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<div class="section" id="equilibrium-statistical-mechanics-summary">
<h1>Equilibrium Statistical Mechanics Summary<a class="headerlink" href="#equilibrium-statistical-mechanics-summary" title="Permalink to this headline">¶</a></h1>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last">This is a review of equilibrium statistical mechanics. Though I called it a review, it is more like a list of keywords at this moment.</p>
</div>
<div class="section" id="review-of-thermodynamics">
<h2>Review of Thermodynamics<a class="headerlink" href="#review-of-thermodynamics" title="Permalink to this headline">¶</a></h2>
<ol class="arabic">
<li><p class="first">Description of States in statistical mechanmics: thermodynamical quantities as macroscopic state;</p>
</li>
<li><p class="first">Kinematics: equation of state; <a class="reference internal" href="../thermodynamics/summary.html#thermodynamical-potentials"><span class="std std-ref">Thermodynamic Potentials</span></a>.</p>
<div class="figure align-center" id="figure-1">
<a class="reference internal image-reference" href="../_images/thermodynamicPotentials.png"><img alt="../_images/thermodynamicPotentials.png" src="../_images/thermodynamicPotentials.png" style="width: 100%;" /></a>
<p class="caption"><span class="caption-number">Fig. 6 </span><span class="caption-text">The relationship between different thermodynamic potentials. There are three different couplings and five different potentials. For more details please read vocabulary <a class="reference internal" href="../thermodynamics/summary.html#thermodynamical-potentials"><span class="std std-ref">Thermodynamic Potentials</span></a> .</span></p>
</div>
</li>
<li><p class="first">First principles: <a class="reference internal" href="../thermodynamics/summary.html#laws-of-thermodynamics"><span class="std std-ref">The Laws of Four</span></a></p>
</li>
<li><p class="first">Dynamics: Phase transition; Stability; Response</p>
</li>
</ol>
</div>
<div class="section" id="description-of-the-microstates">
<span id="discription-of-microstates"></span><span id="mu-space-and-gamma-space"></span><h2>Description of the Microstates<a class="headerlink" href="#description-of-the-microstates" title="Permalink to this headline">¶</a></h2>
<p id="index-0">For a system with <span class="math notranslate nohighlight">\(N\)</span> particles of <span class="math notranslate nohighlight">\(r\)</span> degrees of freedom, we could always describe the microstates of the system by looking at the state of each particle. There are at least two different point of views, the <span class="math notranslate nohighlight">\(\mu\)</span> space (mu space) and the <span class="math notranslate nohighlight">\(\Gamma\)</span> space (Gamma space).</p>
<p>The <span class="math notranslate nohighlight">\(\mu\)</span> space is a <span class="math notranslate nohighlight">\(r\)</span> dimensional space where each dimension corresponds to one degree of freedom of the particle. Thus a point in the <span class="math notranslate nohighlight">\(\mu\)</span> space represents a the state of one particle. To represent the microstate of the whole system, we need <span class="math notranslate nohighlight">\(N\)</span> points in the <span class="math notranslate nohighlight">\(\mu\)</span> space.</p>
<p>The <span class="math notranslate nohighlight">\(\Gamma\)</span> space is a <span class="math notranslate nohighlight">\(rN\)</span> dimensional space. In the <span class="math notranslate nohighlight">\(\Gamma\)</span> space, we have a holistic view. Each point in the <span class="math notranslate nohighlight">\(\Gamma\)</span> space represents the state of all the particles. For example, we use the first <span class="math notranslate nohighlight">\(r\)</span> dimensions out of the <span class="math notranslate nohighlight">\(rN\)</span> dimension to represent the state of the first particle, the next <span class="math notranslate nohighlight">\(r\)</span> dimensions to represent the state of the second particle, and so on.</p>
<div class="toggle admonition">
<p class="first admonition-title">Why Distingushing between Microstates and Macrostates</p>
<p class="last">In physical systems, we observe limited quantities regarding the internal structure. If we take the Bayesian point of view, we have the freedom to choose the amount of information we would like to use as priors. In statistical mechanics, macrostates is related to our view of the priors.</p>
</div>
</div>
<div class="section" id="what-is-statistical-mechanics-1">
<span id="what-is-statistical-mechanics"></span><h2>What is Statistical Mechanics<a class="headerlink" href="#what-is-statistical-mechanics-1" title="Permalink to this headline">¶</a></h2>
