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thermo-chaos-ja/_sources/gibbs-measures-and-srb-measures.txt

 .. math::
 
    \Psi_N(\beta, p)
-   = \sum_j \left[ \pseq \beta N E_N(\xk[0][j]) \pseq \ln \pseq \right]
+   = \sum_j \left[ \pseq \beta N E_N(\xk[0][j]) + \pseq \ln \pseq \right]
 
 を最小化する分布 :math:`\Pseq[] = \arg \min_p \Psi_N(\beta, p)`
 として導くことが出来る
 
 より詳しくは, Gibbs 測度 :math:`\mu_\beta` から導かれる分布 :math:`\Pseq` は
 
-.. math:: \Pseq = \mu_\beta(J_j^{(N)}) = c_j^{(N)} \PseqL
+.. math::
    :label: gibbs-measure-canonical-prob
 
+   \Pseq
+   = \mu_\beta(J_j^{(N)})
+   = c_j^{(N)} \PseqL
+   = c_j^{(N)} \exp \left( - N \Ftop(\beta) - \beta N E_N(\xk[0][j]) \right)
+
 .. math::
 
-   \PseqL = \exp \left( - N \Ftop(\beta) + \beta N E_N(\xk[0][j]) \right)
+   \PseqL = \exp \left( - N \Ftop(\beta) - \beta N E_N(\xk[0][j]) \right)
 
 
 の形で書け, :math:`c_j^{(N)}` は :math:`N` に依存しない定数で
           [Bowen2008]_ に証明がまとまっているのを見つけたが,関数解析
           の知識が必要.
 
-.. todo:: SRB 測度が自然不変測度と一致することを示す.
-          :math:`c_j^{(N)}` が自然不変密度 :math:`\rho` に収束する
-          ことを示しても, :math:`\Pseq` が不変であることを示すの
-          には不十分では?
-          [Beck1993]_ だけだと説明不足なので他の文献を使う.
-          Gibbs 測度について理解すれば自然に分かるか?
-
 (stub)
 
 .. math::
 Perron-Frobenius 演算子との関係
 -------------------------------
 
-.. todo:: 式 :eq:`gibbs-measure-canonical-prob` と
-          Perron-Frobenius 演算子との関係 (:math:`c_j^{(N)} \to \rho`)
-          を書く.
+この節では, 式 :eq:`gibbs-measure-canonical-prob` が不変測度となる
+要請から, :math:`c_j^{(N)}` の値を決定する.
 
-.. note::
+拡大的 (expanding) 力学系 ならば,マルコフ分割が存在する [#]_ ので,
+:math:`N`-シリンダー :math:`J_j^{(N)}` の逆像は,
+:math:`(N+1)`-シリンダー :math:`J_{k_l}^{(N+1)}` の和で表せる.
 
-   :math:`\beta=1` の時 :math:`\PseqL` は
-   :math:`\frac{l_j^{(N)}}{\sum_j l_j^{(N)}}`
-   でスケールする.
-   :math:`\rho(x)` が密度で
-   :math:`\Pseq = \rho(x) \PseqL` なら, :math:`\PseqL`
-   は :math:`\l_j^{(N)}` であるべきでは?
+.. math::
 
-   :ref:`topological-pressure-and-length-scale` で導いた
-   関係式は,あくまでスケーリング
+   J_j^{(N)} = \bigcup_{\tau=1}^l J_{k_\tau}^{(N)}
 
-   .. math:: P(\xk[0][j]) \sim
-             \frac{\pob[q]{l_j^{(N)}}}{\sum_j \pob[q]{l_j^{(N)}}}
+つまり, Gibbs 測度 :math:`\mu_\beta` には
 
-   であることに注意. :math:`\sim` は定数倍の違いを許すので,
-   :math:`\PseqL = \l_j^{(N)}` でなくても問題無い.
+.. math::
 
