From d6e77dad1078f8316daeb4a614c4b7875d92bb77 Mon Sep 17 00:00:00 2001 From: Peter Eastman Date: Sun, 19 Jul 2015 10:36:13 -0700 Subject: [PATCH] Created figure showing phases of water, plus a few minor edits --- source/conf.py | 4 +- source/images/water.svg | 2875 +++++++++++++++++++++++++++++++++++ source/index.rst | 2 +- source/phasetransitions.rst | 14 +- 4 files changed, 2888 insertions(+), 7 deletions(-) create mode 100644 source/images/water.svg diff --git a/source/conf.py b/source/conf.py index 7638655..d049401 100644 --- a/source/conf.py +++ b/source/conf.py @@ -51,7 +51,7 @@ # General information about the project. project = u'Introduction to Statistical Mechanics' -copyright = u'2014, Peter Eastman' +copyright = u'2014-2015, Peter Eastman' # The version info for the project you're documenting, acts as replacement for # |version| and |release|, also used in various other places throughout the @@ -117,7 +117,7 @@ # The name for this set of Sphinx documents. If None, it defaults to # " v documentation". -#html_title = None +html_title = project # A shorter title for the navigation bar. Default is the same as html_title. #html_short_title = None diff --git a/source/images/water.svg b/source/images/water.svg new file mode 100644 index 0000000..8f598c0 --- /dev/null +++ b/source/images/water.svg @@ -0,0 +1,2875 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/source/index.rst b/source/index.rst index 4f3110e..0eb33d7 100644 --- a/source/index.rst +++ b/source/index.rst @@ -1,7 +1,7 @@ Introduction to Statistical Mechanics ##################################### -Copyright © 2014 by Peter Eastman +Copyright © 2014-2015 by Peter Eastman This work may be freely distributed under the terms of the `Creative Commons Attribution-NoDerivatives 4.0 International License `_. diff --git a/source/phasetransitions.rst b/source/phasetransitions.rst index a66c8ab..c9c5b9c 100644 --- a/source/phasetransitions.rst +++ b/source/phasetransitions.rst @@ -10,9 +10,15 @@ A Qualitative Description of Phase Transitions ============================================== As a familiar example, consider the behavior of water as you change the temperature. It can exist in liquid, solid -(ice), and gas (steam) phases. Figure ??? shows what each of these phases looks like at the molecular level. Let's +(ice), and gas (steam) phases. :numref:`Figure,water` shows what each of these phases looks like at the molecular level. Let's consider each of them in turn. + +.. figure:: images/water.* + :align: center + + :autonumber:`Figure,water`. Water molecules arranged into solid, liquid, and gas phases + In the solid phase, the molecules form a crystal. They are arranged in a regular pattern that repeats over and over again. Each molecule is tightly held in place by hydrogen bonds to its neighboring molecules. This makes it a very low energy state. To move a molecule from its place in the crystal, you need to break those hydrogen bonds. On the other hand, it @@ -25,7 +31,7 @@ bonds and forming new ones as they move around. Consequently, the liquid phase but also higher entropy. In the gas phase, the molecules are no longer packed together. They spread apart to fill whatever space is available -to them. This means there are very few hydrogens bonds, so the energy is much higher. But because each molecules has +to them. This means there are very few hydrogens bonds, so the energy is much higher. But because each molecule has so much more freedom of movement, the entropy is also much higher. (This description is a bit of a simplification. For example, water actually has at least 15(!) different solid phases, @@ -406,7 +412,7 @@ Mean Field Theory Now consider the case where :math:`J \ne 0`. In principle we just need to write down the free energy and find the state that minimizes it for every value of :math:`H` and :math:`T`. That's easier said than done! In one dimension, the -calculation is reasonably straightforward. In two dimensions it is extreme difficult. In three dimensions, it has +calculation is reasonably straightforward. In two dimensions it is extremely difficult. In three dimensions, it has never been successfully solved. Sadly, this is typical of most real world physics problems: they are too complicated to solve exactly. We need to use a different approach: either make simplifying assumptions that let us solve them approximately, or use a computer to solve them numerically. @@ -503,4 +509,4 @@ paramagnetic. This is qualitatively very different from the predictions of mean In two dimensions where each spin has four neighbors, it does much better. Mean field theory gives a qualitatively correct description of the phase diagram. Its quantitative predictions are not exactly right, but they are still in -the correct general range. In three dimensions they are even closer. +the correct general range. In three dimensions where each spin has six neighbors, they are even closer.