Редактирование: Статистические характеристики дискретных сред

[[ТМ|Кафедра ТМ]] > [[Научный справочник]] > [[Механика]] > [[Механика дискретных сред | МДС]] > [[Статистические характеристики дискретных сред | Статистические характеристики ДС]] <HR>
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''Deriving macroscopic phenomenological laws of irreversible thermodynamics from simple microscopic models is one of the tasks of non-equilibrium statistical mechanics.'' [http://www.sciencedirect.com/science/article/pii/S0370157302005586 Lepri, Livi & Politi - Physics Reports, 2003], ([http://arxiv.org/pdf/cond-mat/0112193.pdf arXiv]).

== Обозначения и терминология ==

{| class="wikitable"
|-
! Обозначение
! Русское название
! English name
|-
| <math>X</math>
| [https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0 Случайная величина]
| [https://en.wikipedia.org/wiki/Random_variable Random variable]
|-
| <math>X(t)</math>
| [https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D1%8B%D0%B9_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81%D1%81 Случайный процесс]
| [https://en.wikipedia.org/wiki/Stochastic_process Stochastic process]
|-
| <math>X_1, X_2, \dots, X_N</math>
| Случайный вектор [https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0#.D0.9F.D1.80.D0.B8.D0.BC.D0.B5.D1.80.D1.8B]
| [https://en.wikipedia.org/wiki/Multivariate_random_variable Multivariate random variable] (random vector)
|-
| <math>F(x) = \mathbb{P}( X \leqslant x )</math>
| [https://ru.wikipedia.org/wiki/%D0%9A%D1%83%D0%BC%D1%83%D0%BB%D1%8F%D1%82%D0%B8%D0%B2%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D1%8F Функция распределения]
| [https://en.wikipedia.org/wiki/Cumulative_distribution_function Cumulative distribution function]
|-
| <math>f(x) = F'(x)</math>
| [https://ru.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%8C_%D0%B2%D0%B5%D1%80%D0%BE%D1%8F%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%B8 Плотность распределения]
| [https://en.wikipedia.org/wiki/Probability_density_function Probability density function] (distribution density)
|-
| <math>\left<X\right> = \int_{-\infty}^\infty x f(x)dx</math>
| [https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BE%D0%B6%D0%B8%D0%B4%D0%B0%D0%BD%D0%B8%D0%B5 Математическое ожидание]
| [https://en.wikipedia.org/wiki/Expected_value Expected value] (mathematical expectation)
|-
| <math>\varphi(s) = \left<e^{isX}\right> = \int_{-\infty}^\infty e^{isx} f(x)dx</math>
| [https://ru.wikipedia.org/wiki/%D0%A5%D0%B0%D1%80%D0%B0%D0%BA%D1%82%D0%B5%D1%80%D0%B8%D1%81%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%BE%D0%B9_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D1%8B Характеристическая функция]
| [https://en.wikipedia.org/wiki/Characteristic_function_%28probability_theory%29 Characteristic function]
|-
| <math>M(s) = \left<e^{sX}\right> = \int_{-\infty}^\infty e^{sx} f(x)dx</math>
| [https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4%D1%8F%D1%89%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82%D0%BE%D0%B2 Производящая функция моментов]
| [https://en.wikipedia.org/wiki/Moment-generating_function Moment-generating function]
|-
| <math>\nu_n = \left<X^n\right> = \int_{-\infty}^\infty x^n f(x)dx = \left.\frac{d^n}{ds^n} M(s)\right|_{s=0}</math>
| Начальный момент [https://ru.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82%D1%8B_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%BE%D0%B9_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D1%8B]
| Raw moment [https://en.wikipedia.org/wiki/Moment_%28mathematics%29]
|-
| <math>\mu_n = \left<(X-\nu_1)^n\right> = \int_{-\infty}^\infty (x-\nu_1)^n f(x)dx</math>
| Центральный момент [https://ru.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82%D1%8B_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%BE%D0%B9_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D1%8B]
| Central moment [https://en.wikipedia.org/wiki/Moment_%28mathematics%29]
|-
| <math>\eta_n = \frac{\mu_n}{\sigma^n} = \lambda_{n/2}</math>
| Нормированный момент
| [https://en.wikipedia.org/wiki/Standardized_moment Standardized moment]
|-
| <math>\kappa_n = \left.\frac{d^n}{ds^n}\ln M(s)\right|_{s=0} = (-i)^n\left.\frac{d^n}{ds^n}\ln \varphi(s)\right|_{s=0}</math>
| [https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D1%83%D0%B8%D0%BD%D0%B2%D0%B0%D1%80%D0%B8%D0%B0%D0%BD%D1%82 Полуинвариант] (кумулянт)
| [https://en.wikipedia.org/wiki/Cumulant Cumulant]
|-
| <math>\sigma^2 = \mu_2 = \kappa_2 </math>
| [https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BF%D0%B5%D1%80%D1%81%D0%B8%D1%8F_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%BE%D0%B9_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D1%8B Дисперсия] 
| [https://en.wikipedia.org/wiki/Variance Variance]
|-
| <math>\sigma = \sqrt{\mu_2}</math>
| [https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D0%BD%D0%B4%D0%B0%D1%80%D1%82%D0%BD%D0%BE%D0%B5_%D0%BE%D1%82%D0%BA%D0%BB%D0%BE%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5 Среднеквадратическое отклонение]
| [https://en.wikipedia.org/wiki/Standard_deviation Standard deviation]
|-
| <math>\gamma_1 = \frac{\mu_3}{\sigma^3} = \frac{\kappa_3}{\kappa_2^{3/2}}</math>
| [https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D1%8D%D1%84%D1%84%D0%B8%D1%86%D0%B8%D0%B5%D0%BD%D1%82_%D0%B0%D1%81%D0%B8%D0%BC%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%B8 Коэффициент асимметрии]
| [https://en.wikipedia.org/wiki/Skewness Skewness]
|-
| <math>\gamma_2 = \frac{\mu_4}{\sigma^4}-3 = \frac{\kappa_4}{\kappa_2^2} = \lambda_2 - 3</math>
| [https://ru.wikipedia.org/wiki/%D0%AD%D0%BA%D1%81%D1%86%D0%B5%D1%81%D1%81 Коэффициент эксцесса]
| [https://en.wikipedia.org/wiki/Kurtosis Kurtosis] (excess kurtosis)
|-
| <math>\eta_4 = \frac{\mu_4}{\sigma^4} = \lambda_2</math>
| 4-й нормированный момент [https://ru.wikipedia.org/wiki/%D0%AD%D0%BA%D1%81%D1%86%D0%B5%D1%81%D1%81]
| Historical kurtosis [https://en.wikipedia.org/wiki/Kurtosis] 
|-
| <math>f(x) = \frac{1}{\sigma\sqrt{2\pi}}\; e^{ -\frac{(x-\nu)^2}{2\sigma^2} }</math>
| Плотность [https://ru.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5 нормального распределения]
| [https://en.wikipedia.org/wiki/Normal_distribution Normal distribution] density
|}