-   追記: :math:`\PseqL = \exp(\cdots)` は無次元数なので
-   :math:`\Pseq = \rho(x) \l_j^{(N)}` になるはずが無い. じゃあ
-   :math:`\Pseq = \mu_\beta(J_j^{(N)}) = \mu_\beta(\{ x_j \}) = \rho(x_j)`
-   ということか? もしくは, シリンダーが縮む速度が一様で,
-   両辺にかかっている微小区間 (シリンダーの長さ) が割りきれる,とか?
+   \mu_\beta(J_j^{(N)})
+   = \mu_\beta(J_{k_1}^{(N)}) + \cdots
+   + \mu_\beta(J_{k_l}^{(N)})
+   = \sum_{\tau=1}^l \mu_\beta(J_{k_\tau}^{(N)})
+
+の関係式が成立する. 式 :eq:`gibbs-measure-canonical-prob` をこの式に
+代入すれば次の関係式が得られる.
+
+.. math::
+
+   c_j^{(N)} & \exp \left( - N \Ftop(\beta) - \beta N E_N(\xk[0][j]) \right) \\
+   = \sum_{\tau=1}^l
+   c_{k_\tau}^{(N+1)} &
+   \exp \left( - (N+1) \Ftop(\beta)
+               - \beta (N+1) E_{N+1}(\xk[0][k_\tau]) \right)
+
+両辺を :math:`\exp(- N \Ftop(\beta))` で割って,
+
+.. math::
+   :label: cj-relation
+
+   &
+   c_j^{(N)} \exp \left( \beta N E_N(\xk[0][j]) \right) \\
+   &= \exp (- \Ftop(\beta))
+   \sum_{\tau=1}^l c_{k_\tau}^{(N+1)}
+   \exp \left( \beta (N+1) E_{N+1}(\xk[0][k_\tau]) \right)
+
+
+ここで,軌跡の局所拡大率 :math:`E_N(x_0)` の定義
+(参照: :ref:`topological-pressure`)
+
+.. math::
+
+   E_N(x_0) := \ooN \ln |{f^N}'(x_0)| = \ooN \sum_{n=0}^{N-1} \ln |f'(x_n)|
+
+を思い出せば,
+
+.. math::
+
+   \exp \left( - \beta N E_N(x_0) \right)
+   &= \exp \left( - \beta \sum_{n=0}^{N-1} \ln |f'(x_n)| \right) \\
+   &= \left( \prod_{n=0}^{N-1} |f'(x_n)| \right)^{-\beta} \\
+   &= \left| {f^N}'(x_0) \right|^{-\beta}
+
+.. math::
+
+   \exp \left( - \beta (N+1) E_{N+1}(x_0) \right)
+   = \left| {f^{N+1}}'(x_0) \right|^{-\beta}
+
+と書き直すことが出来る.
+
+
+:math:`N`-シリンダー :math:`J_j^{(N)}` は :math:`N \to \infty` で
+一点に収束するから, :math:`N`-シリンダーの集合上の関数 :math:`c_j^{(N)}`
+は, 相空間上の関数とみなせる. この関数を :math:`\rho(x)` とおく [#]_ .
+また, :math:`\xk[0][j]` や :math:`\xk[0][k_\tau]` は
+:math:`N`-シリンダー の収束先で置き換えられる.
+まとめると, :math:`N \to \infty` で
+
+.. math::
+
+   \xk[0][j] &\to y \\
+   \xk[0][k_\tau] &\to x_\tau \\
+   c_j^{(N)} &\to \rho(y) \\
+   c_{k_\tau}^{(N+1)} &\to \rho(x_\tau)
+
+となる. ここで, :math:`(N+1)`-シリンダー は :math:`N`-シリンダー
+の逆像だったから, :math:`f(x_\tau) = y` が成り立つことに注意.
+
+以上を 式 :eq:`cj-relation` に代入して,
+
+.. math::
+
+   \rho(y) \left| {f^N}'(y) \right|^{-\beta}
+   = \exp (- \Ftop(\beta))
+   \sum_{\tau=1}^l \rho(x_\tau)
+   {\underbrace{ \left| {f^{N+1}}'(x_\tau)
+   \right|}_{= |f'(x_\tau) {f^N}'(y)|}}^{-\beta}
+
+を得る. この式を整理すれば
+
+.. math::
+   :label: re-gene-perron-frobenius-eq
+
+   \rho(y)
+   = \exp (- \Ftop(\beta))
+   \sum_{\tau=1}^l \rho(x_\tau) \left| f'(x_\tau) \right|^{-\beta}
+
+となる. この式 :eq:`re-gene-perron-frobenius-eq` は,
+一般化した Perron-Frobenius の式である.
+:math:`\beta=1` を代入すれば,
+
+.. math::
+   :label: re-perron-frobenius-eq
+
+   \rho(y) = \exp (\kappa) \sum_{\tau=1}^l
+   \rho(x_\tau) \left| f'(x_\tau) \right|^{-1}
+
+となる. これは, 流出 (escape) のある場合の Perron-Frobenius の式
+とみなすことが出来る. 流出が無い場合は, :math:`\kappa=0`
+通常の Perron-Frobenius の式 を得る. つまり, :math:`\rho(x)` は
+自然不変密度 であることが予想される.
+
+自然不変密度 であるためには,規格化(相空間全体で積分した値が1)されてな
+ければならないから, :math:`\rho(x) \to A \rho_\beta(x)` (:math:`A`: 定数)
+と置き換えて,
+
+.. math:: \lim_{N\to\infty} c_j^{(N)} = A \rho_\beta(\xk[0][j])
+
+を得る.
+
+
+.. [#] らしい.
+
+   .. todo:: 「拡大的 (expanding) 力学系 ならば,マルコフ分割が存在する」
+             について,補足する.
+
+.. [#] 実は, :math:`\rho(y)` を 自然不変密度 とみなした場合
+       (この節の後の方でそうなることを示す), :math:`c_j^{(N)} \to \rho(y)`
+       とおくことは間違いである. 正しくは,自然不変密度の定数倍
+       に収束する, つまり
+       :math:`c_j^{(N)} \to \rho(y) \times \text{const.}` とおくべきである.
+       ただし,式 :eq:`re-gene-perron-frobenius-eq` では両辺に
+       :math:`\rho(y)` が出てくるので,この定数倍の違いは
+       その後の議論に影響を及ぼさない.
+
+       Perron-Frobenius の式で求められるのは固有値1に対応する固有関数で
+       あり,その関数が確率密度であるには,別に規格化の条件が必要だった
+       ことを思い出せば,これは当然である.
+
+       なぜ :math:`\rho(y)` を自然不変密度とみなした時に
+       :math:`c_j^{(N)} \to \rho(y)` が正しく無いか?
+       その議論のために,相空間 :math:`X` を別の軸で測り直した場合に
+       何が起こるかを考えてみよう. ここで新しい相空間 :math:`Z` は元の
+       相空間の定数(:math:`=A`)倍で, :math:`z=Ax` (:math:`x \in X`,
+       :math:`z \in Z`) とする.
+
+       この変換では力学系に変化を及ぼさないので, :math:`Z` 上の
+       シリンダーは,元のシリンダーと定数倍だけ位置と大きさが変化する
+       だけであり, :math:`c_j^{(N)}` の値は変わらない.
+
+       一方, :math:`Z` 上の自然不変密度 :math:`\sigma` は
+       :math:`\sigma(z) = \rho(z/A) / A` となる (:math:`Z` 上で,
+       :math:`B \rho(z/A)` を積分して1となるような :math:`B` を求めよ).
+
+       ところが,
+       :math:`c_j^{(N)} \to \rho(x)` と
+       :math:`c_j^{(N)} \to \sigma(z) = \rho(z/A) / A` が両方成り立つ
+       ことは :math:`A=1` の場合以外ありえない.
+       このことから, :math:`c_j^{(N)}` は自然不変密度の定数倍に収束する
+       ということが分かる.
+
+       また,次のように考えることも出来る :math:`\rho(x)` は
+       [1/長さ] の次元を持つが, :math:`c_j^{(N)}` は, 式
+       :eq:`re-gene-perron-frobenius-eq` を見れば無次元である
+       ことが分かる (左辺の :math:`\Pseq` も右辺の :math:`\exp (...)`
+       も無次元である).よって, :math:`c_j^{(N)}` が単位の変換
+       (上の :math:`z=Ax` に相当) に関して不変であるためには,
+       :math:`\rho(x)` に長さの次元のある定数をかけて無次元化
+       しなくてはならない.
+
+
+SRB 測度 と 自然不変測度
+------------------------
+
+さて, Gibbs 測度 :math:`\mu_\beta` が不変である条件から :math:`\Pseq`
+(式 :eq:`gibbs-measure-canonical-prob`) は,
+
+.. math::
+
+   \Pseq = \mu_\beta(J_j^{(N)}) = A \rho_\beta(\xk[0][j]) \PseqL
+
+とかけることが分かった. しかし, この式からは Gibbs 測度 :math:`\mu_\beta`
+と対応する密度 :math:`\rho_\beta` の関係が分かりづらい.
+測度と密度ならば
+
+.. math::
+
+   \mu_\beta(J_j^{(N)}) = \rho_\beta(\xk[0][j]) l_j^{(N)}
+
+の関係が成り立っているはずである
+(ここで, :math:`l_j^{(N)} = |J_j^{(N)}|` は, :math:`N`-シリンダー
+:math:`J_j^{(N)}` の長さ.
+:ref:`topological-pressure-and-length-scale` 参照).
+
+:math:`\beta=1` の場合にこの関係が成り立つことを示す.
+
+測度が不変であるという条件の無い場合のカノニカル分布
+
+.. math::
+
+   \PseqL = \exp \left( - N \Ftop(\beta) - \beta N E_N(\xk[0][j]) \right)
+
+は, :math:`N \to \infty` で
+
+.. math::
+
+   \PseqL = \frac{\pob{l_j^{(N)}}}{\sum_{j'} \pob{l_{j'}^{(N)}}}
+
+と近似出来たことを思い出そう (:ref:`appendix-pseq-scale-l` 参照).
+この近似を用いれば,
+
+.. math::
+
+   \Pseq = A \rho_\beta(\xk[0][j])
+   \frac{\pob{l_j^{(N)}}}{\sum_{j'} \pob{l_{j'}^{(N)}}}
+
+とかける. 規格化条件
+
+.. math:: \sum_j \Pseq = 1
+
+から :math:`\beta=1` の場合の定数 :math:`A` を求めると,
+
+.. math::
+
+   A =
+   \frac{\sum_{j'} l_{j'}^{(N)}}
+   {\sum_j \rho_\beta(\xk[0][j]) l_j^{(N)}}
+   = \sum_{j'} l_{j'}^{(N)}
+
+となる. ここで,
+
+.. math::
+
+   \sum_j \rho_\beta(\xk[0][j]) l_j^{(N)} \simeq \int \rho_\beta(x) dx = 1
+
+を用いた. よって, :math:`\beta=1` の場合,
+
+.. math::
+
+   \Pseq = \mu_1(J_j^{(N)}) = \rho_1(\xk[0][j]) l_j^{(N)}
+
+が成り立つことが分かる. つまり, SRB 測度 と 自然不変測度 は一致する.
+

thermo-chaos-ja/_sources/index.txt

-================
- カオスの熱力学
-================
+=========================
+ カオスの熱力学 (勉強中)
+=========================
 
