├── docs ├── .nojekyll ├── objects.inv ├── _static │ ├── up.png │ ├── down.png │ ├── file.png │ ├── minus.png │ ├── plus.png │ ├── comment.png │ ├── footerbg.png │ ├── headerbg.png │ ├── middlebg.png │ ├── up-pressed.png │ ├── ajax-loader.gif │ ├── comment-close.png │ ├── dialog-note.png │ ├── dialog-todo.png │ ├── dialog-topic.png │ ├── down-pressed.png │ ├── transparent.gif │ ├── comment-bright.png │ ├── dialog-seealso.png │ ├── dialog-warning.png │ ├── ie6.css │ ├── pygments.css │ ├── epub.css │ ├── pyramid.css │ ├── doctools.js │ ├── water_phase_diagram.svg │ ├── underscore.js │ └── basic.css ├── _sources │ ├── index.txt │ ├── introduction.txt │ └── largenumbers.txt ├── genindex.html ├── search.html ├── introduction.html ├── _images │ ├── oscillator_trajectory.svg │ ├── water_phase_diagram.svg │ └── ising_phase_diagram.svg └── index.html ├── .gitignore ├── source ├── images │ ├── water.pdf │ ├── binomial.pdf │ ├── gaussian.pdf │ ├── ising_model.pdf │ ├── mean_field.pdf │ ├── phase_space.pdf │ ├── ising_phase_diagram.pdf │ ├── water_phase_diagram.pdf │ ├── oscillator_trajectory.pdf │ ├── oscillator_trajectory.svg │ ├── water_phase_diagram.svg │ └── ising_phase_diagram.svg ├── index.rst ├── numsec.py ├── autonumber.py ├── introduction.rst ├── conf.py └── largenumbers.rst ├── LICENSE.md ├── README.md ├── Makefile └── make.bat /docs/.nojekyll: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | build/* 2 | **/__pycache__/* 3 | -------------------------------------------------------------------------------- /docs/objects.inv: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/peastman/statmech/HEAD/docs/objects.inv 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https://raw.githubusercontent.com/peastman/statmech/HEAD/source/images/water_phase_diagram.pdf -------------------------------------------------------------------------------- /source/images/oscillator_trajectory.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/peastman/statmech/HEAD/source/images/oscillator_trajectory.pdf -------------------------------------------------------------------------------- /LICENSE.md: -------------------------------------------------------------------------------- 1 | This work may be freely distributed under the terms of the 2 | [Creative Commons Attribution-NoDerivatives 4.0 International License](http://creativecommons.org/licenses/by-nd/4.0). 3 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | Introduction to Statistical Mechanics 2 | ===================================== 3 | 4 | This is an introductory textbook on statistical mechanics. It is suitable for an undergraduate physics major studying 5 | the subject for the first time. The only required prerequisites are multivariable calculus and classical mechanics. 6 | You can read it online at https://peastman.github.io/statmech/. 7 | 8 | It is written in [Sphinx](http://sphinx-doc.org) format. To build it as HTML, type 9 | 10 | make html 11 | 12 | To build it as a PDF, type 13 | 14 | make latexpdf 15 | 16 | The generated content will be in build/html or build/latex, respectively. 17 | -------------------------------------------------------------------------------- /source/index.rst: -------------------------------------------------------------------------------- 1 | ##################################### 2 | Introduction to Statistical Mechanics 3 | ##################################### 4 | 5 | Copyright © 2014-2015 by Peter Eastman 6 | 7 | This work may be freely distributed under the terms of the 8 | `Creative Commons Attribution-NoDerivatives 4.0 International License `_. 9 | The source can be found at https://github.com/peastman/statmech. 10 | 11 | .. toctree:: 12 | :numbered: 13 | :maxdepth: 2 14 | 15 | introduction.rst 16 | statisticaldescription.rst 17 | largenumbers.rst 18 | interpretation.rst 19 | thermodynamics.rst 20 | phasetransitions.rst 21 | friction.rst 22 | phasespace.rst 23 | -------------------------------------------------------------------------------- /docs/_sources/index.txt: -------------------------------------------------------------------------------- 1 | ##################################### 2 | Introduction to Statistical Mechanics 3 | ##################################### 4 | 5 | Copyright © 2014-2015 by Peter Eastman 6 | 7 | This work may be freely distributed under the terms of the 8 | `Creative Commons Attribution-NoDerivatives 4.0 International License `_. 9 | The source can be found at https://github.com/peastman/statmech. 10 | 11 | .. toctree:: 12 | :numbered: 13 | :maxdepth: 2 14 | 15 | introduction.rst 16 | statisticaldescription.rst 17 | largenumbers.rst 18 | interpretation.rst 19 | thermodynamics.rst 20 | phasetransitions.rst 21 | friction.rst 22 | phasespace.rst 23 | -------------------------------------------------------------------------------- /docs/_static/ie6.css: -------------------------------------------------------------------------------- 1 | * html img, 2 | * html .png{position:relative;behavior:expression((this.runtimeStyle.behavior="none")&&(this.pngSet?this.pngSet=true:(this.nodeName == "IMG" && this.src.toLowerCase().indexOf('.png')>-1?(this.runtimeStyle.backgroundImage = "none", 3 | this.runtimeStyle.filter = "progid:DXImageTransform.Microsoft.AlphaImageLoader(src='" + this.src + "',sizingMethod='image')", 4 | this.src = "_static/transparent.gif"):(this.origBg = this.origBg? this.origBg :this.currentStyle.backgroundImage.toString().replace('url("','').replace('")',''), 5 | this.runtimeStyle.filter = "progid:DXImageTransform.Microsoft.AlphaImageLoader(src='" + this.origBg + "',sizingMethod='crop')", 6 | this.runtimeStyle.backgroundImage = "none")),this.pngSet=true) 7 | );} 8 | -------------------------------------------------------------------------------- /source/numsec.py: -------------------------------------------------------------------------------- 1 | """ 2 | Changes section references to be the section number 3 | instead of the title of the section. 4 | """ 5 | 6 | from docutils import nodes 7 | import sphinx.domains.std 8 | 9 | class CustomStandardDomain(sphinx.domains.std.StandardDomain): 10 | 11 | def __init__(self, env): 12 | env.settings['footnote_references'] = 'superscript' 13 | sphinx.domains.std.StandardDomain.__init__(self, env) 14 | 15 | def resolve_xref(self, env, fromdocname, builder, 16 | typ, target, node, contnode): 17 | res = super(CustomStandardDomain, self).resolve_xref(env, fromdocname, builder, 18 | typ, target, node, contnode) 19 | 20 | if res is None: 21 | return res 22 | 23 | if typ == 'ref' and not node['refexplicit']: 24 | docname, labelid, sectname = self.data['labels'].get(target, ('','','')) 25 | res['refdocname'] = docname 26 | 27 | return res 28 | 29 | def doctree_resolved(app, doctree, docname): 30 | secnums = app.builder.env.toc_secnumbers 31 | for node in doctree.traverse(nodes.reference): 32 | if 'refdocname' in node: 33 | refdocname = node['refdocname'] 34 | if refdocname in secnums: 35 | secnum = secnums[refdocname] 36 | emphnode = node.children[0] 37 | textnode = emphnode.children[0] 38 | 39 | toclist = app.builder.env.tocs[refdocname] 40 | anchorname = None 41 | for refnode in toclist.traverse(nodes.reference): 42 | if refnode.astext() == textnode.astext(): 43 | anchorname = refnode['anchorname'] 44 | if anchorname is None: 45 | continue 46 | linktext = '.'.join(map(str, secnum[anchorname])) 47 | node.replace(emphnode, nodes.Text(linktext)) 48 | 49 | def setup(app): 50 | app.override_domain(CustomStandardDomain) 51 | app.connect('doctree-resolved', doctree_resolved) 52 | 53 | -------------------------------------------------------------------------------- /source/autonumber.py: -------------------------------------------------------------------------------- 1 | from docutils.parsers.rst import roles 2 | from docutils.nodes import Text, reference, section, label 3 | from sphinx.roles import XRefRole 4 | 5 | class autonumber(label): 6 | pass 7 | 8 | class autonumber_ref(reference): 9 | pass 10 | 11 | def autonumber_role(name, rawtext, text, lineno, inliner, options={}, content=[]): 12 | return ([autonumber(text=text)], []) 13 | 14 | def doctree_resolved(app, doctree, docname): 15 | index = {}; 16 | refTable = {} 17 | if app.config.autonumber_by_chapter: 18 | # Record the number of each chapter 19 | 20 | env = app.builder.env 21 | sectionNumbers = {} 22 | for doc in env.toc_secnumbers: 23 | sections = env.toc_secnumbers[doc] 24 | for sectionId in sections: 25 | sectionNumbers[sectionId[1:]] = sections[sectionId] 26 | lastChapter = -1 27 | 28 | # Assign numbers to all the autonumbered objects. 29 | 30 | for node in doctree.traverse(autonumber): 31 | category = node.astext().split(',')[0] 32 | if category in index: 33 | nextNumber = index[category]+1 34 | else: 35 | nextNumber = 1 36 | if app.config.autonumber_by_chapter: 37 | parent = node.parent 38 | chapter = None 39 | while chapter is None: 40 | if isinstance(parent, section): 41 | chapter = parent 42 | parent = parent.parent 43 | chapter = sectionNumbers[chapter.attributes['ids'][0]][0] 44 | if chapter != lastChapter: 45 | index = {} 46 | nextNumber = 1 47 | newNode = Text('%s %d-%d' % (category, chapter, nextNumber)) 48 | lastChapter = chapter 49 | else: 50 | newNode = Text('%s %d' % (category, nextNumber)) 51 | index[category] = nextNumber 52 | refTable[node.astext()] = newNode 53 | node.parent.replace(node, newNode) 54 | 55 | # Replace references with the name of the referenced object 56 | 57 | for ref_info in doctree.traverse(autonumber_ref): 58 | target = ref_info['reftarget'] 59 | if target not in refTable: 60 | raise ValueError('Unknown target for autonumber reference: '+target) 61 | ref_info.replace_self(Text(refTable[target].astext())) 62 | 63 | def setup(app): 64 | app.add_config_value('autonumber_by_chapter', True, False) 65 | roles.register_local_role('autonumber', autonumber_role) 66 | app.add_node(autonumber) 67 | app.add_node(autonumber_ref) 68 | app.add_role('autonumref', XRefRole(nodeclass=autonumber_ref)) 69 | app.connect('doctree-resolved', doctree_resolved) 70 | 71 | -------------------------------------------------------------------------------- /source/introduction.rst: -------------------------------------------------------------------------------- 1 | Introduction 2 | ############ 3 | 4 | Depending on how you look at it, statistical mechanics is either the least fundamental or most fundamental of all fields 5 | of physics. That is because it is not really science at all. It is pure mathematics. 6 | 7 | In other subjects, you learn about "natural laws": Newton's second law, Maxwell's equations, Schrödinger's equation, 8 | etc. These laws are not derived from anything else. They were discovered experimentally and then assumed to reflect 9 | fundamental aspects of reality. But statistical mechanics does not involve any natural laws of this sort. Instead, it 10 | is a set of techniques that can be applied to nearly any physical system, no matter what laws that system obeys. That 11 | is why I call it the most fundamental field of physics. New theories may replace old ones, and natural laws may turn 12 | out to be merely approximations to deeper laws. But statistical mechanics remains valid through it all, and whatever 13 | new laws are discovered, it will almost certainly work just as well with them as it did with the old ones. 14 | 15 | Statistical mechanics was developed in the second half of the 19th century. It was primarily the work of Ludwig 16 | Boltzmann, who personally published more than a hundred papers on the subject during his lifetime. Other scientists 17 | also contributed to it, of course, especially James Clerk Maxwell (the same one Maxwell's equations are named after) and 18 | Josiah Willard Gibbs. 