/
basics.html
494 lines (437 loc) · 26.9 KB
/
basics.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="X-UA-Compatible" content="IE=Edge" />
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<title>Basics of Statistical Mechanics — Statistical Physics Notes</title>
<link rel="stylesheet" href="../_static/basic.css" type="text/css" />
<link rel="stylesheet" href="../_static/pygments.css" type="text/css" />
<link rel="stylesheet" href="../_static/modify.css" type="text/css" />
<link rel="stylesheet" href="../_static/bootswatch-3.3.4/readable/bootstrap.min.css" type="text/css" />
<link rel="stylesheet" href="../_static/bootstrap-sphinx.css" type="text/css" />
<link rel="stylesheet" href="../_static/modify.css" type="text/css" />
<script type="text/javascript" id="documentation_options" data-url_root="../" src="../_static/documentation_options.js"></script>
<script type="text/javascript" src="../_static/jquery.js"></script>
<script type="text/javascript" src="../_static/underscore.js"></script>
<script type="text/javascript" src="../_static/doctools.js"></script>
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<script type="text/javascript" src="../_static/js/jquery-1.11.0.min.js"></script>
<script type="text/javascript" src="../_static/js/jquery-fix.js"></script>
<script type="text/javascript" src="../_static/bootstrap-3.3.4/js/bootstrap.min.js"></script>
<script type="text/javascript" src="../_static/bootstrap-sphinx.js"></script>
<link rel="index" title="Index" href="../genindex.html" />
<link rel="search" title="Search" href="../search.html" />
<link rel="next" title="\(\Gamma\) Space and \(\mu\) Space" href="gamma-space-mu-space.html" />
<link rel="prev" title="Equilibrium Statistical Mechanics Summary" href="summary.html" />
<meta charset='utf-8'>
<meta http-equiv='X-UA-Compatible' content='IE=edge,chrome=1'>
<meta name='viewport' content='width=device-width, initial-scale=1.0, maximum-scale=1'>
<meta name="apple-mobile-web-app-capable" content="yes">
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/font-awesome/4.5.0/css/font-awesome.min.css">
<script type="text/x-mathjax-config">
MathJax.Hub.Config({ TeX: { extensions: ["color.js","cancel.js", "AMSmath.js", "AMSsymbols.js"] }});
</script>
<script type="text/x-mathjax-config">
MathJax.Hub.Register.StartupHook("TeX Jax Ready",function () {
MathJax.Hub.Insert(MathJax.InputJax.TeX.Definitions.macros,{
cancel: ["Extension","cancel"]
});
});
</script>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
tex2jax: {
skipTags: ['script', 'noscript', 'style', 'textarea', 'pre', 'code']
}
});
</script>
<script type="text/x-mathjax-config">
MathJax.Hub.Queue(function() {
var all = MathJax.Hub.getAllJax(), i;
for(i=0; i < all.length; i += 1) {
all[i].SourceElement().parentNode.className += ' has-jax';
}
});
</script>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
tex2jax: {
processEscapes: true
}
});
</script>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
"fast-preview": {
Chunks: {EqnChunk: 10000, EqnChunkFactor: 1, EqnChunkDelay: 0},
color: "inherit!important",
updateTime: 30, updateDelay: 6,
messageStyle: "none",
disabled: false
}
});
</script>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
TeX: {
Macros: {
overlr: ['\\overset\\leftrightarrow{\#1}',1],
overl: ['\\overset\leftarrow{\#1}',1],
overr: ['\\overset\rightarrow{\#1}',1],
bra: ['\\left\\langle \#1\\right|',1],
ket: ['\\left| \#1\\right\\rangle',1],
braket: ['\\langle \#1 \\mid \#2 \\rangle',2],
avg: ['\\left< \#1 \\right>',1],
slashed: ['\\cancel{\#1}',1],
bold: ['\\boldsymbol{\#1}',1],
sech: ['\\operatorname{sech}{\#1}',1],
csch: ['\\operatorname{csch}{\#1}',1],
Tr: ['\\operatorname{Tr}{\#1}',1]
}
}
});
</script>
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<!-- Toggle Environment Begin -->
<script type="text/javascript">
$(document).ready(function() {
$(".toggle > *").hide();
$(".toggle .admonition-title").show();
$(".toggle .admonition-title").click(function() {
$(this).parent().children().not(".admonition-title").toggle(400);
$(this).parent().children(".admonition-title").toggleClass("open");
})
});
</script>
<!-- Toggle Environment End -->
<!-- Google tag (gtag.js) -->