 目標:
 
-- 基本的に [Beck1993]_ の内容の再構成.他の文献も参照しつつ,捕捉を加えたい.
-- 計算の難易度は数学が専門でなくても読める程度に.
-  できればどのあたりが数学的に厳密ではないかにも触れたい.
-- 章はできるだけ独立させつつ,どんな流れで読むべきかも示す.
+- 基本的に Thermodynamics of Chaotic Systems (Beck & Schlögl) [Beck1993]_
+  の内容の再構成.
+- 他の文献も参照しつつ,捕捉を加えたい.
 - まずは Kolmogorov-Sinai エントロピーと Lyapunov 指数の関係について.
 
+.. - 計算の難易度は数学が専門でなくても読める程度に.
+..   できればどのあたりが数学的に厳密ではないかにも触れたい.
+.. - 章はできるだけ独立させつつ,どんな流れで読むべきかも示す.
+
 目次:
 
 .. toctree::
    refs
    todolist
 
+* 変更履歴:
+  `tkf / thermo-chaos-ja / overview --- Bitbucket
+  <https://bitbucket.org/tkf/thermo-chaos-ja/>`_
 * :ref:`genindex`
 * :ref:`search`

thermo-chaos-ja/_sources/topological-pressure-and-dynamical-renyi-entropies.txt

 より (右辺の定数は式 :eq:`sum-pob-pseq-ftop` の :math:`\sim` より),
 :math:`N \to \infty` で,
 
-.. math:: K(\beta, q) = \ooomb [\Ftop(q \beta) - \beta \Ftop(q)]
+.. math:: K(\beta, q) = \ooomb \, [\Ftop(q \beta) - \beta \Ftop(q)]
 
 を得る. :math:`q=1` を代入すれば,
 
-.. math:: K(\beta) = \ooomb [\Ftop(\beta) - \beta \Ftop(1)]
+.. math:: K(\beta) = \ooomb \, [\Ftop(\beta) - \beta \Ftop(1)]
    :label: k_beta_1
 
 である. :math:`\beta=1` について計算するために :math:`\Ftop(\beta)`

thermo-chaos-ja/_sources/topological-pressure.txt

 .. index:: 位相圧力, topological pressure
+.. _topological-pressure:
 
 位相圧力
 ========
 :math:`N`-シリンダー からは初期値のとり方は一意には決まらないので,
 :math:`\xk[0][j] \in J_j` を満たす初期値を適当に選ぶ.
 
-.. [#] 実はアンサンブル :math:`\{\xk\}` のとり方は:math:`N`-シリンダーの集合
+.. [#] 実はアンサンブル :math:`\{\xk\}` のとり方は :math:`N`-シリンダーの集合
        だけではない.
 
        .. todo:: アンサンブル :math:`\{\xk\}` のとり方は色々あって,
 :eq:`local-expansion-rate` を用いて,
 
 .. math::
+   :label: local-expansion-rate-and-cylinder
 
    \frac{l_j^{(N+1)}}{|A_{j_{[-1]}}|}
    = |{f^N}'(\xk[0][j])|^{-1}
 が分かる.つまり, :math:`N \to \infty` の極限で :math:`\Ztop_N(\beta)`
 と :math:`\ZtopL_N(\beta)` は区別する必要が無い.
 
-また, :math:`\sum_j \pob{\frac{l_j^{(N+1)}}{|A_{j_{[-1]}}|}}`
-ではなく :math:`\pob{\frac{l_j^{(N+1)}}{|A_{j_{[-1]}}|}}` について
-同様の議論をすることにより, :math:`N \to \infty` の極限で
+また,同様の議論から, :math:`N \to \infty` の極限で
 位相圧力 の カノニカル分布 :math:`P(\xk[0][j])` は
 
-.. math:: P(\xk[0][j]) \sim \frac{\pob[q]{l_j^{(N)}}}{\sum_j \pob[q]{l_j^{(N)}}}
+.. math:: P(\xk[0][j]) \sim \frac{\pob{l_j^{(N)}}}{\sum_j \pob{l_j^{(N)}}}
    :label: pseq-scale-l
 
-とスケールすることが分かる.
+とスケールすることが分かる. 具体的な計算は, :ref:`appendix-pseq-scale-l`
+を参照.
 
