	O
more	O
probable	O
.	O
</s>
<s>
If	O
we	O
were	O
to	O
run	O
two	O
simulations	O
at	O
temperatures	O
separated	O
by	O
a	O
ΔT	O
,	O
we	O
would	O
find	O
that	O
if	O
ΔT	O
is	O
small	O
enough	O
,	O
then	O
the	O
energy	O
histograms	B-Algorithm
obtained	O
by	O
collecting	O
the	O
values	O
of	O
the	O
energies	O
over	O
a	O
set	O
of	O
Monte	O
Carlo	O
steps	O
N	O
will	O
create	O
two	O
distributions	O
that	O
will	O
somewhat	O
overlap	O
.	O
</s>
<s>
The	O
overlap	O
can	O
be	O
defined	O
by	O
the	O
area	O
of	O
the	O
histograms	B-Algorithm
that	O
falls	O
over	O
the	O
same	O
interval	O
of	O
energy	O
values	O
,	O
normalized	O
by	O
the	O
total	O
number	O
of	O
samples	O
.	O
</s>
<s>
This	O
can	O
be	O
ensured	O
by	O
appropriately	O
choosing	O
regular	O
Monte	O
Carlo	O
updates	O
or	O
parallel	B-Algorithm
tempering	I-Algorithm
updates	O
with	O
probabilities	O
that	O
are	O
independent	O
of	O
the	O
configurations	O
of	O
the	O
two	O
systems	O
or	O
of	O
the	O
Monte	O
Carlo	O
step	O
.	O
</s>
<s>
By	O
a	O
careful	O
choice	O
of	O
temperatures	O
and	O
number	O
of	O
systems	O
one	O
can	O
achieve	O
an	O
improvement	O
in	O
the	O
mixing	O
properties	O
of	O
a	O
set	O
of	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulations	I-Algorithm
that	O
exceeds	O
the	O
extra	O
computational	O
cost	O
of	O
running	O
parallel	O
simulations	O
.	O
</s>
<s>
Set	O
up	O
is	O
important	O
as	O
there	O
must	O
be	O
a	O
practical	O
histogram	B-Algorithm
overlap	O
to	O
achieve	O
a	O
reasonable	O
probability	O
of	O
lateral	O
moves	O
.	O
</s>
<s>
The	O
parallel	B-Algorithm
tempering	I-Algorithm
method	O
can	O
be	O
used	O
as	O
a	O
super	O
simulated	B-Algorithm
annealing	I-Algorithm
that	O
does	O
not	O
need	O
restart	O
,	O
since	O
a	O
system	O
at	O
high	O
temperature	O
can	O
feed	O
new	O
local	O
optimizers	O
to	O
a	O
system	O
at	O
low	O
temperature	O
,	O
allowing	O
tunneling	O
between	O
metastable	O
states	O
and	O
improving	O
convergence	O
to	O
a	O
global	O
optimum	O
.	O
</s>