== Полезные формулы ==

* Распределение случайной величины <math>Y</math>, являющейся функцией случайной величины <math>X</math>
::<math>
    F_y\bigl(y\bigr) = F_x\bigl(x(y)\bigr) ,\qquad
    f_y\bigl(y\bigr) = f_x\bigl(x(y)\bigr)\,x'(y).
</math>

*Нормированные моменты [https://ru.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5 нормальной случайной величины] 
::<math>\lambda_n = \eta_{2n} = \frac{\mu_{2n}}{\sigma^{2n}} = (2n-1)!!\qquad (</math>нечетные моменты равны нулю<math>)</math>.

*[https://ru.wikipedia.org/wiki/%D0%AD%D0%BA%D1%81%D1%86%D0%B5%D1%81%D1%81 Коэффициент эксцесса] суммы независимых случайных величин
::<math>
    \gamma_2\left(\sum_{n=1}^N X_n\right) 
    = \frac{\sum_{n=1}^N\sigma^4(X_n)\,\gamma_2(X_n)}{\left(\sum_{n=1}^N\sigma^2(X_n)\right)^2}
    = \frac1{N^2}\sum_{n=1}^N\gamma_2(X_n)
    = \frac1{N}\gamma_2(X_n),
</math>
: где второе равенство выполняется для случайных величин с равной дисперсией, а третье равенство — для равнораспределенных случайных величин.

