For a countable set of states and an evolving distribution on , we may define the **entropy** at time by

The **H-Theorem** asserts that (with some assumptions) ; the entropy is always non-decreasing. Calculating this quantity, we see

Since we see we have

We assume describes a **Continuous Time Markov Chain**. That is, we assume there exists transition rates for distinct such that evolves according to:

If we write

we may write the above as

If we consider the as an infinite dimensional matrix and similarly consider as an infinite dimensional vector, we may write the above compactly in matrix notation:

.

We also assume the chain is **time reversible.** That is, there exists a distribution on such that the **detailed balance property** holds:

.** **

For example,suppose was finite and the rates were symmetric. Then the corresponding chain is time-reversible and the requisite is the uniform distribution on . It is easy to see the H-theorem holds in this context, since we haveso that

Note that in this sum, both the terms

and

occur. Grouping them together, and summing over , we get:

Each term in the sum is non-negative, so that .

We wish to consider the general time reversible case. Instead of being symmetric, we have

Then

so that

and as before, we observe the terms

and

occur in the above summation. Therefore

## Leave a Reply