 
 .. index:: 流出率, escape rate
 .. [#] 日本語の訳語は知らないので今適当に考えた.
        正しい訳語があれば置き換える予定.
        中国語だと「逃脱率」というようである.
+
+
+.. _appendix-pseq-scale-l:
+
+補足: 位相圧力のカノニカル分布のシリンダー長による近似
+------------------------------------------------------
+
+位相圧力 のカノニカル分布
+
+.. math:: P(\xk[0][j]) := \frac{1}{\Ztop_N(\beta)}
+                          \exp \left( - \beta N E_N(\xk[0][j]) \right)
+
+は, 自由エネルギー
+
+.. math::
+
+   \Psi_N(\beta, p)
+   = \sum_j \left[ \pseq \beta N E_N(\xk[0][j]) + \pseq \ln \pseq \right]
+
+を最小化する確率分布であり, 自由エネルギー と 分配関数 :math:`\Ztop_N`
+には
+
+.. math::
+
+   \min_p \Psi_N(\beta, p) = \Psi_N(\beta, P) = \Psi_N(\beta)
+   = - \ln \Ztop_N(\beta)
+
+なる関係があった. 位相圧力 は, この 自由エネルギー (の符号を反転した値)
+の1ステップあたりの大きさ
+
+.. math:: \Ftop(\beta) = - \limooN \Psi_N(\beta)
+
+で定義されていた.
+
+ここで, カノニカル分布が, :math:`\ZtopL_N` と :math:`\Ztop_N` の
+場合と同様に, :math:`N \to \infty` で
+
+.. math::
+
+   \tilde P(\xk[0][j]) = \frac{\pob{l_j^{(N)}}}{\sum_j \pob{l_j^{(N)}}}
+
+とスケールすることを示す.
+
+式 :eq:`local-expansion-rate`, :eq:`local-expansion-rate-and-cylinder`
+を思い出せば,
+
+.. math::
+
+   E_N(x_0) = \ooN \ln |{f^N}'(x_0)|
+   = - \ooN \ln \frac{l_j^{(N+1)}}{|A_{j_{[-1]}}|}
+
+とかけるので, 自由エネルギーは以下のように計算できる.
+
+.. math::
+
+   \Psi_{N-1}(\beta, \tilde P)
+   &= \sum_j \left[ \tilde P(\xk[0][j]) \beta N E_{N-1}(\xk[0][j])
+                    + \tilde P(\xk[0][j]) \ln \tilde P(\xk[0][j]) \right] \\
+   &= \sum_j \tilde P(\xk[0][j])
+      \left[ - \beta \ln \frac{l_j^{(N)}}{|A_{j_{[-1]}}|}
+             + \ln \frac{\pob{l_j^{(N)}}}{\sum_{j'} \pob{l_{j'}^{(N)}}}
+      \right] \\
+   &= \sum_j \tilde P(\xk[0][j])
+      \left[ \beta \ln |A_{j_{[-1]}}|
+             - \ln \sum_{j'} \pob{l_{j'}^{(N)}}
+      \right] \\
+   &= \sum_j \tilde P(\xk[0][j])
+      \left[ \beta \ln |A_{j_{[-1]}}|
+             - \ln \ZtopL_N(\beta)
+      \right]
+
+
+ここで, :math:`\ZtopL_N` の時と同様に, 分割のセル数は有限だから,
+:math:`|A_{j_{[-1]}}|` は最小値と最大値が存在して
+:math:`c_1 \le |A_{j_{[-1]}}| \le c_2` と挟み込めることを利用すれば,
+:math:`[...]` 内の :math:`j` 依存の項をなくすことが出来る.
+:math:`\beta \ge 0` の場合 (:math:`\beta < 0` の場合は :math:`c_1` と
+:math:`c_2` を入れ替える), :math:`\Psi_{N-1}(\beta, \tilde P)` は
+
+.. math::
+
+   \beta \ln c_1
+   - \ln \ZtopL_N(\beta)
+   \le
+   \Psi_{N-1}(\beta, \tilde P)
+   \le
+   \beta \ln c_2
+   - \ln \ZtopL_N(\beta)
+
+と評価出来る. :math:`N-1` で割って :math:`N \to \infty` での振る舞いを
+考えれば,
+
+.. math::
+
+   \underbrace{ \ooN[N-1] \beta \ln c_1 }_{\to 0}
+   \underbrace{ - \ooN[N-1] \ln \ZtopL_N(\beta)
+                }_{\to - \Ftop(\beta)}
+   \le \\
+   \ooN[N-1] \Psi_{N-1}(\beta, \tilde P)
+   \le \\
+   \underbrace{ \ooN[N-1] \beta \ln c_2 }_{\to 0}
+   \underbrace{ - \ooN[N-1] \ln \ZtopL_N(\beta)
+                }_{\to - \Ftop(\beta)}
+
+より,
+
+.. math::
+
+   \ooN[N-1] \Psi_{N-1}(\beta, \tilde P)
+   \to - \Ftop(\beta)
+
+となることが分かる.
+
+よって, :math:`N \to \infty` で :math:`\tilde P` と :math:`P`
+から,同じ位相圧力 :math:`\Ftop(\beta)` が計算出来るので,
+位相圧力を計算することが目的ならばこの二つは区別する必要が無い.

