From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- source/know/concept/detailed-balance/index.md | 232 ++++++++++++++++++++++++++ 1 file changed, 232 insertions(+) create mode 100644 source/know/concept/detailed-balance/index.md (limited to 'source/know/concept/detailed-balance') diff --git a/source/know/concept/detailed-balance/index.md b/source/know/concept/detailed-balance/index.md new file mode 100644 index 0000000..b7d5386 --- /dev/null +++ b/source/know/concept/detailed-balance/index.md @@ -0,0 +1,232 @@ +--- +title: "Detailed balance" +date: 2021-11-27 +categories: +- Physics +- Mathematics +- Stochastic analysis +layout: "concept" +--- + +Consider a system that can be regarded as a +[Markov process](/know/concept/markov-process/), +which means that its components (e.g. particles) are transitioning +between a known set of states, +with no history-dependence and no appreciable influence from interactions. + +At equilibrium, the principle of **detailed balance** then says that +for all states, the rate of leaving that state is exactly equal to +the rate of entering it, for every possible transition. +In effect, such a system looks "frozen" to an outside observer, +since all net transition rates are zero. + +We will focus on the case where both time and the state space are continuous. +Given some initial conditions, +assume that a component's trajectory can be described +as an [Itō diffusion](/know/concept/ito-calculus/) $X_t$ +with a time-independent drift $f$ and intensity $g$, +and with a probability density $\phi(t, x)$ governed by the +[forward Kolmogorov equation](/know/concept/kolmogorov-equations/) +(in 3D): + +$$\begin{aligned} + \pdv{\phi}{t} + = - \nabla \cdot \big( \vb{u} \phi - D \nabla \phi \big) +\end{aligned}$$ + +We start by demanding **stationarity**, +which is a weaker condition than detailed balance. +We want the probability $P$ of being in an arbitrary state volume $V$ +to be constant in time: + +$$\begin{aligned} + 0 + = \pdv{}{t}P(X_t \in V) + = \pdv{}{t}\int_V \phi \dd{V} + = \int_V \pdv{\phi}{t} \dd{V} +\end{aligned}$$ + +We substitute the forward Kolmogorov equation, +and apply the divergence theorem: + +$$\begin{aligned} + 0 + = - \int_V \nabla \cdot \big( \vb{u} \phi - D \nabla \phi \big) \dd{V} + = - \oint_{\partial V} \big( \vb{u} \phi - D \nabla \phi \big) \cdot \dd{\vb{S}} +\end{aligned}$$ + +In other words, the "flow" of probability *into* the volume $V$ +is equal to the flow *out of* $V$. +If such a probability density exists, +it is called a **stationary distribution** $\phi(t, x) = \pi(x)$. +Because $V$ was arbitrary, $\pi$ can be found by solving: + +$$\begin{aligned} + 0 + = - \nabla \cdot \big( \vb{u} \pi - D \nabla \pi \big) +\end{aligned}$$ + +Therefore, stationarity means that the state transition rates are constant. +To get detailed balance, however, we demand that +the transition rates are zero everywhere: +the probability flux through an arbitrary surface $S$ must vanish +(compare to closed surface integral above): + +$$\begin{aligned} + 0 + = - \int_{S} \big( \vb{u} \phi - D \nabla \phi \big) \cdot \dd{\vb{S}} +\end{aligned}$$ + +And since $S$ is arbitrary, this is only satisfied if the flux is trivially zero +(the above justification can easily be repeated in 1D, 2D, 4D, etc.): + +$$\begin{aligned} + \boxed{ + 0 = \vb{u} \phi - D \nabla \phi + } +\end{aligned}$$ + +This is a stronger condition that stationarity, +but fortunately often satisfied in practice. + +The fact that a system in detailed balance appears "frozen" +implies it is **time-reversible**, +meaning its statistics are the same for both directions of time. +Formally, given two arbitrary functions $h(x)$ and $k(x)$, +we have the property: + +$$\begin{aligned} + \boxed{ + \mathbf{E}\big[ h(X_0) \: k(X_t) \big] + = \mathbf{E}\big[ h(X_t) \: k(X_0) \big] + } +\end{aligned}$$ + +
+ + + +
+ + + +## References +1. U.H. Thygesen, + *Lecture notes on diffusions and stochastic differential equations*, + 2021, Polyteknisk Kompendie. -- cgit v1.2.3