几何视角下的热力学

Thermodynamics and geometry

几何和热力学中的联系长期没有受到足够的注意.本文用几何分析了热力学中的三个问题.第一,从数学上切触几何理解汤姆孙-贝特洛原理,并直接提炼出热力学第三定律;第二,如何从二维曲面的高斯曲率研究气液临界点附近的形状,研究van der Waals方程中a、b系数是否和温度有关这一问题;第三,涨落理论可以有度规描述并具有标量曲率,并利用van der Waals方程展示了标量曲率如何刻画分子之间的相互作用.因此,通过挖掘热力学问题的几何内涵不但可以解决热力学疑难问题,还可以发现新的研究方向.

The interesting connections between thermodynamics and geometry have all but vanished from the literature.Three thermodynamic issues are treated with geometry.The first is how to geometrically interpret the Thomsen-Berthelot principle,leading to the third law of thermodynamics.The second is why the van der Waals parameters a and b depend on the temperature once the equation of state is viewed as a two-dimensional surface.The third is that the thermodynamic fluctuations permit metric representations associated with the Riemann geometry,and the van der Waals equation of state is used to demonstrate the scalar curvature measures the interactions.Attempts to understand the thermodynamics within geometry do not only offer an effective teaching and learning,but also may be helpful to find new research areas.