<script async src="https://www.googletagmanager.com/gtag/js?id=G-B0WKN82ZGD"></script>
<script>
window.dataLayer = window.dataLayer || [];
function gtag(){dataLayer.push(arguments);}
gtag('js', new Date());
gtag('config', 'G-B0WKN82ZGD');
</script>
</head><body>
<div id="navbar" class="navbar navbar-inverse navbar-default ">
<div class="container">
<div class="navbar-header">
<!-- .btn-navbar is used as the toggle for collapsed navbar content -->
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".nav-collapse">
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="../index.html">
Statistical Physics</a>
<span class="navbar-text navbar-version pull-left"><b></b></span>
</div>
<div class="collapse navbar-collapse nav-collapse">
<ul class="nav navbar-nav">
<li class="dropdown globaltoc-container">
<a role="button"
id="dLabelGlobalToc"
data-toggle="dropdown"
data-target="#"
href="../index.html">Chapters <b class="caret"></b></a>
<ul class="dropdown-menu globaltoc"
role="menu"
aria-labelledby="dLabelGlobalToc"><ul>
<li class="toctree-l1"><a class="reference internal" href="../vocabulary/index.html">Vocabulary and Program</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../vocabulary/functions.html">Functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../vocabulary/stability-analysis.html">Stability Analysis</a></li>
<li class="toctree-l2"><a class="reference internal" href="../vocabulary/integrals.html">Integrals</a></li>
<li class="toctree-l2"><a class="reference internal" href="../vocabulary/transforms.html">Transforms</a></li>
<li class="toctree-l2"><a class="reference internal" href="../vocabulary/green.html">Green’s Function</a></li>
<li class="toctree-l2"><a class="reference internal" href="../vocabulary/computations.html">Computations</a></li>
</ul>
</li>
</ul>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../thermodynamics/index.html">Thermodynamics</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../thermodynamics/summary.html">Summary</a></li>
<li class="toctree-l2"><a class="reference internal" href="../thermodynamics/topics/index.html">Topics on Thermodynamics</a></li>
</ul>
</li>
</ul>
<ul class="current">
<li class="toctree-l1 current"><a class="reference internal" href="index.html">Equilibrium Statistical Mechanics</a><ul class="current">
<li class="toctree-l2"><a class="reference internal" href="summary.html">Equilibrium Statistical Mechanics Summary</a></li>
<li class="toctree-l2 current"><a class="current reference internal" href="#">Basics of Statistical Mechanics</a></li>
<li class="toctree-l2"><a class="reference internal" href="gamma-space-mu-space.html"><span class="math notranslate nohighlight">\(\Gamma\)</span> Space and <span class="math notranslate nohighlight">\(\mu\)</span> Space</a></li>
<li class="toctree-l2"><a class="reference internal" href="macroscopic-states-and-microscopic-states.html">Macroscopic States and Microscropic State</a></li>
<li class="toctree-l2"><a class="reference internal" href="most-probable-distribution.html">Most Probable Distribution</a></li>
<li class="toctree-l2"><a class="reference internal" href="ho-dos.html">Harmonic Oscillator and Density of States</a></li>
<li class="toctree-l2"><a class="reference internal" href="gibbs-mixing-paradox.html">Gibbs Mixing Paradox</a></li>
<li class="toctree-l2"><a class="reference internal" href="observables.html">Observables in Statistical Physics</a></li>
<li class="toctree-l2"><a class="reference internal" href="debye-model.html">Debye Model</a></li>
<li class="toctree-l2"><a class="reference internal" href="phase-transitions.html">Phase Transitions</a></li>
<li class="toctree-l2"><a class="reference internal" href="gas-revisited.html">Gas Revisited</a></li>
<li class="toctree-l2"><a class="reference internal" href="ising-model.html">Ising Model</a></li>
<li class="toctree-l2"><a class="reference internal" href="systematic-view.html">A More Systematic View</a></li>
<li class="toctree-l2"><a class="reference internal" href="monte-carlo.html">Monte Carlo Method</a></li>
<li class="toctree-l2"><a class="reference internal" href="ensembles.html">Ensembles</a></li>
<li class="toctree-l2"><a class="reference internal" href="topics/index.html">Topics on Equilibrium Statistical Mechanics</a></li>