<p>Physical systems are usually composed of a large amount of particles. In principle, we could calculate the observable quantities if we know the exact motions of the particles. For example, we only need the momentum transfer per unit area to know the pressure of the gas and momentum transfer could be calculated if we know the motion of the particles.</p>
<p>This method is obviously unrealistic given the number of particles that we are dealing with. Alternatively, we could figure out the probabilities of each possible values of the observable quantities, i.e., the probability of the system being on each point in the <span class="math notranslate nohighlight">\(\Gamma\)</span> space. For each microscopic state, we could calculate the thermodynamic observables corresponding to it.</p>
<p>However, this approach requires a first principle that we could use to figure out the distribution of the observables <span class="math notranslate nohighlight">\(\{\mathscr O_i\}\)</span>, i.e., <span class="math notranslate nohighlight">\(p(\{\mathscr O_i\})\)</span>. More regoriously, it is expected that we derive a theory that tells us the conditional probability <span class="math notranslate nohighlight">\(p(\{\mathscr O_i\} \vert t, \{m_i, r_i\})\)</span> where <span class="math notranslate nohighlight">\(\{m_i, r_i\}\)</span> is a set of features that are defined by the materials, the enviroment and the restrictions, <span class="math notranslate nohighlight">\(t\)</span> is time.</p>
<div class="note admonition">
<p class="first admonition-title">A Bayesian View</p>
<p>In Bayesian statistics,</p>
<div class="math notranslate nohighlight">
\[p(\{\mathscr O_i\}, \{m_i, r_i\}) = p(\{\mathscr O_i\} \mid \{m_i, r_i\}) p(\{m_i, r_i\} ) = p(\{m_i, r_i\}\mid \{\mathscr O_i\} ) p(\{\mathscr O_i\}).\]</div>
<p><span class="math notranslate nohighlight">\(p(\{\mathscr O_i\})\)</span> is the prior distribution and is observed in experiments.</p>
<p>Ideally, it would be a perfect model if we determine the joint distribution <span class="math notranslate nohighlight">\(p(\{\mathscr O_i\}, \{m_i, r_i\})\)</span>. The marginalized distribution <span class="math notranslate nohighlight">\(p(\{\mathscr O_i\}) = \int p(\{\mathscr O_i\}, \{m_i, r_i\}) \mathrm d\{m_i, r_i\}\)</span> corresponds to our observations. This joint distribution connects the microscopic view and the macroscopic view. However, it is utterly impossible to calculate the details of the probabilities for areal-world statistical system. First, we have no information of the initial state. We have to introduce the stochastic processes to describe the states. Secondly, this joint probability often becomes very hard to compute when we introduce interactions between the particles.</p>
<p>Nevertheless, we can still perform some analysis using approximations. If we ask for the probability of the microscopic states for given macroscopic observables, we have the inference</p>
<div class="math notranslate nohighlight">
\[p(\{m_i, r_i\}\mid \{\mathscr O_i\} ) = \frac{p(\{\mathscr O_i\} \mid \{m_i, r_i\}) p(\{m_i, r_i\} ) }{ p(\{\mathscr O_i\}) },\]</div>
<p>where <span class="math notranslate nohighlight">\(p(\{\mathscr O_i\} \mid \{m_i, r_i\})\)</span> is given by a physics model such as momentum transfer as pressure of ideal gas, <span class="math notranslate nohighlight">\(p(\{m_i, r_i\}\)</span> is given by some prior knowledge such as Boltzmann’s equal a priori probability, <span class="math notranslate nohighlight">\(p(\{\mathscr O_i\})\)</span> is from observation.</p>
<p>Computation aside, this formalism brings in the question of how our statistical theory of matter can be validated. There are two sides in the theory: A statistical model that predicts the most prominent values of the observables as well as the confidence, and the sampled probability distributions of observables from our experiments. To validate the statistical model, we perform some kind of hypothesis test.</p>
<p class="last">On the other hand, real-world statistical physics deals with a huge amount of particles which leads to an extremely narrow confidence interval. We can simply match the results without considering the fluctuations.</p>
</div>
<p>For example, the Boltzmann theory assume equal a priori probabilities for the microstates. In Boltzmann theory, we need two aspects of knowledge to understand the statistical system.</p>
<ol class="arabic simple">
<li>The distribution of the mirostates, which has been assumed to be equal.</li>