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 <span id="gibbs-measures-and-srb-measures"></span><span id="index-0"></span><h1>Gibbs 測度 と SRB 測度<a class="headerlink" href="#gibbs-srb" title="Permalink to this headline">¶</a></h1>
 <p>位相圧力の カノニカル分布 <img class="math" src="_images/math/6361e0be0623d1684f984ed504c95583b0f86836.png" alt="P(\xk)"/> は,自由エネルギー</p>
 <div class="math">
-<p><img src="_images/math/9a1d106a024e87d0ad7f95aa2eca524f756a83bd.png" alt="\Psi_N(\beta, p)
-= \sum_j \left[ \pseq \beta N E_N(\xk[0][j]) \pseq \ln \pseq \right]" /></p>
+<p><img src="_images/math/052a8082f64632bc29d97c89dd5790189abc2e6f.png" alt="\Psi_N(\beta, p)
+= \sum_j \left[ \pseq \beta N E_N(\xk[0][j]) + \pseq \ln \pseq \right]" /></p>
 </div><p>を最小化する分布 <img class="math" src="_images/math/2d3226aa67748139b156df813d7827b188cf6f60.png" alt="\Pseq[] = \arg \min_p \Psi_N(\beta, p)"/>
 として導くことが出来る
 (<a class="reference internal" href="canonical-distributions.html#principle-of-minimum-free-energy"><em>自由エネルギー最小の原理</em></a> を参照).
 と呼ばれる.</p>
 <p>より詳しくは, Gibbs 測度 <img class="math" src="_images/math/e65fb3ce55995c08991faa44d2cfbe6c99762181.png" alt="\mu_\beta"/> から導かれる分布 <img class="math" src="_images/math/3f4bd5d3efbe2041065e5f18823c7cc307e52031.png" alt="\Pseq"/> は</p>
 <div class="math" id="equation-gibbs-measure-canonical-prob">
-<p><span class="eqno">(1)</span><img src="_images/math/b99846cfb4ee5258971ef3c736d4e07a14a4c813.png" alt="\Pseq = \mu_\beta(J_j^{(N)}) = c_j^{(N)} \PseqL" /></p>
+<p><span class="eqno">(1)</span><img src="_images/math/1cb62bf1ecea6380c7d76592fb87c19606e5b6e4.png" alt="\Pseq
+= \mu_\beta(J_j^{(N)})
+= c_j^{(N)} \PseqL
+= c_j^{(N)} \exp \left( - N \Ftop(\beta) - \beta N E_N(\xk[0][j]) \right)" /></p>
 </div><div class="math">
-<p><img src="_images/math/1909fe16791d2fffb530f8ff72ad1ba3848e542f.png" alt="\PseqL = \exp \left( - N \Ftop(\beta) + \beta N E_N(\xk[0][j]) \right)" /></p>
+<p><img src="_images/math/d0a3b221f5ffd60ae0eecd830ec7a8eb58c51c02.png" alt="\PseqL = \exp \left( - N \Ftop(\beta) - \beta N E_N(\xk[0][j]) \right)" /></p>
 </div><p>の形で書け, <img class="math" src="_images/math/b27a916783f6e75bec525e8ea688b3d4e7c8cdb1.png" alt="c_j^{(N)}"/> は <img class="math" src="_images/math/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" alt="N"/> に依存しない定数で
 <img class="math" src="_images/math/fe0e6580c5f53d78727ecd4fd3954d318002e929.png" alt="c_1 \le c_j^{(N)} \le c_2"/> のように バウンドされている.
 <img class="math" src="_images/math/27e5ae1bb3f3d8bfa8e3a7c70fddb47516422373.png" alt="\PseqL"/> は制約なしの場合のカノニカル分布である.</p>
 <a class="reference internal" href="refs.html#bowen2008">[Bowen2008]</a> に証明がまとまっているのを見つけたが,関数解析
 の知識が必要.</p>
 </div>
-<div class="admonition-todo admonition " id="index-2">
-<p class="first admonition-title">Todo</p>
-<p class="last">SRB 測度が自然不変測度と一致することを示す.
-<img class="math" src="_images/math/b27a916783f6e75bec525e8ea688b3d4e7c8cdb1.png" alt="c_j^{(N)}"/> が自然不変密度 <img class="math" src="_images/math/0027034d8a10372a06deaf4f4084c01956587479.png" alt="\rho"/> に収束する
-ことを示しても, <img class="math" src="_images/math/3f4bd5d3efbe2041065e5f18823c7cc307e52031.png" alt="\Pseq"/> が不変であることを示すの
-には不十分では?
-<a class="reference internal" href="refs.html#beck1993">[Beck1993]</a> だけだと説明不足なので他の文献を使う.
-Gibbs 測度について理解すれば自然に分かるか?</p>
-</div>
 <p>(stub)</p>
 <div class="math">
 <p><img src="_images/math/406eb865522d0e241a0fb4be77299cab19fa2d0a.png" alt="\Ftop(\beta)
 = \sup_\mu \lim_{N \to \infty} \left(- \frac{1}{N} \Psi_N(\beta, p) \right)" /></p>
 </div><div class="section" id="perron-frobenius">
 <h2>Perron-Frobenius 演算子との関係<a class="headerlink" href="#perron-frobenius" title="Permalink to this headline">¶</a></h2>
-<div class="admonition-todo admonition " id="index-3">
+<p>この節では, 式 <a href="#equation-gibbs-measure-canonical-prob">(1)</a> が不変測度となる
+要請から, <img class="math" src="_images/math/b27a916783f6e75bec525e8ea688b3d4e7c8cdb1.png" alt="c_j^{(N)}"/> の値を決定する.</p>
+<p>拡大的 (expanding) 力学系 ならば,マルコフ分割が存在する <a class="footnote-reference" href="#id5" id="id3">[1]</a> ので,
+<img class="math" src="_images/math/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" alt="N"/>-シリンダー <img class="math" src="_images/math/e66a36b723f7bd5ebc39ab9ff8aa41843855af65.png" alt="J_j^{(N)}"/> の逆像は,
+<img class="math" src="_images/math/f7a1d178284ff1610746d7b8039b834e68949b19.png" alt="(N+1)"/>-シリンダー <img class="math" src="_images/math/723d7b078db6ac4338ca430ec702c48d46c70c1d.png" alt="J_{k_l}^{(N+1)}"/> の和で表せる.</p>
+<div class="math">
+<p><img src="_images/math/78b784199d88c35837103a4601b18196d44ab10d.png" alt="J_j^{(N)} = \bigcup_{\tau=1}^l J_{k_\tau}^{(N)}" /></p>
+</div><p>つまり, Gibbs 測度 <img class="math" src="_images/math/e65fb3ce55995c08991faa44d2cfbe6c99762181.png" alt="\mu_\beta"/> には</p>
+<div class="math">
+<p><img src="_images/math/21a80f84f7190d247e558fba39f0960ae890ea21.png" alt="\mu_\beta(J_j^{(N)})
+= \mu_\beta(J_{k_1}^{(N)}) + \cdots
++ \mu_\beta(J_{k_l}^{(N)})
+= \sum_{\tau=1}^l \mu_\beta(J_{k_\tau}^{(N)})" /></p>
+</div><p>の関係式が成立する. 式 <a href="#equation-gibbs-measure-canonical-prob">(1)</a> をこの式に
+代入すれば次の関係式が得られる.</p>
+<div class="math">
+<p><img src="_images/math/03fe014103d76ba81dcaa18d30c81e63aebb0bf5.png" alt="c_j^{(N)} &amp; \exp \left( - N \Ftop(\beta) - \beta N E_N(\xk[0][j]) \right) \\
+= \sum_{\tau=1}^l
+c_{k_\tau}^{(N+1)} &amp;
+\exp \left( - (N+1) \Ftop(\beta)
+            - \beta (N+1) E_{N+1}(\xk[0][k_\tau]) \right)" /></p>
+</div><p>両辺を <img class="math" src="_images/math/1bd775a1228c29313e5c50ca6a2dfa5e0e2e0de1.png" alt="\exp(- N \Ftop(\beta))"/> で割って,</p>
+<div class="math" id="equation-cj-relation">
+<p><span class="eqno">(2)</span><img src="_images/math/c169f4d122dcb08e4462b6df9364de724edfd216.png" alt="&amp;
+c_j^{(N)} \exp \left( \beta N E_N(\xk[0][j]) \right) \\
+&amp;= \exp (- \Ftop(\beta))
+\sum_{\tau=1}^l c_{k_\tau}^{(N+1)}