</ul>
</li>
</ul>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../nonequilibrium/index.html">Stochastic And Non-Equilibrium Statistical Mechanics</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../nonequilibrium/important-questions.html">Important Questions of Statistical Mechanics</a></li>
<li class="toctree-l2"><a class="reference internal" href="../nonequilibrium/master-eqn.html">Master Equation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../nonequilibrium/solving-master-eqn.html">Solving Master Equations</a></li>
<li class="toctree-l2"><a class="reference internal" href="../nonequilibrium/master-eqn-examples.html">Examples of Master Equations</a></li>
<li class="toctree-l2"><a class="reference internal" href="../nonequilibrium/smoluchowski-eqn.html">Smoluchowski Equation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../nonequilibrium/effect-of-defects.html">Effect of Defects</a></li>
<li class="toctree-l2"><a class="reference internal" href="../nonequilibrium/brownian-motion.html">Brownian Motion</a></li>
<li class="toctree-l2"><a class="reference internal" href="../nonequilibrium/quantum-master-eqn.html">Quantum Master Equation</a></li>
</ul>
</li>
</ul>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../topics/index.html">Some Topics in Statistical Mechanics</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../topics/epidemics/index.html">Epidemics</a></li>
<li class="toctree-l2"><a class="reference internal" href="../topics/information-theory-and-statistical-mechanics.html">Information Theory and Statistical Mechanics</a></li>
</ul>
</li>
</ul>
</ul>
</li>
<li class="dropdown">
<a role="button"
id="dLabelLocalToc"
data-toggle="dropdown"
data-target="#"
href="#">Sections <b class="caret"></b></a>
<ul class="dropdown-menu localtoc"
role="menu"
aria-labelledby="dLabelLocalToc"><ul>
<li><a class="reference internal" href="#">Basics of Statistical Mechanics</a><ul>
<li><a class="reference internal" href="#phase-space">Phase Space</a></li>
<li><a class="reference internal" href="#boltzmann-factor">Boltzmann Factor</a></li>
<li><a class="reference internal" href="#partition-function">Partition Function</a></li>
<li><a class="reference internal" href="#magnetization">Magnetization</a></li>
<li><a class="reference internal" href="#heat-capacity">Heat Capacity</a></li>
<li><a class="reference internal" href="#specific-heat">Specific Heat</a></li>
<li><a class="reference internal" href="#importance-of-dimensions">Importance of Dimensions</a></li>
</ul>
</li>
</ul>
</ul>
</li>
<li>
<a href="summary.html" title="Previous Chapter: Equilibrium Statistical Mechanics Summary"><span class="glyphicon glyphicon-chevron-left visible-sm"></span><span class="hidden-sm hidden-tablet">« Equilibrium S...</span>
</a>
</li>
<li>
<a href="gamma-space-mu-space.html" title="Next Chapter: \(\Gamma\) Space and \(\mu\) Space"><span class="glyphicon glyphicon-chevron-right visible-sm"></span><span class="hidden-sm hidden-tablet">\(\Gamma\) Sp... »</span>
</a>
</li>
</ul>
<form class="navbar-form navbar-right" action="../search.html" method="get">
<div class="form-group">
<input type="text" name="q" class="form-control" placeholder="Search" />
</div>
<input type="hidden" name="check_keywords" value="yes" />
<input type="hidden" name="area" value="default" />
</form>
</div>
</div>
</div>
<div class="container">
<div class="row">
<div class="col-md-12 content">
<div class="section" id="basics-of-statistical-mechanics">
<span id="index-0"></span><h1>Basics of Statistical Mechanics<a class="headerlink" href="#basics-of-statistical-mechanics" title="Permalink to this headline">¶</a></h1>
<div class="section" id="phase-space">
<h2>Phase Space<a class="headerlink" href="#phase-space" title="Permalink to this headline">¶</a></h2>
<p>Why are statistics important? In statistical physics, we are in fact dealing with DoFs in the order of Avogadro’s number, most of the time. If we are going to calculate the dynamics of these systems by combining all the dynamics of each particles, it becomes rather difficult. It already takes a huge amount of storage to store one screenshot of the system. Assuming each DoF take only 8 bits, we need <span class="math notranslate nohighlight">\(10^23\)</span> bytes that is <span class="math notranslate nohighlight">\(10^17\)</span> GB. It seems to be impossible to store this one snapshot of the system. It is time to steer away from this Newtonian approach.</p>
