physics

Logarithmic Sobolev inequalities

Logarithmic Sobolev inequalities is a physics 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 Logarithmic Sobolev inequalities rather than just read about it. In short: In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f {\displaystyle f} , its logarithm, and its gradient ∇ f {\displaystyle \nabla f} . These inequalities were discovered and named by Leonard Gross, who established them in dimension-independent form, in the context of constructive quantum field theory.

Key takeaways

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

Reference excerpt

In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f {\displaystyle f} , its logarithm, and its gradient ∇ f {\displaystyle \nabla f} . These inequalities were discovered and named by Leonard Gross, who established them in dimension-independent form, in the context of constructive quantum field theory. Similar results were discovered by other mathematicians before and many variations on such inequalities are known. Gross proved the inequality:

∫ R n | f ( x ) | 2 log ⁡ | f ( x ) | d ν ( x ) ≤ ∫ R n | ∇ f ( x ) | 2 d ν ( x ) + ‖ f ‖ 2 2 log ⁡ ‖ f ‖ 2 , {\displaystyle \int _{\mathbb {R} ^{n}}{\big |}f(x){\big |}^{2}\log {\big |}f(x){\big |}\,d\nu (x)\leq \int _{\mathbb {R} ^{n}}{\big |}\nabla f(x){\big |}^{2}\,d\nu (x)+\|f\|_{2}^{2}\log \|f\|_{2},}

where ‖ f ‖ 2 {\displaystyle \|f\|_{2}} is the L 2 ( ν ) {\displaystyle L^{2}(\nu )} -norm of f {\displaystyle f} , with ν {\displaystyle \nu } being standard Gaussian measure on R n . {\displaystyle \mathbb {R} ^{n}.} Unlike classical Sobolev inequalities, Gross's log-Sobolev inequality does not have any dimension-dependent constant, which makes it applicable in the infinite-dimensional limit.

Entropy functional Define the entropy functional Ent μ ⁡ ( f ) = ∫ ( f ln ⁡ f ) d μ − ∫ f ln ⁡ ( ∫ f d μ ) d μ {\displaystyle \operatorname {Ent} _{\mu }(f)=\int (f\ln f)d\mu -\int f\ln \left(\int fd\mu \right)d\mu } This is equal to the (unnormalized) KL divergence by Ent μ ⁡ ( f ) = D K L ( f d μ ‖ ( ∫ f d μ ) d μ ) {\textstyle \operatorname {Ent} _{\mu }(f)=D_{KL}(fd\mu \|(\int fd\mu )d\mu )} . A probability measure μ {\displaystyle \mu } on R n {\displaystyle \mathbb {R} ^{n}} is said to satisfy the log-Sobolev inequality with constant C > 0 {\displaystyle C>0} if for any smooth function f

Ent μ ⁡ ( f 2 ) ≤ C ∫ R n | ∇ f ( x ) | 2 d μ ( x ) , {\displaystyle \operatorname {Ent} _{\mu }(f^{2})\leq C\int _{\mathbb {R} ^{n}}{\big |}\nabla f(x){\big |}^{2}\,d\mu (x),}

Variants

Notes

References Tao, Terence (2012). Topics in random matrix theory. Graduate studies in mathematics. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-7430-1. Blower, G. (2009). Random matrices: high dimensional phenomena. London Mathematical Society lecture note series. Cambridge, New York: Cambridge University Press. ISBN 978-0-521-13312-8. Gross, Leonard (1975a), "Logarithmic Sobolev inequalities", American Journal of Mathematics, 97 (4): 1061–1083, doi:10.2307/2373688, JSTOR 2373688 Gross, Leonard (1975b), "Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form", Duke Mathematical Journal, 42 (3): 383–396, doi:10.1215/S0012-7094-75-04237-4

Worked examples

Example 1 — a first encounter with Logarithmic Sobolev inequalities

Start with the simplest possible case. Write down what Logarithmic Sobolev inequalities claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, 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 Logarithmic Sobolev inequalities 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 Logarithmic Sobolev inequalities 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 Logarithmic Sobolev inequalities

In research: Logarithmic Sobolev inequalities appears in physics 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 Logarithmic Sobolev inequalities 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: Logarithmic Sobolev inequalities is common in secondary-school and first-year university syllabi. It links to neighbouring topics Axiomatic quantum field theory, Logarithms, Sobolev spaces, so understanding it makes those chapters shorter.
In everyday life: Look for Logarithmic Sobolev inequalities 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 Logarithmic Sobolev inequalities in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Logarithmic Sobolev inequalities 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 Logarithmic Sobolev inequalities out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Logarithmic Sobolev inequalities in simple terms?

Why does Logarithmic Sobolev inequalities matter?

Because it connects several physics 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 Logarithmic Sobolev inequalities?

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 Logarithmic Sobolev inequalities.