Changeset 4c08e69 in sasmodels for doc/guide/pd/polydispersity.rst

Timestamp:

Jan 9, 2018 2:14:34 PM (6 years ago)

Author:

Paul Kienzle <pkienzle@…>

Branches:

master, core_shell_microgels, magnetic_model, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests

Children:

c11d09f

Parents:

22279a4 (diff), 791281c (diff)
Note: this is a merge changeset, the changes displayed below correspond to the merge itself.
Use the (diff) links above to see all the changes relative to each parent.

Message:

Merge branch 'master' into boltzmann

File:

• doc/guide/pd/polydispersity.rst (modified) (4 diffs)

doc/guide/pd/polydispersity.rst

@@ -42 +42 @@
 calculations are generally more robust with more data points or more angles.
 
-The following five distribution functions are provided:
+The following distribution functions are provided:
 
 *  *Rectangular Distribution*
+*  *Uniform Distribution*
 *  *Gaussian Distribution*
 *  *Lognormal Distribution*
 *  *Schulz Distribution*
 *  *Array Distribution*
+*  *Boltzmann Distribution*
 
 These are all implemented as *number-average* distributions.
@@ -85 +87 @@
     Rectangular distribution.
 
+Uniform Distribution
+^^^^^^^^^^^^^^^^^^^^^^^^
+
+The Uniform Distribution is defined as
+
+    .. math::
+
+        f(x) = \frac{1}{\text{Norm}}
+        \begin{cases}
+          1 & \text{for } |x - \bar x| \leq \sigma \\
+          0 & \text{for } |x - \bar x| > \sigma
+        \end{cases}
+
+    where $\bar x$ is the mean of the distribution, $\sigma$ is the half-width, and
+    *Norm* is a normalization factor which is determined during the numerical
+    calculation.
+
+    Note that the polydispersity is given by
+
+    .. math:: \text{PD} = \sigma / \bar x
+
+    .. figure:: pd_uniform.jpg
+
+        Uniform distribution.
+
 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
 
@@ -183 +210 @@
 ^^^^^^^^^^^^^^^^^^
 
-This user-definable distribution should be given as as a simple ASCII text
+This user-definable distribution should be given as a simple ASCII text
 file where the array is defined by two columns of numbers: $x$ and $f(x)$.
 The $f(x)$ will be normalized to 1 during the computation.
@@ -202 +229 @@
 given for the model will have no affect, and will be ignored when computing
 the average.  This means that any parameter with an array distribution will
-not be fittable.
+not be fitable.
+
+.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
+
+Boltzmann Distribution
+^^^^^^^^^^^^^^^^^^^^^^
+
+The Boltzmann Distribution is defined as
+
+.. math::
+
+    f(x) = \frac{1}{\text{Norm}}
+           \exp\left(-\frac{ | x - \bar x | }{\sigma}\right)
+
+where $\bar x$ is the mean of the distribution and *Norm* is a normalization
+factor which is determined during the numerical calculation.
+The width is defined as
+
+.. math:: \sigma=\frac{k T}{E}
+
+which is the inverse Boltzmann factor,
+where $k$ is the Boltzmann constant, $T$ the temperature in Kelvin and $E$ a
+characteristic energy per particle.
+
+.. figure:: pd_boltzmann.jpg
+
+    Boltzmann distribution.
 
 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