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(A) Partition Function Subject PHYSICS Paper VI Paper Code PHY522 Topic Statistical Physics Dr. Mahesh Chandra Mishra Associate professor of physics Millat ollege, Darbhanga PG SEMESTER:- II, PAPER: - VI PHYSICS CHE522: STATISTICAL PHYSICS M.Sc Physics:Second Semester STATISTICAL PHYSICS Paper: VI Paper Code: PHY 522 UNIT-I Objective of statistical macrostates, microstates, phase space and ensembles. Ergodic hypothesis Postulates of equal a priori probability and equality of ensemble average and time average Boltzman’s postulates of entropy Counting the number of microstates in phase space Entropy of ideal gas SackurTetrode equation and Gibb’s paradox. Liouville’s theorem. UNIT-II System in contact with a heat reservoir Expression of entropy Canonical partition function Helmholtz free energy Fluctuation of internal energy. System in contact with a particle reservoir Chemical potential Grand canonical partition function and grand potential fluctuation of particle number Chemical potential of ideal gas. UNIT-III Mean field theory and Vanderwaal’s equation of state Density matrix Quantum Liouville theorem PG SEMESTER:- II, PAPER: - VI PHYSICS CHE522: STATISTICAL PHYSICS Density matrices for microcanonical canonical and grand canonical systems Simple examples of density matrices- one electron in a magnetic field,, particle in a box; Identical particles- B-E and F-D distributions UNIT-IV Equation of state Bose condensation Equation of state of ideal Fermi gas Fermi gas at finite T Ising model Partition function for one dimensional case Chemical equilibrium and Saha ionisation formula. PG SEMESTER:- II, PAPER: - VI PHYSICS CHE522: STATISTICAL PHYSICS UNIT - I PG SEMESTER:- II, PAPER: - VI PHYSICS CHE522: STATISTICAL PHYSICS Statistical Mechanics Statistical Physics: It is the study of macroscopic parameters of a system in equilibrium with the help of microscopic properties of its constituent particles using the laws of mechanics. It is different from thermodynamical approach as in thermodynamics, macroscopic system (bulk matter) in equilibrium is studied with the help of macroscopic properties. The macroscopic behavior is related to bulk properties i.e. large scale whereas microscopic behavior is related to individual particles Classification of Statistical Physics The study of statistical physics is mainly classified into two categories (I) Classical Statistics or Maxwell-Boltzmann (M.B.) Statistics (II) Quantum Statistics or Bose-Einstein (B.E.) and Fermi-Dirac (F.D.) Statistics. (I) Classical Statistics: It is based on the classical results of Maxwell’s laws of distribution of molecular velocities and Boltzmann theorem relating entropy and probability. So it is also known as Maxwell-Boltzmann (M.B.) Statistics. M.B. Statistics deals with the distinguishable identical particles of any spin. (II) Quantum Statistics ; It was developed by Bose, Einstein, Fermi and Dirac. So it is also known as Bose-Einstein (B.E.) and Fermi-Dirac (F.D.) Statistics. The B.E. statistics deals with the indistinguishable identical particles having zero or integral spin.The F.D. Statistics deals with the indistinguishable identical particles having half-integral spin.The particles obeying B.E. Statistics are called bosons whereas those obeying F.D. Statistics are called fermions and obey Pauli exclusion principle.Classical statistics is only a limiting case of quantum statistics. PG SEMESTER:- II, PAPER: - VI PHYSICS CHE522: STATISTICAL PHYSICS Objectives of Statistical Mechanics: The main objective of statistical mechanics is to establish the relation between the macroscopic behaviour of the substance in terms of microscopic behavior of particles. Macrostates and Microstates: Macrostates – A macrostate of the ensemble may be defined by the specification of phase points in each cell. Microstates – A microstate of the ensemble may be defined by the specification of the individual position of phase points for each system or molecule of the ensemble There may be many different microstates which may correspond to the same macrostate. Understanding macrostate with example To clearly understand macrostate, we consider an example. Let four distinguishable particles a, b, c & d to be distributed into two exactly similar boxes. When any particle is thrown, there is ½ probability of going it into either of the two boxes. There are five different ways of distribution like (0,4), (1,3), (2,2), (3,1) & (4,0) i.e. zero particle in first box and 4 particles in second box as one way, one particle in first box and three particles in second box as second way and so on. So the total no. of macrostates is five. For n no. of particles the total no. of macrostates is (n+1) Understanding microstate with example Each distinct arrangement of particle is known as microstate of the system. In the same example, first arrangement is (0,4) has zero particle in first box and all the four particles