Thermodynamic proofs of algebraic inequalities

Zylka, Ch.; Vojta, G.

doi:10.1016/0375-9601(91)91086-s

Cited by 6 publications

(2 citation statements)

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“…Perdang [5] emphasized that Landsbergʼs approach is not a novel technique, putting his result in a historical perspective and concluding that this type of physical approach has been operating in the past at two levels: (i) it served the purpose of suggesting new mathematical results and (ii) the physical approach also provides simpler and shorter proofs of already known theorems. Thereafter, Zylka and Vojta [6] showed that the physical procedure introduced by Sommerfeld [7] and Landsberg [8] in order to derive algebraic inequalities from the laws of thermodynamics is not only intuitively reasonable but mathematically correct. They arrived at this conclusion by means of the so-called majorization technique.…”

Section: Introductionmentioning

confidence: 99%

“…Although the problem of the connection between the first and second laws of thermodynamics and some algebraic inequalities has been extensively discussed [1][2][3][4][5][6][7][8][9][10], here we present another interesting case that has to do with the theory of norms. The application of this approach may be of interest for undergraduate and graduate courses in physics and mathematics, and also for research as it might bring new tools to solve some otherwise unclear problems, such as the equilibrium temperature of many infinite-heat-capacity bodies.…”

Section: Introductionmentioning

confidence: 99%

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Equivalent norms in ${{\mathbb{R}}}^{n}$ from thermodynamical laws

González-Ayala¹,

Angulo‐Brown²

2015

Eur. J. Phys.

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In 1978, Landsberg proposed an elegant way of obtaining the inequality between arithmetic and geometric mean by using the first and second laws of thermodynamics. This result opened a debate on the logic legitimacy of this procedure to obtain some mathematical truths. Although this discussion can not be considered completed, the Landsberg approach has shown a great richness in obtaining many algebraic inequalities. In the present article we apply the Landsberg method to some properties of normed spaces trough a vector space of temperatures. In this way, the result that establishes the equivalence between all p-norms in the space n  and the minimal constant that guaranties this fact are readily found. Geometrical surfaces stemming from energy conservation are a consequence of this interpretation. In this manner, an application for thermal equilibrium of n reservoirs is suggested as an example of the contribution that the theory of norms may offer to physical problems when almost ideal heat baths are systems with a heat capacity of the form C aT m = with m  ¥. For this class of reservoirs the problem of obtaining the final temperature of two thermal baths with infinite or nearly infinite heat capacities in thermal contact is unambiguously solved. This might be the case in some systems which produce an incredibly large heat exchange due to small variations of temperature.

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