+\exp \left( \beta (N+1) E_{N+1}(\xk[0][k_\tau]) \right)" /></p>
+</div><p>ここで,軌跡の局所拡大率 <img class="math" src="_images/math/c70490da1875e52fb5841f57bb2db28109d9d711.png" alt="E_N(x_0)"/> の定義
+(参照: <a class="reference internal" href="topological-pressure.html#topological-pressure"><em>位相圧力</em></a>)</p>
+<div class="math">
+<p><img src="_images/math/8f989732de66911f3339d7c0732a725bff7cf287.png" alt="E_N(x_0) := \ooN \ln |{f^N}'(x_0)| = \ooN \sum_{n=0}^{N-1} \ln |f'(x_n)|" /></p>
+</div><p>を思い出せば,</p>
+<div class="math">
+<p><img src="_images/math/63055077ffb824e7a75d35993019953acd8c2b6d.png" alt="\exp \left( - \beta N E_N(x_0) \right)
+&amp;= \exp \left( - \beta \sum_{n=0}^{N-1} \ln |f'(x_n)| \right) \\
+&amp;= \left( \prod_{n=0}^{N-1} |f'(x_n)| \right)^{-\beta} \\
+&amp;= \left| {f^N}'(x_0) \right|^{-\beta}" /></p>
+</div><div class="math">
+<p><img src="_images/math/1e1e539334c84102b028c7124c58749745f5d829.png" alt="\exp \left( - \beta (N+1) E_{N+1}(x_0) \right)
+= \left| {f^{N+1}}'(x_0) \right|^{-\beta}" /></p>
+</div><p>と書き直すことが出来る.</p>
+<p><img class="math" src="_images/math/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" alt="N"/>-シリンダー <img class="math" src="_images/math/e66a36b723f7bd5ebc39ab9ff8aa41843855af65.png" alt="J_j^{(N)}"/> は <img class="math" src="_images/math/587deaa115238bac53970a4d6e0a68e303834080.png" alt="N \to \infty"/> で
+一点に収束するから, <img class="math" src="_images/math/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" alt="N"/>-シリンダーの集合上の関数 <img class="math" src="_images/math/b27a916783f6e75bec525e8ea688b3d4e7c8cdb1.png" alt="c_j^{(N)}"/>
+は, 相空間上の関数とみなせる. この関数を <img class="math" src="_images/math/920d76ff898b5f4c18aaae835de2e472405e3133.png" alt="\rho(x)"/> とおく <a class="footnote-reference" href="#id6" id="id4">[2]</a> .
+また, <img class="math" src="_images/math/7de1372f34f371588b83c4c48cf130bc57fb0071.png" alt="\xk[0][j]"/> や <img class="math" src="_images/math/2161d18703853671aad734259e69d4c40e09b928.png" alt="\xk[0][k_\tau]"/> は
+<img class="math" src="_images/math/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" alt="N"/>-シリンダー の収束先で置き換えられる.
+まとめると, <img class="math" src="_images/math/587deaa115238bac53970a4d6e0a68e303834080.png" alt="N \to \infty"/> で</p>
+<div class="math">
+<p><img src="_images/math/76c16efd342502d16506de4189a820b1149c3aa7.png" alt="\xk[0][j] &amp;\to y \\
+\xk[0][k_\tau] &amp;\to x_\tau \\
+c_j^{(N)} &amp;\to \rho(y) \\
+c_{k_\tau}^{(N+1)} &amp;\to \rho(x_\tau)" /></p>
+</div><p>となる. ここで, <img class="math" src="_images/math/f7a1d178284ff1610746d7b8039b834e68949b19.png" alt="(N+1)"/>-シリンダー は <img class="math" src="_images/math/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" alt="N"/>-シリンダー
+の逆像だったから, <img class="math" src="_images/math/6e5d1bcbc9ac74a12b5f6732d7255669b54d25bd.png" alt="f(x_\tau) = y"/> が成り立つことに注意.</p>
+<p>以上を 式 <a href="#equation-cj-relation">(2)</a> に代入して,</p>
+<div class="math">
+<p><img src="_images/math/727d90f54cc65f0ded1d7de3d623df4b236e7de2.png" alt="\rho(y) \left| {f^N}'(y) \right|^{-\beta}
+= \exp (- \Ftop(\beta))
+\sum_{\tau=1}^l \rho(x_\tau)
+{\underbrace{ \left| {f^{N+1}}'(x_\tau)
+\right|}_{= |f'(x_\tau) {f^N}'(y)|}}^{-\beta}" /></p>
+</div><p>を得る. この式を整理すれば</p>
+<div class="math" id="equation-re-gene-perron-frobenius-eq">
+<p><span class="eqno">(3)</span><img src="_images/math/3d0f68d1c2f146de16682d2141bb0156e7574bfb.png" alt="\rho(y)
+= \exp (- \Ftop(\beta))
+\sum_{\tau=1}^l \rho(x_\tau) \left| f'(x_\tau) \right|^{-\beta}" /></p>
+</div><p>となる. この式 <a href="#equation-re-gene-perron-frobenius-eq">(3)</a> は,
+一般化した Perron-Frobenius の式である.
+<img class="math" src="_images/math/bd0841c3936bc875fefe9192ca5209033f156466.png" alt="\beta=1"/> を代入すれば,</p>
+<div class="math" id="equation-re-perron-frobenius-eq">
+<p><span class="eqno">(4)</span><img src="_images/math/3625825400e147582bebfe4df6dea665ba91754b.png" alt="\rho(y) = \exp (\kappa) \sum_{\tau=1}^l
+\rho(x_\tau) \left| f'(x_\tau) \right|^{-1}" /></p>
+</div><p>となる. これは, 流出 (escape) のある場合の Perron-Frobenius の式
+とみなすことが出来る. 流出が無い場合は, <img class="math" src="_images/math/0544f690b92d76645e7f4c401c3086069dfc815a.png" alt="\kappa=0"/>
+通常の Perron-Frobenius の式 を得る. つまり, <img class="math" src="_images/math/920d76ff898b5f4c18aaae835de2e472405e3133.png" alt="\rho(x)"/> は
+自然不変密度 であることが予想される.</p>
+<p>自然不変密度 であるためには,規格化(相空間全体で積分した値が1)されてな
+ければならないから, <img class="math" src="_images/math/5fcdb312d37614acb5d27fd0b5ea94b24c609228.png" alt="\rho(x) \to A \rho_\beta(x)"/> (<img class="math" src="_images/math/019e9892786e493964e145e7c5cf7b700314e53b.png" alt="A"/>: 定数)
+と置き換えて,</p>
+<div class="math">
+<p><img src="_images/math/8c62f01c5480a07e810336df9caad8a3ab37d407.png" alt="\lim_{N\to\infty} c_j^{(N)} = A \rho_\beta(\xk[0][j])" /></p>
+</div><p>を得る.</p>
+<table class="docutils footnote" frame="void" id="id5" rules="none">
+<colgroup><col class="label" /><col /></colgroup>
+<tbody valign="top">
+<tr><td class="label"><a class="fn-backref" href="#id3">[1]</a></td><td><p class="first">らしい.</p>
+<div class="admonition-todo last admonition " id="index-2">
 <p class="first admonition-title">Todo</p>
-<p class="last">式 <a href="#equation-gibbs-measure-canonical-prob">(1)</a> と
-Perron-Frobenius 演算子との関係 (<img class="math" src="_images/math/e8ab9a5573da814c76be98f39b98d652cd9e5735.png" alt="c_j^{(N)} \to \rho"/>)
-を書く.</p>
+<p class="last">「拡大的 (expanding) 力学系 ならば,マルコフ分割が存在する」
+について,補足する.</p>
 </div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p><img class="math" src="_images/math/bd0841c3936bc875fefe9192ca5209033f156466.png" alt="\beta=1"/> の時 <img class="math" src="_images/math/27e5ae1bb3f3d8bfa8e3a7c70fddb47516422373.png" alt="\PseqL"/> は
-<img class="math" src="_images/math/5bd6848f7a482e711d1e7546ce4c1bd7b88965d8.png" alt="\frac{l_j^{(N)}}{\sum_j l_j^{(N)}}"/>
-でスケールする.
-<img class="math" src="_images/math/920d76ff898b5f4c18aaae835de2e472405e3133.png" alt="\rho(x)"/> が密度で
