Добрый день!
Есть у меня идея как из свойств событий и причинных связей между ними можно было бы построить интересную физику. Идея, конечно, сырая и содержит много предположений, требующих доказательства. Но все равно хотелось бы вынести ее на обсуждение. Без обратной связи трудно двигаться дальше.
Часть 1:
В теории относительности используется понятие ''пространство событий''. Рассмотрим пространство событий

с евклидовой метрикой. Предположим, что точки этого пространства - элементарные события, могут переходить из состояния ''не произошло'' в состояние ''произошло'' и обратно случайно с течением времени

. Вероятность такого перехода зависит от наличия в некоторой окрестности уже произошедших событий.
Введем плотность событий в точке пространства событий в момент времени

как предел отношения объема

, занимаемого произошедшими элементарными событиями внутри некоторой окрестности этой точки, к объему

этой окрестности, при стремлении его к нулю.

Выведем уравнение эволюции плотности событий. Рассмотрим плотность событий в одномерном пространстве событий

. Тогда в точке

в момент времени

плотность событий будет отличаться от плотности событий в момент времени

на величину, пропорциональную

и зависящую от значения плотности в момент

в этой точке и ее ближайших окрестностях. Зависимость от значения плотности в окрестностях зададим пропорциональную значению плотности на расстоянии


Используя формулу Тейлора, получим для плотности событий уравнение

, где

и

.
[Я.Б.Зельдович, Р. А.Д. Мышкис. Элементы математической физики. 2008 г. ФИЗМАТЛИТ - гл.V §1 стр.242]
Функция

определяет зависимость скорости изменения плотности от текущей плотности в точке. Эта функция должна удовлетворять условиям

В качестве такой функции можно выбрать

. В этом случае небольшие стохастические флуктуации плотности будут рассасываться.
![$$
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}; %x^2*(1-x)
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$$ $$
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$$](https://dxdy-01.korotkov.co.uk/f/8/d/9/8d9cbaeb152395bbacd5af15ba2ab06082.png)
Рассмотрим, для примера, одномерную задачу

Эволюция флуктуаций разного начального размера представлена на следующих рисунках (при

,

)
![$$
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$$](https://dxdy-04.korotkov.co.uk/f/3/d/3/3d35cf28c94232cdd256421b0bb804df82.png)
Эта задача имеет решение в виде движущегося фронта превращения событий (волны переключения), см.
[Колмогоров А.Н., Петровский И.Г., Пискунов Н.С. Исследование уравнения диффузии, соединенной с возрастанием количества вещества, и его применение к одной биологической проблеме. 1937 г. Бюл. МГУ. Математика и механика - №6 т.1 стр.1-26]
[Канель Я.И. О стабилизации решений задачи Коши для уравнений, встречающихся в теории горения. 1962 г. Математический сборник - т.59 (101) стр.245-288]
[Полянин А.Д., Вязьмин А.В., Журов А.И., Казенин Д.А. Справочник по точным решениям уравнений тепло- и массопереноса. 1998 г. ''ФАКТОРИАЛ'', МОСКВА гл.3 стр.233]
Решением этого уравнения будет

Примем

.Скорость движения фронта в сторону положительного

будет

.
![$$
\tikz[scale=0.2,>=latex]{
\draw[->] (-16,0) -- (16,0) node[right] {$x$};
\draw[->] (0,-0.2) -- (0,11) node[above] {$\alpha_0$};
\foreach \x in {-15,-14,...,15}
\draw[shift={(\x,0)}] (0pt,0pt) -- (0pt,-5pt);
\foreach \y/\ytext in {10/1}
\draw[shift={(0,\y)}] (-5pt,0pt) -- (0pt,0pt) node[left] {$\ytext$};
\draw[smooth, thick, black, domain=-15:15] plot coordinates{
(-15.,9.99)(-13.5,9.99)(-12.,9.98)(-10.5,9.95)(-9.,9.89)(-7.5,9.77)(-6.,9.53)(-4.5,9.05)(-3.,8.18)(-1.5,6.79)(0.,5.)(1.5,3.21)(3.,1.82)(4.5,0.95)(6.,0.47)(7.5,0.23)(9.,0.11)(10.5,0.05)(12.,0.02)(13.5,0.01)(15.,0.01)
}; %1/(1+exp(sqrt(1/2)*\x))
\draw[->, very thick] (0,5) -- (5,5) node[right] {$c$};
}
$$ $$
\tikz[scale=0.2,>=latex]{
\draw[->] (-16,0) -- (16,0) node[right] {$x$};
\draw[->] (0,-0.2) -- (0,11) node[above] {$\alpha_0$};
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}; %1/(1+exp(sqrt(1/2)*\x))
\draw[->, very thick] (0,5) -- (5,5) node[right] {$c$};
}
$$](https://dxdy-02.korotkov.co.uk/f/9/7/5/975c27bf56ba0e59f3b6e0be21ed9a5f82.png)
Вообще говоря, фронт устойчив к небольшим возмущениям, например возникающим в результате случайных флуктуаций плотности событий. Но допустим, что уравнение для плотности событий имеет такую нелинейность, что у него есть решения в виде плоского фронта с устойчивыми возмущениями в виде солитона.
Можно предположить, что скорость движения таких возмущений вдоль фронта не может превышать