19 | 20 | It grew out of thermodynamics, an earlier theory that described the behavior of a mysterious substance called "heat" or 21 | "caloric". Thermodynamics was a physical theory of the more conventional sort. It involved natural laws discovered by 22 | experiment, and made no claims about why those laws happened to hold. Caloric was supposed to be a substance much like 23 | other forms of matter. But this view turned out to be incorrect. Heat is actually an emergent phenomenon: a 24 | mathematical quantity defined in terms of a more detailed theory (the movement of individual atoms). The "laws" of 25 | thermodynamics can be derived from that deeper theory by applying statistical techniques. If you ignore the details of 26 | how each atom is moving, you find they collectively behave in a way that resembles a continuous fluid. That is what 27 | statistical mechanics is all about: deriving high level descriptions by starting from lower level ones and then 28 | averaging out lots of details. 29 | 30 | Is it possible other theories could be explained in the same way? That is an open question, and a fascinating one. 31 | Many physicists suspect gravity is an emergent phenomenon, that it arises from the collective behavior of some deeper 32 | degrees of freedom. Statistical interpretations have also been proposed for quantum mechanics. These are all very 33 | speculative, of course, and they could easily turn out to be wrong. But they also could easily turn out to be right. 34 | Based on what we know today, it is entirely possible that the very structure of spacetime is a consequence of 35 | statistical mechanics. 36 | -------------------------------------------------------------------------------- /docs/_sources/introduction.txt: -------------------------------------------------------------------------------- 1 | Introduction 2 | ############ 3 | 4 | Depending on how you look at it, statistical mechanics is either the least fundamental or most fundamental of all fields 5 | of physics. That is because it is not really science at all. It is pure mathematics. 6 | 7 | In other subjects, you learn about "natural laws": Newton's second law, Maxwell's equations, Schrödinger's equation, 8 | etc. These laws are not derived from anything else. They were discovered experimentally and then assumed to reflect 9 | fundamental aspects of reality. But statistical mechanics does not involve any natural laws of this sort. Instead, it 10 | is a set of techniques that can be applied to nearly any physical system, no matter what laws that system obeys. That 11 | is why I call it the most fundamental field of physics. New theories may replace old ones, and natural laws may turn 12 | out to be merely approximations to deeper laws. But statistical mechanics remains valid through it all, and whatever 13 | new laws are discovered, it will almost certainly work just as well with them as it did with the old ones. 14 | 15 | Statistical mechanics was developed in the second half of the 19th century. It was primarily the work of Ludwig 16 | Boltzmann, who personally published more than a hundred papers on the subject during his lifetime. Other scientists 17 | also contributed to it, of course, especially James Clerk Maxwell (the same one Maxwell's equations are named after) and 18 | Josiah Willard Gibbs. 19 | 20 | It grew out of thermodynamics, an earlier theory that described the behavior of a mysterious substance called "heat" or 21 | "caloric". Thermodynamics was a physical theory of the more conventional sort. It involved natural laws discovered by 22 | experiment, and made no claims about why those laws happened to hold. Caloric was supposed to be a substance much like 23 | other forms of matter. But this view turned out to be incorrect. Heat is actually an emergent phenomenon: a 24 | mathematical quantity defined in terms of a more detailed theory (the movement of individual atoms). The "laws" of 25 | thermodynamics can be derived from that deeper theory by applying statistical techniques. If you ignore the details of 26 | how each atom is moving, you find they collectively behave in a way that resembles a continuous fluid. That is what 27 | statistical mechanics is all about: deriving high level descriptions by starting from lower level ones and then 28 | averaging out lots of details. 29 | 30 | Is it possible other theories could be explained in the same way? That is an open question, and a fascinating one. 31 | Many physicists suspect gravity is an emergent phenomenon, that it arises from the collective behavior of some deeper 32 | degrees of freedom. Statistical interpretations have also been proposed for quantum mechanics. These are all very 33 | speculative, of course, and they could easily turn out to be wrong. But they also could easily turn out to be right. 34 | Based on what we know today, it is entirely possible that the very structure of spacetime is a consequence of 35 | statistical mechanics. 36 | -------------------------------------------------------------------------------- /docs/genindex.html: -------------------------------------------------------------------------------- 1 | 2 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Index — Introduction to Statistical Mechanics 11 | 12 | 13 | 14 | 15 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 34 | 35 | 36 | 37 | 38 | 47 | 48 |
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86 | 95 | 99 | 100 | -------------------------------------------------------------------------------- /docs/_static/pygments.css: -------------------------------------------------------------------------------- 1 | .highlight .hll { background-color: #ffffcc } 2 | .highlight { background: #eeffcc; } 3 | .highlight .c { color: #408090; font-style: italic } /* Comment */ 4 | .highlight .err { border: 1px solid #FF0000 } /* Error */ 5 | .highlight .k { color: #007020; font-weight: bold } /* Keyword */ 6 | .highlight .o { color: #666666 } /* Operator */ 7 | .highlight .cm { color: #408090; font-style: italic } /* Comment.Multiline */ 8 | .highlight .cp { color: #007020 } /* Comment.Preproc */ 9 | .highlight .c1 { color: #408090; font-style: italic } /* Comment.Single */ 10 | .highlight .cs { color: #408090; background-color: #fff0f0 } /* Comment.Special */ 11 | .highlight .gd { color: #A00000 } /* Generic.Deleted */ 12 | .highlight .ge { font-style: italic } /* Generic.Emph */ 13 | .highlight .gr { color: #FF0000 } /* Generic.Error */ 14 | .highlight .gh { color: #000080; font-weight: bold } /* Generic.Heading */ 15 | .highlight .gi { color: #00A000 } /* Generic.Inserted */ 16 | .highlight .go { color: #333333 } /* Generic.Output */ 17 | .highlight .gp { color: #c65d09; font-weight: bold } /* Generic.Prompt */ 18 | .highlight .gs { font-weight: bold } /* Generic.Strong */ 19 | .highlight .gu { color: #800080; font-weight: bold } /* Generic.Subheading */ 20 | .highlight .gt { color: #0044DD } /* Generic.Traceback */ 21 | .highlight .kc { color: #007020; font-weight: bold } /* Keyword.Constant */ 22 | .highlight .kd { color: #007020; font-weight: bold } /* Keyword.Declaration */ 23 | .highlight .kn { color: #007020; font-weight: bold } /* Keyword.Namespace */ 24 | .highlight .kp { color: #007020 } /* Keyword.Pseudo */ 25 | .highlight .kr { color: #007020; font-weight: bold } /* Keyword.Reserved */ 26 | .highlight .kt { color: #902000 } /* Keyword.Type */ 27 | .highlight .m { color: #208050 } /* Literal.Number */ 28 | .highlight .s { color: #4070a0 } /* Literal.String */ 29 | .highlight .na { color: #4070a0 } /* Name.Attribute */ 30 | .highlight .nb { color: #007020 } /* Name.Builtin */ 31 | .highlight .nc { color: #0e84b5; font-weight: bold } /* Name.Class */ 32 | .highlight .no { color: #60add5 } /* Name.Constant */ 33 | .highlight .nd { color: #555555; font-weight: bold } /* Name.Decorator */ 34 | .highlight .ni { color: #d55537; font-weight: bold } /* Name.Entity */ 35 | .highlight .ne { color: #007020 } /* Name.Exception */ 36 | .highlight .nf { color: #06287e } /* Name.Function */ 37 | .highlight .nl { color: #002070; font-weight: bold } /* Name.Label */ 38 | .highlight .nn { color: #0e84b5; font-weight: bold } /* Name.Namespace */ 39 | .highlight .nt { color: #062873; font-weight: bold } /* Name.Tag */ 40 | .highlight .nv { color: #bb60d5 } /* Name.Variable */ 41 | .highlight .ow { color: #007020; font-weight: bold } /* Operator.Word */ 42 | .highlight .w { color: #bbbbbb } /* Text.Whitespace */ 43 | .highlight .mb { color: #208050 } /* Literal.Number.Bin */ 44 | .highlight .mf { color: #208050 } /* Literal.Number.Float */ 45 | .highlight .mh { color: #208050 } /* Literal.Number.Hex */ 46 | .highlight .mi { color: #208050 } /* Literal.Number.Integer */ 47 | .highlight .mo { color: #208050 } /* Literal.Number.Oct */ 48 | .highlight .sb { color: #4070a0 } /* Literal.String.Backtick */ 49 | .highlight .sc { color: #4070a0 } /* Literal.String.Char */ 50 | .highlight .sd { color: #4070a0; font-style: italic } /* Literal.String.Doc */ 51 | .highlight .s2 { color: #4070a0 } /* Literal.String.Double */ 52 | .highlight .se { color: #4070a0; font-weight: bold } /* Literal.String.Escape */ 53 | .highlight .sh { color: #4070a0 } /* Literal.String.Heredoc */ 54 | .highlight .si { color: #70a0d0; font-style: italic } /* Literal.String.Interpol */ 55 | .highlight .sx { color: #c65d09 } /* Literal.String.Other */ 56 | .highlight .sr { color: #235388 } /* Literal.String.Regex */ 57 | .highlight .s1 { color: #4070a0 } /* Literal.String.Single */ 58 | .highlight .ss { color: #517918 } /* Literal.String.Symbol */ 59 | .highlight .bp { color: #007020 } /* Name.Builtin.Pseudo */ 60 | .highlight .vc { color: #bb60d5 } /* Name.Variable.Class */ 61 | .highlight .vg { color: #bb60d5 } /* Name.Variable.Global */ 62 | .highlight .vi { color: #bb60d5 } /* Name.Variable.Instance */ 63 | .highlight .il { color: #208050 } /* Literal.Number.Integer.Long */ -------------------------------------------------------------------------------- /docs/search.html: -------------------------------------------------------------------------------- 1 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Search — Introduction to Statistical Mechanics 10 | 11 | 12 | 13 | 14 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 32 | 33 | 34 | 35 | 36 | 37 | 40 | 41 | 42 | 43 | 44 | 45 | 54 | 55 |