== Научные разделы, связанные со статистическим описанием дискретных сред  ==

* [https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%B2%D0%B5%D1%80%D0%BE%D1%8F%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%B5%D0%B9 Теория вероятностей] и [https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0 Математическая статистика]
** [https://ru.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BD%D1%82%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0 Центральная предельная теорема]
* Стохастическое исчисление
** [https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D0%BE%D1%85%D0%B0%D1%81%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%B4%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5 Стохастическое дифференциальное уравнение]
*** [https://ru.wikipedia.org/wiki/%D3%F0%E0%E2%ED%E5%ED%E8%E5_%CB%E0%ED%E6%E5%E2%E5%ED%E0 Уравнение Ланжевена]
** [https://ru.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%A4%D0%BE%D0%BA%D0%BA%D0%B5%D1%80%D0%B0_%E2%80%94_%D0%9F%D0%BB%D0%B0%D0%BD%D0%BA%D0%B0 Уравнение Фоккера — Планка]
** [https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D0%BE%D1%85%D0%B0%D1%81%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%B8%D1%81%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D0%98%D1%82%D0%BE#.D0.A1.D1.82.D0.BE.D1.85.D0.B0.D1.81.D1.82.D0.B8.D1.87.D0.B5.D1.81.D0.BA.D0.B0.D1.8F_.D0.BF.D1.80.D0.BE.D0.B8.D0.B7.D0.B2.D0.BE.D0.B4.D0.BD.D0.B0.D1.8F Стохастическое исчисление Ито]
** Математические модели: [https://ru.wikipedia.org/wiki/%D0%91%D0%B5%D0%BB%D1%8B%D0%B9_%D1%88%D1%83%D0%BC Белый шум], [https://ru.wikipedia.org/w/index.php?title=%D0%92%D0%B8%D0%BD%D0%B5%D1%80%D0%BE%D0%B2%D1%81%D0%BA%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81%D1%81&stable=0#.D0.A1.D0.B2.D0.BE.D0.B9.D1.81.D1.82.D0.B2.D0.B0_.D0.B2.D0.B8.D0.BD.D0.B5.D1.80.D0.BE.D0.B2.D1.81.D0.BA.D0.BE.D0.B3.D0.BE_.D0.BF.D1.80.D0.BE.D1.86.D0.B5.D1.81.D1.81.D0.B0 Винеровский процесс], [https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%BE%D0%B5_%D0%B1%D0%BB%D1%83%D0%B6%D0%B4%D0%B0%D0%BD%D0%B8%D0%B5 Случайное блуждание], [https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%80%D1%82%D0%B8%D0%BD%D0%B3%D0%B0%D0%BB Мартингал]
** Физические модели: [https://ru.wikipedia.org/wiki/%D0%91%D1%80%D0%BE%D1%83%D0%BD%D0%BE%D0%B2%D1%81%D0%BA%D0%BE%D0%B5_%D0%B4%D0%B2%D0%B8%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5 Броуновское движение]
** [http://synset.com/ru/%D0%A1%D1%82%D0%BE%D1%85%D0%B0%D1%81%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%BC%D0%B8%D1%80 С.С. Степанов. Стохастический мир]
* [https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 Статистическая механика]
** [https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D0%93%D0%B8%D0%B1%D0%B1%D1%81%D0%B0 Распределение Гиббса]
** [https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D0%91%D0%BE%D0%BB%D1%8C%D1%86%D0%BC%D0%B0%D0%BD%D0%B0 Статистика Максвелла — Больцмана]
** [https://ru.wikipedia.org/w/index.php?title=%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%9B%D0%B8%D1%83%D0%B2%D0%B8%D0%BB%D0%BB%D1%8F Теорема Лиувилля]
** [http://ru.wikipedia.org/wiki/%D0%92%D0%B8%D1%80%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0 Теорема о вириале]
** [https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BE_%D1%80%D0%B0%D0%B2%D0%BD%D0%BE%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B8#.D0.A2.D0.B5.D0.BF.D0.BB.D0.BE.D1.91.D0.BC.D0.BA.D0.BE.D1.81.D1.82.D1.8C_.D1.82.D0.B2.D1.91.D1.80.D0.B4.D1.8B.D1.85_.D1.82.D0.B5.D0.BB Теорема о равнораспределении] кинетической энергии по степеням свободы
** [https://ru.