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Logical relations among different definitions of “extensity”
Volume 107A, number 8
PHYSICS LETTERS
25 February 1985
LOGICAL RELATIONS AMONG D I F F E R E N T DEFINITIONS OF "EXTENSIVITY" J. DUNNING-DAVIES
Department of Applied Mathematics, The University, Hull, UK and P.T. LANDSBERG
Faculty of Mathematical Studies, The University, Southampton, UK Received 28 November 1984 The word "extensive" has been given various meanings in the literature. Four definitions in particular are considered here, three from thermodynamics and one from statistical mechanics. The logical relations between them are given. For large systems, the statistical mechanical definition is most suitable as it implies the other three, but the latter are useful also if this restriction does not hold.
Introduction. In a recent article [ I ] , it was shown that the various definitions o f an extensive quantity appearing in standard thermodynamics texts are not equivalent. Furthermore, none was found equivalent to the usual statistical mechanical meaning of the term. Suppose the quantity f to be a function of the vect o r x which has components Xl, x2, x 3 so that f(x) ~ f(x 1, X 2, X3), then f is said to be an extensive quantity if 1. its value in a composite system equals the sum of its values in each o f the subsystems; that is,
[(x + y ) = f ( x ) + [@) , or
2. its value is halved when a system in equilibrium is divided into two equal parts; that is,
f(x) - - f ( x i , X2, X3) ~ x3f(Pl,
P2)'
or more accurately, if the limit lim
x 3 1 f ( x l ' x2' x3) =f(Pl, P 2 ) ,
XI,X2,X3 ---~ ,o 1 ,,o2 f i x e d
exists and depends on the intensive quantities P l and P2" Here intensive quantities are those which are independent of the size of the system. The relations between these definitions, as discussed in ref. [1 ] and obtained here, are summarised in fig. 1. Note that, although not stated explicitly in ref. [1 ] while definition 4 is obviously a statement for large systems only, this constraint is not part of any of the other definitions. Also, it is assumed that the systems considered are in thermodynamic equilibrium.
/{2x) = 2/1x), or
3. it is homogeneous o f degree one; that is, f(Kv) = Xf(x)
for all X > 0 .
Finally, the statistical mechanical definition o f an extensive quantity states that 4. for large x 1, x 2, x3 with fixed p 1 = x 1/x 3 and 192 = X2/X3, the quantity f i s extensive if
0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Additional implication. Firstly, suppose the function f obeys definition 3, then i f x and y are two parallel vectors such that y = kx, f ( x ) + f ( y ) = f ( x ) + f(Xx) = (1 + X)f(x) = f ( ( 1 + X)x) = f ( x + y ) .
383
Volume 107A, number 8
PtlYSICS LETTERS 3+
1÷C
~2
1
,,~ \
\
P
+
C
-~ 3
2 I
4/ //
4
Fig. 1. Implications between the definitions of an extensive quantity. Here C is the assumption that the function f occurring in the definitions is continuous in its arguments. For the implications indicated by dashed arrows, attention is restricted in definitions 1, 2 and 3 to large systems. Only equilibrium systems are considered. Hence, if P is the added assumption that the vectors x and y are parallel, 3 + P--+ 1 . Also, it follows immediately that 2+C+P-+I
.
It has been shown already [1], that, provided attention is restricted to large systems, definition 4 implies the validity of the other three definitions.
384
25 February 1985
To see the complete link between 4 and the remaining definitions, it must be noted that 4 is composed of two parts: If the function f i s extensive according to 4 then (a) f is homogeneous of degree one and (b) in the limit a s x l , x 2 , x 3 ~ o o the ratios p I = X l /X 3 , P2 = x 2/x 3 remain fixed. It is seen immediately that, provided attention is restricted to large systems, 4 may be deduced from 3 it" the ratios P l = Xl/X3 and P2 = x 2 / x 3 are fixed. Hence, for ~the special case of large systems, 3 implies 4. Since tile relations between 1 and 3 and between 2 and 3 are k n o w n (see fig. 1), this completes the examination of the links between the various definitions in use for the term "extensive". Hence, it may be concluded that, for large systems, the statistical mechanical definition is most suitable as it implies the other three, but the latter are useful also if this restriction does not hold. Notably, 3 + P and 1 + C imply all the other three definitions. Reference 11] J. Dunning-Davies, Phys. Lett. 94A (1983) 346.