-<img class="math" src="_images/math/1076c25b2cfe4817e68d9317fd8489dc9fa75a3a.png" alt="\Pseq = \rho(x) \PseqL"/> なら, <img class="math" src="_images/math/27e5ae1bb3f3d8bfa8e3a7c70fddb47516422373.png" alt="\PseqL"/>
-は <img class="math" src="_images/math/9f9143a54ed9220a9eeae64bd8cde83778ab9161.png" alt="\l_j^{(N)}"/> であるべきでは?</p>
-<p><a class="reference internal" href="topological-pressure.html#topological-pressure-and-length-scale"><em>幾何的な意味</em></a> で導いた
-関係式は,あくまでスケーリング</p>
+</td></tr>
+</tbody>
+</table>
+<table class="docutils footnote" frame="void" id="id6" rules="none">
+<colgroup><col class="label" /><col /></colgroup>
+<tbody valign="top">
+<tr><td class="label"><a class="fn-backref" href="#id4">[2]</a></td><td><p class="first">実は, <img class="math" src="_images/math/f3e17c000d283d99cf4c9d02a982b79d33d0163a.png" alt="\rho(y)"/> を 自然不変密度 とみなした場合
+(この節の後の方でそうなることを示す), <img class="math" src="_images/math/73631a4c969266ef138ffcfa7cc62cb3751cb36d.png" alt="c_j^{(N)} \to \rho(y)"/>
+とおくことは間違いである. 正しくは,自然不変密度の定数倍
+に収束する, つまり
+<img class="math" src="_images/math/4aa704cb480179fc9d1cc8ff69605b84dbfc3686.png" alt="c_j^{(N)} \to \rho(y) \times \text{const.}"/> とおくべきである.
+ただし,式 <a href="#equation-re-gene-perron-frobenius-eq">(3)</a> では両辺に
+<img class="math" src="_images/math/f3e17c000d283d99cf4c9d02a982b79d33d0163a.png" alt="\rho(y)"/> が出てくるので,この定数倍の違いは
+その後の議論に影響を及ぼさない.</p>
+<p>Perron-Frobenius の式で求められるのは固有値1に対応する固有関数で
+あり,その関数が確率密度であるには,別に規格化の条件が必要だった
+ことを思い出せば,これは当然である.</p>
+<p>なぜ <img class="math" src="_images/math/f3e17c000d283d99cf4c9d02a982b79d33d0163a.png" alt="\rho(y)"/> を自然不変密度とみなした時に
+<img class="math" src="_images/math/73631a4c969266ef138ffcfa7cc62cb3751cb36d.png" alt="c_j^{(N)} \to \rho(y)"/> が正しく無いか?
+その議論のために,相空間 <img class="math" src="_images/math/6a47ca0fe7cb276abc022af6ac88ddae1a9d6894.png" alt="X"/> を別の軸で測り直した場合に
+何が起こるかを考えてみよう. ここで新しい相空間 <img class="math" src="_images/math/3ead47fb9fb4a4c273feee398f72ff2a09702b84.png" alt="Z"/> は元の
+相空間の定数(<img class="math" src="_images/math/9e87f6139f9e102f6c6ee9dec8393daff1e9d24f.png" alt="=A"/>)倍で, <img class="math" src="_images/math/14f348add0a5b53ecb889d7d2d7bc692273fcca5.png" alt="z=Ax"/> (<img class="math" src="_images/math/805adb67e19c27850f15dc11d7cbaf8748f9459e.png" alt="x \in X"/>,
+<img class="math" src="_images/math/a027278f665e3510809fb33fd15bd60ece45f21a.png" alt="z \in Z"/>) とする.</p>
+<p>この変換では力学系に変化を及ぼさないので, <img class="math" src="_images/math/3ead47fb9fb4a4c273feee398f72ff2a09702b84.png" alt="Z"/> 上の
+シリンダーは,元のシリンダーと定数倍だけ位置と大きさが変化する
+だけであり, <img class="math" src="_images/math/b27a916783f6e75bec525e8ea688b3d4e7c8cdb1.png" alt="c_j^{(N)}"/> の値は変わらない.</p>
+<p>一方, <img class="math" src="_images/math/3ead47fb9fb4a4c273feee398f72ff2a09702b84.png" alt="Z"/> 上の自然不変密度 <img class="math" src="_images/math/fa35d9fc104207e09a712110ac81612c5b279a6c.png" alt="\sigma"/> は
+<img class="math" src="_images/math/08e93f3697193dc9ecdc7462cdcec4cf07ce94bb.png" alt="\sigma(z) = \rho(z/A) / A"/> となる (<img class="math" src="_images/math/3ead47fb9fb4a4c273feee398f72ff2a09702b84.png" alt="Z"/> 上で,
+<img class="math" src="_images/math/574f50037b0ab544e404c759439b008861f1665e.png" alt="B \rho(z/A)"/> を積分して1となるような <img class="math" src="_images/math/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png" alt="B"/> を求めよ).</p>
+<p>ところが,
+<img class="math" src="_images/math/8eaafa9311c2555cca9c2ffc09ab024ae010720e.png" alt="c_j^{(N)} \to \rho(x)"/> と
+<img class="math" src="_images/math/89fd6087839f5d25b6eb92c31da059da4a61bc71.png" alt="c_j^{(N)} \to \sigma(z) = \rho(z/A) / A"/> が両方成り立つ
+ことは <img class="math" src="_images/math/f18428217dcd07944b2adbebcafd0978587a770a.png" alt="A=1"/> の場合以外ありえない.
+このことから, <img class="math" src="_images/math/b27a916783f6e75bec525e8ea688b3d4e7c8cdb1.png" alt="c_j^{(N)}"/> は自然不変密度の定数倍に収束する
+ということが分かる.</p>
+<p class="last">また,次のように考えることも出来る <img class="math" src="_images/math/920d76ff898b5f4c18aaae835de2e472405e3133.png" alt="\rho(x)"/> は
+[1/長さ] の次元を持つが, <img class="math" src="_images/math/b27a916783f6e75bec525e8ea688b3d4e7c8cdb1.png" alt="c_j^{(N)}"/> は, 式
+<a href="#equation-re-gene-perron-frobenius-eq">(3)</a> を見れば無次元である
+ことが分かる (左辺の <img class="math" src="_images/math/3f4bd5d3efbe2041065e5f18823c7cc307e52031.png" alt="\Pseq"/> も右辺の <img class="math" src="_images/math/7d4f52db3f954e14a84ed30b32ce271b7206377a.png" alt="\exp (...)"/>
+も無次元である).よって, <img class="math" src="_images/math/b27a916783f6e75bec525e8ea688b3d4e7c8cdb1.png" alt="c_j^{(N)}"/> が単位の変換
+(上の <img class="math" src="_images/math/14f348add0a5b53ecb889d7d2d7bc692273fcca5.png" alt="z=Ax"/> に相当) に関して不変であるためには,
+<img class="math" src="_images/math/920d76ff898b5f4c18aaae835de2e472405e3133.png" alt="\rho(x)"/> に長さの次元のある定数をかけて無次元化
+しなくてはならない.</p>
+</td></tr>
+</tbody>
+</table>
+</div>
+<div class="section" id="srb">
+<h2>SRB 測度 と 自然不変測度<a class="headerlink" href="#srb" title="Permalink to this headline">¶</a></h2>
+<p>さて, Gibbs 測度 <img class="math" src="_images/math/e65fb3ce55995c08991faa44d2cfbe6c99762181.png" alt="\mu_\beta"/> が不変である条件から <img class="math" src="_images/math/3f4bd5d3efbe2041065e5f18823c7cc307e52031.png" alt="\Pseq"/>
+(式 <a href="#equation-gibbs-measure-canonical-prob">(1)</a>) は,</p>
 <div class="math">
-<p><img src="_images/math/fbf88b91d1592372578b2fa62fcf6600dedf58d3.png" alt="P(\xk[0][j]) \sim
-\frac{\pob[q]{l_j^{(N)}}}{\sum_j \pob[q]{l_j^{(N)}}}" /></p>
-</div><p>であることに注意. <img class="math" src="_images/math/e55156a4008b0944ad00d5bc71bc5aa6315aabb7.png" alt="\sim"/> は定数倍の違いを許すので,
-<img class="math" src="_images/math/c4540948a3ded3dff109c67f8bffa89f0874ab02.png" alt="\PseqL = \l_j^{(N)}"/> でなくても問題無い.</p>
-<p class="last">追記: <img class="math" src="_images/math/93e91f2dda2010245534255c10980aacd8b59c4e.png" alt="\PseqL = \exp(\cdots)"/> は無次元数なので