<li>How the energy of combinations of single particles are calculated. For example, this refers to the calculation of the energy levels in quantum mechanics.</li>
</ol>
</div>
<div class="section" id="the-two-approaches-of-statistical-mechanics">
<h2>The Two Approaches of Statistical Mechanics<a class="headerlink" href="#the-two-approaches-of-statistical-mechanics" title="Permalink to this headline">¶</a></h2>
<p>The <strong>probability distribution of the microscopic states</strong> of the system, <span class="math notranslate nohighlight">\(p(\{O_i\})\)</span>, is needed to estimate the observables <span class="math notranslate nohighlight">\(\{O_i\}\)</span>. For example, to estimate the energy of the system, we take the statistical average using the distribution <span class="math notranslate nohighlight">\(\int E p(E) \mathrm dE\)</span>.</p>
<p>However the microscopic state of the system is not known in general. We have to apply some assumptions and tricks.</p>
<p>There are two famous approaches developed in statistical mechanics. The Boltzmann’s approach is utilizing the most probable distributions while the Gibbs’ approach is using ensembles. They do not only differ from the way of estimating the probabilities of the states but also differ philosophically.</p>
<div class="figure align-center" id="figure-2">
<a class="reference internal image-reference" href="../_images/BoltzmannVSGibbs.png"><img alt="../_images/BoltzmannVSGibbs.png" src="../_images/BoltzmannVSGibbs.png" style="width: 100%;" /></a>
<p class="caption"><span class="caption-number">Fig. 7 </span><span class="caption-text">Modeling of the two theories. Refer to <a class="reference internal" href="most-probable-distribution.html#most-probable-distribution"><span class="std std-ref">Most Probable Distribution</span></a>.</span></p>
</div>
<div class="section" id="boltzmann-statistics">
<span id="summary-boltzmann-statistics"></span><h3>Boltzmann Statistics<a class="headerlink" href="#boltzmann-statistics" title="Permalink to this headline">¶</a></h3>
<p>As mentioned in <a class="reference internal" href="#discription-of-microstates"><span class="std std-ref">Description of the Microstates</span></a>, many microstates have the same observables such as energy <span class="math notranslate nohighlight">\(E\)</span>. For each value of energy, we could figure out the number of microstates, the distribution of microstates <span class="math notranslate nohighlight">\(\Omega(E, \cdots)\)</span>. What makes this distribution powerful is that we could figure out the total number of microstates for this distribution by integrating or summing up for all energies <span class="math notranslate nohighlight">\(\int \Omega(E, \cdots) \mathrm d E \mathrm d\cdots\)</span>. The total number of microstates is closely related the the probability of this distribution as will be discussed below. Meanwhile, we could calculate the thermodynamic observables using the distribution.</p>
<p>In statistical physics, we will be focusing on the <strong>distribution of the microstates</strong> with respect to thermodynamic variables.</p>
<p>In Boltzmann statistics, we follow these guidelines.</p>
<ol class="arabic simple">
<li>Two postulates:<ol class="arabic">
<li>Occurrence of state in phase space ( <a class="reference internal" href="most-probable-distribution.html#equal-a-prior-probability"><span class="std std-ref">Equal A Prior Probability</span></a> ): all microstates have the same probabilities of occurence; This means that the most probable distribution for different energy <span class="math notranslate nohighlight">\(\Omega(E, \cdots)\)</span> should have the <strong>largest</strong> total number of microstates, <span class="math notranslate nohighlight">\(\int \Omega(E, \cdots) \mathrm d E \mathrm d\cdots\)</span>.</li>
<li>The most probable energy state is the state that an equilibrium system is staying at. This means that the most probable distribution discussed in 1 will be the actual distribution of the system. This postulate is not precise but there is a reason why it works. The distribution of the energy states is an extremely sharp peak at the most probable state.</li>
</ol>
</li>
<li>We find the most probable distrinution by maximizing the total number of microstates. Boltzmann distribution and Boltzmann factor is derived from this.</li>
<li>Partition function makes it easy to calculate the observables.<ol class="arabic">
<li>Density of state <span class="math notranslate nohighlight">\(g(E)\)</span> ;</li>
<li>Partition function <span class="math notranslate nohighlight">\(Z = \int g(E) \exp(-\beta E) \mathrm dE\)</span>; Variable of integration can be changed;</li>
<li>Systems of 3N DoFs <span class="math notranslate nohighlight">\(Z = Z_1^{3N}\)</span>.</li>