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BF%D0%BE%D1%87%D0%BA%D0%B0_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B9_%D0%91%D0%BE%D0%B3%D0%BE%D0%BB%D1%8E%D0%B1%D0%BE%D0%B2%D0%B0 Цепочка уравнений Боголюбова]
** [http://ru.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D1%8B_%D0%93%D1%80%D0%B8%D0%BD%D0%B0_%E2%80%94_%D0%9A%D1%83%D0%B1%D0%BE#cite_note-.D0.9F.D1.80.D0.BE.D1.85.D0.BE.D1.80.D0.BE.D0.B2.E2.80.941992.E2.80.94.D0.93.D0.A0.D0.98.D0.9D.D0.90_-_.D0.9A.D0.A3.D0.91.D0.9E_.D0.A4.D0.9E.D0.A0.D0.9C.D0.A3.D0.9B.D0.AB.E2.80.94-1 Формулы Грина — Кубо]
** [https://en.wikipedia.org/wiki/Fermi-Pasta-Ulam_problem Fermi–Pasta–Ulam problem]
** [https://en.wikipedia.org/wiki/Toda_lattice Toda lattice]
* [https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0 Статистическая физика]
* [https://ru.wikipedia.org/wiki/%D0%A4%D0%B8%D0%B7%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BA%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D0%BA%D0%B0 Физическая кинетика]
** [https://ru.wikipedia.org/wiki/%D0%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%91%D0%BE%D0%BB%D1%8C%D1%86%D0%BC%D0%B0%D0%BD%D0%B0 Кинетическое уравнение Больцмана]
** [https://ru.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%92%D0%BB%D0%B0%D1%81%D0%BE%D0%B2%D0%B0 Уравнение Власова]
** [https://ru.wikipedia.org/wiki/%D0%A4%D0%BE%D0%BD%D0%BE%D0%BD Фонон]
* [https://ru.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BC%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BA%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D0%BA%D0%B0 Химическая кинетика]
** [https://ru.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%90%D1%80%D1%80%D0%B5%D0%BD%D0%B8%D1%83%D1%81%D0%B0 Уравнение Аррениуса]
* [https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B0 Термодинамика]
* [https://ru.wikipedia.org/wiki/%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%BD%D0%BE%D0%B2%D0%B5%D1%81%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B0 Неравновесная термодинамика]
** [https://en.wikipedia.org/wiki/Onsager_reciprocal_relations Onsager reciprocal relations]
* [https://en.wikipedia.org/wiki/Ergodic_theory Ergodic theory]
** [https://ru.wikipedia.org/wiki/%D0%AD%D1%80%D0%B3%D0%BE%D0%B4%D0%B8%D1%87%D0%BD%D0%BE%D1%81%D1%82%D1%8C Эргодичность], [https://ru.wikipedia.org/wiki/%D0%AD%D1%80%D0%B3%D0%BE%D0%B4%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%B3%D0%B8%D0%BF%D0%BE%D1%82%D0%B5%D0%B7%D0%B0 эргодическая гипотеза], [https://en.wikipedia.org/wiki/Ergodic_process ergodic process]
* Discrete calculus and discrete analysis
** [https://en.wikipedia.org/wiki/Finite_difference Finite difference]
** [https://en.wikipedia.org/wiki/Recurrence_relation#Relationship_to_difference_equations_narrowly_defined Recurrence relation]
** [https://en.wikipedia.org/wiki/Logistic_map Logistic map]
** [https://en.wikipedia.org/wiki/Z-transform Z-transform]
** See also [https://en.wikipedia.org/wiki/Discrete_mathematics#Calculus_of_finite_differences.2C_discrete_calculus_or_discrete_analysis Discrete mathematics]

