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I have read several articles from non-native speakers about phase transitions. They are roughly classified in different orders, depending on the behavior of thermodynamic potentials at the phase transition. Now what is a good way of writing about the different orders:

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Thank you for any advice.

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4  
They're all fine. But be consistent. –  Peter Shor Jun 12 '12 at 12:56
1  
They are all fine, but if you are going to end up with large numbers, I would avoid the third one (where the number is spelled out). Also, I'm not sure that italicizing the number (in the fourth example) makes it any more understandable, so that typographic treatment is unnecessary. –  JLG Jun 12 '12 at 14:20
    
@JLG: Good advice, but for this specific case, nobody talks about 22nd-order phase transitions. Google doesn't return any hits higher than 5th-order. and those are incredibly rare—I wasn't expecting anything higher than 3rd-order. –  Peter Shor Jun 12 '12 at 14:29
    
@PeterShor, Good to know. When deciding on a style, though, it is good to think ahead to all potentialities. –  JLG Jun 12 '12 at 14:33
    
I would say any of the first 3 will be fine. But (at least in a mathematical context) do not use italics for this, so as not to confuse this with the use italics for variables. –  GEdgar Jun 12 '12 at 14:42
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3 Answers

up vote 3 down vote accepted

Options one, two and four are the same, in terms of English language and usage. They display different typographical conventions. One should be consistent in typographical usage. Stylistically the use of superscript ordinals has decreased as more documents have been created by their authors using typewriters or word-processors. They therefore have a more old-fashioned feel, even when professionally typeset. The italic version is rare in my experience, though it may be part of the style-guidelines of some publications.

In continuous prose it is customary to spell out all numbers up to some maximum. The maximum varies depending on the source of advice, but would include two in all cases. I tend to follow the same practice for ordinal numbers, so would use 'second-order', but '22nd-order'.

None of your options would be unclear, or indicate non-first-language use by themselves. So I'd suggest you chose one you are comfortable with and use it consistently.

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I'd quibble with your second paragraph. Yes, for conventional writing, the convention is to spell out smaller numbers -- I've generally heard the rule as "anything up to 20". But I wouldn't necessarily apply this rule to a technical paper. I would be quite surprised to find a calculus book that said, "find the integral of the equation four and three-fifths times x to the power of three plus ..."! –  Jay Jun 12 '12 at 17:00
    
@Jay It's slightly different here: it's a phrase! Even in a technical paper, 1st-order is not necessarily more correct than first-order, or conversely, first-order any more pedantic. –  Kris Jun 13 '12 at 12:06
    
@Kris I didn't mean that "1st-order" was "more correct" than "first-order", but rather that in technical writing, the usual rule about spelling out small numbers does not consistently apply and thus either would be acceptable. –  Jay Jun 13 '12 at 15:58
    
@jay, agreed, although your example is rather dubious (imho), because mathematical equations I wouldn't include as "continuous prose". They have their own particular typesetting conventions. I wouldn't be surprised if the calculus text said "There is nothing special about the number two, however. The process works for arbitrary powers." There is very little that can be said, in my experience, that cannot be made false by sufficiently diligent exception-seeking! :) –  Ian Jul 1 '12 at 21:57
    
@Ian True, there's a difference between an equation and text. But that difference can blur. I might well write in text, "And so the term 3x is found ..." I think we're sort of, kinda agreeing: the more "text-like" it is the more the standard rules apply, the more "equation-like" it is the less so. –  Jay Jul 2 '12 at 13:42
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1, 2, and 3 are all good and valid. I wouldn't use italics as in number 4 unless it was for emphasis -- like in context the reader might think we were talking about the 3rd-order transition and you want to make clear that we are talking about the 2nd.

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Considering the fact that in the context, the range extends from first to fifth at the most, there is no reason at all to use any options other than as:

first-order [second-order ... fifth-order].

Authors may have adopted various alternatives according to their requirements, but that need not worry you.

I can recall standard texts following the above convention, though we would jot down "1st order" in our notes :)

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