a, b, c, d in second box, so the possible arrangement is only one.In (1,3) macrostate, one particle in first box and three particles in second box. PG SEMESTER:- II, PAPER: - VI PHYSICS CHE522: STATISTICAL PHYSICS So arrangements can be (a, bcd), (b, cda), (c, dab), (d, abc). Thus possible arrangements are four. In (2,2) macrostate, 2 particles are in first box and the two particles in second. So arrangements may be (ab, cd), (ac, bd), (ad, bc), (bc, ad), (bd, ac), (cd, ab). Hence possible arrangements are six. Similar to (1,3), (3,1) macrostate can be arranged as (bcd, a) (cda,b) (dab,c), (abc,d) resulting in four possible arrangements. Finally (4,0) macrostate is similar to (0,4) i.e (abcd,0) with possible arrangement only one. Thus in the above example of four particles the total number of microstates is equal to total number of arrangements i.e 1+4+6+4+1=16=24. Hence in general, for a system of n particles, total number of microstates are 2n. Phase space, Ensemble & Ensemble average Phase space:- It is defined as the combination of position space and momentum space. The phase space has six dimensions i.e. three position co-ordinates (x, y, z) and three momentum co-ordinates (px, py, pz). Thus the position of a particle in phase space is specified by a point with six co-ordinates x, y, z, px, py, pz.By these six co-ordinates the complete information like position and momentum of any particle can be obtained. Phase space and Phase cell A small element in phase space is denoted by d and is given by d = (dx dy dz)(dpx dpy dpz) Where, dx, dy, dz, dpx, dpy, dpz are the sides of six dimensional cells. Such cells are called phase cells. According to uncertainty principle – dxdpx ≥ h PG SEMESTER:- II, PAPER: - VI PHYSICS CHE522: STATISTICAL PHYSICS similarly dydpy h & dz dpz h d h3 So, a point in the phase space is like a cell whose minimum volume is ̴ h3. Hence the particle in the phase space can not be considered exactly located at point x, y, z, px, py, pz but within a phase cell centered at that point. The phase space is denoted by space. Ensemble:- It is defined as a collection of large number of macroscopically identical but essentially independent systems. Here macroscopically identical means each of the system constituting an ensemble satisfies the same macroscopic conditions like volume, energy , pressure, temperature, total no. of particles etc. Independent system means the system consisting an ensemble are mutually non-interacting. A system is the collection of a no. of particles. Ensemble Average It is the average at a fixed time over all the elements in an ensemble. This average closely agrees with time average provided (1) The system is a macroscopic system consisting of a large no. of molecules so that the microscopic variables can be truly randomise. (2) The no. of elements forming the ensemble at one time is large so that they can truly represent the range of states accessible to the system over a long period of time. It is to be noted that all members of an ensemble are identical in features like no. of particles N, volume V, energy E etc and are called elements. Kinds of ensembles According to Gibbs, there are three types of ensembles. They are 1. Microcanonical ensemble 2. Canonical ensemble 3. Grand canonical ensemble PG SEMESTER:- II, PAPER: - VI PHYSICS CHE522: STATISTICAL PHYSICS 1. Microcanonical ensemble It is the collection of a large no. of independent systems having the same energy E, Same volume V and same no. of particles N. The individual systems of a microcanonical ensemble are separated by rigid, impermeable and well insulated walls such that the values of E, V and N for a particular system are not affected by the presence of other systems. 2. Canonical ensemble It is the collection of a large no. of independent systems having the same temperature T, same volume V and same no. of identical particles N. The individual systems of a canonical ensemble are separated by rigid, impermeable but conducting walls. The equality of temperature of all the systems can be achieved by bringing each in thermal contact with large heat reservoir at constant temperature T or bringing all the systems in thermal contact with one another PG SEMESTER:- II, PAPER: - VI PHYSICS CHE522: STATISTICAL PHYSICS 3. Grand canonical ensemble It is the collection of a large no. of essentially independent systems having the same temperature T, same volume V and the same chemical potential . The individual systems are separated by rigid, permeable and conducting walls. As the separating walls are conducting and permeable, the exchange of heat energy as well as of that of particles between the systems takes place in such a way that all the systems arrive at common temperature T and chemical potential . PG SEMESTER:- II, PAPER: - VI PHYSICS CHE522: STATISTICAL PHYSICS Comparison of Ensembles Property Microcanonical Canonical Grand Canonical 1.
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