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A generalised mean suggested by the equilibrium temperature
Volume 78A number I
PHYSICS LETTERS
7 July 1980
A GENERALISED MEAN SUGGESTED BY THE EQUILIBRIUM TEMPERATURE P.T. LANDSBERG Department of Mathematics, University of Southampton, UK Received 23 April 1980 Revised manuscript received 5 May 1980
A set of n equilibrating systems reaching a final temperature T from initial temperatures T~are regarded as a physical embodiment of a generalised mean T of items T~.Arithmetic, geometric and harmonic weighted means, and inequalities between them, are obtainable from this formulation.
In view of some interest [1,2] in the letter on a thermodynamic proof of the inequality between arithmetic and geometric mean [3], the following further development may be of interest [4]. Regard the heat capacities C1(t) (i = 1,2, ...) of n bodies merely as real positive functions of a real positive variable t, and replace the entropy factor T by a positive non-decreasing function 0(t). Given the initial temperature, C, and 0, the final temperature T T {C1}, 0; {7~}lin an entropy-conserving equilibrating process is then given by n
~ C 1=1
[C.(t)//(t)~ dt = 0
(1)
~.
Viewed abstractly, T[fC1}, ~ fT1}J is a generalised mean with weighting-type functions C, and 0. The5 choiceconvinces of powersone C1(t) Ap1tm (p,>the 0), arithmetic, 0(t) T soon of this fact, using geometric and harmonic weighted means Tam ~PiT~/~Pi
(m
0,s0),
T[{C~}, 1; {T1}] > T[{C,}, 4; {T~}] (3) Proofof(3). By taking 0(t) outside the integral one can see that —
X(k) =~
k
J” [C~(t)/0(t)] dt T1
k
> [1/~(k)]
E C(t) dt.
(4)
1=1
Choosingk T[{C~},c5; ~T~}],X:=0by eq. (1). Jfk = T[{C~}, 1; {T1}], however, the right-hand side of eq. (4) vanishes by eq. (1), and X>~0. Since dX/dk >0, k in the second case is greater than, or equal to, k in the first case, and this yields eq. (3). physically eq.n(3) states thattoenergy-conserving equi. libration among bodies leads a higher equilibrium temperature than entropy-conserving equilibration. The reason is that in the latter process energy can be extracted from the n-body system for the performance of external work. If we takep 1 p2 = 1 Sidhu’s generalisation is found [21, and is seen to be itself a ...
Tgm (Tfi (m
0,s
Tç2 ...)1~Pi =
1 orm
special case of eqs. (2). For, using the powers given in
—1,s =0), -l
ThE ~EP1/T~/~P~} (m = —l,s = 1). (2) A simple theorem for the new mean, discussed elsewhere [4], is
eqs.(2),eq.(3)yields T~aT~Th. The remark [3] that the original procedure can be generalised is therefore illustrated here and in refs. [2,4]. Other substitutions lead to entirely new inequalities. 29
Volume 78A, number 1
PHYSICS LETTERS
My point that the simple argument of ref. [3] shows the remarkably direct accessibility of a mathematical truth from principles of science, should of course not be construed to imply that a proof was given based on clear axioms, or that the thermodynamic argument is preferable to a formal mathematical proof. In fact, one would expect such arguments to be found occasionally from consistent physical theories, their interest being greatest when the argument is simplest.
30
7 July 1980
References [11 J.P. Abriata, Phys. Lett. 71A (1979) 309. [2~ S.S. Sidhu, Phys. Lett. 76A (1980) 107. [3] P.T. Landsberg, Phys. Lett. 67A (1978) 1; also in: P.T. Landsberg, Thermodynamics and statistical mechanics (Oxford, U.P., 1979) p. 52. [4] P.T. Landsberg, J. Math. Anal. Appl., to be published.
PHYSICS LETTERS
7 July 1980
A GENERALISED MEAN SUGGESTED BY THE EQUILIBRIUM TEMPERATURE P.T. LANDSBERG Department of Mathematics, University of Southampton, UK Received 23 April 1980 Revised manuscript received 5 May 1980
A set of n equilibrating systems reaching a final temperature T from initial temperatures T~are regarded as a physical embodiment of a generalised mean T of items T~.Arithmetic, geometric and harmonic weighted means, and inequalities between them, are obtainable from this formulation.
In view of some interest [1,2] in the letter on a thermodynamic proof of the inequality between arithmetic and geometric mean [3], the following further development may be of interest [4]. Regard the heat capacities C1(t) (i = 1,2, ...) of n bodies merely as real positive functions of a real positive variable t, and replace the entropy factor T by a positive non-decreasing function 0(t). Given the initial temperature, C, and 0, the final temperature T T {C1}, 0; {7~}lin an entropy-conserving equilibrating process is then given by n
~ C 1=1
[C.(t)//(t)~ dt = 0
(1)
~.
Viewed abstractly, T[fC1}, ~ fT1}J is a generalised mean with weighting-type functions C, and 0. The5 choiceconvinces of powersone C1(t) Ap1tm (p,>the 0), arithmetic, 0(t) T soon of this fact, using geometric and harmonic weighted means Tam ~PiT~/~Pi
(m
0,s0),
T[{C~}, 1; {T1}] > T[{C,}, 4; {T~}] (3) Proofof(3). By taking 0(t) outside the integral one can see that —
X(k) =~
k
J” [C~(t)/0(t)] dt T1
k
> [1/~(k)]
E C(t) dt.
(4)
1=1
Choosingk T[{C~},c5; ~T~}],X:=0by eq. (1). Jfk = T[{C~}, 1; {T1}], however, the right-hand side of eq. (4) vanishes by eq. (1), and X>~0. Since dX/dk >0, k in the second case is greater than, or equal to, k in the first case, and this yields eq. (3). physically eq.n(3) states thattoenergy-conserving equi. libration among bodies leads a higher equilibrium temperature than entropy-conserving equilibration. The reason is that in the latter process energy can be extracted from the n-body system for the performance of external work. If we takep 1 p2 = 1 Sidhu’s generalisation is found [21, and is seen to be itself a ...
Tgm (Tfi (m
0,s
Tç2 ...)1~Pi =
1 orm
special case of eqs. (2). For, using the powers given in
—1,s =0), -l
ThE ~EP1/T~/~P~} (m = —l,s = 1). (2) A simple theorem for the new mean, discussed elsewhere [4], is
eqs.(2),eq.(3)yields T~aT~Th. The remark [3] that the original procedure can be generalised is therefore illustrated here and in refs. [2,4]. Other substitutions lead to entirely new inequalities. 29
Volume 78A, number 1
PHYSICS LETTERS
My point that the simple argument of ref. [3] shows the remarkably direct accessibility of a mathematical truth from principles of science, should of course not be construed to imply that a proof was given based on clear axioms, or that the thermodynamic argument is preferable to a formal mathematical proof. In fact, one would expect such arguments to be found occasionally from consistent physical theories, their interest being greatest when the argument is simplest.
30
7 July 1980
References [11 J.P. Abriata, Phys. Lett. 71A (1979) 309. [2~ S.S. Sidhu, Phys. Lett. 76A (1980) 107. [3] P.T. Landsberg, Phys. Lett. 67A (1978) 1; also in: P.T. Landsberg, Thermodynamics and statistical mechanics (Oxford, U.P., 1979) p. 52. [4] P.T. Landsberg, J. Math. Anal. Appl., to be published.