Mathematics as the Language of Physics.

Transcription

1 Mathematics as the Language of Physics. J. Dunning-Davies Deartment of Physics University of Hull Hull HU6 7RX England. Abstract. Courses in mathematical methods for hysics students are not known for including too much in the way of mathematical rigour and in some ways understandably so. However the conditions under which some quite commonly used mathematical exressions are valid can be of great imortance in some hysical circumstances. Here one such exression which figures frequently in the maniulations leading to the isothermal comressibility aearing in formulae is examined as an illustrative examle.

2 Mathematics is undoubtedly a subject which deserves to be studied in its own right. As such it rovides a suerb exercise for the mind and can easily be seen to lead to eole ossessed of a flexible thinking which may be alied in myriad areas. However mathematics also has a vital role to lay as the language of hysics. As such the art it lays must be subservient to the hysics but having said that its role is no less imortant than that it enjoys when it is studied as a subject in its own right. In both roles the conditions affecting the validity of results are equally imortant and it is unfortunate that this asect of mathematics so imortant to the ure mathematician should aear to be overlooked on so many occasions within hysics. One mathematical result which illustrates this to erfection comes from the realms of artial differentiation. Rather than use general mathematical symbols as variables consider the case of an ideal gas for which the equation of state allows the ressure to be written as a function of the volume and absolute temerature ; that is = ( ). For this function d d d. If is now considered as a function of and it follows that d d d d. Comaring coefficients of d on both sides of this equation leads to 0 or more familiarly 1. (1) his result receives quite widesread use in various areas of hysics articularly in thermodynamics and statistical mechanics. However it is vitally imortant to note that the result is valid subject to some simle straightforward conditions mathematical conditions which in this case translate easily into obvious hysical requirements. he derivation is clearly deendent on the ressure being a definite function of both the volume and the absolute temerature. his condition simle though it seems is not always met in hysical situations. For examle as has been mentioned reviously 1 the ressure of a gas becomes indeendent of its volume below the condensation temerature and so below that temerature result (1) would not be alicable. If in fact the result is used it can lead to incorrect conclusions as has been shown 1 quite clearly for the case of an ideal relativistic Bose gas. he case of the Bose gas to which reference is made raises questions immediately when use is made of the isothermal comressibility in examinations of a system. Again as has been shown 1 from the definition of the grand artition function it follows directly that the mean square relative fluctuation in number of articles is given by

3 N N k N. () N N By using (1) above together with a consequence of the Gibbs-Duhem equation 3 it follows that where N N k 1 N is the isothermal comressibility. As has been shown reviously 1 using (3) to evaluate the mean square relative fluctuations in number of articles below the condensation temerature can lead to errors. On the other hand () is a generally valid result. his examle is cited merely to draw attention to a very real roblem within hysics and that is that quite often results are derived or quoted with no mention of their ranges of validity. If such ranges are not mentioned the imlication must be that the results are valid quite generally. his lack of mention of ranges of validity should ose no roblem to the rofessional hysicist although quite understandably it does on occasions. However for the student the osition is left unclear. tudents do not have the exerience or exertise to realise that some results are valid quite generally while others are not. Frequently though not always mathematical rigour is not a notion overstressed in undergraduate hysics courses. Hence it becomes more and more imortant to make it absolutely clear when results are valid generally and when not. As indicated already one articular area of concern surrounds results deendent on the exression for the isothermal comressibility. Firstly its definition; is it 4 1 N 1 N or 5 1? N Firstly the conditions under which these two exressions are actually the same need to be noted and one is that N be both finite and non-zero. Also as is seen the imortant oint arising when the ressure becomes indeendent of the volume is more readily visible via the first of these definitions; it is somewhat masked in the second. However as was seen to occur when considering mean square fluctuations in number of articles the isothermal comressibility often occurs in forms of formulae modified by use of equations of the form of (1) above. It doesn t regularly aear in the initial forms of formulae as is illustrated by the examle mentioned above and by considering the equation giving the difference between the constant ressure and constant volume heat caacities: If the entroy is assumed to be a function of both the absolute temerature and the volume it follows that k d d d (3) 3

4 4 from which it is readily seen that. By using a Maxwell relation this is seen to be equivalent to. By using (1) above this becomes which may be written C C where 1 is the coefficient of volume exansion at constant ressure. Yet again the isothermal comressibility aears in an exression following use of equation (1) and so in situations occurring below the condensation temerature where the ressure becomes indeendent of the volume great care must be taken when considering use of this final form of the result since as has been mentioned already but cannot be overstressed below the condensation temerature the ressure becomes indeendent of the volume and so use of (1) is not allowed and exressions derived using it must be invalid for that temerature region. Attention has been restricted here solely to roblems arising from use of equation (1) and in articular when that relation is used to introduce the isothermal comressibility into equations. he reason for doing that is clear in that the isothermal comressibility is a quantity whose value may more readily be determined than the values of those quantities it has relaced. However the above illustrates the need for extreme caution when using results involving this quantity. Nevertheless this is merely an examle; the need for caution when utilising mathematical results is always resent. he one comforting thought for the hysicist is that when some of the mathematical results become invalid the hysical situation existing is often unusual also in the above case of the Bose gas a change of hase has occurred and it is obvious from everyday examles such as the transitions between the various states of water that eculiar things are haening hysically at hase transitions.

5 References. 1. J. Dunning-Davies; 1968 Il Nuovo Cimento 57B L. D. Landau and E. M. Lifshitz; 1959 tatistical Physics (Pergamon Press London) 3. J. Dunning-Davies: 1968 Il Nuovo Cimento 53B P.. Landsberg; 1961 hermodynamics with Quantum tatistical Illustrations (Interscience Publishers London) 5. C. J. Adkins; 1968 Equilibrium hermodynamics (McGraw-Hill London) 5