<div class="figure align-center" id="figure-1">
<a class="reference internal image-reference" href="../_images/newtonian-dream.svg"><img alt="Newton's Dream" src="../_images/newtonian-dream.svg" width="60%" /></a>
<p class="caption"><span class="caption-number">Fig. 8 </span><span class="caption-text">Newton’s plan of mechanics. Mechanics was in the center of all physics.</span></p>
</div>
<p>What is mechanics? Mechanics is the thinking that deals with dynamics of object in the following way:</p>
<ul class="simple">
<li>Description of initial state;</li>
<li>Time evolution of the system;</li>
<li>Extraction of observables.</li>
</ul>
<p>The initial state of an object is exactly what the name indicates. In a coordinate system approach, the initial state should include the coordinates and the time derivative of the coordinates since we are interested in dynamics. In fact, the coordinates and time derivative of them form a phase space. In general, a vector in phase space is a complete description of the state of an object. Thus time evolution of the object is simply the motion of state vector in phase space. Finally, we will do whatever is needed to extract observables. For example, we could trivially use the projection of points in phase space to get the position or velocity.</p>
<p>The problem arise when it comes to a system with a large amount of particles. As mentioned previously, it’s in general not possible to record all these DoFs. We need a completely new scheme.</p>
</div>
<div class="section" id="boltzmann-factor">
<h2>Boltzmann Factor<a class="headerlink" href="#boltzmann-factor" title="Permalink to this headline">¶</a></h2>
<div class="math notranslate nohighlight">
\[\text{probability of a point in phase space} = \exp(-\frac{E}{k_B T})\]</div>
<p>Boltzmann factor gives us the (not normalized) probability of the system staying on a phase space state with energy <span class="math notranslate nohighlight">\(E\)</span>.</p>
<div class="note admonition">
<p class="first admonition-title">Why Boltzmann Factor</p>
<p>Why does Boltzmann factor appear a lot in equilibrium statistical mechanics? Equilibrium of the system means when we add infinitesimal amount of energy to the whole thing including system and reservoir, a characteristic quantity <span class="math notranslate nohighlight">\(C(E) = C_S C_R\)</span> won’t change. That is the system and the reservoir will have the same changing rate of the characteristic quantity when energy is changed, i.e.,</p>
<div class="math notranslate nohighlight">
\[\frac{\partial \ln C_S}{\partial E_S} = - \frac{\partial \ln C_R}{\partial E_R} .\]</div>
<p>We have <span class="math notranslate nohighlight">\(\mathrm dE_1 = -\mathrm dE_2\)</span> in a equilibrium state. They should both be a constant, which we set to <span class="math notranslate nohighlight">\(\beta\)</span>. Finally we have something like</p>
<div class="math notranslate nohighlight">
\[\frac{\partial \ln C_S}{\partial E_S} = \beta\]</div>
<p>which will give us a Boltzmann factor there.</p>
<p class="last">This is just a very simple procedure to show that Boltzmann factor is kind of a natural factor in equilibrium system.</p>
</div>
</div>
<div class="section" id="partition-function">
<h2>Partition Function<a class="headerlink" href="#partition-function" title="Permalink to this headline">¶</a></h2>
<p>For a given Hamiltonian H, the (classical) partition function Z is</p>
<div class="math notranslate nohighlight">
\[Z = \int d p \int d x e^{-\beta H}\]</div>
<p>A simple example is the Harmonic Oscillator,</p>
<div class="math notranslate nohighlight">
\[H = \frac{p^2}{2m} + \frac{1}{2} q x^2\]</div>
<p>The partition function</p>
<div class="math notranslate nohighlight">
\[Z = \int e^{-\beta p^2/(2m)} d q \int e^{-\beta \frac{1}{2} q x^2 } d x = 2\pi \sqrt{m/q} \frac{1}{\beta}\]</div>
<p>Energy</p>
<div class="math notranslate nohighlight">
