Maximum term method

Key takeaways

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

Reference excerpt

The maximum-term method is a consequence of the large numbers encountered in statistical mechanics. It states that under appropriate conditions the logarithm of a summation is essentially equal to the logarithm of the maximum term in the summation. These conditions are (see also proof below) that (1) the number of terms in the sum is large and (2) the terms themselves scale exponentially with this number. A typical application is the calculation of a thermodynamic potential from a partition function. These functions often contain terms with factorials n ! {\displaystyle n!} which scale as n 1 / 2 n n / e n {\displaystyle n^{1/2}n^{n}/e^{n}} (Stirling's approximation).

Example

lim M → ∞ ln ⁡ ( ∑ N = 1 M N ! ) ln ⁡ M ! = 1 {\displaystyle \lim _{M\rightarrow \infty }{\cfrac {\ln \left({\sum _{N=1}^{M}N!}\right)}{\ln {M!}}}=1\ }

Proof Consider the sum

S = ∑ N = 1 M T N {\displaystyle S=\sum _{N=1}^{M}T_{N}\ }

where T N {\displaystyle T_{N}} >0 for all N. Since all the terms are positive, the value of S must be greater than the value of the largest term, T max {\displaystyle T_{\max }} , and less than the product of the number of terms and the value of the largest term. So we have

T max ≤ S ≤ M T max . {\displaystyle T_{\max }\leq S\leq MT_{\max }.\ }

Taking logarithm gives

ln ⁡ T max ≤ ln ⁡ S ≤ ln ⁡ T max + ln ⁡ M . {\displaystyle \ln T_{\max }\leq \ln S\leq \ln T_{\max }+\ln M.\ }

As frequently happens in statistical mechanics, we assume that T max {\displaystyle T_{\max }} will be O ( ln ⁡ M ! ) = O ( e M ) {\displaystyle O(\ln M!)=O(e^{M})} : see Big O notation. Here we have

O ( M ) ≤ ln ⁡ S ≤ O ( M ) + ln ⁡ M ⇒ 1 ≤ ln ⁡ S O ( M ) ≤ 1 + ln ⁡ M O ( M ) = 1 + o ( 1 ) {\displaystyle O(M)\leq \ln S\leq O(M)+\ln M\qquad \Rightarrow 1\leq {\frac {\ln S}{O(M)}}\leq 1+{\frac {\ln M}{O(M)}}=1+o(1)}

For large M, ln ⁡ M {\displaystyle \ln M} is negligible with respect to M itself, and so ln ⁡ M / O ( e M ) ∈ o ( 1 ) {\displaystyle \ln M/O(e^{M})\in o(1)} . Then, we can see that ln S is bounded from above and below by ln ⁡ T max {\displaystyle \ln T_{\max }} , and so

ln ⁡ S O ( ln ⁡ T max ) = 1 {\displaystyle {\frac {\ln S}{O(\ln T_{\max })}}=1\ }

References D.A. McQuarrie, Statistical Mechanics. New York: Harper & Row, 1976. T.L. Hill, An Introduction to Statistical Thermodynamics. New York: Dover Publications, 1987

Worked examples

Example 1 — a first encounter with Maximum term method

Start with the simplest possible case. Write down what Maximum term method 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 Maximum term method 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 Maximum term method 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 Maximum term method

In research: Maximum term method 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 Maximum term method 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: Maximum term method is common in secondary-school and first-year university syllabi. It links to neighbouring topics Physical chemistry, Physical chemistry stubs, Statistical mechanics, so understanding it makes those chapters shorter.
In everyday life: Look for Maximum term method 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.

How to study Maximum term method in 20 minutes

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

Frequently asked questions

What is Maximum term method in simple terms?

The maximum-term method is a consequence of the large numbers encountered in statistical mechanics. It states that under appropriate conditions the logarithm of a summation is essentially equal to the logarithm of the maximum term in the summation.

Why does Maximum term method 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 Maximum term method?

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 Maximum term method.