الفهرس 
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Abstract
In this thesis we studied the critical behavior of some classical models, especially on
two models: the generalized van der Walls model of real fluid and Ising model of
general spin. We completed studies in five chapters and we summarized our results
in the the current dissertation as follows.
In chapter1, We reviewed the subject critical events and phase transitions to
provide an overview of the necessary scientific basis of my work The theory of critical
events and phase transitions was discussed, as well as the classification of phase transitions
based on the loss of analyticity of thermodynamic functions and the importance
of critical point exponents. We explained the significance of providing exact solutions
to models so that phenomena such that phase transitions and critical phenomena can
be considered within the realm of statistical mechanics. Finally, we introduced the
objectives of the thesis. After this general introductory, we discussed the main results in the following chapters.
In chapter 2, We briefly described the van der Walls model and gave a survey of
thermodynamic functions as well as the model’s pivotal point. We presented the most
common modifications of the model.We generalized the standard two-parameter van
der walls model by setting another three parameters and calculated exactly the solution
of the cubic equation in the general form. We investigated the critical behavior
of some thermodynamic properties near to the critical point on two separate critical
routes, such as isothermal compressibility, isobaric expansion coefficient, and isobaric
heat capacity (isobarc path and isochore path). We obtained the critical exponents
in the two path and verified that the critical exponents satisfied the scaling laws.
In chapter 3, we considered the Ising model to describe the critical behavior of the
magnetic systems. We studied the one-dimensional Ising of spin-1/2 with various
boundary conditions and calculated exactly the Scaling functions of finite size Entropy,
magnetization, susceptibility, free energy, and the correlation function are only
a few examples. In the absence (presence) of an external field, the transfer matrix
technique is used. We discovered that the system’s finite size and enforced boundary
conditions influence the thermodynamical properties of the one-dimensional model.
The critical behaviour of these thermodynamic characteristics was explored, and the
critical exponents in the thermodynamic limit Nrightarrowinfty were calculated
and the scaling laws were verified.In chapter 4,We applied universal s-spin to the one-dimensional Ising. Using the
transfer matrix technique, the model was solved analytically. By computing the maximal
eigenvalue of the transfer matrix for some values of the order spin s, we were
able to obtain exact results for free energy, internal energy, entropy, magnetization,
and magnetic susceptibility for free energy, internal energy, entropy, magnetization,
and magnetic susceptibility. For some values of spin-s (1/2, 1 and 3/2) we explored
the asymptotic behaviour of these properties as a function of temperature when the
temperature T → 0, i.e, close to the critical temperature Tc and when T → ∞. The
behavior of the magnetization and the susceptibility analysed in relation to temperature
and magnetic field, and the critical point Tc was examined. Our findings are
in line with those obtained previously for the one-dimensional Ising system. The
outlines further work and recommendations to do in this research:
1- We can transform the standard van der Walls model into the to investigate
non-Gaussian metrics of particle number fluctuations, a large canonical ensemble was
used.
2- We can study the quantum statistical generalization of van der Waals equation
and its application.
3- When there is a magnetic field present, we can investigate The Ising model,
which includes the nearest-neighbor and next-nearest-neighbor variables.
4- We can study Ising model Monte-Carlo simulations.