\[E = \frac{1}{Z} \int \int e^{-\beta p^2/(2m)} e^{-\beta \frac{1}{2} q x^2 } H d p d x = \cdots = k_B T\]</div>
<p>(This result is obvious if we think about equipartition theorem.)</p>
<p>A more clever approach for the energy is to take the derivative of partition function over <span class="math notranslate nohighlight">\(\beta\)</span>, which exactly is</p>
<div class="math notranslate nohighlight">
\[\langle E \rangle = -\frac{\partial }{\partial \beta } \ln Z\]</div>
<p>In our simple case,</p>
<div class="math notranslate nohighlight">
\[\ln Z = -\frac{\partial}{\partial \beta} \left(\ln (k_B T) + \mathrm{Some Constant} \right)= k_B T\]</div>
<p>This is the power of partition function. To continue the SHO example, we find the specific heat is</p>
<div class="math notranslate nohighlight">
\[C = k_B\]</div>
<div class="toggle admonition">
<p class="first admonition-title">Does This Result Depend on SHO</p>
<p class="last">This result has nothing to do with the detail of the SHO, no matter what mass they have, no matter what potential constant <span class="math notranslate nohighlight">\(q\)</span> they have, no matter what kind of initial state they have. All the characteristic quantities of SHO are irrelevant. Why? Mathematically, it’s because we have Gaussian integral here. <strong>But what is the physics behind this?</strong> Basicly this classical limit is a high temperature limit.</p>
</div>
</div>
<div class="section" id="magnetization">
<h2>Magnetization<a class="headerlink" href="#magnetization" title="Permalink to this headline">¶</a></h2>
<p>We have such a result in an experiment of magnetization with N magnetic dipoles in 1D.</p>
<img alt="../_images/magnetizationExp.jpg" class="align-center" src="../_images/magnetizationExp.jpg" />
<p>How can we describe this with a theory?</p>
<p>It’s not possible to describe the system by writing down the dynamics of each magnetic dipole. So we have to try some macroscpic view of the system. Probability theory is a great tool for this. The probability of a dipole on a energy state <span class="math notranslate nohighlight">\(E_i\)</span> is</p>
<div class="math notranslate nohighlight">
\[P(E_i) = \frac{\exp(-\beta E_i)}{\sum_{i=1}^{n} \exp(-\beta E_i)} .\]</div>
<p>So the megnetization in this simple case is</p>
<div class="math notranslate nohighlight">
\[M = (\mu N e^{\beta \mu B} - \mu N e^{-\beta \mu B})/(\exp(\beta \mu B) + \exp(-\beta \mu B)) = \mu N \tanh (\beta \mu B)\]</div>
<div class="toggle admonition">
<p class="first admonition-title">Python Code</p>
<p>Use ipython notebook to display this result. The original notebook can be downloaded from <a class="reference external" href="http://emptymalei.github.io/StatisticalPhysics/equilibrium/display.ipynb">here</a>. (Just put the link to <a class="reference external" href="http://nbviewer.ipython.org">nbviewer</a> and everyone can view online.)</p>
<div class="code python highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="n">pylab</span> <span class="n">inline</span>
<span class="kn">from</span> <span class="nn">pylab</span> <span class="kn">import</span> <span class="o">*</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Populating</span> <span class="n">the</span> <span class="n">interactive</span> <span class="n">namespace</span> <span class="kn">from</span> <span class="nn">numpy</span> <span class="ow">and</span> <span class="n">matplotlib</span>
</pre></div>
</div>
<div class="code python highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">x</span><span class="o">=</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
<span class="n">y</span><span class="o">=</span><span class="n">tanh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
</pre></div>
</div>
<div class="code python last highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">figure</span><span class="p">()</span>
<span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="s1">'r'</span><span class="p">)</span>
<span class="n">xlabel</span><span class="p">(</span><span class="s1">'External Magnetic Field'</span><span class="p">)</span>
<span class="n">ylabel</span><span class="p">(</span><span class="s1">'M'</span><span class="p">)</span>
<span class="n">title</span><span class="p">(</span><span class="s1">'Tanh theory'</span><span class="p">)</span>