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93 | 102 | 106 | 107 | -------------------------------------------------------------------------------- /docs/_static/epub.css: -------------------------------------------------------------------------------- 1 | /* 2 | * default.css_t 3 | * ~~~~~~~~~~~~~ 4 | * 5 | * Sphinx stylesheet -- default theme. 6 | * 7 | * :copyright: Copyright 2007-2015 by the Sphinx team, see AUTHORS. 8 | * :license: BSD, see LICENSE for details. 9 | * 10 | */ 11 | 12 | @import url("basic.css"); 13 | 14 | /* -- page layout ----------------------------------------------------------- */ 15 | 16 | body { 17 | font-family: {{ theme_bodyfont }}; 18 | font-size: 100%; 19 | background-color: {{ theme_footerbgcolor }}; 20 | color: #000; 21 | margin: 0; 22 | padding: 0; 23 | } 24 | 25 | div.document { 26 | background-color: {{ theme_sidebarbgcolor }}; 27 | } 28 | 29 | div.documentwrapper { 30 | float: left; 31 | width: 100%; 32 | } 33 | 34 | div.bodywrapper { 35 | margin: 0 0 0 230px; 36 | } 37 | 38 | div.body { 39 | background-color: {{ theme_bgcolor }}; 40 | color: {{ theme_textcolor }}; 41 | padding: 0 20px 30px 20px; 42 | } 43 | 44 | {%- if theme_rightsidebar|tobool %} 45 | div.bodywrapper { 46 | margin: 0 230px 0 0; 47 | } 48 | {%- endif %} 49 | 50 | div.footer { 51 | color: {{ theme_footertextcolor }}; 52 | width: 100%; 53 | padding: 9px 0 9px 0; 54 | text-align: center; 55 | font-size: 75%; 56 | } 57 | 58 | div.footer a { 59 | color: {{ theme_footertextcolor }}; 60 | text-decoration: underline; 61 | } 62 | 63 | div.related { 64 | background-color: {{ theme_relbarbgcolor }}; 65 | line-height: 30px; 66 | color: {{ theme_relbartextcolor }}; 67 | } 68 | 69 | div.related a { 70 | color: {{ theme_relbarlinkcolor }}; 71 | } 72 | 73 | div.sphinxsidebar { 74 | {%- if theme_stickysidebar|tobool %} 75 | top: 30px; 76 | bottom: 0; 77 | margin: 0; 78 | position: fixed; 79 | overflow: auto; 80 | height: auto; 81 | {%- endif %} 82 | {%- if theme_rightsidebar|tobool %} 83 | float: right; 84 | {%- if theme_stickysidebar|tobool %} 85 | right: 0; 86 | {%- endif %} 87 | {%- endif %} 88 | } 89 | 90 | {%- if theme_stickysidebar|tobool %} 91 | /* this is nice, but it it leads to hidden headings when jumping 92 | to an anchor */ 93 | /* 94 | div.related { 95 | position: fixed; 96 | } 97 | 98 | div.documentwrapper { 99 | margin-top: 30px; 100 | } 101 | */ 102 | {%- endif %} 103 | 104 | div.sphinxsidebar h3 { 105 | font-family: {{ theme_headfont }}; 106 | color: {{ theme_sidebartextcolor }}; 107 | font-size: 1.4em; 108 | font-weight: normal; 109 | margin: 0; 110 | padding: 0; 111 | } 112 | 113 | div.sphinxsidebar h3 a { 114 | color: {{ theme_sidebartextcolor }}; 115 | } 116 | 117 | div.sphinxsidebar h4 { 118 | font-family: {{ theme_headfont }}; 119 | color: {{ theme_sidebartextcolor }}; 120 | font-size: 1.3em; 121 | font-weight: normal; 122 | margin: 5px 0 0 0; 123 | padding: 0; 124 | } 125 | 126 | div.sphinxsidebar p { 127 | color: {{ theme_sidebartextcolor }}; 128 | } 129 | 130 | div.sphinxsidebar p.topless { 131 | margin: 5px 10px 10px 10px; 132 | } 133 | 134 | div.sphinxsidebar ul { 135 | margin: 10px; 136 | padding: 0; 137 | color: {{ theme_sidebartextcolor }}; 138 | } 139 | 140 | div.sphinxsidebar a { 141 | color: {{ theme_sidebarlinkcolor }}; 142 | } 143 | 144 | div.sphinxsidebar input { 145 | border: 1px solid {{ theme_sidebarlinkcolor }}; 146 | font-family: sans-serif; 147 | font-size: 1em; 148 | } 149 | 150 | {% if theme_collapsiblesidebar|tobool %} 151 | /* for collapsible sidebar */ 152 | div#sidebarbutton { 153 | background-color: {{ theme_sidebarbtncolor }}; 154 | } 155 | {% endif %} 156 | 157 | /* -- hyperlink styles ------------------------------------------------------ */ 158 | 159 | a { 160 | color: {{ theme_linkcolor }}; 161 | text-decoration: none; 162 | } 163 | 164 | a:visited { 165 | color: {{ theme_visitedlinkcolor }}; 166 | text-decoration: none; 167 | } 168 | 169 | a:hover { 170 | text-decoration: underline; 171 | } 172 | 173 | {% if theme_externalrefs|tobool %} 174 | a.external { 175 | text-decoration: none; 176 | border-bottom: 1px dashed {{ theme_linkcolor }}; 177 | } 178 | 179 | a.external:hover { 180 | text-decoration: none; 181 | border-bottom: none; 182 | } 183 | 184 | a.external:visited { 185 | text-decoration: none; 186 | border-bottom: 1px dashed {{ theme_visitedlinkcolor }}; 187 | } 188 | {% endif %} 189 | 190 | /* -- body styles ----------------------------------------------------------- */ 191 | 192 | div.body h1, 193 | div.body h2, 194 | div.body h3, 195 | div.body h4, 196 | div.body h5, 197 | div.body h6 { 198 | font-family: {{ theme_headfont }}; 199 | background-color: {{ theme_headbgcolor }}; 200 | font-weight: normal; 201 | color: {{ theme_headtextcolor }}; 202 | border-bottom: 1px solid #ccc; 203 | margin: 20px -20px 10px -20px; 204 | padding: 3px 0 3px 10px; 205 | } 206 | 207 | div.body h1 { margin-top: 0; font-size: 200%; } 208 | div.body h2 { font-size: 160%; } 209 | div.body h3 { font-size: 140%; } 210 | div.body h4 { font-size: 120%; } 211 | div.body h5 { font-size: 110%; } 212 | div.body h6 { font-size: 100%; } 213 | 214 | a.headerlink { 215 | color: {{ theme_headlinkcolor }}; 216 | font-size: 0.8em; 217 | padding: 0 4px 0 4px; 218 | text-decoration: none; 219 | } 220 | 221 | a.headerlink:hover { 222 | background-color: {{ theme_headlinkcolor }}; 223 | color: white; 224 | } 225 | 226 | div.body p, div.body dd, div.body li { 227 | text-align: justify; 228 | line-height: 130%; 229 | } 230 | 231 | div.admonition p.admonition-title + p { 232 | display: inline; 233 | } 234 | 235 | div.admonition p { 236 | margin-bottom: 5px; 237 | } 238 | 239 | div.admonition pre { 240 | margin-bottom: 5px; 241 | } 242 | 243 | div.admonition ul, div.admonition ol { 244 | margin-bottom: 5px; 245 | } 246 | 247 | div.note { 248 | background-color: #eee; 249 | border: 1px solid #ccc; 250 | } 251 | 252 | div.seealso { 253 | background-color: #ffc; 254 | border: 1px solid #ff6; 255 | } 256 | 257 | div.topic { 258 | background-color: #eee; 259 | } 260 | 261 | div.warning { 262 | background-color: #ffe4e4; 263 | border: 1px solid #f66; 264 | } 265 | 266 | p.admonition-title { 267 | display: inline; 268 | } 269 | 270 | p.admonition-title:after { 271 | content: ":"; 272 | } 273 | 274 | pre { 275 | padding: 5px; 276 | background-color: {{ theme_codebgcolor }}; 277 | color: {{ theme_codetextcolor }}; 278 | line-height: 120%; 279 | border: 1px solid #ac9; 280 | border-left: none; 281 | border-right: none; 282 | } 283 | 284 | code { 285 | background-color: #ecf0f3; 286 | padding: 0 1px 0 1px; 287 | font-size: 0.95em; 288 | } 289 | 290 | th { 291 | background-color: #ede; 292 | } 293 | 294 | .warning code { 295 | background: #efc2c2; 296 | } 297 | 298 | .note code { 299 | background: #d6d6d6; 300 | } 301 | 302 | .viewcode-back { 303 | font-family: {{ theme_bodyfont }}; 304 | } 305 | 306 | div.viewcode-block:target { 307 | background-color: #f4debf; 308 | border-top: 1px solid #ac9; 309 | border-bottom: 1px solid #ac9; 310 | } 311 | -------------------------------------------------------------------------------- /Makefile: -------------------------------------------------------------------------------- 1 | # Makefile for Sphinx documentation 2 | # 3 | 4 | # You can set these variables from the command line. 5 | SPHINXOPTS = 6 | SPHINXBUILD = sphinx-build 7 | PAPER = 8 | BUILDDIR = build 9 | 10 | # User-friendly check for sphinx-build 11 | ifeq ($(shell which $(SPHINXBUILD) >/dev/null 2>&1; echo $$?), 1) 12 | $(error The '$(SPHINXBUILD)' command was not found. Make sure you have Sphinx installed, then set the SPHINXBUILD environment variable to point to the full path of the '$(SPHINXBUILD)' executable. Alternatively you can add the directory with the executable to your PATH. If you don't have Sphinx installed, grab it from http://sphinx-doc.org/) 13 | endif 14 | 15 | # Internal variables. 16 | PAPEROPT_a4 = -D latex_paper_size=a4 17 | PAPEROPT_letter = -D latex_paper_size=letter 18 | ALLSPHINXOPTS = -d $(BUILDDIR)/doctrees $(PAPEROPT_$(PAPER)) $(SPHINXOPTS) source 19 | # the i18n builder cannot share the environment and doctrees with the others 20 | I18NSPHINXOPTS = $(PAPEROPT_$(PAPER)) $(SPHINXOPTS) source 21 | 22 | .PHONY: help clean html dirhtml singlehtml pickle json htmlhelp qthelp devhelp epub latex latexpdf text man changes linkcheck doctest gettext 23 | 24 | help: 25 | @echo "Please use \`make ' where is one of" 26 | @echo " html to make standalone HTML files" 27 | @echo " dirhtml to make HTML files named index.html in directories" 28 | @echo " singlehtml to make a single large HTML file" 29 | @echo " pickle to make pickle files" 30 | @echo " json to make JSON files" 31 | @echo " htmlhelp to make HTML files and a HTML help project" 32 | @echo " qthelp to make HTML files and a qthelp project" 33 | @echo " devhelp to make HTML files and a Devhelp project" 34 | @echo " epub to make an epub" 35 | @echo " latex to make LaTeX files, you can set PAPER=a4 or PAPER=letter" 36 | @echo " latexpdf to make LaTeX files and run them through pdflatex" 37 | @echo " latexpdfja to make LaTeX files and run them through platex/dvipdfmx" 38 | @echo " text to make text files" 39 | @echo " man to make manual pages" 40 | @echo " texinfo to make Texinfo files" 41 | @echo " info to make Texinfo files and run them through makeinfo" 42 | @echo " gettext to make PO message catalogs" 43 | @echo " changes to make an overview of all changed/added/deprecated items" 44 | @echo " xml to make Docutils-native XML files" 45 | @echo " pseudoxml to make pseudoxml-XML files for display purposes" 46 | @echo " linkcheck to check all external links for integrity" 47 | @echo " doctest to run all doctests embedded in the documentation (if enabled)" 48 | 49 | clean: 50 | rm -rf $(BUILDDIR)/* 51 | 52 | html: 53 | $(SPHINXBUILD) -b html $(ALLSPHINXOPTS) $(BUILDDIR)/html 54 | @echo 55 | @echo "Build finished. The HTML pages are in $(BUILDDIR)/html." 56 | 57 | dirhtml: 58 | $(SPHINXBUILD) -b dirhtml $(ALLSPHINXOPTS) $(BUILDDIR)/dirhtml 59 | @echo 60 | @echo "Build finished. The HTML pages are in $(BUILDDIR)/dirhtml." 61 | 62 | singlehtml: 63 | $(SPHINXBUILD) -b singlehtml $(ALLSPHINXOPTS) $(BUILDDIR)/singlehtml 64 | @echo 65 | @echo "Build finished. The HTML page is in $(BUILDDIR)/singlehtml." 66 | 67 | pickle: 68 | $(SPHINXBUILD) -b pickle $(ALLSPHINXOPTS) $(BUILDDIR)/pickle 69 | @echo 70 | @echo "Build finished; now you can process the pickle files." 71 | 72 | json: 73 | $(SPHINXBUILD) -b json $(ALLSPHINXOPTS) $(BUILDDIR)/json 74 | @echo 75 | @echo "Build finished; now you can process the JSON files." 76 | 77 | htmlhelp: 78 | $(SPHINXBUILD) -b htmlhelp $(ALLSPHINXOPTS) $(BUILDDIR)/htmlhelp 79 | @echo 80 | @echo "Build finished; now you can run HTML Help Workshop with the" \ 81 | ".hhp project file in $(BUILDDIR)/htmlhelp." 82 | 83 | qthelp: 84 | $(SPHINXBUILD) -b qthelp $(ALLSPHINXOPTS) $(BUILDDIR)/qthelp 85 | @echo 86 | @echo "Build finished; now you can run "qcollectiongenerator" with the" \ 87 | ".qhcp project file in $(BUILDDIR)/qthelp, like this:" 88 | @echo "# qcollectiongenerator $(BUILDDIR)/qthelp/StatisticalMechanics.qhcp" 89 | @echo "To view the help file:" 90 | @echo "# assistant -collectionFile $(BUILDDIR)/qthelp/StatisticalMechanics.qhc" 91 | 92 | devhelp: 93 | $(SPHINXBUILD) -b devhelp $(ALLSPHINXOPTS) $(BUILDDIR)/devhelp 94 | @echo 95 | @echo "Build finished." 96 | @echo "To view the help file:" 97 | @echo "# mkdir -p $$HOME/.local/share/devhelp/StatisticalMechanics" 98 | @echo "# ln -s $(BUILDDIR)/devhelp $$HOME/.local/share/devhelp/StatisticalMechanics" 99 | @echo "# devhelp" 100 | 101 | epub: 102 | $(SPHINXBUILD) -b epub $(ALLSPHINXOPTS) $(BUILDDIR)/epub 103 | @echo 104 | @echo "Build finished. The epub file is in $(BUILDDIR)/epub." 105 | 106 | latex: 107 | $(SPHINXBUILD) -b latex $(ALLSPHINXOPTS) $(BUILDDIR)/latex 108 | @echo 109 | @echo "Build finished; the LaTeX files are in $(BUILDDIR)/latex." 110 | @echo "Run \`make' in that directory to run these through (pdf)latex" \ 111 | "(use \`make latexpdf' here to do that automatically)." 112 | 113 | latexpdf: 114 | $(SPHINXBUILD) -b latex $(ALLSPHINXOPTS) $(BUILDDIR)/latex 115 | @echo "Running LaTeX files through pdflatex..." 116 | $(MAKE) -C $(BUILDDIR)/latex all-pdf 117 | @echo "pdflatex finished; the PDF files are in $(BUILDDIR)/latex." 118 | 119 | latexpdfja: 120 | $(SPHINXBUILD) -b latex $(ALLSPHINXOPTS) $(BUILDDIR)/latex 121 | @echo "Running LaTeX files through platex and dvipdfmx..." 122 | $(MAKE) -C $(BUILDDIR)/latex all-pdf-ja 123 | @echo "pdflatex finished; the PDF files are in $(BUILDDIR)/latex." 124 | 125 | text: 126 | $(SPHINXBUILD) -b text $(ALLSPHINXOPTS) $(BUILDDIR)/text 127 | @echo 128 | @echo "Build finished. The text files are in $(BUILDDIR)/text." 129 | 130 | man: 131 | $(SPHINXBUILD) -b man $(ALLSPHINXOPTS) $(BUILDDIR)/man 132 | @echo 133 | @echo "Build finished. The manual pages are in $(BUILDDIR)/man." 134 | 135 | texinfo: 136 | $(SPHINXBUILD) -b texinfo $(ALLSPHINXOPTS) $(BUILDDIR)/texinfo 137 | @echo 138 | @echo "Build finished. The Texinfo files are in $(BUILDDIR)/texinfo." 139 | @echo "Run \`make' in that directory to run these through makeinfo" \ 140 | "(use \`make info' here to do that automatically)." 141 | 142 | info: 143 | $(SPHINXBUILD) -b texinfo $(ALLSPHINXOPTS) $(BUILDDIR)/texinfo 144 | @echo "Running Texinfo files through makeinfo..." 145 | make -C $(BUILDDIR)/texinfo info 146 | @echo "makeinfo finished; the Info files are in $(BUILDDIR)/texinfo." 