so far, we have talked about the statistical mechanics as a physical theory. the statistical mechanics also has a mathematical side. to calculate the probability distribution, the statistical mechanics needs to do a lot of sampling. the sampling is done by taking many states of the system and calculating all the relevant energies of the system in these states.
prof. kivelson explained that the statistical mechanics can also be used to calculate the entropy of a system. the entropy is an important parameter of a system and it is determined by the probability distribution. the statistical mechanics also provides us with the expression for the pressure. this is the second law of thermodynamics that tells us that the pressure should be zero at zero temperature. at a certain temperature, there is a maximum pressure called the critical point.
where r is the distance from the center of the guest molecule to the center of the water box. the number of water molecules at a distance r from the guest molecule is calculated using a spherical shell technique, where the radius is calculated using the distance between the center of the guest molecule and the center of the water box (35.5 å). the total number of water molecules in the system is calculated using the volume of the box and the number of guest molecules. the center of the shell was shifted to the center of the guest molecule to ensure that water molecules within 5.0 å of the guest molecule were counted in the calculations of and, ensuring a linear distance dependence of the potential. the relative dielectric permittivity of the solvent is set to 66.6 in the calculation of and to 78.4 in the calculation of, as appropriate for the spc water model 85 and to 78.4 in the calculation of, as appropriate for water, 56 to account for the inaccurate dielectric permittivity of the spc water model. the calculation of the charging free energies of the guest molecule is similar, with the relative dielectric permittivity of the solvent set to 78.4 in the calculation of,, and in the calculation of, as appropriate for water, 56 to account for the inaccurate dielectric permittivity of the spc water model. to ensure that water molecules are counted only once in the calculations of and, the number of water molecules within a spherical shell was used, similar to the calculations of and, where is the radius of the shell. the number of water molecules at a distance r from the guest molecule was calculated using a spherical shell technique, where the radius is calculated using the distance between the center of the guest molecule and the center of the water box (35. the total number of water molecules in the system was calculated using the volume of the box and the number of guest molecules. the relative dielectric permittivity of the solvent is set to 78.
quantum statistics is discussed in phys*3240, statistical physics i. topics include the grand canonical ensemble, fermi-dirac and bose-einstein statistical distributions, the occupation number formalism, and quantum fluctuations of the number of particles.
in phys*3230, the hamiltonian formulation of classical mechanics is given. the course introduces the concepts of momentum, energy, forces, action, and phase space. the course also discusses the concepts of lagrangian and euler-lagrange equations and their solutions. motion in a potential is discussed.
a discussion of the general theory of irreversible thermodynamics and the first and second laws of thermodynamics are given in phys*3230, statistical mechanics i. the course discusses the concept of work, the form of the first law of thermodynamics, the importance of heat transfer, entropy, and gibbs free energy and its equilibrium value. in the final semester of the course, the second law of thermodynamics is discussed in a quantum statistical mechanical context. special emphasis is placed on the entropy of a system in equilibrium. the role of quantum mechanics in the second law is also discussed.
this course is a continuation of the study of the laws of statistical mechanics and thermodynamics begun in phys*3240, statistical physics i. statistical physics is the study of the physical properties of systems consisting of a very large number of atoms, molecules, or other particles. in spite of the enormous complexity of macroscopic bodies when viewed from an atomistic viewpoint these bodies obey quite definite laws. macroscopic observable quantities such as temperature and pressure are averages over microscopic properties and the macroscopic laws which these quantities obey are of a statistical nature. the objectives of this course are to develop an understanding of the statistical nature of the laws of thermodynamics, to examine the basic theory of statistical mechanics and to apply this theory to a wide variety of interesting problems.