Categories: Mathematics, Physics, Stochastic analysis.

Detailed balance

Consider a system that can be regarded as a 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 \(X_t\) with a time-independent drift \(f\) and intensity \(g\), and with a probability density \(\phi(t, x)\) governed by the forward Kolmogorov equation (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.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.
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