147 | 148 | gettext: 149 | $(SPHINXBUILD) -b gettext $(I18NSPHINXOPTS) $(BUILDDIR)/locale 150 | @echo 151 | @echo "Build finished. The message catalogs are in $(BUILDDIR)/locale." 152 | 153 | changes: 154 | $(SPHINXBUILD) -b changes $(ALLSPHINXOPTS) $(BUILDDIR)/changes 155 | @echo 156 | @echo "The overview file is in $(BUILDDIR)/changes." 157 | 158 | linkcheck: 159 | $(SPHINXBUILD) -b linkcheck $(ALLSPHINXOPTS) $(BUILDDIR)/linkcheck 160 | @echo 161 | @echo "Link check complete; look for any errors in the above output " \ 162 | "or in $(BUILDDIR)/linkcheck/output.txt." 163 | 164 | doctest: 165 | $(SPHINXBUILD) -b doctest $(ALLSPHINXOPTS) $(BUILDDIR)/doctest 166 | @echo "Testing of doctests in the sources finished, look at the " \ 167 | "results in $(BUILDDIR)/doctest/output.txt." 168 | 169 | xml: 170 | $(SPHINXBUILD) -b xml $(ALLSPHINXOPTS) $(BUILDDIR)/xml 171 | @echo 172 | @echo "Build finished. The XML files are in $(BUILDDIR)/xml." 173 | 174 | pseudoxml: 175 | $(SPHINXBUILD) -b pseudoxml $(ALLSPHINXOPTS) $(BUILDDIR)/pseudoxml 176 | @echo 177 | @echo "Build finished. The pseudo-XML files are in $(BUILDDIR)/pseudoxml." 178 | -------------------------------------------------------------------------------- /make.bat: -------------------------------------------------------------------------------- 1 | @ECHO OFF 2 | 3 | REM Command file for Sphinx documentation 4 | 5 | if "%SPHINXBUILD%" == "" ( 6 | set SPHINXBUILD=sphinx-build 7 | ) 8 | set BUILDDIR=build 9 | set ALLSPHINXOPTS=-d %BUILDDIR%/doctrees %SPHINXOPTS% source 10 | set I18NSPHINXOPTS=%SPHINXOPTS% source 11 | if NOT "%PAPER%" == "" ( 12 | set ALLSPHINXOPTS=-D latex_paper_size=%PAPER% %ALLSPHINXOPTS% 13 | set I18NSPHINXOPTS=-D latex_paper_size=%PAPER% %I18NSPHINXOPTS% 14 | ) 15 | 16 | if "%1" == "" goto help 17 | 18 | if "%1" == "help" ( 19 | :help 20 | echo.Please use `make ^` where ^ is one of 21 | echo. html to make standalone HTML files 22 | echo. dirhtml to make HTML files named index.html in directories 23 | echo. singlehtml to make a single large HTML file 24 | echo. pickle to make pickle files 25 | echo. json to make JSON files 26 | echo. htmlhelp to make HTML files and a HTML help project 27 | echo. qthelp to make HTML files and a qthelp project 28 | echo. devhelp to make HTML files and a Devhelp project 29 | echo. epub to make an epub 30 | echo. latex to make LaTeX files, you can set PAPER=a4 or PAPER=letter 31 | echo. text to make text files 32 | echo. man to make manual pages 33 | echo. texinfo to make Texinfo files 34 | echo. gettext to make PO message catalogs 35 | echo. changes to make an overview over all changed/added/deprecated items 36 | echo. xml to make Docutils-native XML files 37 | echo. pseudoxml to make pseudoxml-XML files for display purposes 38 | echo. linkcheck to check all external links for integrity 39 | echo. doctest to run all doctests embedded in the documentation if enabled 40 | goto end 41 | ) 42 | 43 | if "%1" == "clean" ( 44 | for /d %%i in (%BUILDDIR%\*) do rmdir /q /s %%i 45 | del /q /s %BUILDDIR%\* 46 | goto end 47 | ) 48 | 49 | 50 | %SPHINXBUILD% 2> nul 51 | if errorlevel 9009 ( 52 | echo. 53 | echo.The 'sphinx-build' command was not found. Make sure you have Sphinx 54 | echo.installed, then set the SPHINXBUILD environment variable to point 55 | echo.to the full path of the 'sphinx-build' executable. Alternatively you 56 | echo.may add the Sphinx directory to PATH. 57 | echo. 58 | echo.If you don't have Sphinx installed, grab it from 59 | echo.http://sphinx-doc.org/ 60 | exit /b 1 61 | ) 62 | 63 | if "%1" == "html" ( 64 | %SPHINXBUILD% -b html %ALLSPHINXOPTS% %BUILDDIR%/html 65 | if errorlevel 1 exit /b 1 66 | echo. 67 | echo.Build finished. The HTML pages are in %BUILDDIR%/html. 68 | goto end 69 | ) 70 | 71 | if "%1" == "dirhtml" ( 72 | %SPHINXBUILD% -b dirhtml %ALLSPHINXOPTS% %BUILDDIR%/dirhtml 73 | if errorlevel 1 exit /b 1 74 | echo. 75 | echo.Build finished. The HTML pages are in %BUILDDIR%/dirhtml. 76 | goto end 77 | ) 78 | 79 | if "%1" == "singlehtml" ( 80 | %SPHINXBUILD% -b singlehtml %ALLSPHINXOPTS% %BUILDDIR%/singlehtml 81 | if errorlevel 1 exit /b 1 82 | echo. 83 | echo.Build finished. The HTML pages are in %BUILDDIR%/singlehtml. 84 | goto end 85 | ) 86 | 87 | if "%1" == "pickle" ( 88 | %SPHINXBUILD% -b pickle %ALLSPHINXOPTS% %BUILDDIR%/pickle 89 | if errorlevel 1 exit /b 1 90 | echo. 91 | echo.Build finished; now you can process the pickle files. 92 | goto end 93 | ) 94 | 95 | if "%1" == "json" ( 96 | %SPHINXBUILD% -b json %ALLSPHINXOPTS% %BUILDDIR%/json 97 | if errorlevel 1 exit /b 1 98 | echo. 99 | echo.Build finished; now you can process the JSON files. 100 | goto end 101 | ) 102 | 103 | if "%1" == "htmlhelp" ( 104 | %SPHINXBUILD% -b htmlhelp %ALLSPHINXOPTS% %BUILDDIR%/htmlhelp 105 | if errorlevel 1 exit /b 1 106 | echo. 107 | echo.Build finished; now you can run HTML Help Workshop with the ^ 108 | .hhp project file in %BUILDDIR%/htmlhelp. 109 | goto end 110 | ) 111 | 112 | if "%1" == "qthelp" ( 113 | %SPHINXBUILD% -b qthelp %ALLSPHINXOPTS% %BUILDDIR%/qthelp 114 | if errorlevel 1 exit /b 1 115 | echo. 116 | echo.Build finished; now you can run "qcollectiongenerator" with the ^ 117 | .qhcp project file in %BUILDDIR%/qthelp, like this: 118 | echo.^> qcollectiongenerator %BUILDDIR%\qthelp\StatisticalMechanics.qhcp 119 | echo.To view the help file: 120 | echo.^> assistant -collectionFile %BUILDDIR%\qthelp\StatisticalMechanics.ghc 121 | goto end 122 | ) 123 | 124 | if "%1" == "devhelp" ( 125 | %SPHINXBUILD% -b devhelp %ALLSPHINXOPTS% %BUILDDIR%/devhelp 126 | if errorlevel 1 exit /b 1 127 | echo. 128 | echo.Build finished. 129 | goto end 130 | ) 131 | 132 | if "%1" == "epub" ( 133 | %SPHINXBUILD% -b epub %ALLSPHINXOPTS% %BUILDDIR%/epub 134 | if errorlevel 1 exit /b 1 135 | echo. 136 | echo.Build finished. The epub file is in %BUILDDIR%/epub. 137 | goto end 138 | ) 139 | 140 | if "%1" == "latex" ( 141 | %SPHINXBUILD% -b latex %ALLSPHINXOPTS% %BUILDDIR%/latex 142 | if errorlevel 1 exit /b 1 143 | echo. 144 | echo.Build finished; the LaTeX files are in %BUILDDIR%/latex. 145 | goto end 146 | ) 147 | 148 | if "%1" == "latexpdf" ( 149 | %SPHINXBUILD% -b latex %ALLSPHINXOPTS% %BUILDDIR%/latex 150 | cd %BUILDDIR%/latex 151 | make all-pdf 152 | cd %BUILDDIR%/.. 153 | echo. 154 | echo.Build finished; the PDF files are in %BUILDDIR%/latex. 155 | goto end 156 | ) 157 | 158 | if "%1" == "latexpdfja" ( 159 | %SPHINXBUILD% -b latex %ALLSPHINXOPTS% %BUILDDIR%/latex 160 | cd %BUILDDIR%/latex 161 | make all-pdf-ja 162 | cd %BUILDDIR%/.. 163 | echo. 164 | echo.Build finished; the PDF files are in %BUILDDIR%/latex. 165 | goto end 166 | ) 167 | 168 | if "%1" == "text" ( 169 | %SPHINXBUILD% -b text %ALLSPHINXOPTS% %BUILDDIR%/text 170 | if errorlevel 1 exit /b 1 171 | echo. 172 | echo.Build finished. The text files are in %BUILDDIR%/text. 173 | goto end 174 | ) 175 | 176 | if "%1" == "man" ( 177 | %SPHINXBUILD% -b man %ALLSPHINXOPTS% %BUILDDIR%/man 178 | if errorlevel 1 exit /b 1 179 | echo. 180 | echo.Build finished. The manual pages are in %BUILDDIR%/man. 181 | goto end 182 | ) 183 | 184 | if "%1" == "texinfo" ( 185 | %SPHINXBUILD% -b texinfo %ALLSPHINXOPTS% %BUILDDIR%/texinfo 186 | if errorlevel 1 exit /b 1 187 | echo. 188 | echo.Build finished. The Texinfo files are in %BUILDDIR%/texinfo. 189 | goto end 190 | ) 191 | 192 | if "%1" == "gettext" ( 193 | %SPHINXBUILD% -b gettext %I18NSPHINXOPTS% %BUILDDIR%/locale 194 | if errorlevel 1 exit /b 1 195 | echo. 196 | echo.Build finished. The message catalogs are in %BUILDDIR%/locale. 197 | goto end 198 | ) 199 | 200 | if "%1" == "changes" ( 201 | %SPHINXBUILD% -b changes %ALLSPHINXOPTS% %BUILDDIR%/changes 202 | if errorlevel 1 exit /b 1 203 | echo. 204 | echo.The overview file is in %BUILDDIR%/changes. 205 | goto end 206 | ) 207 | 208 | if "%1" == "linkcheck" ( 209 | %SPHINXBUILD% -b linkcheck %ALLSPHINXOPTS% %BUILDDIR%/linkcheck 210 | if errorlevel 1 exit /b 1 211 | echo. 212 | echo.Link check complete; look for any errors in the above output ^ 213 | or in %BUILDDIR%/linkcheck/output.txt. 214 | goto end 215 | ) 216 | 217 | if "%1" == "doctest" ( 218 | %SPHINXBUILD% -b doctest %ALLSPHINXOPTS% %BUILDDIR%/doctest 219 | if errorlevel 1 exit /b 1 220 | echo. 221 | echo.Testing of doctests in the sources finished, look at the ^ 222 | results in %BUILDDIR%/doctest/output.txt. 223 | goto end 224 | ) 225 | 226 | if "%1" == "xml" ( 227 | %SPHINXBUILD% -b xml %ALLSPHINXOPTS% %BUILDDIR%/xml 228 | if errorlevel 1 exit /b 1 229 | echo. 230 | echo.Build finished. The XML files are in %BUILDDIR%/xml. 231 | goto end 232 | ) 233 | 234 | if "%1" == "pseudoxml" ( 235 | %SPHINXBUILD% -b pseudoxml %ALLSPHINXOPTS% %BUILDDIR%/pseudoxml 236 | if errorlevel 1 exit /b 1 237 | echo. 238 | echo.Build finished. The pseudo-XML files are in %BUILDDIR%/pseudoxml. 239 | goto end 240 | ) 241 | 242 | :end 243 | -------------------------------------------------------------------------------- /docs/_static/pyramid.css: -------------------------------------------------------------------------------- 1 | /* 2 | * pyramid.css_t 3 | * ~~~~~~~~~~~~ 4 | * 5 | * Sphinx stylesheet -- pylons theme. 6 | * 7 | * :copyright: Copyright 2007-2015 by the Sphinx team, see AUTHORS. 8 | * :license: BSD, see LICENSE for details. 9 | * 10 | */ 11 | 12 | @import url("basic.css"); 13 | 14 | /* -- page layout ----------------------------------------------------------- */ 15 | 16 | body { 17 | font-family: "Nobile", sans-serif; 18 | font-size: 100%; 19 | background-color: #393939; 20 | color: #ffffff; 21 | margin: 0; 22 | padding: 0; 23 | } 24 | 25 | div.documentwrapper { 26 | float: left; 27 | width: 100%; 28 | } 29 | 30 | div.bodywrapper { 31 | margin: 0 0 0 230px; 32 | } 33 | 34 | hr { 35 | border: 1px solid #B1B4B6; 36 | } 37 | 38 | div.document { 39 | background-color: #eee; 40 | } 41 | 42 | div.header { 43 | width:100%; 44 | background: #f4ad32 url(headerbg.png) repeat-x 0 top; 45 | border-bottom: 2px solid #ffffff; 46 | } 47 | 48 | div.logo { 49 | text-align: center; 50 | padding-top: 10px; 51 | } 52 | 53 | div.body { 54 | background-color: #ffffff; 55 | color: #3E4349; 56 | padding: 0 30px 30px 30px; 57 | font-size: 1em; 58 | border: 2px solid #ddd; 59 | border-right-style: none; 60 | overflow: auto; 61 | } 62 | 63 | div.footer { 64 | color: #ffffff; 65 | width: 100%; 66 | padding: 13px 0; 67 | text-align: center; 68 | font-size: 75%; 69 | background: transparent; 70 | clear:both; 71 | } 72 | 73 | div.footer a { 74 | color: #ffffff; 75 | text-decoration: none; 76 | } 77 | 78 | div.footer a:hover { 79 | color: #e88f00; 80 | text-decoration: underline; 81 | } 82 | 83 | div.related { 84 | line-height: 30px; 85 | color: #373839; 86 | font-size: 0.8em; 87 | background-color: #eee; 88 | } 89 | 90 | div.related a { 91 | color: #1b61d6; 92 | } 93 | 94 | div.related ul { 95 | padding-left: 240px; 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319 | } 320 | 321 | table.highlighttable td { 322 | padding: 0; 323 | } 324 | 325 | a em.std-term { 326 | color: #007f00; 327 | } 328 | 329 | a:hover em.std-term { 330 | text-decoration: underline; 331 | } 332 | 333 | .download { 334 | font-family: "Nobile", sans-serif; 335 | font-weight: normal; 336 | font-style: normal; 337 | } 338 | 339 | code.xref { 340 | font-weight: normal; 341 | font-style: normal; 342 | } 343 | 344 | div.code-block-caption { 345 | background-color: #ddd; 346 | color: #222; 347 | } -------------------------------------------------------------------------------- /docs/introduction.html: -------------------------------------------------------------------------------- 1 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 1. Introduction — Introduction to Statistical Mechanics 10 | 11 | 12 | 13 | 14 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 35 | 36 | 37 | 38 | 39 | 54 | 55 |
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61 |

