Add a section at the end of the manual about stochastic forces, integrators, and

random numbers.  Add 2 citations about Langevin synchronization.  Convert the
references.bib file to UNIX line endings.

docs-source/usersguide/theory.rst

@@ -1379,3 +1379,65 @@ specific types of rules.  They are:
 .. math::
    \mathbf{r}=\mathbf{o}+p_1\mathbf{\hat{x}}+p_2\mathbf{\hat{y}}+p_3\mathbf{\hat{z}}
 ..
+
+Random Numbers with Stochastic Integrators and Forces
+*****************************************************
+
+OpenMM includes many stochastic integrators and forces that make extensive use
+of random numbers. It is impossible to generate truly random numbers on a
+computer like you would with a dice roll or coin flip in real life---instead
+programs rely on pseudo-random number generators (PRNGs) that take some sort of
+initial "seed" value and steps through a sequence of seemingly random numbers.
+
+The exact implementation of the PRNGs is not important (in fact, each platform
+may have its own PRNG whose performance is optimized for that hardware).  What
+*is* important, however, is that the PRNG must generate a uniform distribution
+of random numbers between 0 and 1. Random numbers drawn from this distribution
+can be manipulated to yield random integers in a desired range or even a random
+number from a different type of probability distribution function (e.g., a
+normal distribution).
+
+What this means is that the random numbers used by integrators and forces within
+OpenMM cannot have any discernible pattern to them.  Patterns can be induced in
+PRNGs in two principle ways:
+    1. The PRNG uses a bad algorithm with a short period.
+    2. Two PRNGs are started using the same seed
+
+All PRNG algorithms in common use are periodic---that is their sequence of
+random numbers repeats after a given *period*, defined by the number of "unique"
+random numbers in the repeating pattern.  As long as this period is longer than
+the total number of random numbers your application requires (preferably by
+orders of magnitude), the first problem described above is avoided. All PRNGs
+employed by OpenMM have periods far longer than any current simulation can cycle
+through.
+
+Point two is far more common in biomolecular simulation, and can result in very
+strange artifacts that may be difficult to detect. For example, with Langevin
+dynamics, two simulations that use the same sequence of random numbers appear to
+synchronize in their global movements.\ :cite:`Uberuaga2004`\
+:cite:`Sindhikara2009` It is therefore very important that the stochastic forces
+and integrators in OpenMM generate unique sequences of pseudo-random numbers not
+only within a single simulation, but between two different simulations of the
+same system as well (including any restarts of previous simulations).
+
+Every stochastic force and integrator that does (or could) make use of random
+numbers has two instance methods attached to it: ``getRandomNumberSeed()`` and
+``setRandomNumberSeed(int seed)``. If you set a unique random seed for two
+different simulations (or different forces/integrators if applicable), OpenMM
+guarantees that the generated sequences of random numbers will be different (by
+contrast, no guarantee is made that the same seed will result in identical
+random number sequences).
+
+Since breaking simulations up into pieces and/or running multiple replicates of
+a system to obtain more complete statistics is common practice, a new strategy
+has been employed for OpenMM versions 6.3 and later with the aim of trying to
+ensure that each simulation will be started with a unique random seed. A random
+seed value of 0 (the default) will cause a unique random seed to be generated
+when a new ``Context`` is instantiated.
+
+Prior to the introduction of this feature, deserializing a serialized
+``System`` XML file would result in each stochastic force or integrator being
+assigned the same random seed as the original instance that was serialized. If
+you use a ``System`` XML file generated by a version of OpenMM older than 6.3
+to start a new simulation, you should manually set the random number seed of
+each stochastic force or integrator to 0 (or another unique value).