== Разное ==

* [https://ru.wikipedia.org/wiki/%D0%92%D0%B5%D1%80%D0%BE%D1%8F%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE Вероятностное пространство]
* [https://en.wikipedia.org/wiki/Differential_entropy Differential entropy]
* Непрерывные распределения вероятности, которые могут выступать обобщениями [https://ru.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5 нормального распределения]
** [https://en.wikipedia.org/wiki/Pearson_distribution Pearson distribution] — a four-parametric family of probability distributions that extend the normal law to include different skewness and kurtosis values [https://en.wikipedia.org/wiki/Normal_distribution#Extensions].
** [https://en.wikipedia.org/wiki/Student%27s_t-distribution Student's t-distribution] — a one-parametric family of simmetric probability distributions estimating the mean of a normally distributed population (can be generalized to a three-parametric [https://en.wikipedia.org/wiki/Location-scale_family location-scale family]).   
** [https://en.wikipedia.org/wiki/Generalized_normal_distribution Generalized normal distribution] — two families (symmetric and non-symmetric) of probability distributions adding a shape parameter to the normal distribution. 
** См. также [https://en.wikipedia.org/wiki/List_of_probability_distributions#Supported_on_the_whole_real_line List of probability distributions] and [https://en.wikipedia.org/wiki/Relationships_among_probability_distributions#Reciprocal_of_a_random_variable Relationships among probability distributions].
* [https://en.wikipedia.org/wiki/R_%28programming_language%29 R (programming language)] — free software programming language and software environment for statistical computing and graphics.

== Литература ==

* [http://www.williamhoover.info/ Hoover W.G.] '''Computational Statistical Mechanics.''' ''Series "Studies in Modern Thermodynamics".'' Elsevier Science Publisher, Amsterdam, 1991, 324 pp. (Download pdf: [http://williamhoover.info/book.pdf 23 Mb], [http://www.williamhoover.info/ download page])
* [http://ru.wikipedia.org/wiki/%D0%93%D0%B8%D0%B1%D0%B1%D1%81,_%D0%94%D0%B6%D0%BE%D0%B7%D0%B0%D0%B9%D1%8F_%D0%A3%D0%B8%D0%BB%D0%BB%D0%B0%D1%80%D0%B4 Гиббс Дж.В.] '''Основные принципы статистической механики''' ''(излагаемые со специальным применением к рациональному обоснованию термодинамики)''. М.-Л.: ОГИЗ, 1946. (Скачать djvu: [http://eqworld.ipmnet.ru/ru/library/books/Gibbs1946ru.djvu 2.5 Mb], [http://eqworld.ipmnet.ru/ru/library/physics/statphys.htm страница загрузки]).
: [http://en.wikipedia.org/wiki/Josiah_Willard_Gibbs#Scientific_recognition Gibbs, Josiah Willard] (1902). '''Elementary Principles in Statistical Mechanics''', ''developed with especial reference to the rational foundation of thermodynamics.'' New York: Charles Scribner's Sons. (Download djvu: 19 Mb, [http://bookfi.org/book/499416 download page]).

*Борн М. «Непрерывность, детерминизм, реальность» в книге «Размышления и воспоминания физика». М.: Мир, 1977. стр.162-187. (Скачать djvu: [http://padabum.com/x.php?id=28958 2.4 Mb], [http://padabum.com/d.php?id=28958 страница загрузки]).
:Born M. «Continuity, determinism and reality», Kongelige Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser, Bind 30, Nr.2, (1955) 1-26.  
:— Впервые рассмотрена (согласно [https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0]) классическая статистическая механика одной частицы (1955 г.) 