1. Introduction

62 |

Depending on how you look at it, statistical mechanics is either the least fundamental or most fundamental of all fields 63 | of physics. That is because it is not really science at all. It is pure mathematics.

64 |

In other subjects, you learn about “natural laws”: Newton’s second law, Maxwell’s equations, Schrödinger’s equation, 65 | etc. These laws are not derived from anything else. They were discovered experimentally and then assumed to reflect 66 | fundamental aspects of reality. But statistical mechanics does not involve any natural laws of this sort. Instead, it 67 | is a set of techniques that can be applied to nearly any physical system, no matter what laws that system obeys. That 68 | is why I call it the most fundamental field of physics. New theories may replace old ones, and natural laws may turn 69 | out to be merely approximations to deeper laws. But statistical mechanics remains valid through it all, and whatever 70 | new laws are discovered, it will almost certainly work just as well with them as it did with the old ones.

71 |

Statistical mechanics was developed in the second half of the 19th century. It was primarily the work of Ludwig 72 | Boltzmann, who personally published more than a hundred papers on the subject during his lifetime. Other scientists 73 | also contributed to it, of course, especially James Clerk Maxwell (the same one Maxwell’s equations are named after) and 74 | Josiah Willard Gibbs.

75 |

It grew out of thermodynamics, an earlier theory that described the behavior of a mysterious substance called “heat” or 76 | “caloric”. Thermodynamics was a physical theory of the more conventional sort. It involved natural laws discovered by 77 | experiment, and made no claims about why those laws happened to hold. Caloric was supposed to be a substance much like 78 | other forms of matter. But this view turned out to be incorrect. Heat is actually an emergent phenomenon: a 79 | mathematical quantity defined in terms of a more detailed theory (the movement of individual atoms). The “laws” of 80 | thermodynamics can be derived from that deeper theory by applying statistical techniques. If you ignore the details of 81 | how each atom is moving, you find they collectively behave in a way that resembles a continuous fluid. That is what 82 | statistical mechanics is all about: deriving high level descriptions by starting from lower level ones and then 83 | averaging out lots of details.

84 |

Is it possible other theories could be explained in the same way? That is an open question, and a fascinating one. 85 | Many physicists suspect gravity is an emergent phenomenon, that it arises from the collective behavior of some deeper 86 | degrees of freedom. Statistical interpretations have also been proposed for quantum mechanics. These are all very 87 | speculative, of course, and they could easily turn out to be wrong. But they also could easily turn out to be right. 88 | Based on what we know today, it is entirely possible that the very structure of spacetime is a consequence of 89 | statistical mechanics.

