Consider a system that can be regarded as a
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 diffusionXt
with a time-independent drift f and intensity g,
and with a probability density ϕ(t,x) governed by the
forward Kolmogorov equation
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:
We substitute the forward Kolmogorov equation,
and apply the divergence theorem:
In other words, the “flow” of probability into the volume V
is equal to the flow out ofV.
If such a probability density exists,
it is called a stationary distributionϕ(t,x)=π(x).
Because V was arbitrary, π can be found by solving:
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):
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.):
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:
Consider the following weighted inner product,
whose weight function is a stationary distribution π
satisfying detailed balance,
where L^ is the Kolmogorov operator:
Where we used the forward Kolmogorov equation
and the definition of an adjoint operator.
Therefore, since the expectation E
does not explicitly depend on t (only implicitly via Xt),
we can naively move the differentiation inside
(only valid within E):