<li>Macroscopic observables are calculated by taking specific transformations such as derivatives of the partition function.</li>
</ol>
</li>
<li>Observable<ol class="arabic" start="0">
<li>Assumptions about free energy <span class="math notranslate nohighlight">\(A = - k_B T\ln Z\)</span>; Combine this with thermodynamics potential relations we can calculate entropy then everything.</li>
<li>Internal energy <span class="math notranslate nohighlight">\(U = \avg{E} = - \partial_\beta \ln Z\)</span>; All quantities can be extracted from partition function except those serve as variables of internal energy.</li>
<li>Heat capacity <span class="math notranslate nohighlight">\(C = \partial_T U\)</span></li>
</ol>
</li>
</ol>
</div>
<div class="section" id="gibbs-ensemble-theory">
<h3>Gibbs Ensemble Theory<a class="headerlink" href="#gibbs-ensemble-theory" title="Permalink to this headline">¶</a></h3>
<ol class="arabic simple">
<li>Ensembles</li>
<li>Density of states; Liouville equation; Von Neumann equation</li>
<li>Equilibrium</li>
<li>Three ensembles</li>
<li>Observables</li>
</ol>
</div>
<div class="section" id="boltzmann-factor">
<h3>Boltzmann Factor<a class="headerlink" href="#boltzmann-factor" title="Permalink to this headline">¶</a></h3>
<p>Boltzmann factor appears many times in thermodynamics and statistical mechanics. In Boltzmann’s most probable theory, ensemble theory, etc.</p>
</div>
</div>
<div class="section" id="applications-of-these-theories">
<h2>Applications of These Theories<a class="headerlink" href="#applications-of-these-theories" title="Permalink to this headline">¶</a></h2>
<div class="section" id="oscillators">
<h3>Oscillators<a class="headerlink" href="#oscillators" title="Permalink to this headline">¶</a></h3>
<p>Theories of chains of oscillators in different dimensions are very useful. In fact the fun thing is, most of the analytically solvable models in physics are harmonic oscillators.</p>
<p>A nice practice for this kind of problem is to calculate the heat capacity of diatom chain. A chain of N atom with alternating mass M and m interacting only through nearest neighbors.</p>
<p>The plan for this problem is</p>
<ol class="arabic simple">
<li>Write down the equation of motion for the whole system;</li>
<li>Fourier transform the system to decouple the modes (by finding the eigen modes);</li>
<li>Solve the eigen modes;</li>
<li>Calculate the partition function of each mode;</li>
<li>Sum over each mode.</li>
</ol>
<p>Problem is, we usually can not solve the problem exactly. So we turn to Debye theory. Debye theory assumes continuous spectrum even though our boundary condition quantizes the spectrum. So we need to turn the summation into integration using DoS using any of the several ways of obtaining DoS. Finally we analyze the different limits to get the low temperature or high temperature behavior.</p>
<div class="admonition hint">
<p class="first admonition-title">Hint</p>
<p class="last">Here are several methods to obtain DoS. <strong>To do!</strong></p>
</div>
</div>
<div class="section" id="heat-capacity">
<h3>Heat Capacity<a class="headerlink" href="#heat-capacity" title="Permalink to this headline">¶</a></h3>
<ol class="arabic simple">
<li>Classical theory: equipartition theorem;</li>
<li>Einstein theory: all modes of oscillations are the same;</li>
<li>Debye theory: difference between modes of oscillations are considered.</li>
</ol>
</div>
<div class="section" id="gibbs-mixing-paradox">
<h3>Gibbs Mixing Paradox<a class="headerlink" href="#gibbs-mixing-paradox" title="Permalink to this headline">¶</a></h3>
<p><a class="reference internal" href="gibbs-mixing-paradox.html#gibbs-mixing-paradox"><span class="std std-ref">Gibbs Mixing Paradox</span></a> is important for the coming in of quantum statistical mechanics.</p>
</div>
<div class="section" id="mean-field-theory">
<h3>Mean Field Theory<a class="headerlink" href="#mean-field-theory" title="Permalink to this headline">¶</a></h3>
<p><a class="reference internal" href="phase-transitions.html#mean-field-theory"><span class="std std-ref">Mean Field Thoery</span></a> is the idea of treating interaction between particles as interactions between particles and a mean field.</p>
</div>
<div class="section" id="van-der-waals-gas">
<h3>Van der Waals Gas<a class="headerlink" href="#van-der-waals-gas" title="Permalink to this headline">¶</a></h3>
<p><a class="reference internal" href="phase-transitions.html#van-der-waals-gas"><span class="std std-ref">Van der Waals Gas</span></a> can be derived using Mayer expansion and Leonard-Jones potential.</p>
</div>
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