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121 | 136 | 140 | 141 | -------------------------------------------------------------------------------- /docs/_static/doctools.js: -------------------------------------------------------------------------------- 1 | /* 2 | * doctools.js 3 | * ~~~~~~~~~~~ 4 | * 5 | * Sphinx JavaScript utilities for all documentation. 6 | * 7 | * :copyright: Copyright 2007-2015 by the Sphinx team, see AUTHORS. 8 | * :license: BSD, see LICENSE for details. 9 | * 10 | */ 11 | 12 | /** 13 | * select a different prefix for underscore 14 | */ 15 | $u = _.noConflict(); 16 | 17 | /** 18 | * make the code below compatible with browsers without 19 | * an installed firebug like debugger 20 | if (!window.console || !console.firebug) { 21 | var names = ["log", "debug", "info", "warn", "error", "assert", "dir", 22 | "dirxml", "group", "groupEnd", "time", "timeEnd", "count", "trace", 23 | "profile", "profileEnd"]; 24 | window.console = {}; 25 | for (var i = 0; i < names.length; ++i) 26 | window.console[names[i]] = function() {}; 27 | } 28 | */ 29 | 30 | /** 31 | * small helper function to urldecode strings 32 | */ 33 | jQuery.urldecode = function(x) { 34 | return decodeURIComponent(x).replace(/\+/g, ' '); 35 | }; 36 | 37 | /** 38 | * small helper function to urlencode strings 39 | */ 40 | jQuery.urlencode = encodeURIComponent; 41 | 42 | /** 43 | * This function returns the parsed url parameters of the 44 | * current request. 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to show the default. 14 | 15 | import sys 16 | import os 17 | 18 | # If extensions (or modules to document with autodoc) are in another directory, 19 | # add these directories to sys.path here. If the directory is relative to the 20 | # documentation root, use os.path.abspath to make it absolute, like shown here. 21 | #sys.path.insert(0, os.path.abspath('.')) 22 | 23 | sys.path.insert(0, os.path.abspath('.')) 24 | 25 | # -- General configuration ------------------------------------------------ 26 | 27 | # If your documentation needs a minimal Sphinx version, state it here. 28 | #needs_sphinx = '1.0' 29 | 30 | # Add any Sphinx extension module names here, as strings. They can be 31 | # extensions coming with Sphinx (named 'sphinx.ext.*') or your custom 32 | # ones. 33 | extensions = [ 34 | 'sphinx.ext.todo', 35 | 'sphinx.ext.mathjax', 36 | 'numsec', 37 | 'autonumber' 38 | ] 39 | 40 | # Add any paths that contain templates here, relative to this directory. 41 | templates_path = ['_templates'] 42 | 43 | # The suffix of source filenames. 44 | source_suffix = '.rst' 45 | 46 | # The encoding of source files. 47 | #source_encoding = 'utf-8-sig' 48 | 49 | # The master toctree document. 50 | master_doc = 'index' 51 | 52 | # General information about the project. 53 | project = u'Introduction to Statistical Mechanics' 54 | copyright = u'2014-2015, Peter Eastman' 55 | 56 | # The version info for the project you're documenting, acts as replacement for 57 | # |version| and |release|, also used in various other places throughout the 58 | # built documents. 59 | # 60 | # The short X.Y version. 61 | version = '' 62 | # The full version, including alpha/beta/rc tags. 63 | release = '' 64 | 65 | # The language for content autogenerated by Sphinx. Refer to documentation 66 | # for a list of supported languages. 67 | #language = None 68 | 69 | # There are two options for replacing |today|: either, you set today to some 70 | # non-false value, then it is used: 71 | #today = '' 72 | # Else, today_fmt is used as the format for a strftime call. 73 | #today_fmt = '%B %d, %Y' 74 | 75 | # List of patterns, relative to source directory, that match files and 76 | # directories to ignore when looking for source files. 77 | exclude_patterns = [] 78 | 79 | # The reST default role (used for this markup: `text`) to use for all 80 | # documents. 81 | #default_role = None 82 | 83 | # If true, '()' will be appended to :func: etc. cross-reference text. 84 | #add_function_parentheses = True 85 | 86 | # If true, the current module name will be prepended to all description 87 | # unit titles (such as .. function::). 88 | #add_module_names = True 89 | 90 | # If true, sectionauthor and moduleauthor directives will be shown in the 91 | # output. They are ignored by default. 92 | #show_authors = False 93 | 94 | # The name of the Pygments (syntax highlighting) style to use. 95 | pygments_style = 'sphinx' 96 | 97 | # A list of ignored prefixes for module index sorting. 98 | #modindex_common_prefix = [] 99 | 100 | # If true, keep warnings as "system message" paragraphs in the built documents. 101 | #keep_warnings = False 102 | 103 | 104 | # -- Options for HTML output ---------------------------------------------- 105 | 106 | # The theme to use for HTML and HTML Help pages. See the documentation for 107 | # a list of builtin themes. 108 | html_theme = 'pyramid' 109 | 110 | # Theme options are theme-specific and customize the look and feel of a theme 111 | # further. For a list of options available for each theme, see the 112 | # documentation. 113 | #html_theme_options = {} 114 | 115 | # Add any paths that contain custom themes here, relative to this directory. 116 | #html_theme_path = [] 117 | 118 | # The name for this set of Sphinx documents. If None, it defaults to 119 | # " v documentation". 120 | html_title = project 121 | 122 | # A shorter title for the navigation bar. Default is the same as html_title. 123 | #html_short_title = None 124 | 125 | # The name of an image file (relative to this directory) to place at the top 126 | # of the sidebar. 127 | #html_logo = None 128 | 129 | # The name of an image file (within the static path) to use as favicon of the 130 | # docs. This file should be a Windows icon file (.ico) being 16x16 or 32x32 131 | # pixels large. 132 | #html_favicon = None 133 | 134 | # Add any paths that contain custom static files (such as style sheets) here, 135 | # relative to this directory. They are copied after the builtin static files, 136 | # so a file named "default.css" will overwrite the builtin "default.css". 137 | html_static_path = ['_static'] 138 | 139 | # Add any extra paths that contain custom files (such as robots.txt or 140 | # .htaccess) here, relative to this directory. These files are copied 141 | # directly to the root of the documentation. 142 | #html_extra_path = [] 143 | 144 | # If not '', a 'Last updated on:' timestamp is inserted at every page bottom, 145 | # using the given strftime format. 146 | #html_last_updated_fmt = '%b %d, %Y' 147 | 148 | # If true, SmartyPants will be used to convert quotes and dashes to 149 | # typographically correct entities. 150 | #html_use_smartypants = True 151 | 152 | # Custom sidebar templates, maps document names to template names. 153 | #html_sidebars = {} 154 | 155 | # Additional templates that should be rendered to pages, maps page names to 156 | # template names. 157 | #html_additional_pages = {} 158 | 159 | # If false, no module index is generated. 160 | #html_domain_indices = True 161 | 162 | # If false, no index is generated. 163 | #html_use_index = True 164 | 165 | # If true, the index is split into individual pages for each letter. 166 | #html_split_index = False 167 | 168 | # If true, links to the reST sources are added to the pages. 169 | #html_show_sourcelink = True 170 | 171 | # If true, "Created using Sphinx" is shown in the HTML footer. Default is True. 172 | #html_show_sphinx = True 173 | 174 | # If true, "(C) Copyright ..." is shown in the HTML footer. Default is True. 175 | #html_show_copyright = True 176 | 177 | # If true, an OpenSearch description file will be output, and all pages will 178 | # contain a tag referring to it. The value of this option must be the 179 | # base URL from which the finished HTML is served. 180 | #html_use_opensearch = '' 181 | 182 | # This is the file name suffix for HTML files (e.g. ".xhtml"). 183 | #html_file_suffix = None 184 | 185 | # Output file base name for HTML help builder. 186 | htmlhelp_basename = 'StatisticalMechanics' 187 | 188 | 189 | # -- Options for LaTeX output --------------------------------------------- 190 | 191 | latex_elements = { 192 | 'releasename': '' 193 | 194 | # The paper size ('letterpaper' or 'a4paper'). 195 | #'papersize': 'letterpaper', 196 | 197 | # The font size ('10pt', '11pt' or '12pt'). 198 | #'pointsize': '10pt', 199 | 200 | # Additional stuff for the LaTeX preamble. 201 | #'preamble': '', 202 | } 203 | 204 | # Grouping the document tree into LaTeX files. List of tuples 205 | # (source start file, target name, title, 206 | # author, documentclass [howto, manual, or own class]). 207 | latex_documents = [ 208 | ('index', 'StatisticalMechanics.tex', u'Introduction to Statistical Mechanics', 209 | u'Peter Eastman', 'manual'), 210 | ] 211 | 212 | # The name of an image file (relative to this directory) to place at the top of 213 | # the title page. 214 | #latex_logo = None 215 | 216 | # For "manual" documents, if this is true, then toplevel headings are parts, 217 | # not chapters. 218 | #latex_use_parts = False 219 | 220 | # If true, show page references after internal links. 221 | #latex_show_pagerefs = False 222 | 223 | # If true, show URL addresses after external links. 224 | #latex_show_urls = False 225 | 226 | # Documents to append as an appendix to all manuals. 227 | #latex_appendices = [] 228 | 229 | # If false, no module index is generated. 230 | #latex_domain_indices = True 231 | 232 | 233 | # -- Options for manual page output --------------------------------------- 234 | 235 | # One entry per manual page. List of tuples 236 | # (source start file, name, description, authors, manual section). 237 | man_pages = [ 238 | ('index', 'statisticalmechanics', u'Introduction to Statistical Mechanics', 239 | [u'Peter Eastman'], 1) 240 | ] 241 | 242 | # If true, show URL addresses after external links. 243 | #man_show_urls = False 244 | 245 | 246 | # -- Options for Texinfo output ------------------------------------------- 247 | 248 | # Grouping the document tree into Texinfo files. List of tuples 249 | # (source start file, target name, title, author, 250 | # dir menu entry, description, category) 251 | texinfo_documents = [ 252 | ('index', 'StatisticalMechanics', u'Introduction to Statistical Mechanics', 253 | u'Peter Eastman', 'StatisticalMechanics', 'One line description of project.', 254 | 'Miscellaneous'), 255 | ] 256 | 257 | # Documents to append as an appendix to all manuals. 258 | #texinfo_appendices = [] 259 | 260 | # If false, no module index is generated. 261 | #texinfo_domain_indices = True 262 | 263 | # How to display URL addresses: 'footnote', 'no', or 'inline'. 264 | #texinfo_show_urls = 'footnote' 265 | 266 | # If true, do not generate a @detailmenu in the "Top" node's menu. 267 | #texinfo_no_detailmenu = False 268 | 269 | 270 | # -- Options for Epub output ---------------------------------------------- 271 | 272 | # Bibliographic Dublin Core info. 273 | epub_title = u'Introduction to Statistical Mechanics' 274 | epub_author = u'Peter Eastman' 275 | epub_publisher = u'Peter Eastman' 276 | epub_copyright = u'2014, Peter Eastman' 277 | 278 | # The basename for the epub file. It defaults to the project name. 279 | #epub_basename = u'Statistical Mechanics' 280 | 281 | # The HTML theme for the epub output. Since the default themes are not optimized 282 | # for small screen space, using the same theme for HTML and epub output is 283 | # usually not wise. This defaults to 'epub', a theme designed to save visual 284 | # space. 285 | #epub_theme = 'epub' 286 | 287 | # The language of the text. It defaults to the language option 288 | # or en if the language is not set. 289 | #epub_language = '' 290 | 291 | # The scheme of the identifier. Typical schemes are ISBN or URL. 292 | #epub_scheme = '' 293 | 294 | # The unique identifier of the text. This can be a ISBN number 295 | # or the project homepage. 296 | #epub_identifier = '' 297 | 298 | # A unique identification for the text. 299 | #epub_uid = '' 300 | 301 | # A tuple containing the cover image and cover page html template filenames. 302 | #epub_cover = () 303 | 304 | # A sequence of (type, uri, title) tuples for the guide element of content.opf. 305 | #epub_guide = () 306 | 307 | # HTML files that should be inserted before the pages created by sphinx. 308 | # The format is a list of tuples containing the path and title. 309 | #epub_pre_files = [] 310 | 311 | # HTML files shat should be inserted after the pages created by sphinx. 312 | # The format is a list of tuples containing the path and title. 313 | #epub_post_files = [] 314 | 315 | # A list of files that should not be packed into the epub file. 316 | epub_exclude_files = ['search.html'] 317 | 318 | # The depth of the table of contents in toc.ncx. 319 | #epub_tocdepth = 3 320 | 321 | # Allow duplicate toc entries. 322 | #epub_tocdup = True 323 | 324 | # Choose between 'default' and 'includehidden'. 325 | #epub_tocscope = 'default' 326 | 327 | # Fix unsupported image types using the PIL. 328 | #epub_fix_images = False 329 | 330 | # Scale large images. 331 | #epub_max_image_width = 0 332 | 333 | # How to display URL addresses: 'footnote', 'no', or 'inline'. 334 | #epub_show_urls = 'inline' 335 | 336 | # If false, no index is generated. 337 | #epub_use_index = True 338 | -------------------------------------------------------------------------------- /docs/_static/basic.css: -------------------------------------------------------------------------------- 1 | /* 2 | * basic.css 3 | * ~~~~~~~~~ 4 | * 5 | * Sphinx stylesheet -- basic theme. 6 | * 7 | * :copyright: Copyright 2007-2015 by the Sphinx team, see AUTHORS. 8 | * :license: BSD, see LICENSE for details. 9 | * 10 | */ 11 | 12 | /* -- main layout ----------------------------------------------------------- */ 13 | 14 | div.clearer { 15 | clear: both; 16 | } 17 | 18 | /* -- relbar ---------------------------------------------------------------- */ 19 | 20 | div.related { 21 | width: 100%; 22 | font-size: 90%; 23 | } 24 | 25 | div.related h3 { 26 | display: none; 27 | } 28 | 29 | div.related ul { 30 | margin: 0; 31 | padding: 0 0 0 10px; 32 | list-style: none; 33 | } 34 | 35 | div.related li { 36 | display: inline; 37 | } 38 | 39 | div.related li.right { 40 | float: right; 41 | margin-right: 5px; 42 | } 43 | 44 | /* -- sidebar --------------------------------------------------------------- */ 45 | 46 | div.sphinxsidebarwrapper { 47 | padding: 10px 5px 0 10px; 48 | } 49 | 50 | div.sphinxsidebar { 51 | float: left; 52 | width: 230px; 53 | margin-left: -100%; 54 | font-size: 90%; 55 | } 56 | 57 | div.sphinxsidebar ul { 58 | list-style: none; 59 | } 60 | 61 | div.sphinxsidebar ul ul, 62 | div.sphinxsidebar ul.want-points { 63 | margin-left: 20px; 64 | list-style: square; 65 | } 66 | 67 | div.sphinxsidebar ul ul { 68 | margin-top: 0; 69 | margin-bottom: 0; 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581 | } 582 | 583 | /* -- printout stylesheet --------------------------------------------------- */ 584 | 585 | @media print { 586 | div.document, 587 | div.documentwrapper, 588 | div.bodywrapper { 589 | margin: 0 !important; 590 | width: 100%; 591 | } 592 | 593 | div.sphinxsidebar, 594 | div.related, 595 | div.footer, 596 | #top-link { 597 | display: none; 598 | } 599 | } -------------------------------------------------------------------------------- /docs/index.html: -------------------------------------------------------------------------------- 1 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Introduction to Statistical Mechanics — Introduction to Statistical Mechanics 10 | 11 | 12 | 13 | 14 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 34 | 35 | 36 | 37 | 38 | 50 | 51 |
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Introduction to Statistical Mechanics

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Copyright © 2014-2015 by Peter Eastman

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This work may be freely distributed under the terms of the 60 | Creative Commons Attribution-NoDerivatives 4.0 International License. 61 | The source can be found at https://github.com/peastman/statmech.

