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manual/develop/en/dla/users-manual/output.html

@@ -151,6 +151,9 @@ <h3><span class="section-number">4.6.1.1. </span>Notations<a class="headerlink"
 <section id="main-results">
 <h2><span class="section-number">4.6.2. </span>Main results<a class="headerlink" href="#main-results" title="Permalink to this heading">¶</a></h2>
 <p>Main results are written in a file with the name specified by <code class="docutils literal notranslate"><span class="pre">outfile</span></code> keyword in the input parameter file.</p>
+<p>NOTICE: In general, Monte Carlo simulations have systematic errors of <span class="math notranslate nohighlight">\(O(1/N)\)</span> with respect to the number of samples <span class="math notranslate nohighlight">\(N\)</span> (<code class="docutils literal notranslate"><span class="pre">nmcs</span></code>) for the expectation values including nonlinear functions of the sample average like the specific heat and the susceptibility.
+For example, in the region where the specific heat becomes very small, e.g., in the low-temperature region below the energy gap, the calculated value may be negative.
+For precise analysis, we need to take into account not only the statistical errors but also the systematic errors.</p>
 <dl>
 <dt><code class="docutils literal notranslate"><span class="pre">sign</span></code></dt><dd><p>The sign of the weights.</p>
 <p><span class="math notranslate nohighlight">\(\sum_i W_i / \sum_i |W_i\)</span></p>
@@ -163,9 +166,6 @@ <h2><span class="section-number">4.6.2. </span>Main results<a class="headerlink"
 </dd>
 <dt><code class="docutils literal notranslate"><span class="pre">spe</span></code></dt><dd><p>The specific heat</p>
 <p><span class="math notranslate nohighlight">\(\displaystyle C_V \equiv \frac{\partial \epsilon}{\partial T} = \frac{1}{N_s T^2} \left[\left\langle\left(E_0 - TN_v\right)^2\right\rangle - \left\langle\left(E_0 - TN_v\right)\right\rangle^2 - T^2\left\langle N_v \right\rangle\right]\)</span></p>
-<p>NOTICE: In quantum Monte Carlo simulations, the specific heat is calculated with poor accuracy compared with other physical quantities.
-Moreover, the systematic error <span class="math notranslate nohighlight">\(1/N\)</span> also appears with respect to the number of samples <span class="math notranslate nohighlight">\(N\)</span> (<code class="docutils literal notranslate"><span class="pre">nmcs</span></code>).
-Especially, in the region where the specific heat becomes very small, e.g., in the low-temperature region below the energy gap, the calculated value may be negative.</p>
 </dd>
 <dt><code class="docutils literal notranslate"><span class="pre">som</span></code></dt><dd><p>The ratio of the specific heat and the temperature.</p>
 <p><span class="math notranslate nohighlight">\(\displaystyle \gamma \equiv \frac{C_V}{T} = \beta C_V\)</span></p>

manual/develop/jp/dla/users-manual/output.html

@@ -149,6 +149,9 @@ <h2><span class="section-number">4.6.1. </span>フォーマット<a class="heade
 <section id="id2">
 <h2><span class="section-number">4.6.2. </span>メイン出力<a class="headerlink" href="#id2" title="この見出しへのパーマリンク">¶</a></h2>
 <p>メイン出力ファイルは, 入力パラメータファイルの <code class="docutils literal notranslate"><span class="pre">outfile</span></code> キーワードで指定した名前で出力されます.</p>
+<p>NOTICE: 一般にモンテカルロ法では, 比熱や感受率など, サンプル平均の非線形関数を含むような量の期待値には, モンテカルロサンプル数 <span class="math notranslate nohighlight">\(N\)</span> (<code class="docutils literal notranslate"><span class="pre">nmcs</span></code>)に対して <span class="math notranslate nohighlight">\(O(1/N)\)</span> の系統誤差も含まれます.
+たとえば, エネルギーギャップ以下の極低温領域など, 比熱の値が非常に小さくなるような場合には計算結果が負になることがあります.
+精密な解析には統計誤差だけではなく, 系統誤差にも注意する必要があります.</p>
 <dl>
 <dt><code class="docutils literal notranslate"><span class="pre">sign</span></code></dt><dd><p>重みの符号.</p>
 <p><span class="math notranslate nohighlight">\(\frac{\sum_i W_i }{ \sum_i |W_i| }\)</span>, ここで <span class="math notranslate nohighlight">\(i\)</span> はモンテカルロサンプルの番号.</p>
@@ -161,8 +164,6 @@ <h2><span class="section-number">4.6.2. </span>メイン出力<a class="headerli
 </dd>
 <dt><code class="docutils literal notranslate"><span class="pre">spe</span></code></dt><dd><p>比熱.</p>
 <p><span class="math notranslate nohighlight">\(\displaystyle C_V \equiv \frac{\partial \epsilon}{\partial T} = \frac{1}{N_s T^2} \left[\left\langle\left(E_0 - TN_v\right)^2\right\rangle - \left\langle\left(E_0 - TN_v\right)\right\rangle^2 - T^2\left\langle N_v \right\rangle\right]\)</span></p>
-<p>NOTICE: 量子モンテカルロ法において, 比熱の計算は他の物理量と比べて精度が悪く, サンプル数 <span class="math notranslate nohighlight">\(N\)</span> (<code class="docutils literal notranslate"><span class="pre">nmcs</span></code>)に対して <span class="math notranslate nohighlight">\(1/N\)</span> の系統誤差も現れます.
-特に, エネルギーギャップ以下の極低温領域など, 比熱の値が非常に小さくなるような場合には, 計算結果が負になることがあります.</p>
 </dd>
 <dt><code class="docutils literal notranslate"><span class="pre">som</span></code></dt><dd><p>比熱と温度の比.</p>
 <p><span class="math notranslate nohighlight">\(\displaystyle \gamma \equiv \frac{C_V}{T} = \beta C_V\)</span></p>