. Существование предельной скорости приводит нас к СТО.
В рассмотренном уравнении скорость изменения плотности в точке

линейно зависит от

и эта зависимость имеет одну точку равновесия при

. Мы можем взять зависимость не линейную, а, например, в виде

В этом случае точки устойчивого равновесия будут при

. Возмущение фронта, в вершине которого выполняется вышеуказанное равенство, будет устойчивым и обладать ''упругостью'', т.е. при небольших искажениях формы - будет стремиться ее восстановить. Назовем такие устойчивые возмущения ''частицами''.
![$$
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(-3.6,-5.69)(-2.8,-7.55)
(-2.,-10.56)(-1.6,-12.57)(-1.2,-14.85)(-0.8,-17.17)(-0.4,-19.13)
(0.,-20.)
(0.4,-19.13)(0.8,-17.17)(1.2,-14.85)(1.6,-12.57)(2.,-10.56)
(2.8,-7.55)(3.6,-5.69)
(4.4,-4.56)(5.2,-3.84)(6.,-3.33)(8.,-2.5)(10.,-2.)
(12.,-1.67)(14.,-1.43)(16.,-1.25)(18.,-1.11)(20.,-1.)
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$$ $$
\tikz[scale=0.15,>=latex]{
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\draw[->] (0,-22) -- (0,2) node[above] {$x_0$};
\foreach \x in {-20,-15,...,20}
\draw[shift={(\x,0)}] (0pt,0pt) -- (0pt,-5pt);
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\draw[shift={(0,\y)}] (-5pt,0pt) -- (0pt,0pt);
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(0.,-20.)
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(2.8,-7.55)(3.6,-5.69)
(4.4,-4.56)(5.2,-3.84)(6.,-3.33)(8.,-2.5)(10.,-2.)
(12.,-1.67)(14.,-1.43)(16.,-1.25)(18.,-1.11)(20.,-1.)
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$$](https://dxdy-02.korotkov.co.uk/f/9/8/c/98c5749f214d8ffcfe046cda593b629482.png)
Очевидно, размер частицы (и радиус кривизны в вершине частицы) ограничен снизу шириной фронта. Поскольку радиус кривизны ограничен снизу, то

ограничено сверху и максимальное

, возможно, определяет количество поколений частиц.
Двигающийся фронт переключения будет на своем пути натыкаться на случайные флуктуации плотности событий, которые, сливаясь с фронтом, будут приводить к возникновению небольших возмущений на фронте и в его кривизне, а соответственно и в метрике заметаемой местом слияния фронтов гиперповерхности. Эти возмущения будут быстро рассасываться. Тем не менее, метрика гиперповерхности будет иметь стохастические флуктуации, а двигающаяся вместе с фронтом частица будет на этих возмущениях испытывать стохастические изменения траектории. Это в соответствии с подходами стохастической интерпретации квантовой механики приводит нас к квантовой механике и квантовой электродинамике.
[Nelson E. Derivation of the Schrödinger equation from Newtonian Mechanics. 1966 г. Physical Review v.150 n.4]
[Nelson E. Quantum fluctuations. 1985 г. Princeton, New Jersey]
[Luis de la Peña and Ana Maria Cetto. The Quantum Dice. An Introduction to Stochastic Electrodynamics. 1996 г. Instituto de Fisica, Mexico]
[Намсрай Х. Стохастическая механика. 1981 г. ''Физика элементарных частиц и атомного ядра'' том 12, вып. 5]