-<img class="math" src="_images/math/998e1715e7c1c30a57552de0e939d4c61c09fb3e.png" alt="\Pseq = \rho(x) \l_j^{(N)}"/> になるはずが無い. じゃあ
-<img class="math" src="_images/math/f2be30fea02e9dec11c2590c2863718d46dd0097.png" alt="\Pseq = \mu_\beta(J_j^{(N)}) = \mu_\beta(\{ x_j \}) = \rho(x_j)"/>
-ということか? もしくは, シリンダーが縮む速度が一様で,
-両辺にかかっている微小区間 (シリンダーの長さ) が割りきれる,とか?</p>
-</div>
+<p><img src="_images/math/37b5456c6b4047dfd1bf40c8f5d0d251ab3d1495.png" alt="\Pseq = \mu_\beta(J_j^{(N)}) = A \rho_\beta(\xk[0][j]) \PseqL" /></p>
+</div><p>とかけることが分かった. しかし, この式からは Gibbs 測度 <img class="math" src="_images/math/e65fb3ce55995c08991faa44d2cfbe6c99762181.png" alt="\mu_\beta"/>
+と対応する密度 <img class="math" src="_images/math/5f78aa2a65a28beb854b9385ad6e39b0ae93ad33.png" alt="\rho_\beta"/> の関係が分かりづらい.
+測度と密度ならば</p>
+<div class="math">
+<p><img src="_images/math/9c6191bf394cb26490e16fad4616a941faf49650.png" alt="\mu_\beta(J_j^{(N)}) = \rho_\beta(\xk[0][j]) l_j^{(N)}" /></p>
+</div><p>の関係が成り立っているはずである
+(ここで, <img class="math" src="_images/math/d4884ba10b103c8dc1e46e0b4cafbb2a434c2442.png" alt="l_j^{(N)} = |J_j^{(N)}|"/> は, <img class="math" src="_images/math/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" alt="N"/>-シリンダー
+<img class="math" src="_images/math/e66a36b723f7bd5ebc39ab9ff8aa41843855af65.png" alt="J_j^{(N)}"/> の長さ.
+<a class="reference internal" href="topological-pressure.html#topological-pressure-and-length-scale"><em>幾何的な意味</em></a> 参照).</p>
+<p><img class="math" src="_images/math/bd0841c3936bc875fefe9192ca5209033f156466.png" alt="\beta=1"/> の場合にこの関係が成り立つことを示す.</p>
+<p>測度が不変であるという条件の無い場合のカノニカル分布</p>
+<div class="math">
+<p><img src="_images/math/d0a3b221f5ffd60ae0eecd830ec7a8eb58c51c02.png" alt="\PseqL = \exp \left( - N \Ftop(\beta) - \beta N E_N(\xk[0][j]) \right)" /></p>
+</div><p>は, <img class="math" src="_images/math/587deaa115238bac53970a4d6e0a68e303834080.png" alt="N \to \infty"/> で</p>
+<div class="math">
+<p><img src="_images/math/6c9765b9a6765f108f3e72b4624dfdcfbd8a5b9b.png" alt="\PseqL = \frac{\pob{l_j^{(N)}}}{\sum_{j'} \pob{l_{j'}^{(N)}}}" /></p>
+</div><p>と近似出来たことを思い出そう (<a class="reference internal" href="topological-pressure.html#appendix-pseq-scale-l"><em>補足: 位相圧力のカノニカル分布のシリンダー長による近似</em></a> 参照).
+この近似を用いれば,</p>
+<div class="math">
+<p><img src="_images/math/86f29b839670981912d65af15fbec3bae01d0f73.png" alt="\Pseq = A \rho_\beta(\xk[0][j])
+\frac{\pob{l_j^{(N)}}}{\sum_{j'} \pob{l_{j'}^{(N)}}}" /></p>
+</div><p>とかける. 規格化条件</p>
+<div class="math">
+<p><img src="_images/math/886528b6a3df43c668c811de0b47e35bea333676.png" alt="\sum_j \Pseq = 1" /></p>
+</div><p>から <img class="math" src="_images/math/bd0841c3936bc875fefe9192ca5209033f156466.png" alt="\beta=1"/> の場合の定数 <img class="math" src="_images/math/019e9892786e493964e145e7c5cf7b700314e53b.png" alt="A"/> を求めると,</p>
+<div class="math">
+<p><img src="_images/math/b0d9b2fc78b789483f1ee650bd37f6bb8e2895a9.png" alt="A =
+\frac{\sum_{j'} l_{j'}^{(N)}}
+{\sum_j \rho_\beta(\xk[0][j]) l_j^{(N)}}
+= \sum_{j'} l_{j'}^{(N)}" /></p>
+</div><p>となる. ここで,</p>
+<div class="math">
+<p><img src="_images/math/f064ee6e4b7e9c3f52b5dc183134c1bc10536b8a.png" alt="\sum_j \rho_\beta(\xk[0][j]) l_j^{(N)} \simeq \int \rho_\beta(x) dx = 1" /></p>
+</div><p>を用いた. よって, <img class="math" src="_images/math/bd0841c3936bc875fefe9192ca5209033f156466.png" alt="\beta=1"/> の場合,</p>
+<div class="math">
+<p><img src="_images/math/17c92287568d6b46f76d99fbff2e41931a428bed.png" alt="\Pseq = \mu_1(J_j^{(N)}) = \rho_1(\xk[0][j]) l_j^{(N)}" /></p>
+</div><p>が成り立つことが分かる. つまり, SRB 測度 と 自然不変測度 は一致する.</p>
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   <div class="section" id="id1">
-<h1>カオスの熱力学<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h1>
+<h1>カオスの熱力学 (勉強中)<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h1>
 <p>目標:</p>
 <ul class="simple">
-<li>基本的に <a class="reference internal" href="refs.html#beck1993">[Beck1993]</a> の内容の再構成.他の文献も参照しつつ,捕捉を加えたい.</li>
-<li>計算の難易度は数学が専門でなくても読める程度に.
-できればどのあたりが数学的に厳密ではないかにも触れたい.</li>
-<li>章はできるだけ独立させつつ,どんな流れで読むべきかも示す.</li>
+<li>基本的に Thermodynamics of Chaotic Systems (Beck &amp; Schlögl) <a class="reference internal" href="refs.html#beck1993">[Beck1993]</a>
+の内容の再構成.</li>
+<li>他の文献も参照しつつ,捕捉を加えたい.</li>
 <li>まずは Kolmogorov-Sinai エントロピーと Lyapunov 指数の関係について.</li>
 </ul>
 <p>目次:</p>
 <li class="toctree-l2"><a class="reference internal" href="topological-pressure.html#id2">定義</a></li>
 <li class="toctree-l2"><a class="reference internal" href="topological-pressure.html#topological-pressure-and-length-scale">幾何的な意味</a></li>
 <li class="toctree-l2"><a class="reference internal" href="topological-pressure.html#escape-rate">流出率</a></li>
+<li class="toctree-l2"><a class="reference internal" href="topological-pressure.html#appendix-pseq-scale-l">補足: 位相圧力のカノニカル分布のシリンダー長による近似</a></li>
 </ul>
 </li>
 <li class="toctree-l1"><a class="reference internal" href="perron-frobenius-operator.html">Perron-Frobenius 演算子</a></li>
 <li class="toctree-l1"><a class="reference internal" href="gibbs-measures-and-srb-measures.html">Gibbs 測度 と SRB 測度</a><ul>
 <li class="toctree-l2"><a class="reference internal" href="gibbs-measures-and-srb-measures.html#perron-frobenius">Perron-Frobenius 演算子との関係</a></li>
+<li class="toctree-l2"><a class="reference internal" href="gibbs-measures-and-srb-measures.html#srb">SRB 測度 と 自然不変測度</a></li>
 </ul>
 </li>
 <li class="toctree-l1"><a class="reference internal" href="topological-pressure-and-dynamical-renyi-entropies.html">位相圧力 と Rényi エントロピーの関係</a></li>
 </ul>
 </div>
 <ul class="simple">
+<li>変更履歴:
+<a class="reference external" href="https://bitbucket.org/tkf/thermo-chaos-ja/">tkf / thermo-chaos-ja / overview &#8212; Bitbucket</a></li>
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thermo-chaos-ja/searchindex.js

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thermo-chaos-ja/todolist.html

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