Analytical Derivation And Application Of The Microstate Sequence Theory In Lattice Models

Lattice model is one of the statistical mechanical models to study the phase transition behavior and state of materials.However,the mathematical and physical depth of lattice model is insufficient.If it is not combined with data fitting,approximate processing and computeraided methods,the research in related fields is often difficult to move on,such as the qualitative and quantitative research of three-dimensional Ising which have been discussed in main body of this paper and the analysis of free volume lattice model mentioned in the outlook section.Therefore,any researcher who attempts to study the lattice model at a non-trivial,fundamental,and more physical level should always follow the basic rules of strict mathematics without any approximate or assumptions.And by increasing the complexity of the model step by step from the relatively simple lattice model may finally approaches the exact solution of the real model of the target system.Based on the consideration of both complexity and analyticity,we propose an equilibrium statistical mechanical approach termed microstate sequence(MSS)theory that is built on a novel idea of arranging all microstates of a discrete thermodynamic system into a sequence with strict “smooth” property for all macro-statistic parameters.We focus on paradigm discrete thermodynamic systems,the Ising model and the Potts model,to demonstrate all essential elements of the MSS theory.In Ising model,a technically convenient screw boundary condition is introduced to enable a unified mapping of the Ising model in arbitrary dimension to an effective Spin chain.Based on this chain physical picture,we present a systematic method to construct MSS for every n>1(n represents the dimensionality)Ising model that accounts for the “long-range” interactions between non-adjacent spins along the chain at a rigorous level.We show that the MSS converges to a continuous microstate variable in the thermodynamic limit.Taking advantage of the similarity of mathematical structures of MSS at two successive spatial dimensions,we provide a concise proof of the second-order phase transition nature of the Ising model in all n>2 dimensions,starting from the well-known exact result for the n=2 Ising model.Due to its distinguished mathematical properties,the MSS theoretical framework holds a promise to facilitate the study of an extended range of discrete thermodynamic systems(e.g.,the Potts model,lattice models for polymers)in the future.At the same time,we also complete the calculation method of MSS to replace the real phase trajectory in study of thermodynamics.Due to unique mathematical properties of MSS,we apply the MSS theoretical framework to other similar physical systems,such as Potts model and Ising model with more complex conditions.At the same time,we also complete the calculation method of MSS.Meanwhile,we further apply the MSS theoretical framework to Ising family models.And the most important one may be the Potts lattice model for the complexity of its phase transition behavior.Our main achievement is to build a deep exploration theory of lattice model and apply it to physical models with different complexity to solve the qualitative and quantitative rules related to phase transition and phase trajectory.The main contents of this paper are as follows:(1)The construction of microstate sequence and its application in Ising model.With the help of spiral boundary conditions,we combine the concepts of density function,pair correlation function and a special function(called Sh function,which is defined by the inevitable continuity of the real phase trajectory)with the concept of energy degeneracy through some methods in number theory,and create a new concept-microstate sequence index(MSSI).The set formed by this variable has a one-dimensional mathematical structure,which is called the microstate sequence(MSS).There is no difference between MSS and real phase trajectory in the study of thermodynamic behavior.Finally,this paper analyze the transition properties of Ising models in arbitrary dimensions and make some calculations of MSS(including transition temperature point)to compare with the experiments.(2)Application of microstate sequence theory in Ising family models.This paper discuss Ising family models in a variety of conditions.The Potts model is the special one among them.Different from the microscopic state sequence of the Ising model,the Sh function in the Potts model is deeply associated with the density function.It means that the symmetry of the Sh function in the Potts model(Sh is not explicitly contained in the energy expression,but is explicitly contained in the neighborhood of the phase point,so it also reflects the characteristics of phase transition behavior)is more likely to be destroyed.In principle,Potts model is “kindlier”for the experimental system than Ising.The MSS of Potts model shows that MSS theory has the possibility to be implemented in more and more complex systems.The other Ising family models turned to be trivial in MSS theory’s description.(3)Analysis of microstate sequence theory.We summarize the advantages and disadvantages of various ideas proposed in the study of microstate sequence theory,especially the effect of Sh function on different systems.This part of the paper also discusses about calculation methods of the new theory.(4)Conclusion and outlook.This chapter summarize the achievements of MSS and discusses the future plan in outlook.