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160 | 172 | 176 | 177 | -------------------------------------------------------------------------------- /source/largenumbers.rst: -------------------------------------------------------------------------------- 1 | .. _very-large-numbers: 2 | 3 | Mathematical Interlude: Very Large Numbers 4 | ########################################## 5 | 6 | Problems in statistical mechanics tend to involve large numbers. Very *very* large numbers. For example, we often 7 | need to average over all the atoms or molecules in a system. For any macroscopic system, that will be on the order of 8 | Avogadro's number: 10\ :sup:`23`, plus or minus a few orders of magnitude. In other cases we average over microstates, 9 | the number of which usually grows *exponentially* in the number of atoms. These are really astronomically large 10 | numbers. 11 | 12 | This has an interesting result. In many cases, we can simply treat averages as exact numbers. Any variation about the 13 | average value is so small, we can completely ignore it. This is a great simplification, and is one of the reasons 14 | statistical mechanics is so successful. 15 | 16 | It also is the reason that, despite its name, statistical mechanics involves very little real statistics. A large part 17 | of the field of statistics deals with the variations about averages: computing the probability of a distribution 18 | producing values in a certain range, determining whether a measured value might plausibly have come from a certain 19 | distribution, etc. But in statistical mechanics, we can usually ignore these questions. The chance of measuring any 20 | value *except* the average one is negligible. 21 | 22 | The main goal of this chapter is to demonstrate this fact. We will do that first through a simple example, and then 23 | through a very general theorem. 24 | 25 | 26 | .. _the-binomial-distribution: 27 | 28 | The Binomial Distribution 29 | ========================= 30 | 31 | Do you ever worry that all the oxygen molecules in the air around you might spontaneously migrate to the other side of 32 | the room, leaving you unable to breathe? It could happen. Each molecule diffuses independently, so there is a small 33 | but finite probability that just by chance, they will all end up in the same half of the room at the same time. 34 | 35 | I hope I am not worrying you. While this is theoretically possible, it is so absurdly improbable that you really can 36 | ignore it. We will first compute just how improbable it is, and then study the statistics of this problem in more 37 | detail. 38 | 39 | Let :math:`N` be the number of oxygen molecules in the room. We will describe the position of each one simply by 40 | whether it is in the left or right half of the room. That gives :math:`2^N` possible arrangements of molecules, every 41 | one of which is equally probable. Of all those arrangements, only one has all the oxygen in the opposite half of the 42 | room from you, so the probability is :math:`1/2^N`. 43 | 44 | Depending on the size of the room you are in, it probably has somewhere in the vicinity of 10\ :sup:`26` oxygen 45 | molecules in it. So the probability of all the oxygen being on the same side of the room at any given moment is 46 | 47 | .. math:: 48 | \frac{1}{2^{10^{26}}} \approx \frac{1}{10^{10000000000000000000000000}} 49 | 50 | That is a very, very large denominator. For comparison here are some much smaller numbers: 51 | 52 | * The age of the universe is on the order of 10\ :sup:`17` seconds. 53 | 54 | * There are somewhere around 10\ :sup:`80` atoms in the visible universe. 55 | 56 | In short, you really do not need to worry about this happening. You would be much better off worrying about more likely 57 | events. For example, that you will be struck by lightning and then, as you try to pick yourself back up, hit by 58 | a meteor falling from space. And then immediately be struck by lightning a second time just for good measure. That is 59 | the sort of thing that is far more likely to happen to you. 60 | 61 | Perhaps you are still not reassured. You may have realized that even if only 90% of the oxygen moved to the other side 62 | of the room, that would still leave you in a rather awkward position. We really want to work out the entire probability 63 | distribution. Let :math:`m` be the number of oxygen molecules in the left half of the room and :math:`N-m` the number 64 | in the right half. Assuming the energy of each molecule is independent of which side of the room it is on, and that the 65 | oxygen molecules are far enough apart that we can ignore their interactions (both reasonable assumptions), the 66 | probability is simply proportional to :math:`\Omega(m)`. Let us consider how that varies with :math:`m`. 67 | 68 | * There is precisely one arrangement of molecules that has all of them on the left side of the room, so 69 | :math:`\Omega(0)=1`. 70 | 71 | * There are :math:`N` possible ways to have exactly one molecule on the left side of the room (any of the :math:`N` 72 | molecules could be the one), so :math:`\Omega(1)=N`. 73 | 74 | * To get two molecules on the left side of the room, we have :math:`N` choices for the first molecule and :math:`N-1` 75 | for the second molecule. But now we have counted each state twice: for any pair of molecules, it does not matter which 76 | order we choose them in. So :math:`\Omega(2)=N(N-1)/2`. 77 | 78 | Continuing like this, we find the general form: 79 | 80 | .. math:: 81 | \Omega(m) = \frac{N!}{m!(N-m)!} 82 | :label: binomialomega 83 | 84 | The probability is then 85 | 86 | .. math:: 87 | p(m) = \frac{1}{2^N} \frac{N!}{m!(N-m)!} 88 | :label: binomialprobability 89 | 90 | This is a special case of the *binomial distribution*. (In the general case, the probability for a particle to be on 91 | the left side of the room could be different from its probability to be on the right side, but for this example we can 92 | ignore that.) It is shown in :autonumref:`Figure,binomial` for several values of :math:`N`. Notice how, as :math:`N` 93 | increases, the probability becomes steadily more concentrated around its midpoint. 94 | 95 | .. figure:: images/binomial.* 96 | :align: center 97 | :scale: 80 98 | 99 | :autonumber:`Figure,binomial`. The binomial distribution for several values of :math:`N`. To make them easier to 100 | compare, all the curves have been normalized to go from 0 to 1 along each axis. 101 | 102 | The name comes from the fact that it has the same form as the terms in the *binomial series*: 103 | 104 | .. math:: 105 | (a+b)^N = \sum_{m=0}^N \frac{N!}{m!(N-m)!}a^Nb^{N-m} 106 | :label: binomialseries 107 | 108 | If we let :math:`a=b=1/2`, this simplifies to 109 | 110 | .. math:: 111 | \left(\frac{1}{2}+\frac{1}{2}\right)^N = \sum_{m=0}^N \frac{N!}{m!(N-m)!} \frac{1}{2^N} 112 | 113 | so that 114 | 115 | .. math:: 116 | \sum_{m=0}^N \frac{N!}{m!(N-m)!} = 2^N 117 | :label: binomialsum 118 | 119 | Let us examine this probability distribution a bit further. We can roughly characterize it by calculating its mean and 120 | standard deviation. The mean is easy: every particle has an equal chance to be on each side of the room, so 121 | :math:`\langle m \rangle = N/2`. Another way to think about this is that the distribution is symmetric: for any value 122 | of :math:`m`, :math:`p(m)=p(N-m)`, from which it directly follows that the mean must be :math:`N/2`. 123 | 124 | The standard deviation is a bit more complicated to derive. As a first step, let us first calculate 125 | 126 | .. math:: 127 | \langle m(m-1) \rangle = \frac{1}{2^N} \sum_{m=0}^N m(m-1) \frac{N!}{m!(N-m)!} 128 | 129 | Notice that the first two terms of the sum are both zero, so we can increase the lower bound to begin from 2. This 130 | allows us to then cancel out the factors of :math:`m(m-1)` that appear in both numerator and denominator: 131 | 132 | .. math:: 133 | \langle m(m-1) \rangle &= \frac{1}{2^N} \sum_{m=2}^N \frac{N!}{(m-2)!(N-m)!} \\ 134 | &= \frac{N(N-1)}{2^N} \sum_{m=2}^N \frac{(N-2)!}{(m-2)!(N-m)!} 135 | 136 | Now make two substitutions: define :math:`X=N-2` and :math:`y=m-2`. This simplifies it to: 137 | 138 | .. math:: 139 | \langle m(m-1) \rangle = \frac{N(N-1)}{2^N} \sum_{y=0}^X \frac{X!}{y!(X-y)!} 140 | 141 | We immediately recognize the sum as being the same one that appeared in equation :eq:`binomialsum`. Replacing it by 142 | :math:`2^X=2^{N-2}`, 143 | 144 | .. math:: 145 | \langle m(m-1) \rangle = \frac{N(N-1)2^{N-2}}{2^N} = \frac{N(N-1)}{4} 146 | 147 | We now have all the pieces we need. Recall that 148 | 149 | .. math:: 150 | Var(m) &= \langle m^2 \rangle - \langle m \rangle^2 \\ 151 | &= \langle m(m-1) \rangle + \langle m \rangle- \langle m \rangle^2 \\ 152 | &= \frac{N(N-1)}{4} + \frac{N}{2} - \frac{N^2}{4} \\ 153 | &= \frac{N}{4} 154 | :label: binomialvariance 155 | 156 | The standard deviation is then 157 | 158 | .. math:: 159 | \sigma \equiv \sqrt{Var(m)} = \frac{\sqrt{N}}{2} 160 | :label: binomialstddev 161 | 162 | Let us take a moment to consider these results. The average value is proportional to :math:`N`, while the standard 163 | deviation is proportional to :math:`\sqrt{N}`. In most cases, what we really care about is the ratio of the two. You 164 | want to know what *fraction* of the oxygen is likely to be on one side of the room, not the specific number of 165 | molecules. A number that would be huge in a broom closet would be negligible in an auditorium. Taking the ratio gives 166 | 167 | .. math:: 168 | \frac{\sigma}{\langle m \rangle} = \frac{1}{\sqrt{N}} 169 | :label: binomialfractionaldeviation 170 | 171 | For an average sized room with 10\ :sup:`26` oxygen molecules, that equals 10\ :sup:`-13`. This is the 172 | magnitude of the typical fluctuations, measured as a fraction of the total number of molecules. There are very few 173 | physical quantities that can actually be measured to a precision of 13 significant digits, and this is not one of them. 174 | The random fluctuations in the number of molecules on each side of the room are simply too small to measure. 175 | 176 | 177 | The Central Limit Theorem 178 | ========================= 179 | 180 | Having worked through one example in detail, you might wonder how general our conclusions are. Do other types of 181 | statistical problems behave roughly the same way? The answer is yes. There is a very general theorem which guarantees 182 | that a wide range of quantities will scale in essentially the same way. 183 | 184 | Before presenting that theorem, I first need to introduce one other very important probability distribution, known as 185 | the *Gaussian* or *normal* distribution: 186 | 187 | .. math:: 188 | p(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{\frac{-(x-\mu)^2}{2 \sigma^2}} 189 | :label: gaussianprobability 190 | 191 | It is shown in :autonumref:`Figure,gaussian`. It is not really as complicated as it looks. The basic form is just :math:`e^{-x^2}`. 192 | Replacing :math:`x` by :math:`x-\mu` shifts the mean of the distribution to :math:`\mu`, and then the exponent is scaled 193 | to make the standard deviation equal :math:`\sigma`. The factor in front is just a normalization, to ensure that 194 | 195 | .. math:: 196 | \int_{-\infty}^{\infty} p(x) dx = 1 197 | 198 | .. figure:: images/gaussian.* 199 | :align: center 200 | :scale: 80 201 | 202 | :autonumber:`Figure,gaussian`. A normal distribution with mean of 0 and standard deviation of 1. 203 | 204 | The name "normal distribution" gives a sense of just how important this distribution is in statistics. It comes up 205 | constantly. We will see the reason for this in just a moment. 206 | 207 | In the last section, we considered the number of oxygen molecules :math:`m` in the left half of the room. You can think 208 | of this quantity as a sum of random values: 209 | 210 | .. math:: 211 | m = \sum_{i=1}^{N} x_i 212 | 213 | Each value :math:`x_i` is either 0 or 1 (depending on which half of the room the molecule is in) with equal probability. 214 | Now we will generalize this to let the values be drawn from an arbitrary distribution. That distribution could be 215 | either discrete or continuous. The only thing we will assume about it is that it has a known mean and standard 216 | deviation. There is a remarkable result called the *Central Limit Theorem*: 217 | 218 | .. admonition:: The Central Limit Theorem 219 | 220 | Consider the sum 221 | 222 | .. math:: 223 | S = \sum_{i=1}^N x_i 224 | 225 | where the values :math:`x_i` are independently drawn from a distribution with mean :math:`\mu_x` and standard 226 | deviation :math:`\sigma_x`. In the limit :math:`N \to \infty`, the sum :math:`S` is distributed according to a 227 | normal distribution with mean :math:`\mu = N \mu_x` and standard deviation :math:`\sigma = \sqrt{N} \sigma_x`. 228 | 229 | This theorem dates back to 1718, when Abraham de Moivre presented a proof of a special case of it in his book 230 | *The Doctrine of Chances*. This book was not, as you might guess, an academic treatise for mathematicians. It was a 231 | book for gamblers, discussing how to win at various games of chance. Don't ever let anyone tell you that mathematics 232 | is not useful! 233 | 234 | This is the reason that normal distributions are so important. No matter what distribution you start out with, once 235 | you add enough values together the result will always be normally distributed. How large :math:`N` needs to be varies, 236 | of course, depending on the initial distribution, but it often does not need to be very large. In many cases, summing 237 | over 10 values is already enough to give quite a good approximation to a normal distribution. 238 | 239 | Notice that our main conclusions from the previous section apply just as well to the general case as they did for our 240 | specific example. The standard deviation is proportional to :math:`\sqrt{N}`, the mean is proportional to :math:`N`, 241 | and their ratio scales as :math:`1/\sqrt{N}`. No matter what distribution the individual values come from, once you add 242 | up a macroscopic number of them the fluctuations will be negligibly small. 