* Лукач Е. Характеристические функции. Пер. с анг. 1979. М.: Наука. 424 с. [http://urss.ru/cgi-bin/db.pl?lang=Ru&blang=ru&page=Book&id=24313#FF1 Оглавление]  
:Lukacs, E. (1970) Characteristic Functions (2nd Edition), Griffin, London. (Download djvu: 3.9 Mb, [http://lib.org.by/info/M_Mathematics/MV_Probability/Lukacs%20E.%20Characteristic%20functions%20%282ed.,%20Griffin,%201970%29%28KA%29%28600dpi%29%28T%29%28360s%29_MV_.djvu download page]).
:— A negative result (Theorem 7.3.5): '''The [https://en.wikipedia.org/wiki/Cumulant#cite_ref-3 cumulant] generating function cannot be a finite-order polynomial of degree greater than 2.''' <toggledisplay status=hide showtext="Clarification &gt;&gt;" hidetext="Clarification &lt;&lt;" linkstyle="font-size:default"> (Given the results for the cumulants of the [https://en.wikipedia.org/wiki/Normal_distribution normal distribution], it might be hoped to find families of distributions for which κ<sub>''m''</sub>&nbsp;=&nbsp;κ<sub>''m''+1</sub>&nbsp;=&nbsp;...&nbsp;=&nbsp;0 for some ''m''&nbsp;>&nbsp;3, with the lower-order cumulants (orders 3 to ''m''&nbsp;&minus;&nbsp;1) being non-zero. From the theorem it follows that there are no such distributions. In other words: '''the normal distribution is the only distribution with a finite number (two) of non-zero cumulants'''.)</toggledisplay>  <toggledisplay status=hide showtext="Origin &gt;&gt;" hidetext="Origin &lt;&lt;" linkstyle="font-size:default"> Данное утверждение является следствием теоремы, впервые доказанной [https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%80%D1%86%D0%B8%D0%BD%D0%BA%D0%B5%D0%B2%D0%B8%D1%87,_%D0%AE%D0%B7%D0%B5%D1%84 Юзефом Марцинкевичем], польским математиком, погибшим во время Второй мировой войны: Marcinkiewicz, J. (1938). Sur une propriete de la loi de Gauss. Math. Zeitschr., 44, 612-618 ([http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PPN=PPN266833020_0044&DMDID=DMDLOG_0058&LOGID=LOG_0058&PHYSID=PHYS_0616 read online], download pdf: 397 Kb [http://gdz.sub.uni-goettingen.de/index.php?id=14&PPN=PPN266833020_0044&DMDID=DMDLOG_0059&LOGID=LOG_0059&PHYSID=PHYS_0618&L=0 download page]). Reprinted in J. Marcinkiewicz, Collected Papers. Panstwowe wydawnictwo Naukowe Warszawa, 1964. [http://zbmath.org/?q=an:0124.24103 Abstract.] </toggledisplay>

* [http://eqworld.ipmnet.ru/ru/library/mathematics/probability.htm Теория вероятностей и математическая статистика] на сайте [http://eqworld.ipmnet.ru/indexr.htm EqWorld]
* [http://eqworld.ipmnet.ru/ru/library/physics/statphys.htm Статистическая физика] на сайте [http://eqworld.ipmnet.ru/indexr.htm EqWorld]
* [http://lib.org.by/_djvu/M_Mathematics/MV_Probability/ Probability] на сайте [http://lib.org.by/ Белорусская научная библиотека]
* [http://en.wikipedia.org/wiki/List_of_notable_textbooks_in_statistical_mechanics List of textbooks in statistical mechanics]


== См. также ==

* [[Одномерный кристалл]]
* [[Механика дискретных сред]]

== Приложение к динамике цепочки ==

Рассмотрим одномерную дискретную среду, сотоящую из <math>N</math> частиц. Обозначим <math>u_n</math> — некоторую характеристику частицы, например ее перемещение. Введем среднее значение характеристики как 

: <math>\left<u_n\right> = \sum_{n=1}^Nu_n</math>

и среднее значение степени <math>k</math>

: <math>\left<u_n^k\right> = \sum_{n=1}^Nu_n^k</math>.

