Geometrical Aspect Of Compressibility Critical Exponent

In physics,the use of geometric methods is extremely extensive and far-reaching.One of the famous results is Noether’s theorem,which shows that conservation laws can be derived from the symmetries of physical systems.And after decades of development,the introduction of gauge symmetry successfully unifies the three most fundamental interactions other than gravity,which is known as the Standard Model.And the theory that describes gravity,the core mathematics of general relativity,is also Riemannian geometry.This shows that geometric methods are at the heart of physical theory.In thermodynamics and statistical physics,the main tools are mathematical analysis and mathematical statistics,and geometric methods are rarely used.But Ruppeiner’s introduction of Riemann geometry in 1979 which named Ruppeiner geometry to study the fluctuation theory shows the importance of geometric methods in the field of thermodynamics.Based on the idea of introducing geometric methods into the field of thermodynamics,this paper studies the relationship between the Gaussian curvature of the surface of the equation of state and the critical exponent γby using differential geometry,hoping to serve as inspiration for understanding thermodynamics and statistical physics from a geometric perspective.In the theory of differential geometry,according to what we said earlier,for an arbitrary surface,the most critical quantities are those invariants under the coordinate transformation,that is,the Gaussian curvature and the mean curvature.Among them,for an intrinsic system,that is,when only the first basic form of the surface is given,only the Gaussian curvature of the surface needs to be studied.In thermodynamics,there are three basic thermodynamic functions,namely the equation of state,the internal energy function and the entropy function,which correspond to the state functions introduced by the zeroth law of thermodynamics,the first law of thermodynamics and the second law of thermodynamics,respectively.Knowing these three functions,other thermodynamic functions can be derived from the Maxwell relation and Legendre transformation,so the equilibrium properties of the system can be completely determined.The equation of state is a state function introduced by the zeroth law of thermodynamics,which is simple yet fundamental and important.For the simplest p-V-T system,the equation of state can be written as,geometrically,a smooth two-dimensional surface in a three-dimensional flat space.Therefore,at each point it has its Gaussian curvature and its mean curvature,the variation of these two curvatures just reflects the state change of the system.In particular,for the most important critical point in phase transition research,it is experimentally observed that near the critical point,the time scale of the system approaching from the temperature or volume direction to the critical point is quite long,which is referred to as the phenomenon of critical slowing,means the thermodynamic properties of the critical point are independent of the path.Meanwhile,critical phenomena can be described by critical exponents.Therefore,the theory of differential geometry can be used to study the properties of critical points and critical exponents.The following three parts are the main content of this thesis:First,the Gaussian curvature and the average curvature of the surface of the equation of state at the critical point are solved and calculated,and the physical meaning of the Gaussian curvature must be less than or equal to zero is pointed out.Subsequently,the specific numerical values of the Gaussian curvature and the average curvature of various empirical and semi-empirical equation of state surfaces proposed since 1873 are calculated at the critical point,and it is pointed out that the Gaussian curvature value can be used to judge the validity of these equations of state.A basis for,and the Gaussian curvature of the real gas-liquid phase transition is 0,that is,the assumption of zero Gaussian curvature.Second,the relationship between the Gaussian curvature of the equation of state at the critical point and the critical exponent γ of the compressibility is analyzed.It is found that whether or not this Gaussian curvature is zero is entirely determined by the compressibility critical exponent γ.As a test,the Gaussian curvature of the ideal Bose gas equation of state at the critical point is directly calculated,and the critical exponent is found to be greater than 1,which is consistent with the experimental results.Third,it is proposed to modify the van der Waals equation according to the zero Gaussian curvature.And according to the modified van der Waals equation,the critical exponent of the compressibility is calculated to be 3,which is consistent with the Gaussian curvature-critical exponent relationship proposed above.All in all,this paper introduces the method of differential geometry to study the Gaussian curvature of the surface of the equation of state at the critical point,and obtains the relationship between the Gaussian curvature and the critical exponent,and points out the geometric meaning of the critical exponent of the compressibility.Furthermore,the critical exponent of the compressibility of the modified van der Waals equation is studied.This paper gives some insights into the introduction of more geometric methods to study the problems of thermodynamics and statistical physics.