mathematics

Kramers–Moyal expansion

Kramers–Moyal expansion is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Kramers–Moyal expansion rather than just read about it. In short: In stochastic processes, the Kramers–Moyal expansion refers to a Taylor series expansion of the master equation, and is named after Hans Kramers and José Enrique Moyal. In many textbooks, the expansion is only used to derive the Fokker–Planck equation, and never used again.

Key takeaways

Kramers–Moyal expansion belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Kramers–Moyal expansion to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Kramers–Moyal expansion from memory before moving on to harder problems.

Reference excerpt

In stochastic processes, the Kramers–Moyal expansion refers to a Taylor series expansion of the master equation, and is named after Hans Kramers and José Enrique Moyal. In many textbooks, the expansion is only used to derive the Fokker–Planck equation, and never used again. In general, continuous stochastic processes are essentially Markovian, and so Fokker–Planck equations are sufficient for studying them. The higher-order Kramers–Moyal expansion only comes into play when the process is jumpy. This usually means it is a Poisson-like process. For a real stochastic process, one can compute its central-moment functions from experimental data on the process, from which one can then compute its Kramers–Moyal coefficients, and thus empirically measure its Kolmogorov forward and backward equations.

Statement Start with the integro-differential master equation

∂ p ( x , t ) ∂ t = ∫ ( p ( x , t | x 0 , t 0 ) p ( x 0 , t 0 ) − p ( x 0 , t 0 | x , t ) p ( x , t ) ) d x 0 {\displaystyle {\frac {\partial p(x,t)}{\partial t}}=\int \left(p(x,t|x_{0},t_{0})p(x_{0},t_{0})-p(x_{0},t_{0}|x,t)p(x,t)\right)dx_{0}}

where p ( x , t | x 0 , t 0 ) {\displaystyle p(x,t|x_{0},t_{0})} is the transition probability function, and p ( x , t ) {\displaystyle p(x,t)} is the probability density at time t {\displaystyle t} . The Kramers–Moyal expansion transforms the above to an infinite order partial differential equation

∂ t p ( x , t ) = ∑ n = 1 ∞ ( − ∂ x ) n [ D n ( x , t ) p ( x , t ) ] {\displaystyle \partial _{t}p(x,t)=\sum _{n=1}^{\infty }(-\partial _{x})^{n}[D_{n}(x,t)p(x,t)]}

and also ∂ t p ( x , t | x 0 , t 0 ) = ∑ n = 1 ∞ ( − ∂ x ) n [ D n ( x , t ) p ( x , t | x 0 , t 0 ) ] {\displaystyle \partial _{t}p(x,t|x_{0},t_{0})=\sum _{n=1}^{\infty }(-\partial _{x})^{n}[D_{n}(x,t)p(x,t|x_{0},t_{0})]}

where D n ( x , t ) {\displaystyle D_{n}(x,t)} are the Kramers–Moyal coefficients, defined by D n ( x , t ) = 1 n ! lim τ → 0 1 τ μ n ( t | x , t − τ ) {\displaystyle D_{n}(x,t)={\frac {1}{n!}}\lim _{\tau \to 0}{\frac {1}{\tau }}\mu _{n}(t|x,t-\tau )} and μ n {\displaystyle \mu _{n}} are the central moment functions, defined by

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Kramers–Moyal expansion

Start with the simplest possible case. Write down what Kramers–Moyal expansion claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Kramers–Moyal expansion before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Kramers–Moyal expansion ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Kramers–Moyal expansion

In research: Kramers–Moyal expansion appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Kramers–Moyal expansion in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Kramers–Moyal expansion is common in secondary-school and first-year university syllabi. It links to neighbouring topics Statistical mechanics, Stochastic calculus, so understanding it makes those chapters shorter.
In everyday life: Look for Kramers–Moyal expansion outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Kramers–Moyal expansion in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Kramers–Moyal expansion means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Kramers–Moyal expansion out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Kramers–Moyal expansion in simple terms?

Why does Kramers–Moyal expansion matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Kramers–Moyal expansion?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Kramers–Moyal expansion.