PHYSICS LETTERS
25 February 1985
LOGICAL RELATIONS AMONG D I F F E R E N T DEFINITIONS OF "EXTENSIVITY" J. DUNNING-DAVIES
Department of Applied Mathematics, The University, Hull, UK and P.T. LANDSBERG
Faculty of Mathematical Studies, The University, Southampton, UK Received 28 November 1984 The word "extensive" has been given various meanings in the literature. Four definitions in particular are considered here, three from thermodynamics and one from statistical mechanics. The logical relations between them are given. For large systems, the statistical mechanical definition is most suitable as it implies the other three, but the latter are useful also if this restriction does not hold.
Introduction. In a recent article [ I ] , it was shown that the various definitions o f an extensive quantity appearing in standard thermodynamics texts are not equivalent. Furthermore, none was found equivalent to the usual statistical mechanical meaning of the term. Suppose the quantity f to be a function of the vect o r x which has components Xl, x2, x 3 so that f(x) ~ f(x 1, X 2, X3), then f is said to be an extensive quantity if 1. its value in a composite system equals the sum of its values in each o f the subsystems; that is,
[(x + y ) = f ( x ) + [@) , or
2. its value is halved when a system in equilibrium is divided into two equal parts; that is,
f(x) - - f ( x i , X2, X3) ~ x3f(Pl,
P2)'
or more accurately, if the limit lim
x 3 1 f ( x l ' x2' x3) =f(Pl, P 2 ) ,
XI,X2,X3 ---~ ,o 1 ,,o2 f i x e d
exists and depends on the intensive quantities P l and P2" Here intensive quantities are those which are independent of the size of the system. The relations between these definitions, as discussed in ref. [1 ] and obtained here, are summarised in fig. 1. Note that, although not stated explicitly in ref. [1 ] while definition 4 is obviously a statement for large systems only, this constraint is not part of any of the other definitions. Also, it is assumed that the systems considered are in thermodynamic equilibrium.
/{2x) = 2/1x), or
3. it is homogeneous o f degree one; that is, f(Kv) = Xf(x)
for all X > 0 .
Finally, the statistical mechanical definition o f an extensive quantity states that 4. for large x 1, x 2, x3 with fixed p 1 = x 1/x 3 and 192 = X2/X3, the quantity f i s extensive if
0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Additional implication. Firstly, suppose the function f obeys definition 3, then i f x and y are two parallel vectors such that y = kx, f ( x ) + f ( y ) = f ( x ) + f(Xx) = (1 + X)f(x) = f ( ( 1 + X)x) = f ( x + y ) .
383
Volume 107A, number 8
PtlYSICS LETTERS 3+
1÷C
~2
1
,,~ \
\
P
+
C
-~ 3
2 I
4/ //
4
Fig. 1. Implications between the definitions of an extensive quantity. Here C is the assumption that the function f occurring in the definitions is continuous in its arguments. For the implications indicated by dashed arrows, attention is restricted in definitions 1, 2 and 3 to large systems. Only equilibrium systems are considered. Hence, if P is the added assumption that the vectors x and y are parallel, 3 + P--+ 1 . Also, it follows immediately that 2+C+P-+I
.
It has been shown already [1], that, provided attention is restricted to large systems, definition 4 implies the validity of the other three definitions.
384
25 February 1985
To see the complete link between 4 and the remaining definitions, it must be noted that 4 is composed of two parts: If the function f i s extensive according to 4 then (a) f is homogeneous of degree one and (b) in the limit a s x l , x 2 , x 3 ~ o o the ratios p I = X l /X 3 , P2 = x 2/x 3 remain fixed. It is seen immediately that, provided attention is restricted to large systems, 4 may be deduced from 3 it" the ratios P l = Xl/X3 and P2 = x 2 / x 3 are fixed. Hence, for ~the special case of large systems, 3 implies 4. Since tile relations between 1 and 3 and between 2 and 3 are k n o w n (see fig. 1), this completes the examination of the links between the various definitions in use for the term "extensive". Hence, it may be concluded that, for large systems, the statistical mechanical definition is most suitable as it implies the other three, but the latter are useful also if this restriction does not hold. Notably, 3 + P and 1 + C imply all the other three definitions. Reference 11] J. Dunning-Davies, Phys. Lett. 94A (1983) 346.