Если интерпретировать <math>u_n</math> как случайную величину, то при достаточно большом <math>N</math> величину <math>\left<u_n^k\right></math> можно называть <math>k</math>-м моментом случайной величины.

== Ссылки ==

* Случайные величины и их характеристики
** [https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0 Случайная величина]
** [https://ru.wikipedia.org/wiki/%D0%92%D0%B5%D1%80%D0%BE%D1%8F%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE Вероятностное пространство]
** [https://ru.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82%D1%8B_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%BE%D0%B9_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D1%8B Моменты случайной величины]
** [https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D1%8B%D0%B9_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81%D1%81 Случайный процесс]
** [https://en.wikipedia.org/wiki/Random_vector Multivariate random variable] (random vector)
== Терминология ==

* '''Начальным''' и '''центральным''' моментом [https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0 случайной величины] <math>\displaystyle X</math> называются, соответственно, величины
:: <math>\nu_k = \mathbb{E}\left[X^k\right], \qquad \mu_k = \mathbb{E}\left[(X - \mathbb{E}X)^k\right]</math>
: где <math>\mathbb{E}</math> — [https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BE%D0%B6%D0%B8%D0%B4%D0%B0%D0%BD%D0%B8%D0%B5 математическое ожидание] случайной величины, <math>k</math> — степень момента.
* <math>\nu_1=\mathbb{E}\left[X\right]</math> — [https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BE%D0%B6%D0%B8%D0%B4%D0%B0%D0%BD%D0%B8%D0%B5 математическое ожидание], <math>\mu_2=\sigma^2</math> — [https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BF%D0%B5%D1%80%D1%81%D0%B8%D1%8F_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%BE%D0%B9_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D1%8B дисперсия], <math>\mu_3/\sigma^3</math> — [https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D1%8D%D1%84%D1%84%D0%B8%D1%86%D0%B8%D0%B5%D0%BD%D1%82_%D0%B0%D1%81%D0%B8%D0%BC%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%B8 коэффициент асимметрии], <math>(\mu_4/\sigma^4-3)</math> — [https://ru.wikipedia.org/wiki/%D0%AD%D0%BA%D1%81%D1%86%D0%B5%D1%81%D1%81 коэффициент эксцесса].

== Словарь ==

* [https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0 Случайная величина] — [https://en.wikipedia.org/wiki/Random_variable Random variable]
* [https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BE%D0%B6%D0%B8%D0%B4%D0%B0%D0%BD%D0%B8%D0%B5 Математическое ожидание] — [https://en.wikipedia.org/wiki/Expected_value Expected value] (mathematical expectation)
* [https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BF%D0%B5%D1%80%D1%81%D0%B8%D1%8F_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%BE%D0%B9_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D1%8B Дисперсия случайной величины] — [https://en.wikipedia.org/wiki/Variance Variance]
* [https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D0%BD%D0%B4%D0%B0%D1%80%D1%82%D0%BD%D0%BE%D0%B5_%D0%BE%D1%82%D0%BA%D0%BB%D0%BE%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5 Среднеквадратическое отклонение] — [https://en.wikipedia.org/wiki/Standard_deviation Standard deviation]
* [https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D1%8D%D1%84%D1%84%D0%B8%D1%86%D0%B8%D0%B5%D0%BD%D1%82_%D0%B0%D1%81%D0%B8%D0%BC%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%B8 Коэффициент асимметрии] — [https://en.wikipedia.org/wiki/Skewness Skewness]
* [https://ru.wikipedia.org/wiki/%D0%AD%D0%BA%D1%81%D1%86%D0%B5%D1%81%D1%81 Коэффициент эксцесса] — [https://en.wikipedia.org/wiki/Kurtosis Kurtosis] 

</toggledisplay>

}}

[[Category: Механика дискретных сред]]