<span class="n">show</span><span class="p">();</span>
</pre></div>
</div>
</div>
<img alt="../_images/display_2_0.png" class="align-center" src="../_images/display_2_0.png" />
<p>This is exactly the thing we saw in the experiment.</p>
<p>This can be classified as a category of problems. In this specific example we see saturation of magnetization. However this is not alway true.</p>
<div class="note admonition">
<p class="first admonition-title">Examples</p>
<p class="last">Examples can be shown here.</p>
</div>
</div>
<div class="section" id="heat-capacity">
<h2>Heat Capacity<a class="headerlink" href="#heat-capacity" title="Permalink to this headline">¶</a></h2>
<p>Another category of problems is temperature related. For example, a study of average energy with change temperature.</p>
<p>For the paramagnetic example, the energy of the system is</p>
<div class="math notranslate nohighlight">
\[E = -(\mu B N e^{\beta \mu B} - \mu N e^{-\beta \mu B})/(\exp(\beta \mu B) + \exp(-\beta \mu B)) = -\mu N B \tanh (\beta \mu B)\]</div>
<p>Obviously, no phase transition would occur. But if we introduce self interactions between dipoles and go to higher dimensions, it’s possible to find phase transitions.</p>
</div>
<div class="section" id="specific-heat">
<h2>Specific Heat<a class="headerlink" href="#specific-heat" title="Permalink to this headline">¶</a></h2>
<div class="math notranslate nohighlight">
\[C = \frac{d}{T}\langle E \rangle\]</div>
<p>Check the behavior of specific heat,</p>
<ol class="arabic simple">
<li>Is there a Discontinuity?</li>
<li>Constant?</li>
<li>Blow up?</li>
<li>Converge?</li>
</ol>
<p>Specific heat can be used for second order phase transition. An simple example of this is Landau theory.</p>
</div>
<div class="section" id="importance-of-dimensions">
<h2>Importance of Dimensions<a class="headerlink" href="#importance-of-dimensions" title="Permalink to this headline">¶</a></h2>
<p><a class="reference external" href="http://nbviewer.ipython.org/github/emptymalei/StatisticalPhysics/blob/master/equilibrium/homework/StatMech_HW1.ipynb">IPython Notebook about heat capacity of systems with different dimensions.</a> .</p>
</div>
</div>
</div>
</div>
</div>
<hr>
<!-- <script src="https://utteranc.es/client.js"
repo="emptymalei/statisticalphysics"
issue-term="title"
label="Comment"
theme="github-light"
crossorigin="anonymous"
async>
</script> -->
<div class="container" style="margin-top:10px;padding-top:10px;padding-bottom:10px;">
<script src="https://giscus.app/client.js"
data-repo="emptymalei/statisticalphysics"
data-repo-id="MDEwOlJlcG9zaXRvcnk1MzM3MjU5NA=="
data-category="Comments"
data-category-id="DIC_kwDOAy5mss4B_dLp"
data-mapping="pathname"
data-reactions-enabled="1"
data-emit-metadata="0"
data-theme="light"
crossorigin="anonymous"
async>
</script>
</div>
<hr>
<div class="container" style="border-top:solid 1px black;margin-top:10px;padding-top:10px;padding-bottom:10px;">
<p class="pull-right">
<a href="#"><span class="label label-info">Back to top</span></a>
</p>
<p>
<span>© <a href="https://leima.is">2021, Lei Ma <span class="glyphicon glyphicon-link" aria-hidden="true"></span></a> </span>|
<span>Created with
<a href="http://sphinx.pocoo.org/">Sphinx</a> and <span class="glyphicon glyphicon-heart" aria-hidden="true"></span>.
</span>|
<span><span class="glyphicon glyphicon-cloud" aria-hidden="true"></span>
<a href="https://github.com/emptymalei/statisticalphysics">Source on GitHub</a>
</span>|
<span><span class="glyphicon glyphicon-book" aria-hidden="true"></span>
<a href="http://physics.leima.is">Physics Notebook</a>
</span>
<span><span class="glyphicon glyphicon-book" aria-hidden="true"></span>
<a href="http://datumorphism.github.io">Datumorphism</a>
</span>
|
<span><span class="glyphicon glyphicon-list" aria-hidden="true"></span>
<a href="/genindex.html">Index</a>
</span> |
<span><span class="glyphicon glyphicon-paperclip" aria-hidden="true"></span>
<a href="../_sources/equilibrium/basics.rst.txt"
rel="nofollow">Page Source</a>
</span>
</p>
<hr>
<script>
window.jQuery || document.write('<script src="_static/js/vendor/jquery-1.9.1.min.js"><\/script>')
</script>
</div>
</body>
</html>