243 | -------------------------------------------------------------------------------- /docs/_sources/largenumbers.txt: -------------------------------------------------------------------------------- 1 | .. _very-large-numbers: 2 | 3 | Mathematical Interlude: Very Large Numbers 4 | ########################################## 5 | 6 | Problems in statistical mechanics tend to involve large numbers. Very *very* large numbers. For example, we often 7 | need to average over all the atoms or molecules in a system. For any macroscopic system, that will be on the order of 8 | Avogadro's number: 10\ :sup:`23`, plus or minus a few orders of magnitude. In other cases we average over microstates, 9 | the number of which usually grows *exponentially* in the number of atoms. These are really astronomically large 10 | numbers. 11 | 12 | This has an interesting result. In many cases, we can simply treat averages as exact numbers. Any variation about the 13 | average value is so small, we can completely ignore it. This is a great simplification, and is one of the reasons 14 | statistical mechanics is so successful. 15 | 16 | It also is the reason that, despite its name, statistical mechanics involves very little real statistics. A large part 17 | of the field of statistics deals with the variations about averages: computing the probability of a distribution 18 | producing values in a certain range, determining whether a measured value might plausibly have come from a certain 19 | distribution, etc. But in statistical mechanics, we can usually ignore these questions. The chance of measuring any 20 | value *except* the average one is negligible. 21 | 22 | The main goal of this chapter is to demonstrate this fact. We will do that first through a simple example, and then 23 | through a very general theorem. 24 | 25 | 26 | .. _the-binomial-distribution: 27 | 28 | The Binomial Distribution 29 | ========================= 30 | 31 | Do you ever worry that all the oxygen molecules in the air around you might spontaneously migrate to the other side of 32 | the room, leaving you unable to breathe? It could happen. Each molecule diffuses independently, so there is a small 33 | but finite probability that just by chance, they will all end up in the same half of the room at the same time. 34 | 35 | I hope I am not worrying you. While this is theoretically possible, it is so absurdly improbable that you really can 36 | ignore it. We will first compute just how improbable it is, and then study the statistics of this problem in more 37 | detail. 38 | 39 | Let :math:`N` be the number of oxygen molecules in the room. We will describe the position of each one simply by 40 | whether it is in the left or right half of the room. That gives :math:`2^N` possible arrangements of molecules, every 41 | one of which is equally probable. Of all those arrangements, only one has all the oxygen in the opposite half of the 42 | room from you, so the probability is :math:`1/2^N`. 43 | 44 | Depending on the size of the room you are in, it probably has somewhere in the vicinity of 10\ :sup:`26` oxygen 45 | molecules in it. So the probability of all the oxygen being on the same side of the room at any given moment is 46 | 47 | .. math:: 48 | \frac{1}{2^{10^{26}}} \approx \frac{1}{10^{10000000000000000000000000}} 49 | 50 | That is a very, very large denominator. For comparison here are some much smaller numbers: 51 | 52 | * The age of the universe is on the order of 10\ :sup:`17` seconds. 53 | 54 | * There are somewhere around 10\ :sup:`80` atoms in the visible universe. 55 | 56 | In short, you really do not need to worry about this happening. You would be much better off worrying about more likely 57 | events. For example, that you will be struck by lightning and then, as you try to pick yourself back up, hit by 58 | a meteor falling from space. And then immediately be struck by lightning a second time just for good measure. That is 59 | the sort of thing that is far more likely to happen to you. 60 | 61 | Perhaps you are still not reassured. You may have realized that even if only 90% of the oxygen moved to the other side 62 | of the room, that would still leave you in a rather awkward position. We really want to work out the entire probability 63 | distribution. Let :math:`m` be the number of oxygen molecules in the left half of the room and :math:`N-m` the number 64 | in the right half. Assuming the energy of each molecule is independent of which side of the room it is on, and that the 65 | oxygen molecules are far enough apart that we can ignore their interactions (both reasonable assumptions), the 66 | probability is simply proportional to :math:`\Omega(m)`. Let us consider how that varies with :math:`m`. 67 | 68 | * There is precisely one arrangement of molecules that has all of them on the left side of the room, so 69 | :math:`\Omega(0)=1`. 70 | 71 | * There are :math:`N` possible ways to have exactly one molecule on the left side of the room (any of the :math:`N` 72 | molecules could be the one), so :math:`\Omega(1)=N`. 73 | 74 | * To get two molecules on the left side of the room, we have :math:`N` choices for the first molecule and :math:`N-1` 75 | for the second molecule. But now we have counted each state twice: for any pair of molecules, it does not matter which 76 | order we choose them in. So :math:`\Omega(2)=N(N-1)/2`. 77 | 78 | Continuing like this, we find the general form: 79 | 80 | .. math:: 81 | \Omega(m) = \frac{N!}{m!(N-m)!} 82 | :label: binomialomega 83 | 84 | The probability is then 85 | 86 | .. math:: 87 | p(m) = \frac{1}{2^N} \frac{N!}{m!(N-m)!} 88 | :label: binomialprobability 89 | 90 | This is a special case of the *binomial distribution*. (In the general case, the probability for a particle to be on 91 | the left side of the room could be different from its probability to be on the right side, but for this example we can 92 | ignore that.) It is shown in :autonumref:`Figure,binomial` for several values of :math:`N`. Notice how, as :math:`N` 93 | increases, the probability becomes steadily more concentrated around its midpoint. 94 | 95 | .. figure:: images/binomial.* 96 | :align: center 97 | :scale: 80 98 | 99 | :autonumber:`Figure,binomial`. The binomial distribution for several values of :math:`N`. To make them easier to 100 | compare, all the curves have been normalized to go from 0 to 1 along each axis. 101 | 102 | The name comes from the fact that it has the same form as the terms in the *binomial series*: 103 | 104 | .. math:: 105 | (a+b)^N = \sum_{m=0}^N \frac{N!}{m!(N-m)!}a^Nb^{N-m} 106 | :label: binomialseries 107 | 108 | If we let :math:`a=b=1/2`, this simplifies to 109 | 110 | .. math:: 111 | \left(\frac{1}{2}+\frac{1}{2}\right)^N = \sum_{m=0}^N \frac{N!}{m!(N-m)!} \frac{1}{2^N} 112 | 113 | so that 114 | 115 | .. math:: 116 | \sum_{m=0}^N \frac{N!}{m!(N-m)!} = 2^N 117 | :label: binomialsum 118 | 119 | Let us examine this probability distribution a bit further. We can roughly characterize it by calculating its mean and 120 | standard deviation. The mean is easy: every particle has an equal chance to be on each side of the room, so 121 | :math:`\langle m \rangle = N/2`. Another way to think about this is that the distribution is symmetric: for any value 122 | of :math:`m`, :math:`p(m)=p(N-m)`, from which it directly follows that the mean must be :math:`N/2`. 123 | 124 | The standard deviation is a bit more complicated to derive. As a first step, let us first calculate 125 | 126 | .. math:: 127 | \langle m(m-1) \rangle = \frac{1}{2^N} \sum_{m=0}^N m(m-1) \frac{N!}{m!(N-m)!} 128 | 129 | Notice that the first two terms of the sum are both zero, so we can increase the lower bound to begin from 2. This 130 | allows us to then cancel out the factors of :math:`m(m-1)` that appear in both numerator and denominator: 131 | 132 | .. math:: 133 | \langle m(m-1) \rangle &= \frac{1}{2^N} \sum_{m=2}^N \frac{N!}{(m-2)!(N-m)!} \\ 134 | &= \frac{N(N-1)}{2^N} \sum_{m=2}^N \frac{(N-2)!}{(m-2)!(N-m)!} 135 | 136 | Now make two substitutions: define :math:`X=N-2` and :math:`y=m-2`. This simplifies it to: 137 | 138 | .. math:: 139 | \langle m(m-1) \rangle = \frac{N(N-1)}{2^N} \sum_{y=0}^X \frac{X!}{y!(X-y)!} 140 | 141 | We immediately recognize the sum as being the same one that appeared in equation :eq:`binomialsum`. Replacing it by 142 | :math:`2^X=2^{N-2}`, 143 | 144 | .. math:: 145 | \langle m(m-1) \rangle = \frac{N(N-1)2^{N-2}}{2^N} = \frac{N(N-1)}{4} 146 | 147 | We now have all the pieces we need. Recall that 148 | 149 | .. math:: 150 | Var(m) &= \langle m^2 \rangle - \langle m \rangle^2 \\ 151 | &= \langle m(m-1) \rangle + \langle m \rangle- \langle m \rangle^2 \\ 152 | &= \frac{N(N-1)}{4} + \frac{N}{2} - \frac{N^2}{4} \\ 153 | &= \frac{N}{4} 154 | :label: binomialvariance 155 | 156 | The standard deviation is then 157 | 158 | .. math:: 159 | \sigma \equiv \sqrt{Var(m)} = \frac{\sqrt{N}}{2} 160 | :label: binomialstddev 161 | 162 | Let us take a moment to consider these results. The average value is proportional to :math:`N`, while the standard 163 | deviation is proportional to :math:`\sqrt{N}`. In most cases, what we really care about is the ratio of the two. You 164 | want to know what *fraction* of the oxygen is likely to be on one side of the room, not the specific number of 165 | molecules. A number that would be huge in a broom closet would be negligible in an auditorium. Taking the ratio gives 166 | 167 | .. math:: 168 | \frac{\sigma}{\langle m \rangle} = \frac{1}{\sqrt{N}} 169 | :label: binomialfractionaldeviation 170 | 171 | For an average sized room with 10\ :sup:`26` oxygen molecules, that equals 10\ :sup:`-13`. This is the 172 | magnitude of the typical fluctuations, measured as a fraction of the total number of molecules. There are very few 173 | physical quantities that can actually be measured to a precision of 13 significant digits, and this is not one of them. 174 | The random fluctuations in the number of molecules on each side of the room are simply too small to measure. 175 | 176 | 177 | The Central Limit Theorem 178 | ========================= 179 | 180 | Having worked through one example in detail, you might wonder how general our conclusions are. Do other types of 181 | statistical problems behave roughly the same way? The answer is yes. There is a very general theorem which guarantees 182 | that a wide range of quantities will scale in essentially the same way. 183 | 184 | Before presenting that theorem, I first need to introduce one other very important probability distribution, known as 185 | the *Gaussian* or *normal* distribution: 186 | 187 | .. math:: 188 | p(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{\frac{-(x-\mu)^2}{2 \sigma^2}} 189 | :label: gaussianprobability 190 | 191 | It is shown in :autonumref:`Figure,gaussian`. It is not really as complicated as it looks. The basic form is just :math:`e^{-x^2}`. 192 | Replacing :math:`x` by :math:`x-\mu` shifts the mean of the distribution to :math:`\mu`, and then the exponent is scaled 193 | to make the standard deviation equal :math:`\sigma`. The factor in front is just a normalization, to ensure that 194 | 195 | .. math:: 196 | \int_{-\infty}^{\infty} p(x) dx = 1 197 | 198 | .. figure:: images/gaussian.* 199 | :align: center 200 | :scale: 80 201 | 202 | :autonumber:`Figure,gaussian`. A normal distribution with mean of 0 and standard deviation of 1. 203 | 204 | The name "normal distribution" gives a sense of just how important this distribution is in statistics. It comes up 205 | constantly. We will see the reason for this in just a moment. 206 | 207 | In the last section, we considered the number of oxygen molecules :math:`m` in the left half of the room. You can think 208 | of this quantity as a sum of random values: 209 | 210 | .. math:: 211 | m = \sum_{i=1}^{N} x_i 212 | 213 | Each value :math:`x_i` is either 0 or 1 (depending on which half of the room the molecule is in) with equal probability. 214 | Now we will generalize this to let the values be drawn from an arbitrary distribution. That distribution could be 215 | either discrete or continuous. The only thing we will assume about it is that it has a known mean and standard 216 | deviation. There is a remarkable result called the *Central Limit Theorem*: 217 | 218 | .. admonition:: The Central Limit Theorem 219 | 220 | Consider the sum 221 | 222 | .. math:: 223 | S = \sum_{i=1}^N x_i 224 | 225 | where the values :math:`x_i` are independently drawn from a distribution with mean :math:`\mu_x` and standard 226 | deviation :math:`\sigma_x`. In the limit :math:`N \to \infty`, the sum :math:`S` is distributed according to a 227 | normal distribution with mean :math:`\mu = N \mu_x` and standard deviation :math:`\sigma = \sqrt{N} \sigma_x`. 228 | 229 | This theorem dates back to 1718, when Abraham de Moivre presented a proof of a special case of it in his book 230 | *The Doctrine of Chances*. This book was not, as you might guess, an academic treatise for mathematicians. It was a 231 | book for gamblers, discussing how to win at various games of chance. Don't ever let anyone tell you that mathematics 232 | is not useful! 233 | 234 | This is the reason that normal distributions are so important. No matter what distribution you start out with, once 235 | you add enough values together the result will always be normally distributed. How large :math:`N` needs to be varies, 236 | of course, depending on the initial distribution, but it often does not need to be very large. In many cases, summing 237 | over 10 values is already enough to give quite a good approximation to a normal distribution. 238 | 239 | Notice that our main conclusions from the previous section apply just as well to the general case as they did for our 240 | specific example. The standard deviation is proportional to :math:`\sqrt{N}`, the mean is proportional to :math:`N`, 241 | and their ratio scales as :math:`1/\sqrt{N}`. No matter what distribution the individual values come from, once you add 242 | up a macroscopic number of them the fluctuations will be negligibly small. 243 | --------------------------------------------------------------------------------